The approximate scattering length a\n\ns\n and approximate healing length ξ (44) for a few typical atoms that undergo BEC [55].
\r\n\tIC offer services both at the macro-scale (country) and at the micro-scale (cities). The stability and the protection of these networks on both scales are a significant task, entrusted to the Operators who operate CI in a concession mandate. Due to the high level of interdependency of CI, which exchange services to each other for their functioning, their management and protection cannot be carried out by a "linearized" strategy (each infrastructure managed and protected independently on the others) due to the presence of tight links which connect each other. CI protection through provision of "smart" properties such as resilience, has become a complex task which must be tackled not only by deploying advanced technological means but also by triggering new management strategies, enabling holistic and global management policies.
\r\n\r\n\tThe book aims at providing an overview of the understanding of complex phenomena taking place on interdependent networks and of the advanced technical solutions related to management, risk analysis and resilience enhancement of networks, either from a theoretical and operational (i.e. with solution related to real or realistic cases) points of view. A large emphasis is provided to the capability opened by the use of field and remote sensing tools for monitoring and assessing risks on CI. The use of comprehensive data set, the access to big data is going to open the way to the realization of new tools for supporting the decision making process needed for both daily and emergency management.
",isbn:"978-1-83962-621-0",printIsbn:"978-1-83962-620-3",pdfIsbn:"978-1-83962-628-9",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"7cfcd62bae8c99be207e18bb73e2a7b1",bookSignature:"Dr. Vittorio Rosato and Dr. Antonio Di Pietro",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10256.jpg",keywords:"Complex systems, Interdependence, Resilience, Cascading effects, Event prediction, Emergency management, Decision support, AI, Field sensors, Remote sensing, IoT, GIS",numberOfDownloads:525,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"June 15th 2020",dateEndSecondStepPublish:"July 6th 2020",dateEndThirdStepPublish:"September 4th 2020",dateEndFourthStepPublish:"November 23rd 2020",dateEndFifthStepPublish:"January 22nd 2021",remainingDaysToSecondStep:"8 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Supervisor and Project Evaluator for EU, for the Italian Ministry of University and Research, and that of Economic Development; Consultant for several Italian Regions and the Italian Ministry of Defense; Coordinator of several National Projects; Co-founder of two SMEs active in software engineering and biotechnology; Author of more than 140 scientific papers in peer reviewed journals and conference proceedings.",coeditorOneBiosketch:"A full researcher at ENEA (Italian Energy, New Technology and Environment Agency) since 2007 and a joined member to the Laboratory for the Analysis and Protection of Critical Infrastructures (APIC) from 2015. Dr. Di Pietro took part in several European and Italian national research projects and acted as an advisor for certain Evaluation Studies commissioned by the EU.",coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"27002",title:"Dr.",name:"Vittorio",middleName:null,surname:"Rosato",slug:"vittorio-rosato",fullName:"Vittorio Rosato",profilePictureURL:"https://mts.intechopen.com/storage/users/27002/images/system/27002.jpg",biography:"Vittorio Rosato received the Laurea degree (M.Sc.) in Physics from the University of Pisa (Italy) and a Ph.D. in Condensed Matter Physics from the University of Nancy (France). He has extensively been working in Computational Physics, particularly in Condensed Matter and Material Science in his positions as Research Assistant at the University College of Wales in Aberystwyth (UK) and at the Centre d'Etudes Nucleaires in Saclay (France). Staff Scientist at ENEA (Italian National Agency for New Technologies, Energy and Sustainable Economic Development) since 1990, he is currently Head of the Laboratory of Analysis and Protection of Critical Infrastructures and Manager of the Italian Node of the European Infrastructure Simulation and Analysis Centre (EISAC.it).\nHis current research activities span from risk analysis to the design of Decision Support Systems for the management of complex technological networks. He acts as Supervisor and Project Evaluator for EU, for the Italian Ministry of University and Research, and that of Economic Development; he is also consultant for several Italian Regions and the Italian Ministry of Defense. He is and has been Coordinator of several National Projects. He is co-founder of two SMEs active in software engineering and biotechnology. He is author of more than 140 scientific papers in peer reviewed journals and conference proceedings.",institutionString:"ENEA (Italian National Agency for New Technologies, Energy and Sustainable Economic Development)",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"0",institution:null}],coeditorOne:{id:"284589",title:"Dr.",name:"Antonio",middleName:null,surname:"Di Pietro",slug:"antonio-di-pietro",fullName:"Antonio Di Pietro",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bReF6QAK/Profile_Picture_1581328351906",biography:"Antonio Di Pietro received the Laurea degree (M.Sc.) in Informatics Engineering from Sapienza University of Rome (Italy) and a Ph.D. in Methodologies for Emergency Management in Critical Infrastructures from Roma Tre University (Rome).\nHe has been working as a full researcher at ENEA (Italian Energy, New Technology and Environment Agency) since 2007 and in 2015 he joined the Laboratory for the Analysis and Protection of Critical Infrastructures (APIC) in the same institution.\nHis research interests include modelling and simulation of critical infrastructures and the development of Decision Support Systems integrating seismic and meteorological natural threat modeling. He is also an Unmanned Aerial Vehicles (UAV) ENAC-certifed pilot to perform critical operations involving aerial photogrammetry tasks for biological and Infrastructure monitoring applications. He took part in several European and Italian national research projects and acted as an advisor in some Evaluation Studies commissioned by the EU. He has also been advisor of several M.Sc. students and also a teacher in several professional courses on Software Engineering and Databases.",institutionString:"ENEA (Italian National Agency for New Technologies, Energy and Sustainable Economic Development)",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"0",institution:null},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"11",title:"Engineering",slug:"engineering"}],chapters:[{id:"74122",title:"Risk Analysis in Early Phase of Complex Infrastructure Projects",slug:"risk-analysis-in-early-phase-of-complex-infrastructure-projects",totalDownloads:87,totalCrossrefCites:0,authors:[null]},{id:"74493",title:"Flood Risk Analysis for Critical Infrastructure Protection: Issues and Opportunities in Less Developed Societies",slug:"flood-risk-analysis-for-critical-infrastructure-protection-issues-and-opportunities-in-less-develope",totalDownloads:50,totalCrossrefCites:0,authors:[null]},{id:"74123",title:"Resilience in Critical Infrastructures: The Role of Modelling and Simulation",slug:"resilience-in-critical-infrastructures-the-role-of-modelling-and-simulation",totalDownloads:78,totalCrossrefCites:0,authors:[null]},{id:"73984",title:"Validation Strategy as a Part of the European Gas Network Protection",slug:"validation-strategy-as-a-part-of-the-european-gas-network-protection",totalDownloads:35,totalCrossrefCites:0,authors:[null]},{id:"74174",title:"Defects Assessment in Subsea Pipelines by Risk Criteria",slug:"defects-assessment-in-subsea-pipelines-by-risk-criteria",totalDownloads:44,totalCrossrefCites:0,authors:[null]},{id:"74240",title:"Analyzing the Cyber Risk in Critical Infrastructures",slug:"analyzing-the-cyber-risk-in-critical-infrastructures",totalDownloads:85,totalCrossrefCites:0,authors:[null]},{id:"74141",title:"Italian Crisis Management in 2020",slug:"italian-crisis-management-in-2020",totalDownloads:49,totalCrossrefCites:0,authors:[null]},{id:"74668",title:"A Strategy to Improve Infrastructure Survivability via Prioritizing Critical Nodes Protection",slug:"a-strategy-to-improve-infrastructure-survivability-via-prioritizing-critical-nodes-protection",totalDownloads:34,totalCrossrefCites:0,authors:[null]},{id:"74143",title:"Resilience of Critical Infrastructures: A Risk Assessment Methodology for Energy Corridors",slug:"resilience-of-critical-infrastructures-a-risk-assessment-methodology-for-energy-corridors",totalDownloads:66,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"205697",firstName:"Kristina",lastName:"Kardum Cvitan",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/205697/images/5186_n.jpg",email:"kristina.k@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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The theoretical framework to describe superfluids was set up in the next decades, and a central concept in this description is the macroscopic (complex-valued) wave function. A normal fluid, when described at the quantum mechanical level, requires a many-body wave function that depends on the coordinates of all the particles in the fluid. A superfluid, however, can be well described by a macroscopic wave function that depends on a single-position coordinate. This wave function has a straightforward ‘hydrodynamic’ interpretation: the density of the superfluid is given by the modulus square of the macroscopic wave function, and the velocity field \n
with m the mass of the atoms. The single-valuedness of the wave function, in conjunction with Eq. (1) leads to remarkable properties for rotating superfluids. Since the phase can wind only in multiples of 2π, circulation is quantized. Contrary to vortices in classical fluids, in a quantum fluid, vortices carry an integer number of circulation quanta—it is not possible to divide a singly-quantized vortex into smaller units. Upon rotation, a superfluid will not behave as a solid body, but will generate a number of singly-quantized vortices that together carry the total circulation of the rotating fluid. In a (neutral) superfluid, vortices with the same sign of the circulation repel due to kinetic energy of the super-flow. This leads to the formation of a lattice of singly quantized vortices, known as an Abrikosov lattice [3].
\nIn helium-II, the quantization of vorticity and the presence of a lattice of singly-quantized vortices were observed by Vinen [4]. His seminal experiments initiated the study of vortices in liquid helium, which still represents a very active and intense field of research. A thorough review of this field can be found in reference [5]. Liquid helium at non-zero temperature is not a ‘pure’ superfluid in the sense that thermal and quantum fluctuations lead to a finite viscosity in the bulk liquid, and this complicates the study of superfluid vortices. However, two decades ago, a new superfluid system became experimentally available: ultra-cold atomic quantum gases. These consist of a cloud of typically 105 atoms that are optically and/or magnetically trapped, and cooled down to nano-kelvin temperatures. The system is very controllable and versatile: temperature, atom number, trapping geometry and dimensionality, and interaction strength are experimentally tunable. Bosonic atoms were cooled down below the critical temperature for Bose-Einstein condensation (BEC) in 1995 (for a review, see reference [6]), and fermionic atoms were cooled down to a regime where the atoms form superfluid pairs in 2003 (reviewed in reference [7]). The tunable interaction strength allows to bring Fermi gases in the regime where they form Cooper pairs, or in the regime where they form strongly bound molecules that Bose condense, or in the crossover regime between those two limits.
\nThe present chapter aims to familiarize researchers in other fields of vortex physics with the ongoing research on vortices specifically in quantum gases, especially Bose-Einstein condensates. First, we present an (non-exhaustive) overview on the experiments related to these superfluid vortices, and in the second part, we outline the theoretical description of vortices in condensates.
\nIn the first few years after the initial observation of Bose-Einstein condensation in atomic gases [8–10], it was not clear how to form vortices in these new superfluids. Early experiments with rotating traps failed to produce the vortices (see the contribution of Ketterle and co-workers in reference [6]), and the question was raised on whether the vortex core might be too small to image. The healing length (and core size) is estimated to be of the order of 100 nm. To make observations of quantum gases, the mainstream method [6] is to shine a laser on the cloud, and image the shadow of that cloud. That way, a map of the column density of the cloud can be made, and vortices with their core parallel to the line of sight should appear as holes in the cloud. In this imaging technique, the trap holding the quantum gas is switched off, so the cloud—initially a micron-sized object—can expand to a size large enough to provide a good resolution. The initial difficulties to observe vortices were surprising since theory predicts that when the trap is switched off the vortex core expands faster than the radius of the atomic cloud [11, 12], and this should facilitate the observation, unless the vortex line twists away from the line of sight. An interferometric technique was proposed [13, 14] to overcome possible difficulties with direct imaging. The phase coherence of the superfluid implies matter-wave interference: when two adjacent condensates are allowed to expand, then a series of parallel matter interference fringes appear in the region of overlap of the expanding clouds. The phase pattern of a cloud containing a vortex is different, and this leads to a ‘fork dislocation’ in the fringe pattern [13]. This idea of interferometry is used in many different branches of physics, e.g. plasma physics to measure the electron density [15, 16] and in the study of optical vortices [17, 18], where the ‘fork dislocation’ is also found. In both cases, this is done by making an interference pattern of the ‘probing beam’ (which probes your object) with a ‘reference beam’. However, note that the interference pattern of two overlapping condensates is a pure quantum mechanical effect [19], while the interference pattern in conventional interferometry is a pure classical interference effect.
\nThe first successful attempt, in 1999, to create a topological excitation similar to a vortex was realized not by rotating the cloud, but by quantum state manipulation [20]. This requires a multi-component condensate, consisting of atoms in different internal states, typically different hyperfine levels, confined to the same-atom trap. In the experiment given in reference [20], initially the trap contains only a non-rotating one-component BEC. A microwave field is then used to couple two hyperfine states and coherently transform a given number of atoms from the first component into the second, so that a BEC can be grown in the second component. An additional laser beam can be used to imprint a phase profile in the second component in such a way that a single quantum of circulation is present in the second-component, ring-like BEC. The core of the vortex-like structure is filled by the atoms still in the (non-rotating) first component. It can be argued that due to the multi-component nature of the condensate, it should be interpreted as a spinor condensate and the resulting object should consequently be classified rather as a skyrmion than as a ‘true’ vortex. Indeed, in these cases, the phase can ‘untwist’ itself, and the excitation is not topologically protected as expected for a quantum vortex [21]. In later experiments using quantum-state manipulation, vortices were created by transferring orbital momentum directly from a laser beam [22]. In other related experiments, the (hyperfine) spin orientation of the atoms in a BEC is manipulated by adiabatically changing an external magnetic field in a position-dependent way to imprint a vortex [23]. In this way, vortices carrying multiple quanta of circulation were also created and their decay into singly quantized vortices was studied [24]. The latest development in vortex physics using internal state manipulation is the possibility to create synthetic magnetic fields [25], i.e. to quantum engineer an effective Lagrangian in which a vector potential term appears, not due to physical rotation or a physical magnetic field, but due to a position-dependent coupling between internal atomic states.
