Open access peer-reviewed chapter - ONLINE FIRST

Superconducting Josephson-Based Metamaterials for Quantum-Limited Parametric Amplification: A Review

By Luca Fasolo, Angelo Greco and Emanuele Enrico

Submitted: June 14th 2019Reviewed: August 22nd 2019Published: October 10th 2019

DOI: 10.5772/intechopen.89305

Downloaded: 12

Abstract

In the last few years, several groups have proposed and developed their own platforms demonstrating quantum-limited linear parametric amplification, with evident applications in quantum information and computation, electrical and optical metrology, radio astronomy, and basic physics concerning axion detection. Here, we propose a short review on the physics behind parametric amplification via metamaterials composed by coplanar waveguides embedding several Josephson junctions. We present and compare different schemes that exploit the nonlinearity of the Josephson current-phase relation to mix the so-called signal, idler, and pump tones. The chapter then presents and compares three different theoretical models, developed in the last few years, to predict the dynamics of these nonlinear systems in the particular case of a 4-wave mixing process and under the degenerate undepleted pump assumption. We will demonstrate that, under the same assumption, all the results are comparable in terms of amplification of the output fields.

Keywords

  • superconductivity
  • metamaterial
  • Josephson effect
  • parametric amplification
  • microwave photonics

1. Introduction

In the last decade, microwave quantum electronics received a substantial boost by the advancements in superconducting circuits and dilution refrigerators technologies. These platforms allow experiments to be easily carried out in the mK regime, where the detection and manipulation of signals in the range 312 GHz reaches energy sensitivities comparable to a single photon [1].

Solid state microwave quantum electronics is founded on a building block that has no analogous in quantum optics: the Josephson junction [2]. This, in fact, is a unique nondissipative and nonlinear component that represents the key element of a large series of quantum experiments.

Furthermore, microwave quantum electronics allows the exploration of the so-called ultrastrong coupling regime [3], hard to be reached in quantum optics, and it is worth mentioning that nonlinear resonator can be exploited to access relativistic quantum effects and quantum vacuum effects. To give an example, the Lamb shift [4] effect has been observed in superconducting artificial atom [5], while the dynamical Casimir effect [6, 7] has been promoted by properly engineered superconducting waveguide [8].

From the very beginning, superconducting electronics has been pushed by the strong interest coming from the quantum computation and information community. However, it has been only recently shown that a new concept of 1D metamaterial with embedded several Josephson junctions enables strong photon-photon on-chip interactions [9], allowing experimentalists to engineer dispersion relations that drive the waves traveling along artificial waveguides [10, 11]. These concepts and technologies allow the control and tunability of the wave mixing process. As an example, a weak signal traveling in a metamaterial can interact with a strong pump tone at a different frequency, activating the so-called parametric amplification [12]. The class of devices where these phenomena are promoted is commonly known as traveling-wave Josephson parametric amplifiers (TWJPA) and represents the solid state analogous to optical χnnonlinear crystals [13].

It has been shown that TWJPAs can act as quantum parametric amplifiers by reaching the so-called quantum limit [14]. With the purpose of a comparison to the state-of-the-art commercially available low-noise amplifiers, these latter can operate at ω/2π=4GHzadding kBTn/ω10noise photons having a noise temperature of Tn=2K, while Josephson-based amplifiers can reduce this added noise up to 1/2photon, or even 0, depending on its working configuration.

The capability to beat the quantum limit is related to the so-called phase-sensitive amplification process, where the metamaterial can operate in degenerate mode (degenerate parametric amplifier, DPA), acting on two waves (signal and idler) at the same frequency (ωs=ωi) by amplifying and de-amplifying their position and momentum quadratures, respectively. In this view, DPA enables the preparation of squeezed states in the microwave regime. Even in the nondegenerate mode (nondegenerate parametric amplifier, NDPA, i.e., ωsωi), the phase-preserving nature of the quantum parametric amplification results in the entanglement condition among the signal and idler generated photons, composing a two-mode squeezed state [15]. It is worth mentioning how such a quantum state is an example of Einstein-Podolsky-Rosen state [16], where correlations between signal and idler are stronger than that allowed by classical theory [17].

It should be evident how superconducting electronics not only has demonstrated to be an ideal platform for microwave quantum parametric amplification but also has pushed forward the research field focusing on the generation of nonclassical radiation with attractive potential applications in metrology and quantum information processing.