\nThe second class of vortex experiments in quantum gases uses setups in which the condensate is physically rotated. Note that the circulation of a vortex and thus the condensate are quantized. This means that a minimal (critical) frequency is needed in order to have a vortex in a quantum gas. As soon as the critical frequency is reached, the rotational frequency of the condensate snaps to this critical frequency, and starts rotating as a whole. Although initially this method was unsuccessful, it is now the most widely used method to create and study vortices in superfluid quantum gases. The first observation of a ‘true’ vortex in quantum gases—distinct from the coreless vortex given in reference [20]—can be found in reference [26]. In that experiment a time-dependent optical potential (a ‘laser spoon’) is used in conjunction with a magnetic trap to stir the condensate, and this proved to be more effective than rotating only the magnetic trap. Below a critical frequency the condensate shakes (surface oscillation modes are present) but it does not rotate (no circulation). Above the critical frequency a singly quantized vortex is nucleated: it arises from a surface mode at the edge of the cloud and moves towards the centre where it remains as long as the cloud is kept in rotation. Using this technique, by rotating faster, an array of vortices was nucleated [27]. Also a ‘field-cooled’ version of this experiment was performed: rather than rotating a condensate, one cools a rotating normal cloud below the critical temperature [28]. When the temperature drops below the critical temperature, a condensate is nucleated in which the circulation ‘snaps’ to the nearest quantized value of circulation, with a corresponding number of vortices appearing. A large number of subsequent experiments have further refined the rotation technique for vortex creation—some noteworthy cases are the investigation of vortex nucleation [29], the development of a purely magnetic technique [30] and the use of a rotating optical lattice [31]. These experiments rely largely on the direct imaging technique, in which the vortex core is visible as a hole in the cloud, and the success of these experiments have dispelled the early doubts that the vortex cores might be too small to observe. Nevertheless, the matter-wave interferometric technique was also used to confirm the presence of the phase singularity, and to make sure that there is indeed circulation around the visible hole in the condensate [32, 33].
\nWith techniques to produce vortices in quantum gases well-established, the path was open to study various aspects of vortex physics, such as the properties and dynamics of single vortex lines [34]. In the next three paragraphs we highlight three specific lines of experimental research that have been pursued very intensively: one that focuses on the properties of the vortex lattice, one that looks for spontaneous generation of vortices and anti-vortices, and finally one that focuses on quantum turbulence.
\nThe equilibrium properties of vortex lattices were studied [35], as well as their formation dynamics [36, 37]. The response of the lattice to external perturbations such as shaking and removal of atoms was investigated, and revealed various unusual vortex configurations, such as giant vortices and vortex sheets. This also allowed to excite the ‘phonons’ or vibrations of the lattice, known as Tkatchenko modes [38]. Increasing the rotation frequency, attempts were made to reach the Landau-level regime, where the number of quantum vortices becomes comparable to the number of particles. Although this regime could not be reached, interesting physics regarding the lattice structure and equilibrium cloud shapes emerged for fast-rotated condensates [39].
\nBy adapting the trapping potential, the atomic cloud can be squeezed into a pancake thin enough to reveal the physics of the two-dimensional (2D) Bose gas—in this case superfluidity is destroyed through the spontaneous appearance of vortices and anti-vortices above a critical temperature, the so-called Kosterlitz-Thouless transition, as given in reference [40]. The creation of vortex-anti-vortex pairs can also be induced in a three-dimensional (3D) cloud by moving an obstacle (such as a blue de-tuned laser beam) through the condensate [41]. The spontaneous generation of vortices and anti-vortices also occurs when a Bose gas is quenched through the phase transition. This quench leads to the nucleation of individual phase-coherent domains with different phases that upon merging lead to phase singularities at the interface of the merged domains. This phenomenon is called the Kibble-Zurek mechanism and with the advent of uniform (‘flat box’) trapping potentials [42] it has become possible to observe this transition in quantum gases [43].
\nBy tilting the vortex array by a small angle around the axis of symmetry, the precession of the array was observed, and Kelvin modes (transverse oscillations of the vortex line) were seen [44]. The technique of tilting the axes of rotation would later prove useful in the studies on quantum turbulence. This is a regime of turbulent flow characterized by a tangle of vortices, and has been extensively studied in helium [45]. The difficulty in studying vortex tangles in liquid helium is that the core size is very small, so it is hard to directly image the vortices. In quantum gases, vortex cores can be readily imaged, and in addition new in situ [46] and tomographic [34] imaging techniques were developed. However, in quantum gases, the ratio between the condensate size and the vortex core is much smaller than in helium (typically it is of the order of 10–50 in BECs), limiting the size of the vortex tangle and inducing edge and finite-size effects. Quantum turbulence was achieved [47] by rotating the condensate around different axes simultaneously, or by rotating in conjunction with shaking. The possibility to flatten the atomic cloud has opened the way to also study quantum turbulence in two dimensions [48]. These techniques offer the potential to resolve many outstanding questions [49] regarding the transition from classical to quantum turbulence, the role of vortex reconnections and the dimensional crossover.
\nThe above examples of recent vortex studies relate to Bose gases. Also in superfluid Fermi gases under rotation, vortex lattices have been observed [50]. The appearance of arrays of singly quantized vortices has been used as a hallmark of superfluidity in these systems. This in turn has been used to investigate how superfluidity breaks down when a population imbalance exists between the two states that combine to form the superfluid pairs [51]. Finally, to conclude this review section, we provide some references to more extensive reviews [52, 53] and a book [54] on vortices in quantum gases.
\nBefore we take a look at vortices, we will briefly discuss and derive the theory that we will use to describe ultra-cold bosonic gases. More elaborated discussions can be found in references [55–57]. The results of this section will be the Gross-Pitaevskii (GP) equation, a hydrodynamic description of the system and the energy of a general ultra-cold, dilute bosonic gas.
\nIn order to give a successful theoretical description of ultra-cold quantum gases, we first need to ask ourselves which framework to use. Is it sufficient to use the classical kinetic theory, or should we resort to quantum theory to get reliable answers? In order to answer this question, we have to look at the de Broglie wavelength[1] - (or thermal wavelength):
\nAs soon as the de Broglie wavelength becomes comparable (or larger) than the typical distance between atoms, we need quantum mechanics. In this case the wavelength associated with a particle will become so large that the quantum mechanical wave functions will overlap.
\nFor typical systems of ultra-cold quantum gases, we deal with clouds of around:
\n105 atoms,
with a scattering length in the order of nm,
trapped in a magneto-optical trap with a typical length scale of μm,
at temperatures of the order of nK.
The typical de Broglie wavelength of this system can now be estimated using the above, this results in:
\nThe distance between the atoms can be estimated from the inverse particle density (n), which yields:
\nAs we see the distance between the atoms is around a factor 1000 smaller than the de Broglie wavelength, which means that we should use quantum mechanics.
\nSince we are working with a gas of N atoms (many body system), we will start in second quantization [59, 60] since it is then easier to write down (and simplify) our Hamiltonian. The most general Hamiltonian for a system with two-body interactions is [61]:
\nwhere \n
For ultra-cold gases the main contribution to the partial wave expansion will come from the s-wave scattering. Since the range of the inter-atomic potential is much smaller than the inter-atomic distance (dilute gases), the potential can be approximated as a contact potential. Using partial wave analysis [62], the two-body potential can now be simply written as [63, 64]:
\nwith a\n\ns\n the (boson-boson[1] -) s-wave scattering length. Physically this s-wave scattering length can be interpreted as the effective radius a\n\ns\n of our interacting particles if they would be approximated as hard spheres.\n[1] -\n
\nA second approximation can be made if we assume that we have a Bose-Einstein condensate (BEC). The mechanism behind Bose-Einstein condensation can be easily seen when we look at the Bose-Einstein distribution:
\nwhere the energy was chosen in such a way that the lowest energy level is situated in ε = 0. This distribution has a divergence in ε = μ and drops exponentially for large values of ε. This means that we can store an infinite amount of particles in the lowest energy level, while we can only have a finite amount in the higher energy levels. The total number of particles in our system of bosons is given by:
\nwhere N\n0 is the number of particles in the ground state, \n
Notice that in order to achieve a BEC we want as many particles as possible, while for our approximation of the interaction potential we want a dilute gas. How can we combine this? The answer is to make the gas ultra-cold, this way the number of particles required to get a BEC becomes small enough to make the gas dilute. If our atomic gas is not diluted, higher order interactions will play a more important role. These higher order interactions will lead to particle being expelled from the atomic gas.
\nCombining both assumptions, it is now possible to simplify our Hamiltonian to make it equal to:
\nIn order to get our equation of motion for our system (9), we use the Heisenberg equation of motion, resulting in:
\nThe above equation (10) is known as the time dependent Gross-Pitaevskii equation (or the nonlinear Schrödinger equation). Note that we substituted the operator \n
Since all expectation values for our system can be rewritten in terms of the density matrix \n
Note that if you want to minimize your energy (9) to find the ground state of your system with a given normalization, you explicitly have to use Lagrange multipliers to fix your normalization. This is because of the fact that our Hamiltonian and thus equations of motion are non-linear. For the ground state this means that you define the Hamiltonian:
\nwhich now contains the chemical potential as a Lagrange multiplier for the number of particles. Doing a functional minimization then yields the differential equation:
\nSo depending on whether you work with a fixed number of particles or a fixed chemical potential you use respectively, Eqs.(12) and (13) or Eqs.(9) and (10). The relation between both views can be easily established by comparison. For the energy we get the known Legendre transform from statistical mechanics:
\nFor the macroscopic wave function, we have an additional phase rotation:
\nWith the above two relations it is possible to switch between fixed number of particles or fixed chemical potential if needed.
\nIn order to simplify the calculations one usually works in the hydrodynamic description. This means that our complex field \n
The field \n
If we now take the Gross-Pitaevskii equation, Eq. (10) and treat it canonically[1] - we get, after simplification:
\nThis is a result known from standard quantum mechanics, where it is seen as a continuity equation since \n
Of course we want to work in the hydrodynamic equations so we will substitute the hydrodynamic description (16) in the continuity equation, Eq. (17) and the definition for the velocity field, Eq. (18). For the continuity equation, Eq. (17) this yields (as expected) that:
\nwhich is known from classical hydrodynamics. For the velocity field (18), we find that
\nso the velocity field is determined entirely by the phase field. Note that our velocity field is also irrotational:
\nmeaning that we will only have irrotational flow patterns.
\nOf course since we have two real fields \n
where the first line (22) is the imaginary part and the second line (23) is the real part of the Gross-Pitaevskii equation, Eq. (10). The imaginary part (22) can be simplified and rewritten in terms of the velocity field and density field, this will yield the continuity equation, Eq. (19). The real part (23) can also be rewritten in terms of the velocity and density fields, this is most easily done by taking the gradient of the equation (23). After simplifying the gradient of the real part (23) we get:
\nwhich is the same as the Euler equation (or frictionless Navier-Stokes equation) from hydrodynamics. In other words, in a BEC we have an irrotational, frictionless flow. By looking at the Euler equation (24), we can identify three kinds of pressure that will occur in our system:
\n\nMechanical pressure: The first pressure we can recognize is the one particle potential \n
\nInteraction pressure: The second pressure is given by the interaction term \n
\nQuantum pressure: The remaining term
\nis known as the quantum pressure. This pressure is a consequence of quantum mechanics where your kinetic energy is proportional to the second derivative of the wave function (the curvature). Note that this pressure is a pure quantum mechanical effect, indeed if we take the limit h → 0 this term becomes equal to zero.
\nSo for our ultra-cold bosonic gases forming a BEC, we get one macroscopic wave function \n
Now that we know the basic physics for a BEC, we can look at vortices in ultra-cold gases. Note that the results discussed here will be general for both vortices in superfluid Bose gases as well as in superfluid Fermi gases.\n[1] - In this section we will not look at the structure of the vortex (this is done in the next section), but we will rather define what we mean with a vortex and what this implies for our vortex flow. By thoroughly studying the definition of a vortex we can already derive a few basic properties of a vortex. The rest of the section is devoted to two analytically solvable systems of a BEC and on how vortices are detected in experiments.
\nIn order to study a vortex, we first have to explain what we mean when we talk about a vortex. Just as in classical fluid dynamics, we start from the definition of the circulation[1] - κ, defined as:
\nwhere C is a closed contour containing the vortex core. For the circulation we then have three possibilities: counter-clockwise rotation (κ > 0), clockwise rotation (κ < 0) and no rotation κ = 0. The value of the circulation κ should be the same for any contour C that encloses the vortex core.[1] - Given the velocity field \n
which is a quantity telling us how the fluid rotates locally. The vorticity follows the vorticity equation which can be derived by taking the curl of the Euler equation, Eq. (24). Applying Stokes’ theorem in Eq. (26), we can relate the circulation κ in terms of the vorticity (24), this results in:
\nmaking the circulation equal to the flux of vorticity. Note that we can use Stokes’ theorem as long as S is a surface without any holes.
\nUsing the above two formulas (26) and (28) we can calculate the circulation of a BEC. To calculate the circulation using the velocity field, we will use the definition of the velocity field in terms of the phase field (20). The circulation then becomes:
\nwhere the gradient theorem was used. In (29) r\n\nb\n is the point where the closed loop begins and r\n\ne\n is the point where the loop ends, although the physical positions are the same the points r\n\nb\n and r\n\ne\n can be different. An example of two points with a different coordinate but the same physical location is given for example when we describe the phase in polar coordinates \n
Since we expect to find the same physics again when we make one round, we also expect our quantum mechanical wave function to be the same up to a global phase that is a multiple of 2π. The fact that we expect to find the same physics again tells us that the phase difference in (29) has to be given by:
\nUsing the above restriction (31) on the phase difference the equation for the circulation (29) then yields:
\nin other words the circulation is quantized in multiples of h / m.
\nThe circulation can now also be calculated by using the formula containing the vorticity field (28), yielding
\nwhere we got zero since the curl of a gradient field is always zero. The fact that our two results for the circulation Eqs. (32) and (33) are in contradiction means we did something (mathematically) wrong. Since Eq. (26) is the definition and the restriction Eq. (31) is exact our result, Eq. (32) should be the right one. The only assumption that we made to go from the definition of the circulation Eq. (26) to the formula with the vorticity field, Eq. (28) is our condensate (and thus the surface we integrate over) has no holes. The fact that the results are different means that it was wrong to assume that the condensate has no holes. Or in other words, having a vortex (so a non-vanishing circulation: n ≠ 0 in Eq. (32)) will create a hole in the condensate. We can even go further and say that it is impossible for a vortex to begin and end in two different points inside of the BEC, and this can be illustrated by contradiction. Suppose we have a vortex that ends (or begins) inside of the BEC, then we can choose a surface S belonging to the contour C in such a way that the surface does not intersect with the vortex, an example is given in \nFigure 1\n. If we can choose our surface as illustrated in \nFigure 1\n, then Stokes’ theorem should hold, leading to a contradiction when calculating the circulation using the two formulas (26) and (28). What we can conclude from this is that vortices extend through the entire BEC,\n[1] - with end and begin points at the end of the condensate.
\nIn blue a cylindrical vortex that ends/begins inside the BEC. In red the contour C and its surface S is chosen in such a way that the surface does not intersect with the vortex core.
From the definition of the circulation we can now make three general conclusions about vortices in ultra-cold gases:
\nVortices will make a hole in the BEC.
The circulation is quantized with quanta h / m.
Vortices cannot have a beginning or ending within the BEC.
The second property is also the reason why vortices in ultra-cold gases are called quantum vortices, and throughout this chapter, we will however drop the prefix quantum. The last property also tells us that vortices have to be formed at the edges of the BEC, since they cannot end the start inside the BEC. This can be seen in experiments where a condensate that is removing its vortices just pushes them to the outside of the condensate, where they leave.
\nIn this chapter we will look at the simplest kind of vortices: vortices which have a cylindrically symmetric. This means that we will be looking at straight vortices with through the centre the (rotational) symmetry axis, for which we will choose the z axis. If our macroscopic wave function \n
where \n
What we see is that the velocity field lies along the ϕ direction, which is indeed a spinning motion in the counter-clockwise direction[1] - for positive values of n (vortices). If n is a negative number, the singularity is called an anti-vortex, and we have a clockwise rotation of the condensate. We also see that the velocity field drops as \n
This tells us that the angular momentum is quantized, every time that we make a vortex the condensate gets an angular momentum kick of the size nℏ. Finally we can also look at the circulation (26) of the wave function \n
Comparing the above result (37) with the circulation of a BEC (32), we can indeed see that the wave function, Eq. (34) can describe a cylindrically symmetric vortex. The number n in the phase factor of the wave function, Eq. (34) is the number of circulation quanta h / m the vortex will have.
\nWith now the macroscopic wave function (34) as a starting point, we can now start our study of vortices in a BEC. In order to determine the vortex structure, we should determine the functional dependence of \n
The first system that can be solved analytically is the homogeneous BEC. For this system we expect that the density is given by:
\nwith \n
Solving the above equation then yields for the chemical potential:
\nThe energy of the condensate can be found by taking the expectation value of the Hamiltonian (9), resulting in:
\nwith V the volume of the BEC and N the total number of particles in the BEC. The above formula for the energy (41) tells us that a BEC likes to spread out and maximize its volume (this decreases the energy). This means that in experiments we will need an external potential \n
which follows from the magneto-optical trap. This harmonic trapping potential leads to cigar-shaped condensates. Another type of confinement potential is the hard wall potential, this trapping potential is realized by using a hollow laser [65], this leads to a potential of the form:
\nIn this chapter, we will mainly focus on the latter hard-wall potential (43) because this yields the simplest calculations and still contains all of the physics. For the harmonic-trapping potential, it is also possible to make simple approximations in order to get easier calculations. The most common approximation made when working with a harmonic trapping potential is the Thomas-Fermi approximation. In the Thomas-Fermi approximation, it is assumed that the kinetic energy is negligible, and this is the case if the number of particles is large enough. In typical experiments this is usually the case.
\nThe next analytic case that will be considered is if we put a hard wall edge in the BEC. In other words we will punch a hole in the BEC and see how it heals. This analytic case is important because it will give us the length scale over which the condensate heals, also called the healing length. This healing length will be useful to use as a typical length scale of the system in order to make numerical equations dimensionless. To introduce the healing length, one usually looks at the Gross-Pitaevskii equation, Eq. (10) and fills in the length scale for which the kinetic energy term becomes equal to the interaction term. When substituting the second derivative in Eq. (10) by \n
for the healing length ξ this solves to:
\nGiven the healing length, we can now again look at a free BEC (in x > 0) in which we place a hard wall potential at x = 0. Filling in the Gross-Pitaevskii equation, Eq. (10) for a 1D BEC now yields:
\nIn the limit x → ∞ we get our homogeneous BEC back, meaning that the chemical potential is given by Eq. (40). If we now define the structure function f(x) as
\nand we fill in the chemical potential, then the Gross-Pitaevskii equation can be written as:
\nwhere we divided the Gross-Pitaevskii equation by gn\n∞ in order to get the healing length. The above non-linear differential equation is solved by substituting a hyperbolic tangent. The solution for the condensate wave function with a hard wall potential is then given by:
\nWhat we see is that the condensate ‘heals’ from a hole over a typical length given by the healing length ξ, and this is also illustrated in \nFigure 2\n.
\nThe healing of a BEC near a hard wall over a typical length of ξ.
This solution for a hard wall edge of the BEC will play an important role when we will be studying vortices. As said at the beginning of this section, a vortex will make a hole in the condensate from which the condensate also heals. It is to be expected that when the condensate heals from a hole of a vortex, it happens in an analogous way as when it heals from a hole due to a hard wall. This is because of the fact that both holes follow the same condensate physics, with the same boundary condition: \n
In essence, every experiment on atomic clouds is done by scattering probe laser light off it. So essentially experimenters will take a ‘picture’ of the condensate by using a laser in either a destructive or non-destructive manner. This ‘picture’ that is obtained is in essence the plot of the superfluid density field \n
This means that the main source of information is given by visual information. For the detection of vortices this is problematic since the healing length is of the order of 10 nm (if we fill in the quantities of the third section). To get an idea of how the healing lengths compare to the scattering lengths for several elements that are used to make a BEC, we have given a few values in \nTable 1\n for typical condensate atoms.
\nCondensate atom | \nScattering length as\n | \nHealing length ξ\n | \nas / ξ\n | \n
---|---|---|---|
Cesium (133Cs) | \n300 | \n3.646 × 102\n | \n1.214 | \n
Rubidium (87Rb) | \n100 | \n6.31 × 102\n | \n6.31 | \n
Sodium (23Na) | \n50 | \n8.92 × 102\n | \n17.84 | \n
Lithium (7Li) | \n30 | \n1.15 × 103\n | \n38.3 | \n
Hydrogen (1H) | \n1 | \n6.31 × 103\n | \n6310 | \n
The approximate scattering length a\n\ns\n and approximate healing length ξ (44) for a few typical atoms that undergo BEC [55].
Both lengths are given in units of a\n0 = 10−8cm (or 0.1 nm), for a typical particle density equal to n\n∞ = 1015cc−1 (or 1021m−3). Note that although the healing length is very small, it is always considerably larger than the s-wave scattering length.
From \nTable 1\n, we see that the healing length is indeed a very small value, which makes vortices hard to detect (and even measure their structure). The common method to detect vortices in a superfluid is to turn off the confining potential. As we know a BEC likes to maximize its volume, so turning off the confinement will lead to an expanding cloud of atoms (also called ‘Ballistic expansion’ by experimenters). Since the BEC expands, the cores of the vortices will start expanding as well, making the vortex cores easier to detect. Typical images that we get from this kind of experiments are given in \nFigure 3\n. As we see the vortices become visible after expansion. The two things that we can see in an experiment are how many vortices we have and how they are ordered.
\nFour experimental outcomes for vortices, depending on how fast the condensate is stirred. From left to right (a–d), the stirring frequency increases, leading to the formation of more vortices in the BEC. The diameter of the cloud in figure (d) is 1 mm after the ballistic expansion, which is a magnification of 20 times. Figures taken from Ref. [67].
During ballistic expansion, we can also extract extra information by using interference of matter waves (condensates). When we make two BECs overlap they will give rise to a matter interference pattern. This means that if we take a reference BEC (without any vortices) and make this interfere with a BEC that contains a vortex, the effect of the vortex can be seen in the interference pattern. The vortex shows up as a fork dislocation, an example of a BEC with one vortex overlapping with a BEC with no vortices is given in \nFigure 4\n. As seen in the figure, the presence of the vortex leads to an irregularity in the interference pattern that is known as a fork dislocation. The fork dislocation was also already known in optical systems where optical vortices are also a field of study [69].
\nTwo BECs in a trap, when the trap is turned off the condensates will expand and eventually overlap leading to an interference pattern. The top figures show two condensates without a vortex, in the bottom figures the left condensate has a vortex. In the interference pattern with the vortex, the fork dislocation becomes apparent. Figure taken from Ref. [68].
The additional information that we get with respect to the simple ballistic expansion is the fact that we can now also see the number of vortex quanta (or circulation quanta) that are present in the vortex. The more circulation quanta a vortex has, the more fringes will be involved in the fork dislocation. The interference pattern of two condensates can also tell us something about the sign of the circulation. A vortex (velocity field gives a counter-clockwise rotation) will have a fork dislocation which is upside down compared to the fork dislocation of an anti-vortex (spinning clockwise).
\nWith two basic solutions for ultra-cold gases and the general properties of a vortex from the previous section we can now continue. In this section we will first look at how vortices appear in a condensate. Afterwards, we will look at the structure of a single vortex and two simple variational models.
\nSince vortices classically appear when we start stirring our system, a rotating condensate might seem a good setting for a vortex system. In atomic gases this rotating system is achieved by stirring the condensate using a (blue de-tuned) laser which acts like a spoon. We will look at a system that rotates with a rotational velocity equal to Ω. Looking back at the previous section it only makes sense that we should be looking for vortices in a rotating condensate. The only way to get a rotating BEC (and hence a non-zero circulation in Eq. (26)) is by having a vortex in the BEC.\n[1] -
\nFrom the third section the energy of a BEC is already known and given by the expectation value of Eq. (9). This leads to three different energy contributions given by:
\nwhere the second equality in Eq. (50) follows from partial integration and the fact that the boundary term drops out.[1] - The time dependency was explicitly left out since we will be looking at stationary solutions for the vortex in the remainder of the chapter.
\nSince for vortices we will be working in the hydrodynamic description, we will substitute the hydrodynamic form of the wave function in Eq. (16) in the different energy contributions in Eqs. (50)–(52), this yields:
\nThe only missing ingredient is the Hamiltonian in a rotating frame of reference, this Hamiltonian is given by:
\nThis additional term \n
The simplest model for a vortex that we can imagine is to represent the vortex by a hollow cylindrical hole. The radius of the vortex hole is chosen to be equal to the healing length[1] - and the height of the vortex given by H. As we know the velocity field of the vortex should be given by Eq. (35), only inside the vortex core there is no condensate and we are free to choose our velocity field. In order to have a correct circulation (even within the vortex core) we assume the velocity field to take a constant value. The simple vortex model is depicted in \nFigure 5\n.
\nThe simplest possible vortex model: a hollow cylindrical vortex core with radius ξ in a cylindrical condensate with height H and radius R, shown in figure (a). Figure (b) shows the corresponding vortex velocity field.
In formulas this means that the simple vortex model is given by:
\nWith this model (and the above formulas) we can now calculate the different energies as given in Eqs. (41–44) for the rotating condensate.
\nSince only the energy difference compared to the homogeneous (vortex free) state is relevant, we will only calculate the energy differences between the homogeneous and vortex state. We will look at a condensate confined by an infinite potential well, so \n
If we have a vortex in our condensate, then we will get a velocity field of the form (58), which will give an extra kinetic energy. The difference kinetic energy (41) then becomes[1] -:
\nWe also have a contribution in energy of the interaction energy, namely the energy that is needed to punch a hole in the BEC. This needed energy is given by
\nwith
\nthe volume of the hole that has been made in the condensate. If we now assume (for simplicity) that[1] - \n
In order to be advantageous, the energy cost of making a vortex from Eq. (62) should be smaller as the energy reduction of making a vortex rotate. In order to calculate the energy reduction of rotating the condensate, we need to calculate the second term of Eq. (56), for this we need the expected angular momentum \n
With Eq. (63) the energy reduction of rotating the condensate is given by:
\nwhere we assumed that Ω is parallel to the vortex core. The fact that Ω is parallel to the vortex core is also needed because otherwise we can have circulation in the BEC without having a vortex, in the previous section we showed this was not possible.
\nA vortex will be formed in the BEC if it is energetically favourable; this means that the cost of making a vortex as in Eq. (62) is smaller than (or equal to) the energy profit or rotating the condensate as in Eq. (64). Setting the energy cost (62) equal to the energy profit (64) we can now calculate the critical frequency:
\nThe critical frequency yields the minimal rotational frequency that is required in order to get a vortex. The critical frequency is plotted as a function of the condensate size in \nFigure 6\n.