2. Historical evolution of the traveling-wave parametric amplifiers

The theory of a new concept of microwave amplifier was developed by Cullen [12] in 1959. In his paper, Cullen showed a novel mechanism of periodic transfer of power between a pump tone and a signal traveling in a transmission line composed of a voltage dependent capacitance per unit length. A nonlinear component of an RLC circuit can change periodically the resonance frequency of the whole system, leading to a novel way of making broadband amplification, the so-called parametric amplification. In Figure 1, we report two toy models for parametric amplification in mechanical systems with their electrical counterparts.

Figure 1.

(a) Sketch of a swing process. An oscillating system at a frequency ω s is excited by parametric amplification via periodical changes of the center of mass position at a frequency ω p = 2 ω s . (b) LC circuit with variable (nonlinear) C and L components. The case in which the capacitance C is periodically changed in time is the circuit analogous to the mechanical system represented in (a), while the case having an oscillating inductance L mimics the condition sketched in (c), consisting in a torque pendulum with variable inertia momentum [18].

One of the first realizations of Cullen’s idea was made by Mavaddat et al. in 1962 [19]. The signal line was basically a low pass filter, in which the shunt elements were similar varactor diodes. There, the nonlinearity was given by the specific capacitance-voltage relation of the varactor diodes, which is highly nonlinear for relative small voltage values. In this pioneering experiment, a gain of 10 dB and a bandwidth of 3 MHz were shown.

After the theorizing and the subsequent discovery of the Josephson effect [2], it was understood that an easy way to embed a nonlinear component into a transmission line and simultaneously reduce losses was to build a nonlinear inductance made of superconducting material, exploiting a Josephson junction as a source of nonlinearity following the vanguard idea by Sweeny and Mahler [13]. There, the parametric amplifier was modeled by a first-order small-signal theory with the same approach adopted to predict the behavior of GaAsFET transmission line amplifiers. The proposed design consisted of a superconducting thin-film niobium transmission line, composed by a coplanar waveguide integrating a large number of Josephson junctions.

The first realization of a traveling-wave parametric amplifier embedding a series of Josephson junctions was possible due to the PARTS process developed at IBM [9]. Exploiting niobium/aluminum technology, Yurke et al. [20] reported the construction and characterization of a coplanar waveguide, in which the central trace was composed by an array of 1000 Josephson junctions. The experiment was there performed in reflection mode, by terminating one end of the device with a short, leading to a relative high gain of 16 dB but a narrow bandwidth of 125 MHz and a noise temperature of 0.5 ± 0.1 K. The mismatch between the theoretical model and the experimental data has resulted in the understanding of a lack of a complete description of the physics behind this device when working in a small-signal regime. The study of the collective behavior of groups of Josephson junctions forming a transmission line has been an active field of study of several theoretical works [21, 22]. Subsequently, the use of numerical analysis [23] helped in clarifying how wave propagation acts inside this kind of transmission line, giving information on cutoff propagation, dispersive behavior, and shock-wave formation. An analytical model of a Josephson traveling-wave amplifier of greater complexity was developed by Yaakobi et al. [24]. There, a transmission line made of a series of capacitively shunted Josephson junctions was considered.

One of the main limitations concerning the maximum achievable gain, common to all the TWJPAs concepts, is represented by the phase mismatch between the different tones into the line. In particular, it has to be noticed that even though the incoming waves can be in phase, photon-photon interactions between different tones (cross-phase modulation) or the same tone (self-phase modulation) lead to a modification of the phase of the traveling tones themselves. Indeed, quantum mechanically speaking, the power transport between the pump and the signal waves takes place through a photon energy conversion between the pump and the signal. This means that for an efficient energy exchange, conservation of both energy and momentum needs to take place. The latter condition is the corpuscular analogous to the phase matching requirement between the different electromagnetic waves. An engineering solution to overcome this problem is represented by the so-called resonant phase matching (RPM) [11]. O’Brien et al. analyzed this method theoretically on a simple transmission line made of a series of Josephson junctions capacitively shunted to ground operating in the so-called 4-wave mixing (4WM) regime. In their model, they shunted the transmission line with several LC resonators with a resonance frequency slightly above the pump tone. Doing this, they were able to show the rise of a stop band in the dispersion relation, which is able to re-phase the pump with the signal tones by changing the pump wave vector, favoring the wave mixing.