\nThe critical frequency in Eq. (65) as a function of R / ξ.
A first thing that can be noted is that the critical frequency given in Eq. (65) is directly proportional to the number of circulation quanta present in the vortex. This means that a n = 1 vortex (one circulation quantum) is the first possible vortex that can be formed. If we now rotate twice as fast we can in principle, a vortex can be formed with n = 2 (two circulation quanta) or we can make two n = 1 vortices. Which one of the two that will be formed depends on the relative energy of both vortex configurations. In order to see which configuration will be beneficial, it is worthwhile to look at how the energy cost (62) and energy reduction (64) depend on the number of circulation quanta n. For the energy cost (62) we see that \n
A second aspect that can be noticed is that from our simple derivation it follows that two vortices will repel each other, while a vortex and anti-vortex will attract each other. The fact that two vortices repel each other can be seen from the fact that if we try to push them in the same place, they will behave like one vortex with the double amount of circulation quanta. As we have seen one vortex with the double amount of circulation quanta is energetically less advantageous than two separate vortices. This means that two vortices will repel each other in order to avoid an energetically less advantageous situation. For a vortex-anti-vortex pair this is different, these will attract in order to reduce the energy. If we have a vortex and an anti-vortex the total circulation equals zero, meaning that the condensate as a whole is not rotating. The fact that the condensate is not rotating means that there is no energy reduction (64), but only an energy cost (62) for making the holes. By pushing the vortex and anti-vortex together, their velocity fields will cancel each other out, leading to an annihilation of both vortices. Interactions between vortex pairs and vortex-anti-vortex pairs will be studied in more detail in the next section.
\nThe results about vortex formation in a BEC can be summarized as:
\nEnergetically is more favourable to only make vortices with one circulation quantum (n = 1).
Two vortices will repel each other.
A vortex and anti-vortex will attract each other.
Since for a condensate it is advantageous to only make n = 1 vortices, we will assume for the remainder of the chapter that n = 1. So we will only look at vortices with one circulation quantum.
\nFor our derivation of the vortex formation we assumed that our vortex core was a cylindrical hole. As it will turn out this is not a bad approximation as long as \n
To determine the vortex structure we start from the Gross-Pitaevskii equation, Eq. (13), where we substitute the Laplacian in cylindrical coordinates \n
In the Gross-Pitaevskii equation, Eq. (66) we can now substitute the general cylindrical symmetric wave function (34). After eliminating the phase factor \n
The extra term next to the potential is the rotational energy due to the vortex flow, note that this term fill force \n
The Gross-Piteavskii equation, Eq. (67) will now be solved for the uniform BEC, so \n
We are now left with an ordinary non-linear differential equation (68) that needs to be solved. This means that the solution will be most likely given by a numerical solution. In order to easily solve the equation of motion (68) numerically, it should be made dimensionless. If we look at the limit \n
where the last equality follows from the solution of a homogeneous BEC in Section 4. We now have two relevant quantities to make the differential equation dimensionless, namely the value at infinity f\n0 and the healing length ξ:
\nUsing the asymptotic value and healing length (70), we can define a dimensionless radial coordinate x and a wave function \n
Substituting our dimensionless quantities (70) and variables (71) into the differential equation (68) yields:
\nThis remaining differential equation, Eq. (72) can now be solved. In order to solve the differential equation, Eq. (72) for the vortex structure, we impose the boundary conditions:
\nThe last boundary condition can also be replaced by
\ndepending on which numerical algorithm is used. In the latter case one should check whether the stable value equals one. In order to solve the equation of motion, Eq. (72) together with the boundary conditions (73) the shooting method can be used. The shooting method is a fast and popular way to solve non-linear ODEs with boundary conditions. Using the shooting method one starts in one of the boundary conditions and then ‘shoots’ out paths at different slopes. In order for a solution to become accepted, it should end within a given tolerance near the other boundary condition. At first the possible paths for \n
The shooting method solution of the ODE (72) with boundary conditions (73). The eventual vortex structure is plotted as a thick red dashed line.
As we see from the above discussion solving a non-linear ODE is quite involved and unstable. A more stable method to find the vortex structure is to numerically minimize the energy functional. In polar coordinates the total energy of the BEC (expectation value of Eq. (9)) is given by:
\nSince we have no ϕ of z dependency in the energy (75), these can be integrated out yielding an energy per height H of the condensate:
\nOne thing that can already be noted is the fact that for large values of ρ we have that f(ρ) becomes constant, this will yield a logarithmic divergence of the integral. In order to get sensible results we can either renormalize the integral by subtracting the ρ → ∞ behaviour, or (as is the case in Eq. (76)) keeping the integration interval finite. In order to simplify the numerics, the energy of the uniform BEC is subtracted from the energy of the cylindrical vortex (76). For a homogeneous BEC we know from Section 4 the energy of a homogeneous BEC (41), per unit length this becomes:
\nwith V the volume of the vortex core. Since the number of particles should be preserved when we go from a uniform BEC to a BEC containing a vortex, the number of particles per unit length N / H should also be preserved. The number of particles per unit length (per cross section of the cylindrical condensate) can be written as ν, related to n\n∞ as:
\nSince in the system with the vortex the number of particles per unit length ν should be the same as in the uniform case, we can now calculate ν using the vortex solution:
\nwhere we added and subtracted the uniform solution in the last step. Using both Eqs. (78) and (79), the energy of a homogeneous BEC per unit length (77) can be rewritten as:
\nwhere the term
\nwas dropped. The term that was dropped is of the order
\nSince we assume that \n
The advantage of writing the energy as Eq. (83) is that the last term now vanishes as ρ → ∞ together with the first term. The form (83) is a convenient form for numerical calculations of the vortex energy. By using our dimensionless quantities (70) and variables (71), the energy (83) can be made dimensionless yielding:
\nNote that the upper boundary of the energy integral equals \n
Note that this is the energy of the cylindrical vortex (62). To get a numerically even nicer form for the energy (83) the first (asymptotic) approximation (85) can be pulled out of the integral. After rewriting the numerically most optimal form is given by:
\nThe main advantage of Eq. (86) is that the integral integrates over \n
The energy in Eq. (86) as a function of R / ξ for the exact solution. The dashed line represents the asymptotic expansion (87).
As we see in \nFigure 8\n, the asymptotic solution of Eq. (87) becomes quite accurate quite fast (already around \n
In the previous we solved for the vortex structure exactly. Once we start looking at more complicated situations this will become quite a time-consuming activity. In order to reduce the numerical work and allow for more analytical work, it is possible to use a variational model. A good variational model should be subjected to the boundary conditions (73). Three possible candidates are given by:
\nIn order to make the calculations as easy as possible, it is wise to choose a variational function consisting of analytical functions, so either Eq. (89) or Eq. (90). Remember from Section 4 that a condensate heals from a hole like a hyperbolic tangent (49), this might be a possible argument to choose (89) as the variational vortex function. In order to determine the values of \n
Given these variational parameters we can now see how they compare to the exact solution. The three variational solutions for Eqs.(88)–(90) are plotted with the exact solution in \nFigure 9\n.
\nThe exact solution for the vortex structure together with the three variational solutions (88)–(90).
For the remainder of this chapter we will work with either the cylindrical vortex core (57) or the hyperbolic tangent (89). This choice of variational model is made because for the hyperbolic tangent it is already proven that it gives good results for the vortex core structure (88). For larger values of R / ξ, the hyperbolic tangent reduces to the cylindrical vortex, as it should since the exact solution does the same, in order to show that the hyperbolic tangent indeed reduces to the cylindrical case.
\nNow that we know how reliable our variational models are and their range of applicability, we can look at three different vortex configurations:
\nA vortex-antivortex system
An off-centre vortex
A vortex-vortex system
This will give us an idea of how stable vortices are in a BEC and how they interact. The interactions between vortices happen because the vortices feel each other’s velocity field, which leads to a Magnus force. For the first part, the easiest variational model (cylindrical hole) is used. Once the forces are known, we can also look at the energy (and thus potential) of the system in a variational manner. For this system, the variational solutions using the hyperbolic tangent are presented.
\nNote that although these methods might seem specific to vortices and hydrodynamics, there are a lot of other applications. An example is given by the application of the Magnus force to model the interaction of the solar wind plasma with the upper layers of planetary atmospheres. In the absence of particle-particle interactions between both plasma media (solar wind and atmosphere), the kinetic momentum of the solar wind will be transferred to the planetary particles through wave-particle interactions. These wave-particle interactions will produce vortex motion, which will cause these Magnus force interactions [70, 71].
\nThe interaction of vortices is because they feel each other’s velocity field. If we have a rotating system (vortex) that moves with a relative velocity with respect to a fluid, it will feel a Magnus force. A nice qualitative picture of the Magnus force is given in \nFigure 10\n. From the image on the right side, it is intuitively clear how the Magnus force will emerge. The rotating cylinder will bend the indecent velocity field. In order to bend the velocity field, an acceleration is needed, which results in a force (second law of Newton). This force will both act on the fluid as well as on the cylinder.
\nLeft: a flow (velocity field) around a rotating (circular) body together with the resulting Magnus force (red), right: the simple qualitative picture.
In order to calculate the force on a vortex, we will use the variational model of the cylindrical vortex core. In this case the force can even be analytically calculated. The starting point for the Magnus force is the Bernoulli equation
\nwhere \n
Note that \n
where the Gauss integration theorem was used. To calculate the Magnus force we now fill in the pressure using the Bernoulli equation, Eq. (95). The density field of our cylindrical vortex is known (57), the velocity field is given by [72]:
\nwhich is a combination of a doublet flow with a vortex flow in a uniform velocity field. For the Bernoulli equation, Eq. (95) we need the square of the velocity, leading to
\nIn order to calculate the Magnus force (96) we will use the first formula, expanding this in the three parts of the cylinder this gives:
\nThe first two integrals in Eq. (99) will be equal to zero since the upper and lower parts of the cylindrical vortex are not inside of the BEC, yielding:
\nthe e\n\nz\n component is zero since this comes from the lower and upper planes which are zero. Note that constant terms (p\n∞,…) do not contribute since they gets averaged out in the integration. Substituting the pressure \n
The Magnus force (per unit of length) can now be rewritten as[1] -\n
\nwhere \n
The first structure that we will study is a vortex and anti-vortex that can feel each other’s velocity field. This is the simplest system that we can have since for this system the condensate does not have to be rotating (total circulation is zero). We look at a vortex-anti-vortex pair where the two vortices are separated by a distance d, a top-view sketch of this system is given in \nFigure 11\n.
\nA top-view sketch of the vortex-anti-vortex system.
In order to determine the force, we need to fill in the formula of the Magnus force (102). What we need to know is the circulation and the translational velocity at which the vortex is moving. The circulation is known since we only look at vortices with one vortex quantum. For the translational velocity we say that the vortices are moving in each other’s velocity field (35), this allows us to estimate \n
As we see from the force on both vortices in the vortex-anti-vortex pair (103), the force points along \n
Given the previous vortex case, we can now look at a vortex that is displaced from the origin with a distance d. A top-view sketch of the system is given in \nFigure 12\n.
\nA top-view sketch of our single vortex system.
Now the translational velocity of the off-centre vortex is given by the rotational velocity of the trapping potential \n
As we see from Eq. (104) the off-centre vortex is being pushed towards the centre of the vortex due to the rotational velocity.
\nNow there is still one problem with our setup (\nFigure 12\n) that comes from the velocity field (35). Since we are working in a cylindrical trap with a hard wall potential, our velocity at the edge of the condensate cannot have a radial component (flow in or out the condensate). In order to get the correct velocity field, we should in principle solve the equations of motion again, given the new boundary conditions. However, since the velocity equation is a linear equation in this case we can apply the method of images. In order to get the right velocity field, that follows the boundary conditions, an anti-vortex is added at a distance b > R from the centre of the trap. The full situation with the image vortex is shown in \nFigure 13\n.
\nA top-view sketch of our single vortex system with an anti-vortex image.
To get a correct estimate of the force on the displaced vortex, we need to determine the position b of the image anti-vortex. To do this, we write down the total velocity field given by:
\nHere, the notation \n
Our Boundary condition (no radial flow) is given by
\nWriting out the left-hand side of the boundary condition Eq. (73) using Eq. (106) then yields:
\nFor the boundary condition Eq. (108) should equal zero, using the fact that on the boundary \n
Solving the second order equation in b then yields two possible solutions:
\nThe first solution is an anti-vortex in the same place as the vortex, hence an annihilation of the vortex. The second solution is a vortex outside of the condensate (since \n
The size of the velocity field \n
Note that in the limit d → 0 we get our single vortex case back, and in the case d → R the vortex and anti-vortex annihilate yielding a zero-velocity field.
\nUsing the above result, we see that in order to get a correct velocity field, we should add an anti-vortex. The addition of the anti-vortex means that our Magnus force gets an extra contribution, pulling the vortex out of the condensate. Since the distance between the vortex and anti-vortex is given by:
\nthe extra term in the Magnus force becomes equal to:
\nCombining both forces on the vortex (rotating condensate and anti-vortex), the total force on the vortex becomes equal to:
\nIn the total force we see a competition between the two forces, by looking for the positions of the vortex d where the force equals zero, we can find the equilibrium positions. Setting Eq. (116) to zero then yields two solutions:
\nThe first solution yields a stable position at the centre of the vortex, while the second solution gives an equilibrium outside of the centre. Note that the second solution only exists if the condensate is rotating fast enough. The second solution is thus some kind of maximum (barrier) needed to keep the vortex inside of the condensate when it is rotating. The minimal velocity needed to keep the vortex inside is given by:
\nNote that this frequency (119) is almost the same as the critical frequency (65). The big difference with the critical frequency is that there we have a length scale (the healing length) that also plays a role. The reason for this difference in frequency is because of the fact that when the Magnus force (102) is calculated, the typical length scale of the vortex (healing length) gets eliminated. This means that the obtained results are excellent as long as the vortices are far enough away from each other. When the vortices do get close to each other, for example if the off centre vortex reaches the edge, we need a more precise (variational or exact) calculation that also takes the vortex structure into account.
\nUsing the total force on the vortex (116) a potential can also be defined as:
\nThis potential (120) is also plotted in \nFigure 14\n. As we see in \nFigure 14\n, as long as the condensate spins fast enough, a vortex will always be pushed towards the centre.
\nThe resulting potential for the off-centre vortex. The black dashed line and dots yield the second stationary position of the off-centre vortex (118).
The last situation we can look at is the vortex-vortex interaction. As we know from the previous section two vortices will repel each other, meaning that now the centre of the trap will not yield a stable solution. As we know from the off-centre vortex, we want our boundary condition to hold Eq. (107), which means that each vortex should have an anti-vortex. A sketch of the two-vortex system is given in \nFigure 15\n.
\nA top-view sketch of the two vortex system with the corresponding anti-vortices.
As earlier, we can now write down the Magnus force of the three different vortices on one of the vortices in the condensate. Since the system is symmetric, it does not matter which vortex is chosen for the calculation; we chose the right one, yielding:
\nAs we see from Eq. (121) the vortices push each other out of the condensate. Since the vortices give a total circulation to the condensate, we should also have a rotation of the superfluid.[1] - Adding the attractive rotational force then yields for the total force on the (right) vortex that:
\nAgain we can look for stable positions of the vortex pair by solving the equation
\nThis equation (123) is analytically solvable by substituting the variable y = d\n2 and solving the remaining equation of third degree by Cardano’s method. This will however lead to a large (non-insightful) equation. For simplicity we assume that \n
The equilibrium position, both exact and the above approximation (124), is plotted in \nFigure 16\n. As we see, the faster the condensate rotates, the more the vortices are pushed together. In the limit \n
The equilibrium position of a vortex-vortex pair.
Notice that this is slightly higher than the critical frequency we would expect from the discussion in Section 5, where we would have:
\nThis overestimation is again an artefact of the fact that the vortex core structure is ignored.
\nAgain it is possible to define a potential which yields:
\nThis potential is plotted for two different rotational frequencies in \nFigure 17\n. In \nFigure 17\n, we see that the vortices indeed repel each other but stay in the trap if the trapping frequency is high enough.
\nThe potential for a vortex-vortex pair for two different frequencies. The solid curves yield the full solution, the dashed curves are the limit d ≪ R.
From the previous sections we now know four variational models for the vortex structure \n
The above macroscopic wave function (128) is not limited to one-vortex states. We can also construct a structure function \n
where \n
with \n
where the sign of κ depends on whether you have a vortex κ > 0 or anti-vortex κ < 0. If we look at a multi-vortex structure with \n
Since we are going to look at stability of the system, we will again add the rotational term according to Eq. (56). For the off-centre vortex and vortex-vortex structures, this will be needed. For the vortex-anti-vortex this will not be needed since the condensate will not be rotating then.
\nFinally since we will be working in polar coordinates, our structure functions \n
Of course in order to do a variational calculation we need an energy functional, this is given by:
\nIn this chapter we look at a homogeneous condensate (\n
Choosing the rotational velocity along the z-axis is possible to do the z integration resulting in:
\nThe above energy can be used for further variational calculations; the calculations themselves are left to the reader.
\nGiven the variational energy (136) it is possible to look at our three different vortex cases again. To conclude the section the different energies for the three-vortex systems above will be calculated variationally. For the different calculations a trap of the size R = 30 was chosen with a healing length of ξ = 1. As a variational model, the hyperbolic tangent was chosen.
\nFor the vortex pair the energy as a function of the distance between the vortices is given in \nFigure 18\n. As we see the vortex-anti-vortex pair attracts as well in the variational model so our results from the simple calculations on the Magnus force still hold. For the off-centre vortex and vortex pair, a rotating condensate is needed. This means that the two integrals of Eq. (136) should be integrated. The first integral yields the energy of the stationary condensate, the second integral yields the energy reduction of rotating the condensate. The total energy is then given by a competition between both energies.
\nThe variational energy for a vortex-anti-vortex pair the solid line gives the full solution, whereas the dashed line yields the asymptotic solution.
The potentials for the off-centre vortex and vortex-vortex systems are given in \nFigure 19\n. As seen on \nFigure 19\n, a repulsive and attractive energy are in competition. Depending on the values of the rotational frequency we will have different situations. The situation for the off-centre vortex is given for a few values of the rotational frequency Ω in \nFigure 20\n. What we can notice in \nFigure 20\n is that for a single vortex we find a non-centred vortex in the stable situation. This is because of the fact that the variational velocity field of the cylindrical model was used for the hyperbolic tangent, yielding small deviations. For the vortex-vortex case the same can be done, resulting in the variational energies as given in \nFigure 21\n. Here we again see that as the rotational velocity increases, the vortices get pushed more towards the centre of the trap, while if the rotational velocity decreases below a certain value the vortices get pushed out of the trap.
\nThe kinetic (a) and rotational (b) energy for the off-center vortex.
The variational energy for the off-centre vortex for several values of Ω. The dashed lines trace the minima and maxima of the potential.
The variational energy for the vortex-vortex for several values of Ω. The dashed lines trace the minima and maxima of the potential.
To conclude the discussion about vortices in BECs we will look at a vortex induced phase transition, the Berezinskii-Kosterlitz-Thouless (BKT) phase transition (Nobel Prize in Physics 2016).
\nVortices carry entropy. The entropy of a given state can be computed from the free energy,
\nbut in this simplified presentation we will use some qualitative arguments to estimate the entropy. The ‘ size’ of a vortex is given by its healing length ξ, and the number of ways that an object of this size can be placed in a condensate of radius R is estimated by \n
A schematic view of a possible microstate contributing to the vortex positional entropy.
A qualitative picture of the BEC-BCS crossover, figure adapted from Ref. [83].
When two vortices are present, or a vortex and an anti-vortex are present, we consider them to be independent. From the previous section, it is clear that this is an approximation that fails when they are close together. However, the number of microstates where this happens is small, and we can still assume the entropy is extensive in the number of vortices:
\nWhen information (or order) is added to the system, the entropy must decrease. For example, we might know that the vortex and anti-vortex are bound, and must be in adjacent patches of \nFigure 22\n. Then the number of microstates that are compatible with this constraint is again proportional to \n
In a three-dimensional Bose gas, superfluidity vanishes above the critical temperature for Bose-Einstein condensation. In the normal state, phase correlations vanish exponentially fast as a function of distance, and phase coherence is absent. In the BEC regime, the phase is locked by long-range off-diagonal order [73], and the phase correlation function is a constant as a function of distance. However, this is not the case for a two-dimensional Bose gas. Due to quantum fluctuations, there is no phase transition to the Bose-Einstein condensation (more accurately: the critical temperature is zero). Nevertheless, this does not mean that phase coherence is lost altogether: the phase correlation function gets a power-law decay rather than the exponentially fast decay in the normal state [74]. This regime is called ‘quasi-condensation’ and allows for superfluidity.
\nObviously, the mechanism whereby superfluidity is destroyed must be different. Berezinskii [75] and independently Kosterlitz and Thouless [76, 77] proposed that phase coherence gets destroyed by the unbinding of vortex-anti-vortex pairs, leading to a sudden proliferation of vortices and anti-vortices that scramble the phase coherence. Here, the results for the interaction energy between vortices and anti-vortices calculated above will be of use. The binding energy of a vortex-anti-vortex pair can be defined as the energy it takes to move a vortex and an anti-vortex that form a pair, away from each other to infinity.
\nHeuristically, the Berezinskii-Kosterlitz-Thouless (BKT) transition can be thought of as a competition between the internal energy and the entropy for vortices. The realized state is that which minimizes the free energy
\nWhen the temperature is low T ≈ 0, the internal energy (here, the binding energy of the vortex-anti-vortex pair) dominates. In this case, the pairs remain bound and no uncompensated phase singularities perturb the coherence. Indeed, as can be seen from \nFigure 18\n, the internal energy is minimal when the vortex and anti-vortex are co-located. The phase windings then cancel out, and U = 0, F = 0. When the temperature is increased, the entropy term TS can become large enough to dominate the internal energy, leading to a transition to the more disordered state. Focusing on the vortex-anti-vortex pair, the disordered state (the state with highest entropy) is that in which the vortex and the anti-vortex are unbound and can be placed freely anywhere in the available volume. In that disordered state, we will also take U to be equal to twice the kinetic energy as in Eq. (62)—the number of microstates where the vortex is close to the anti-vortex is negligible in comparison to the number of microstates when they are well separated, so we can neglect the interaction. This allows to approximate the internal energy of the disordered state by
\nComparing the free energy of the state with bound pairs (F = 0) with the free energy of the disordered state yields the critical temperature where superfluidity vanishes:
\nUsing our earlier results (142) for U and (139) for S, the critical temperature is found to be
\nIn the units used for (143), we can see that the right-hand side indeed describes a temperature, and that it is determined by the superfluid density n\n∞. Above this temperature, it is advantageous for the free energy to spontaneously create and unbind vortex-anti-vortex pairs that scramble the phase and turn the gas into the normal state. As mentioned in the introduction, the BKT transition was observed experimentally [40]. The observed critical phase space density differs from the back of the envelope estimate given here by a factor of about four, mostly due to the fact that the experimental condensate does not have a uniform density, and also due to finite size effects.
\nWith this chapter an introduction to vortices in ultra-cold gases was given. The main focus of the discussion was aimed towards atomic gases of bosons. However the derived (qualitative) results are also applicable to fermionic quantum gases. In this concluding section the main results of the chapter are summarized together with a first look at a condensate of fermionic atoms.
\nConcerning vortices in superfluid atomic gases, the following conclusions can be made:
\nA vortex will always make a hole that extends throughout the entire condensate.
The circulation of a vortex is quantized in multiples of h / m see Eq. (32).
Vortices in a BEC will always carry one quantum of circulation, this is energetically more favourable.
For vortices a minimal rotational frequency (critical frequency) is required as given in Eq. (65).
Two vortices will repel each other.
A vortex and anti-vortex will attract each other.
Rotating a BEC faster will push vortices to the centre of the trap more, for a single vortex.
A single vortex in a trap will be most stable in the centre of the trap, for a single vortex this means that it is stable in the centre of the trap.
For a 2D quasi-condensate superfluidity is destroyed by the unbinding of vortex-anti-vortex pairs that destroy the overall phase coherence. This happens at a certain critical temperature T\n\nC\n.
This chapter was restricted to superfluidity in Bose gases since these already (qualitatively) describe the behaviour of vortices in general superfluids very well. If we are working with fermionic gases a few things change, these are briefly discussed in this section. More elaborate discussions about fermionic quantum gases can be found in references [78–81].
\nFirst of all when we have fermions, we have to take the ‘spin’. With spin we label the different quantum states of our atom, in fermionic atomic gases these will be two different hyperfine states of the considered atom. These spins will be called ‘up’ and ‘down’ as an analogy with the spin known from standard quantum mechanics. In order to fix the number of particles in both spin states, we need two chemical potentials labelled μ\n↑ and μ\n↓. If we now say that we have an ultra-cold gas and thus have s-wave scattering, the Pauli exclusion principle tells us that only opposite spins can interact. Since the s-scattering wave function is symmetric, the spin wave function should be the anti-symmetric singlet state. Applying this to our Hamiltonian we get:
\nGiven the above Hamiltonian (144) atomic Fermi gases can be studied. In order to simplify calculations, one usually defines the average chemical potential μ and polarization ζ as:
\nOf course the question we can ask now is ‘what is the criterion for superfluidity now’? In Section 3, this criterion was given by the Penrose-Onsager criterion which states that the one-body density matrix should have a macroscopic eigenvalue, indicating that there was one state being occupied macroscopically. For fermions, the criterion for bosons was expanded by C.N. Yang, hence named the Penrose-Onsager-Yang criterion. Yang stated that for fermions the second order density matrix should have a macroscopically large eigenvalue, indicating a condensation of fermion pairs. These fermion pairs are also referred to as Cooper pairs, named after L.N. Cooper who first proposed this for electrons [82]. Once this is assumed it is possible to rewrite the entire system in terms of a macroscopic two-body wave function \n
where R is the centre of mass coordinate and \nρ\n is the relative position. So in the end \n
The study of superfluid fermionic atoms opens up a few extra parameters to play with in experiments, which can help explore the physics of ultra-cold gases. The two most prominent are:
\n\nSpin-imbalance: A first parameter to play with is the number of ‘spin-up’ and ‘spin-down’ atoms. If the numbers are unequal (or if the polarisation ζ in Eq. (146) is non-zero), then you will have excess atoms when the condensation occurs. These excess atoms will need to go somewhere. Usually excess atoms are pushed towards the edge of the condensate, or placed inside holes in the condensate, for example vortices. When the number of ‘spin-up’ and ‘spin-down’ atoms is unequal one also talks about pair frustration since the unbound (frustrated atoms) will still try to pair up with other paired atoms.
\n\nBEC-BCS crossover: A second parameter that can be played with is the interaction strength a\n\ns\n, this can also be done with bosons but is more interesting to do with fermions. With fermions it is possible to change the sign of the interaction from negative (repulsive) to positive (attractive). By doing this it is possible to go from a gas of interacting fermions (Cooper pairs or BCS), to a gas of interacting bosons (BEC) by making the interactions so strong that the particles are quasi bosons. The point at which the interaction changes in sign is also referred to as unitatiry. A qualitative picture of the transition form a repulsive interaction (BCS) to an attractive interaction (BEC) is given in \nFigure 23\n.
\nThe authors gratefully acknowledge the extensive, weekly discussions with S.N. Klimin, G. Lombardi, W. Van Alphen and J.P.A. Devreese. Financial support was provided by the Fund for Scientific Research Flanders, through FWO projects: G042915N (Superfluidity and super-conductivity in multi-component quantum condensates) and G011512N (Quantum turbulence in atomic and solid state Bose-Einstein condensates). One of us (N.V.) acknowledges a Ph.D. fellowship of the University of Antwerp (UAntwerp).
\nMaternal infections during pregnancy can have a direct impact on the developing fetus and in some infections can result in fetal demise. It is extremely important to screen women for infections when it is available and practical and to treat when necessary. The current screening tests recommended by the American College of Obstetricians and Gynecologists include rubella, hepatitis B, hepatitis C, human immunodeficiency virus (HIV), Group B streptococcus (GBS), tuberculosis and sexually transmitted infections including syphilis, chlamydia, and gonorrhea if risk factors are present [1]. The incidence of congenital infections in infants varies, with syphilis increasing dramatically from 639 cases in 2016 to over 1300 cases in 2018 in the United States [2]. Additionally, congenital cytomegalovirus, varicella zoster virus and herpes simplex virus diagnoses have increased over the last five decades [3]. Rubella has decreased since the introduction of Rubella immunization; prior to utilization of the immunization, over 100,000 infants were born worldwide with congenital rubella syndrome (CRS). By 2014, a 95% decrease in cases of CRS was observed in countries that followed the immunization schedule [4]. Thus, it is critically important that research efforts continue to prioritize the development of immunizations and treatments plans for all viruses that can result in congenital fetal infection in an attempt to minimize the substantial long-term morbidities that result.
Chorioamnionitis is the term that has been used for decades to describe infection and/or inflammation of the chorion, amnion, or both. This has been further delineated into a “clinical” diagnosis based on maternal symptoms, and a “histological” diagnosis based on the pathology of the placenta following delivery. Clinical signs and symptoms are used to diagnose clinical chorioamnionitis, and include maternal fever, uterine fundal tenderness, maternal and/or fetal tachycardia and purulent amniotic fluid [5]. The most common bacterial organisms to cause chorioamnionitis are Ureaplasma urealyticum and Mycoplasma hominis. Histological chorioamnionitis is diagnosed by observing neutrophil infiltration into the chorion and amnion [6]. The variation in the definition of chorioamnionitis has resulted in confusion in neonatal management as well as difficulty in assessing the long-term impact of chorioamnionitis on development. Therefore, intra-amniotic infection (IAI) has been developed to replace the prior diagnosis of chorioamnionitis [7].
IAI was updated in 2017 by the American College of Obstetricians and Gynecologists into three categories which are readily diagnosed. Isolated maternal fever (IMF) is the first category, in which the mother has a single intrapartum temperature of ≥39.0°C or a temperature of 38.0–38.9°C that persists for 30 min, with treatment recommendations including the consideration of broad-spectrum antibiotics [7, 8]. Given the numerous potential causes of maternal fever, the utilization of antibiotics is at the providers’ discretion. Suspected IAI is diagnosed when the mother has an elevated temperature (≥39.0°C) or a slightly elevated temperature (38.0–38.9°C) along with one of the following risk factors: maternal leukocytosis, purulent cervical drainage or fetal tachycardia [7, 8]. Confirmed IAI is diagnosed with a positive amniotic fluid test or placental pathology demonstrating histologic evidence of infection [7]. Similar to the previously used histological chorioamnionitis, a criticism of this diagnosis is that it is made after the clinical situation has resolved, and thus does not aid in the acute management of the mother or the infant. Both suspected and confirmed IAI diagnoses should result in treatment with intrapartum antibiotics and antipyretics [7].
IAI is present in nearly 50% of very early preterm birth [9], after which multiple complications can occur and a wide array of neonatal morbidities and mortalities are observed. This has led to speculation that IAI is directly impacting the fetal and neonatal development and outcomes, as well as potentially resulting in preterm birth, which then impacts development and outcomes. The majority of studies that have investigated this question utilized diagnoses of chorioamnionitis, which included both clinical and histological cases. Given the variation of diagnoses included in these studies, it is not surprising that the results have also been varied. A large study of 2390 extremely preterm infants (born <27 weeks’ gestational age) from sixteen centers across the United States found infants exposed to histological and clinical chorioamnionitis had an increased risk of cognitive impairment at 18–22 months’ corrected age [10]. A separate study of 350 infants found that while gestational age was significantly lower among those with exposure to histological chorioamnionitis, there was no association with intraventricular hemorrhage, white matter injury around birth, or differences in cognitive or motor outcomes at 18–24 months’ corrected age [11]. Additional studies have found weak causal or associative roles of chorioamnionitis with cerebral palsy risk [12] and no increased risk of white matter injury on magnetic resonance imaging (MRI) following histological chorioamnionitis in premature infants [13]. Additional investigation is required with the new IAI definitions to determine if there are consistent findings with developmental outcomes in those diagnosed with IAI.
TORCH infection is a mnemonic that has classically been used to describe congenital infections that can impact fetal development. In the past, TORCH represented Toxoplasmosis, Other (syphilis, varicella-zoster, parvovirus B19 and newer pathogens such as Zika), Rubella, Cytomegalovirus and Herpes Simplex Virus. However, as more pathogens are being discovered and the “other” category is expanding, some experts feel the mnemonic is not as relevant today.
Toxoplasma gondii is an obligate intracellular protozoan which typically causes mild illness in most immunocompetent individuals [14, 15]. While a large portion of infected children and adults are asymptomatic, Toxoplasmosis is considered one of the major causes of death linked to foodborne illness in the United States. If an immunocompromised individual, pregnant woman, or fetus/infant acquires the infection, there can be severe, even fatal, consequences [14, 15]. Illness can range from non-specific systemic symptoms such as fever, lymphadenopathy and hepatosplenomegaly to congenital toxoplasmosis (CT), which is classically described as a triad of chorioretinitis, intracranial calcifications and hydrocephalus. CT can lead to loss of vision and hearing, decreased cognitive function, and neurodevelopmental delay if untreated [14, 16–18].
T. gondii exists in three forms: tachyzoite, bradyzoite, and sporozoite. The definitive hosts are members of the Felidae family, but warm-blood mammals can also serve as intermediate hosts [17]. Felines can acquire T. gondii through the ingestion of tissue cysts containing bradyzoites in infected prey or through the ingestion of oocysts containing sporozoites in anything contaminated with feces from an infected cat. They can excrete un-sporulated oocysts in their stools 3–30 days after infection and can shed for 7–14 days. If in the right climate (such as warm and humid), the oocysts can sporulate for 1–5 days, after which they can remain infectious for years. If the tissue cysts found in intermediate hosts or the sporulated oocysts are ingested by humans, they transform into active tachyzoites. The tachyzoites then primarily infect the central nervous system, eyes, musculoskeletal system, and placenta by infecting nucleated host cells to bypass the blood brain barrier and placental barricade. Incubation is 7 days with a range of 4–21 days [14, 15, 18].
For pregnant women who have an acute infection with T. gondii, the timing can be crucial and dictates the treatment course. Typically, the earlier in pregnancy that acute infection occurs, the lower the rate of transmission to the fetus. Unfortunately, there is an increased severity of illness if transmission occurs earlier in the pregnancy [14, 15]. The reverse is true for infection later in pregnancy (such as during the third trimester), during which there is a high rate of transmission but with less severe illness in the fetus.
The diagnosis of primary or latent infection is made primarily using serologic tests. Toxoplasma-specific Immunoglobulin G (IgG) and Immunoglobulin M (IgM) can be performed routinely at non-reference laboratories. Any positive IgM results are then submitted to reference laboratories that can perform additional testing for confirmation [18]. If a pregnant woman is found to have acute infection, then an amniocentesis can be performed, and the fluid can be sent for polymerase chain reaction (PCR) testing. If the PCR is negative and the fetus is believed to have not acquired the infection, the next best step is treatment in the mother with spiramycin in an attempt to prevent transmission [14, 15, 17, 18]. If, however, the fetus is thought to be infected, then the mother is started on a combination of pyrimethamine, sulfadiazine, and folinic acid. Spiramycin, a primarily bacteriostatic macrolide that has activity against some gram-negative and gram-positive organisms as well as some spirochetes, is unable to cross the placenta whereas the combination of anti-parasitic medications can cross the placenta and thus can aide in treatment of the fetus [18, 19]. The combination is also used for fetal infection confirmed at or after 18 weeks of gestation or maternal infection acquired during the third trimester [14, 17, 18]. As untreated CT can lead to fetal demise or death within the first few days of life, and chorioretinitis can develop in a significant proportion of infants whose mothers were untreated, it is imperative to diagnose and start treatment in a timely manner [18].
Once an infant with suspected CT is born, he or she should be thoroughly examined and evaluated. Serologies, a complete blood count (CBC), hepatic function tests, blood PCR, urine PCR, cerebrospinal fluid (CSF) PCR, and CSF studies including glucose level, protein, and cell count, should be sent [18]. The newborn should also have ophthalmologic, auditory, and neurologic evaluation including imaging of the brain [18]. Infected infants should receive treatment regardless of any clinically apparent symptoms, as a large proportion of infants with asymptomatic CT at birth go on to develop visual/hearing impairment, learning disabilities, and psychomotor delay [15, 16, 18, 20]. Treatment consists of the same anti-parasitic combination of pyrimethamine, sulfadiazine, and folinic acid [18, 19]. If CSF studies show an elevated protein concentration (greater than 1 g/dL) or there is evidence of severe chorioretinitis, then a corticosteroid such as prednisone is added until there is a decrease in protein concentration in the CSF or resolution of severe chorioretinitis [15, 18, 19]. Treatment is continued at least though 12 months of age, with consideration of shorter treatment duration for infants who remain asymptomatic for the first three months of life [18, 19]. For those infants who are asymptomatic with positive Toxo-specific IgG but negative IgM and Immunoglobulin A (IgA), there should be repeat IgG testing every four to six weeks until disappearance of IgG. There is no clear consensus on the treatment of these infants [18, 19].
Studies looking at the outcomes of infants with CT have shown significantly better neurologic and developmental outcomes in those that were treated than those who were not [21]. It is important to note that compared to their uninfected siblings, the children that received treatment had a lower level of cognitive function though there was no deterioration over time. In terms of ophthalmologic outcomes, it was found that when followed up to 22 years of age, new ocular lesions could be detected in adolescence which points to the importance of continued ophthalmologic evaluation.
Treponema pallidum, a thin, motile spirochete, is the organism that causes syphilis [18], a sexually transmitted infection that can also result in congenital infection to a fetus. While there was initially a decline in the cases of syphilis observed in the United States in 2000–2001, an alarming resurgence has recently been noted. There has been an increase of 72% in the number of reported primary and secondary cases in the United States from 2013 to 2017, with the number of congenital syphilis cases increasing more than 150% from 2013 to 2018 [22–24]. It is thought that the increase in methamphetamine use, having sex with a person who injects drugs, injection drug use and heroin use are the primary factors that are leading to this dramatic increase in syphilis cases [22, 25].
Acquired syphilis is typically divided into three stages: primary, secondary and latent. During the primary stage, painless indurated ulcers form on the skin or mucous membranes of the areas exposed and heal spontaneously in a few weeks. The secondary stage, typically 1–2 months after the primary stage, is characterized by a maculopapular rash that typically includes the palms and soles, lymphadenopathy and mucocutaneous lesions including condylomata lata [18]. Finally, the latent stage occurs when there are no clinical signs or symptoms of infection, but an individual remains seroreactive [18]. T. pallidum can infect the central nervous system (CNS) during any stage, resulting in neurosyphilis. Transmission to the fetus during pregnancy can occur at any point, with primary and secondary syphilis having the highest rates of transmission at 60–100% [18].
It is recommended that all women be screened for syphilis early in pregnancy with a nontreponemal test, with repeat testing later in pregnancy for high risk individuals. These tests include the Venereal Disease Research Laboratory (VDRL) slide test and the rapid plasma reagin (RPR) test [18]. These nontreponemal tests utilize an antigen that reacts in the presence of antibodies (to syphilis). However, given that the antigen is not specific for syphilis and is a component of cell membranes, false positives may result from other infections including varicella and measles, or by tissue damage observed in connective tissue disease and even pregnancy itself [26]. Therefore, a positive nontreponemal test should be followed by a confirmatory test such as fluorescent treponemal antibody absorption (FTA-ABS) or T. pallidum particle agglutination (TP-PA) tests. Additionally, any person found positive for syphilis based on screening and confirmatory testing should also be screened for human immunodeficiency virus (HIV) given the high rate of co-infection.
Treatment for syphilis is parenteral penicillin G; if an individual is allergic to penicillin G, they should undergo desensitization due to the lack of proven efficacy of alternative agents in this setting. Lack of treatment during pregnancy can result in stillbirth and neonatal death in nearly 40% of women with primary and secondary stage disease, 40% of infants being infected and only 20% of infants being healthy and uninfected [27]. Additionally, fetal infection can result in anemia, hepatomegaly and hydrops [24]. Treatment of the infant should not be delayed, as early treatment may prevent neurologic sequelae [24].
A serological diagnosis is made on the infant if the nontreponemal titer (VDRL or RPR) is fourfold higher than that of the mother (both samples should be obtained around the same time), if the nontreponemal titer persists or increases after birth, or if the treponemal antibody titer (FTA-ABS or TP-PA) remains positive at 12–18 months of age. The choice of test on the infant is dependent on the test that the mother had received, as the titers will need to be compared [18]. A complete evaluation, including complete blood cell count (CBC), liver function tests, obtaining cerebrospinal fluid (CSF) to test for VDRL reactivity, ophthalmologic examination and long-bone radiographs to assess abnormal ossification, radiolucencies or dislocation of epiphyses is then needed [28]. Neuroimaging should be considered if there are any concerns for central nervous system involvement [18]. Ten days of treatment with parenteral penicillin G is typically used in infected infants, with close follow up required. Titers should be repeated by 3 months of age and noted to be declining, with nonreactivity noted by 6 months of age [28]. If the mother received appropriate treatment that was administered >4 weeks before delivery, and the infant has a normal physical examination with the titer equal to or less than fourfold the maternal titer, then no evaluation is recommended. However, inadequate treatment in the mother should result in evaluation of the infant and treatment with penicillin G for 10 days [28].
Clinically, nearly half of infants do not have any apparent signs of infection, although bone lesions and hematologic and hepatobiliary abnormalities may be present, with hepatomegaly one of the most common findings [24, 29]. Infants that develop symptoms may have rhinitis in the first week of life, in which persistent white discharge (“snuffles”) occurs which contains spirochetes [29]. Additional symptoms can include generalized lymphadenopathy and a maculopapular rash [29]. Long term outcomes of infants not appropriately treated can include sensorineural hearing loss, interstitial keratitis, secondary glaucoma, corneal scarring, vision impairment, Hutchinson teeth (smaller teeth that are widely spaced with notches), saber shins (sharp anterior bowing of the tibia), frontal bossing, saddle nose, gummas (soft, non-cancerous growth) and scarring [29]. Life-long disabilities can occur in congenital syphilis infections if infants are not appropriately screened and treated [28].
Varicella-zoster virus (VZV) is a herpesvirus that is transmitted by respiratory droplets, direct contact with skin lesions, and transplacentally during pregnancy [30]. Infants that are exposed to VZV during the last few weeks of pregnancy may develop neonatal varicella which can be quite severe; congenital varicella syndrome (CVS) develops in infants exposed during the pregnancy, with the risk being highest if the exposure occurs in the first trimester [30]. Infants exposed after 20 weeks’ gestation only have about 2% chance of developing CVS [31]. Infants with CVS most commonly have skin lesions in a dermatomal distribution followed by neurologic defects, eye disease and skeletal anomalies [31]. Neurologic defects can include cerebral cortical atrophy and ventriculomegaly. Unfortunately, CVS is fatal in about 30% of cases within the first month of life [32].
The monovalent vaccine approved in 1995 and the quadrivalent vaccine introduced in 2005 have impacted the prevalence of congenital infection as seroprotection is nearly 100% after 2 doses of the vaccine [18]. Thus, at this time, CVS is considered an extremely rare disorder.
Human parvovirus B19 is a nonenveloped, single-stranded deoxyribonucleic acid (DNA) virus with humans as the only host [18]. The virus replicates in erythrocyte precursors and is transmitted via respiratory tract secretions, exposure to blood or blood products, and vertically [18]. While it often causes a mild respiratory tract infection with a “slapped cheek” rash, it can be lethal to a fetus, with the risk of death being as high as 10% [33]. The incidence of parvovirus B19 infection during pregnancy is 3–4%, with the transplacental transmission rate approaching 30% [34]. Fortunately, approximately 50–75% of women of reproductive age are immune to parvovirus B19 [35]. The timing of infection during pregnancy does alter the risk of fetal death, with first trimester infections resulting in up to 71% risk of fetal loss [34]. The difficulty in diagnosing the virus during pregnancy arises in the lack of symptoms that most adults experience, and as many as 70% of women would have no symptoms if infected during pregnancy [34]. Arthropathies are one of the most common symptoms and should raise suspicion for possible infection [34]. Additionally, the presence of fetal ascites or pericardial effusions on ultrasound should trigger high suspicion as well [33].
Fetal hydrops, or abnormal accumulation of fluid/edema in two or more compartments, is common in the setting of Parvovirus B19 infection, with a meta-analysis finding a 9.3% pooled incidence, as well as an increased risk of fetal loss, spontaneous abortion and stillbirth [36]. Parvovirus B19 is among the most common causes of non-immune fetal hydrops, and while spontaneous resolution of infection can occur, only about 5% of cases with hydrops will show spontaneous resolution of the infection with disappearance of hydrops on follow up ultrasounds [37].
Severe anemia and thrombocytopenia occur in utero following parvovirus B19 infection, along with myocardial dysfunction [38]. These factors together are likely the etiology of the fetal hydrops. In utero transfusions (IUT) are often necessary and reduce mortality rates when compared to expectant management. A meta-analysis found IUT was performed in 78% of hydropic fetuses compared to 29% of non-hydropic fetuses, with the difference likely due to the hydropic fetuses at higher risk of demise [37]. Complications may occur in up to 5% of cases, especially if the fetus is likely more sensitive to vascular overload [38]. Thus, intrauterine exchange transfusions (IUET) have also been attempted in cases of fetal hydrops in the setting of parvovirus B19 infection. Unfortunately, thus far it results in similar survival rates as IUT and does not seem to be clinically superior as a treatment modality [38].
Longer-term testing reveal abnormal neurodevelopment following intrauterine parvovirus B19 infections in those also diagnosed with hydrops. Brain abnormalities including parenchymal calcifications, venous infarction, arterial infarction, cerebellar hemorrhage, and cortical malformations including diffuse cortical dysplasia and polymicrogyria have been described in congenital parvovirus infections [39]. If there are no abnormalities on imaging and hydrops resolves prior to delivery, one study found normal neurodevelopment in survivors at 1- and 5-year follow-up [40]. While the overall risk of mortality and morbidity are high, there is the potential for a normal outcome in select cases of congenital parvovirus infections.
Zika virus, ZIKV, is an emerging flavivirus that first became apparent internationally after Brazil declared a national public health emergency in 2016 followed by the World Health Organization declaring the outbreak a public health event of international concern [41]. The virus was first identified in 1947 in Uganda, after which cases of human infection have been infrequent and fairly localized [41]. ZIKV is transmitted by infected Aedes spp. mosquitoes, sexual contact and blood transfusions [42]. Around 80% of ZIKV that occur in adults are asymptomatic, with other cases having a mild febrile illness, headache, rash, fever and conjunctivitis [42]. However, severe neurologic sequalae can also occur in adults.
Congenital Zika syndrome (CZS) is variable in the presentation and severity with only a subset of infants that were exposed having apparent signs and symptoms at birth [41]. Infants exposed to ZIKV in utero are expected to survive, however a severe phenotype can result, particularly when exposure occurs in the first trimester [43]. ZIKV replication in brain tissue can continue after birth, and thus infants that are initially asymptomatic may develop symptoms within the first year of life [41]. The phenotype of CZS appears to consist of severe microcephaly and possibly a partially collapsed skull, thin cerebral cortices with subcortical calcifications, macular scarring, congenital contractures and marked early hypertonia [41]. Microcephaly is the most common symptom, occurring in up to 91% of CZS, and is often severe with the mean occipitofrontal head circumference falling 3–4 standard deviations below normal [43]. Both the central and peripheral nervous systems are impacted, with resultant effects on musculoskeletal, auditory and ophthalmologic systems and symptoms including conductive hip dysplasia, abnormal posturing of extremities, conductive hearing loss and abnormalities of the retina and optic nerve [43]. Up to 55% of infants with CZS have structural ocular abnormalities, making visual screening and interventions critically important to occur early in life to allow for neuroplasticity optimizing the outcomes [44]. This has led to the recommendation of any infant with suspected CZS or exposure to ZIKV to have an ocular examination before hospital discharge and again at 3 months of age [44].
A meta-analysis of 42 articles revealed the most common brain abnormalities following ZIKV exposure in utero, including decreased brain volume, increased extra-axial cerebrospinal fluid space, subcortical calcifications, microcephaly, ventriculomegaly, malformation of cortical development, basal ganglia calcifications, and mega cisterna magna [45]. These findings support the concept that ZIKV interferes with normal neuronal migration during development which then impacts the brain development. The major neuronal migration is occurring before the 25th week of gestation, making exposure to the virus in the first and second trimesters the most devastating. Infants with ZIKV exposure and no apparent congenital syndrome are also at risk for abnormal neurodevelopmental outcomes, as evidenced in a recent study of 70 infants followed to age 18 months [46]. These infants had confirmed exposure to ZIKV but no findings to support CZS, and despite the normal head circumference, had subsequent neurodevelopmental deficits develop over the first year of life [46]. As studies continue and longer-term outcomes become known, it is critically important to follow any infant with ZIKV exposure closely.
Rubella is caused by a single stranded ribonucleic acid (RNA) virus which is highly contagious and only transmitted between humans [18, 47]. It is usually spread through respiratory droplets and in most cases will result in a mild viral disease. Symptoms may include fever, rash, malaise and adenopathy. The virus is able to infect cells of the respiratory tract and then spread via the systemic circulation to multiple organ systems, including the placenta [48]. When the infection occurs during pregnancy the virus can be transmitted to the fetus and result in death of the fetus or a range of congenital anomalies known collectively as Congenital Rubella Syndrome (CRS) [18]. The timing of when a pregnant woman contracts the virus appears to be related to the risk of congenital infection and fetal defects. Studies estimate that maternal infection occurring during the first 12 weeks of gestation has roughly a 90% chance of congenital infection with the risk of defects nearly 85% [49]. When congenital infection occurs during the first trimester, hearing defects, heart defects, neurologic damage, and ocular defects appear more commonly. CRS is a combination of these defects but most classically is described as a triad of cataracts, congenital heart disease, and sensorineural deafness [49, 50]. Other manifestations include intrauterine growth restriction (IUGR), hepatomegaly, splenomegaly, thrombocytopenia and dermal erythropoiesis (commonly known as a “blueberry muffin rash”) [18].
Pregnant women in the United States are tested for rubella immunity by serologic screening. Those who have had a natural infection or have received at least one dose of the rubella vaccine tend to have lifelong immunity [18]. Those women who are found to be non-immune should receive one dose of the vaccine after childbirth, as vaccination during pregnancy has theoretical teratogenic risks due to the vaccine being live [18]. If a pregnant woman is exposed to the rubella virus, they should have serologic testing for rubella-specific IgM and IgG. If she is found to have rubella-specific IgG, then she is considered immune. However, if there is no IgG detectable at the time of exposure then convalescent serologies are obtained 3 and 6 weeks after exposure, with IgG reactivity at these time points indicating a recent infection [18]. Unfortunately, there is no treatment for rubella outside of supportive measures.
When congenital infection is suspected, diagnosis can be done by testing for rubella-specific IgM in fetal blood or detection of the virus in amniotic fluid [49]. Postnatally, an enzyme-linked immunosorbent assay (ELISA) can also be done for rubella-specific IgM. If positive, then confirmatory testing is done by reverse transcription polymerase chain reaction (RT-PCR) of nasopharyngeal swabs, urine, or oral fluid [47, 49]. In some infants the virus can be detected in nasopharyngeal secretions and urine for over a year [18, 49]. While there is no treatment for CRS, diagnosis is important in terms of follow up. Due to the risk of cataracts among other ocular abnormalities (including microphthalmia, glaucoma, chorioretinitis), hearing loss, neurologic manifestations (such as developmental delay, autism), and endocrine disorders (including diabetes, thyroid disease) children with CRS must be evaluated periodically for management of these potential complications [48–50]. The introduction of the vaccine has resulted in a significant decline in cases of rubella infection and CRS in the United States, with an average of 14 reported rubella cases a year and 4 CRS cases a year from 2001 to 2004 [51].
Cytomegalovirus (CMV) is a double stranded deoxyribonucleic acid (DNA) virus that is universally found and generally causes mild or subclinical symptoms in most children and adults [18, 52]. It can be transmitted via contact with infected secretions, transfusion of blood products from infected donors, organ transplants from infected individuals, or vertically [18]. When it is vertically transmitted, CMV has the potential to cause severe and permanent sequelae [18, 52, 53]. CMV is known as one of the most common congenital viral infections and is the leading, non-genetic cause of sensorineural hearing loss in children in the United States [18]. It can be transmitted to the fetus by crossing the placenta, through contact of infected cervical secretions during birth, or perinatally by ingestion of breast milk containing the virus [18]. When CMV is transmitted in utero, it can be due to primary maternal infection during pregnancy, reactivation of a prior infection, or reinfection with a different strain despite presence of maternal antibodies [54, 55]. Reactivation and reinfection are more common than a primary infection; however, the latter tends to cause more severe sequelae especially if infection occurs earlier in pregnancy.
Of those infants whose mother had an acute infection during pregnancy, 30–40% will have congenital CMV (cCMV) [18, 55]. Infants with cCMV are symptomatic in 10–15% of the cases, with half to two-thirds of these infants developing sensorineural hearing loss (SNHL) later in life [55]. Symptoms at birth can include thrombocytopenia, hepatomegaly, splenomegaly, microcephaly, periventricular calcifications in the brain, chorioretinitis, hepatitis, and SNHL. Long term outcomes include progressive SNHL and neurodevelopmental delay [18, 53, 55]. Of the infants who are asymptomatic at birth, around 15% will later develop SNHL [18]. Imaging of the fetal brain can be completed in utero via transvaginal ultrasound or with magnetic resonance imaging (MRI). cCMV can result in germinolytic cysts, lenticulostriate vasculopathy, temporal lobe and occipital cysts as well as cerebellar hypoplasia and migrational disorders including polymicrogyria [52]. Periventricular calcifications is the most frequently reported finding on brain imaging of cCMV cases, impacting 34–70% of diagnosed patients [56].
Testing during pregnancy is not routinely done, but serologic testing can be performed if a pregnant woman has been exposed or is suspected of having CMV infection. CMV-specific IgM has low specificity as it can persist for 6–9 month following primary infection and can also be detected during reactivation [54]. CMV IgG avidity index however can be used to confirm primary infection; avidity testing is a method to measure the strength of the bonding between antibodies and the virus. Low avidity would indicate recent infection while high avidity takes time to occur and would indicate a past infection. There is no current recommended treatment for acute CMV infection during pregnancy [18, 54].
There is also no current routine testing for CMV in infants. Some states have mandated targeted CMV screening for those who fail their routine newborn hearing screen, however it is important to note that targeted screening will miss those newborns who are asymptomatic at birth but still at risk for developing SNHL later in life [18]. For symptomatic infants, the diagnosis of cCMV can be made postnatally if testing is done within 3 weeks of birth as to avoid the difficulty of differentiating between intrauterine and perinatal infection [18, 54, 57]. CMV can be isolated from the urine, saliva, respiratory secretions, blood, or cerebrospinal fluid [18]. Viral cultures, rapid shell vial cultures, and PCR can be completed [54]. Treatment for those infants who are symptomatic regardless of CNS involvement includes intravenous ganciclovir or oral valganciclovir [18, 54, 58]. The latter is preferred due to ease of administration as duration of treatment is six months. If there are concerns for abnormal gastrointestinal absorption due to other factors, treatment can be started with IV ganciclovir [54]. Studies have found that those who have anti-viral treatment started within the first month of life have significantly improved audiologic and neurodevelopmental outcomes at 12 and 24 months of age compared to those who do not [53]. Treatment with either valganciclovir or ganciclovir can cause significant neutropenia; absolute neutrophil counts should be monitored weekly for the first six weeks of treatment, followed by screening at eight weeks of treatment, and thereafter monthly for the duration of treatment [54]. Infants with mild symptoms or isolated SNHL are not recommended to receive antiviral treatment at this time due to lack of data in this population [54].
Long term outcomes to consider in children with cCMV include SNHL and neurodevelopmental delay. These children should have frequent audiologic assessments as SNHL can develop and/or progress after the newborn period [54]. While there are no established universal guidelines for hearing evaluation, studies indicate that screening should continue for at least the first four years of life after which late-onset SNHL is seldom seen.
Herpes simplex viruses are large, double-stranded DNA viruses with two types, HSV-1 and HSV-2 [18]. Traditionally, HSV-1 can cause vesicular lesions in areas above the waist while HSV-2 involves areas below the waist. It is, however, becoming increasingly more common to see genital HSV-1 lesions. Both types are able to cause herpetic disease in neonates when acquired from the mother. Transmission can occur during the birthing process via contact with genital lesions, an ascending infection, intrauterine, or postnatally from contact with lesions [18, 52]. A primary genital HSV infection in the mother near delivery has 10–30 times the risk of transmission compared to a recurrent infection. This is thought to be due to lower concentrations of transplacental HSV antibodies in the neonate [18, 59]. Unfortunately, defining an infection as primary versus recurrent may not be straightforward, as women can be asymptomatic and may be unaware that they have had a prior infection with HSV. Furthermore, viral shedding can occur in the absence of clinical symptoms [59].
If a pregnant woman does have genital lesions characteristic of HSV near delivery, then swabs of the lesions can be sent for viral culture and PCR with serologic testing to determine the type. From these results, women can be classified into four different categories: documented first primary infection, documented first episode non-primary infection, assumed first episode (primary or non-primary), or recurrent infection (see Table 1 adapted from Kimberlin et al.).
Diagnostic tests for Herpes simplex virus (HSV) Antibodies and Culture/PCR. This table describes the classification of HSV infection based on culture or PCR test results as well as HSV-1 and HSV-2 antibody test results.
Women classified as having a primary infection or first episode can be treated with oral acyclovir for 7–10 days [18]. Those with a recurrent episode can be treated with the same or higher dose for 5 days [18]. If a woman has a known history of HSV then suppressive therapy should be started at 36 weeks’ gestation to decrease the risk of recurrence at delivery, although this will not entirely suppress shedding [60]. Other preventative methods include avoiding invasive fetal monitoring, such as fetal scalp electrodes, and opting for elective cesarean sections when lesions are present at the time of delivery [52, 60].
Neonatal HSV can have different manifestations. SEM disease includes disease of the skin, eyes and/or mouth; 45% of infants with HSV will have SEM. Another 30% of infants with HSV will have localized central nervous system (CNS) disease with or without skin involvement. The remaining 25% of infants with HSV will have disseminated disease which can involve multiple organs, most commonly the liver and lungs [18]. The onset of disease varies between the different manifestations, with SEM disease presenting at 5–11 days of life, CNS disease presenting between 8 and 17 days of life, and disseminated disease presenting between 10 and 12 days of life [61]. Initial symptoms may be non-specific and include feeding difficulties, lethargy, seizures, suspected sepsis, vesicular rash or severe liver dysfunction, with as many as 30% of infected neonates not having skin lesions [52, 60]. As there can be high morbidity and mortality rates in newborns with HSV, it is imperative to diagnose and initiate treatment as soon as it is suspected [18].
Guidelines have been published on the management of asymptomatic neonates born to women with active genital lesions [59]. In newborns whose mothers have a history of genital HSV prior to pregnancy and present with active lesions at delivery, there is a low risk of transmission. However, the infant should still have surface swabs of the mouth, nasopharynx, conjunctivae, and anus obtained for culture and PCR as well as serum HSV PCR sent at 24 h of life. Waiting to send samples until 24 h of life ensures that any positive results would represent active viral replication in the infant and not maternal contamination [59]. Intravenous acyclovir is not started in this situation unless the infant becomes symptomatic, or the surface swabs and/or serum are positive. This would confirm infection and require a lumbar puncture to obtain cerebrospinal fluid (CSF) for PCR testing. The result of the CSF PCR is key in determining treatment duration. If the CSF and serum HSV PCR are negative, then empiric IV acyclovir is administered for a total of 10 days to prevent progression from infection to disease. If the CSF PCR is positive, then treatment should be administered for 21 days [59]. After the treatment course has completed, a repeat lumbar puncture is necessary in cases of CNS disease to document clearance. If the repeat CSF HSV PCR is still positive, then acyclovir is continued for another 7 days. A repeat lumbar puncture is obtained to show clearance. This process is repeated until the CSF is negative. Any infant who undergoes a treatment course for HSV disease should have suppressive therapy with oral acyclovir for 6 months after the completion of parenteral treatment (see Figures 1 and 2) [59, 62].
Infant evaluation in suspected exposure to Herpes simplex virus (HSV). This flow diagram, adapted from Ref. [59], describes the infant evaluation(s) to complete if there was concern for maternal HSV infection around the time of delivery due to the presence of lesions.
Infant treatment recommendations for suspected congenital Herpes simplex virus (HSV) infection. Tis flow diagram, adapted from Ref. [59], describes treatment regimens based on infant symptoms.
In the case that an asymptomatic neonate is born to a mother with active genital lesions but does not have a history of genital HSV prior to pregnancy, then the importance lies in distinguishing whether it is a primary, non-primary or recurrent infection [59]. The mother should not only have the swabs sent for PCR testing and culture but should also have serum serological tests performed for HSV-1 and HSV-2 antibodies. The infant requires evaluation at 24 h of life with HSV surface cultures and PCR testing of the serum and CSF. The CSF samples should also be sent for cell count and chemistries, with screening serum alanine aminotransferase obtained. IV acyclovir would be started empirically after obtaining the samples at 24 h of age while awaiting results. Once the maternal testing is resulted, maternal classification can then be determined as shown in Table 1. If the mother is deemed to have a first episode primary or non-primary infection, then treatment of the infant would include 10 days of IV acyclovir for a normal evaluation (infant remains asymptomatic, negative CSF and serum HSV PCR, normal CSF indices, and normal serum ALT), 14 days for an abnormal evaluation (positive serum HSV PCR, symptomatic infant, or abnormal ALT) and 21 days for CNS infection (positive CSF PCR or abnormal indices) [59]. A neonate with a positive CSF HSV PCR, regardless of the maternal classification, would be managed as described above for HSV disease. It is important to note that if the infant becomes symptomatic at any point, even prior to the testing obtained at 24 h of life, then immediate evaluation and treatment should be initiated [59]. Other risk factors that may prompt testing and treatment prior the 24 h include: prolonged rupture of membranes (>4–6 h) and prematurity (<37 weeks’ gestation) in the setting of maternal genital lesions characteristic of HSV [59].
Only 10% of infants survive in untreated HSV disseminated disease with 50% of infants surviving in untreated HSV CNS disease [61]. Inadequately treated or untreated HSV SEM disease can progress to either disseminated or CNS disease; those that survive have a significant proportion that show some neurologic sequelae, namely in the form of motor, speech, and developmental delay [61]. Outcomes, especially mortality, improve the earlier that treatment is initiated, making it imperative to evaluate and begin empiric treatment whenever HSV infection is suspected [61]. Oral suppressive therapy has also been shown to improve neurodevelopmental outcomes at 12 months of age compared to those that did not receive long-term antivirals, suggesting that ongoing neurologic injury may occur in infants affected by HSV disease [62].
A review of additional viruses that can impact infants exposed during pregnancy is provided below. These viruses have been associated with a range of adverse outcomes in infants with prenatal/perinatal exposure, however they remain uncommonly diagnosed or the impact on the fetus remains extremely varied. However, given the increased risk of potential adverse outcomes, they are briefly discussed.
The hepatitis E virus (HEV) is a single-stranded RNA virus which is known as a major cause of acute viral hepatitis especially in developing countries through ingestion of contaminated water sources [18, 63]. While it generally causes a mild illness in most adults, pregnant women tend to have more severe disease. Mortality has been observed in pregnant women, especially if infected with genotype 1 [18, 63]. HEV is estimated to be responsible for up to 3000 stillbirths a year in developing countries and can commonly cause preterm delivery in infected mothers with resultant poor neonatal outcomes [63, 64]. When HEV is transmitted vertically, hepatitis can be present from birth and persist throughout the infant’s life but is not known to be associated with congenital anomalies.
Enteroviruses are a group of RNA viruses that can spread between humans via respiratory routes, vertically, and fecal-oral transmission [18]. Symptoms in adults and children can be varied and may include respiratory, dermatologic, neurologic, ocular, cardiac, muscular, and gastrointestinal manifestations [18]. When enterovirus is transmitted vertically or more commonly peripartum, the neonate may remain asymptomatic without sequelae or have severe symptoms including septic shock with multiorgan dysfunction [65]. There is limited evidence to suggest that infection with enterovirus during pregnancy is associated with congenital anomalies or fetal death [65].
Lymphocytic choriomeningitis virus (LCMV) is a single-stranded RNA virus spread by rodents which can cross the placenta; rarely it can be transmitted during delivery by exposure to maternal secretions or blood and cause congenital viral infection [66–68]. Infected pregnant women can have non-specific viral symptoms and may report direct exposure to or the presence of rodents in their homes [66, 68]. Common findings in an infant affected by LCMV are macrocephaly or microcephaly and ocular abnormalities; additionally, neurological abnormalities may be present and include hydrocephalus, periventricular calcifications, seizures, neurodevelopmental sequelae including intellectual disability, or even death [67, 68]. These symptoms suggest a similarity with other congenital infections previously discussed, such as CMV or toxoplasmosis, which may contribute to an underestimation of the prevalence of LCMV when congenital infection is suspected [66, 68].
The West Nile Virus (WNV) is a flavivirus that was initially isolated in 1937 and did not reach the United States until an outbreak in 1999 [69–71]. The primary mode of transmission is through the bite of an infected Culex species mosquito, with individuals ranging from no symptoms to 0.7% of infected individuals developing neuro-invasive disease with encephalitis, meningitis or acute flaccid paralysis possible [69]. There is no specific treatment or vaccine at this time [70]. Case reports of infants born to mothers with WNV have shown an array of outcomes, with follow up at 2–3 years of age not consistently showing any developmental delays [69]. Findings have included chorioretinitis, white-matter loss and cystic changes, and congenital defects such as lissencephaly, polydactyly, aortic coarctation and cleft palate [69]. Additional studies on the impact of infants with exposure during gestation, and longer-term outcomes are needed to truly delineate if WNV results in congenital anomalies.
Human adenoviruses (HAdV) are DNA viruses in the Adenoviridae family, with 7 subgroups and 52 serotypes [72]. While typically the cause of a “cold”, the severity of illness can range from mild to severe with gastroenteritis, pneumonia and neurologic disease possible [73]. Reports have not noted any specific fetal malformations, although infants with positive polymerase chain reaction (PCR) testing had a higher incidence of neural tube defects and echogenic liver lesions with and without hydrops [74].
Many of the maternal infections that previously resulted in significant impact and poor outcomes on the developing fetus have improved as treatments and vaccines have been introduced and refined. However, other pathogens are now becoming more apparent in their impact on fetal development, such as Zika virus. Some infections are declining in incidence, with a resultant decrease in congenital infections (such as the nearly 80% decline in Rubella infections) [51]. Other infections are continuing to increase, with the true impact on society yet to be determined. Thus, it is imperative that we monitor any infections in a pregnant woman, and complete a thorough examination and evaluation of each infant born with the hopes of identifying any abnormalities quickly and improving the outcomes of each infant to the best of our ability.
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