O’Brien’s design was realized not long after [10] using Al technology. In their design, the unit cell of the transmission line was composed by three single nonlinear Josephson cells, the shunt capacitor was made using low-loss amorphous silicon dielectric and a resonator was placed after each group of 17 unit cells. The device showed a maximum gain of 12 dB over a 4 GHz bandwidth centered on 5GHz. Moreover, the authors explain that variations of 2–3 dB in the gain most likely come from imperfect impedance matching between sections and at the level of the bond pads.

A similar design was adopted by Macklin et al. [14] to prove experimentally the capability of a TWJPA combined with the RPM technique to be used as a reliable tool for qubits readout. In this paper, the TWJPA, based on Nb technology and a different RPM periodicity, was first characterized, showing a gain of 20 dB over a 3 GHz bandwidth. Moreover, the quantum efficiency of the amplifier was tested when coupled with a 3D transmon qubit, leading to an efficiency value of 0.49±0.01. A key point of this experiment was the proof that a single TWJPA could be able to perform the readout of more than 20 qubits, thanks to its high dynamic range and multiplexing capabilities. RPM has shown remarkable capabilities and is a promising technique to overcome phase mismatch. It can be implemented in multiple ways [25], by the way, it has to be noticed that this method requires an increase of design complexity, lower tolerances on the constructing parameters, and longer propagation lengths (2 cm–1 m).

Another option to solve the mismatch problem was suggested by Bell and Samlov [26], who proposed a self-phase matching transmission line embedding a series of asymmetric superconducting quantum interference devices (SQUIDs). The remarkable feature of this design is that it does not need any resonant circuit to achieve phase matching. This TWJPA is indeed able to tune the nonlinearity of its SQUIDs just through the use of an external magnetic field. Zhang et al. realized that this design [27] proves the wide tunability on positive and negative values of the Kerr nonlinearity by a magnetic flux and its capability to assist phase matching in the 4WM process. The 4WM process is intrinsically affected by phase mismatch because it takes origin from a cubic (Kerr-like) nonlinearity of the current-phase relation of the SQUIDs composing the TWJPA, getting unwanted effects from self-phase and cross-phase modulations.

Zorin showed [28] that by embedding a chain of rf-SQUIDs into a coplanar waveguide, it is possible to tune both the second and third order nonlinearities of their phase-current relation. This is a totally a novel approach to the TWJPA, since the possibility to use a quadratic term as a source of nonlinearity, allows to work in the 3-Wave Mixing regime (3WM), as theorized by Cullen 57 years before. It is well-known that 3WM has several advantages when compared to 4WM. Firstly, it allows to operate with a minimal phase mismatch. Secondly, it requires a lesser pump power to achieve the same amplification per unit length. Eventually, it separates signal and idler from pump tones, easing the engineering of the experimental setup by removing the requirement of heavy filtering in the middle of the amplification band. A proof of principle based on the Zorin’s layout [29] showed a gain reaching 11 dB over a 3 GHz bandwidth.

A step forward in controlling the metamaterial nonlinearities was attempted by Miano et al. [30] achieving an independent tune of both second and third order terms in the current-phase relation by adjusting the bias current in some inductive circuits surrounding the transmission line. This technology takes the name of symmetric traveling-wave parametric amplifier (STWPA), its peculiarity arising from the symmetric arrangement of the rf-SQUIDs that compose the transmission line. This device concept represents the state-of-the-art in the field, allowing the exploration a wide portion of the control parameters space, leading to a maximum estimated gain of 17 dB and a 4 GHz bandwidth.

3. Theoretical models for a 4WM process in a TWJPA

In the last decade, different theoretical models have been developed to predict the behavior of an electric transmission line containing an array of Josephson junctions, employed as nonlinear elements. In this section, we will focus on those models developed to predict the behavior of a TWJPA in the particular case of a 4WM process, under undepleted degenerate pump approximation (i.e., assuming that the power held by the pump wave is at first approximation constant and larger than the one owned by the signal and the idler). We will firstly focus on the classical theory proposed by Yaakobi et al. in 2013 [24] and O’Brien et al. in 2014 [11], in which the behavior of the transmission line is derived imposing the current conservation in the system. This starting assumption leads to the definition of a partial differential nonlinear equation that can be turned into a system of coupled mode equations, providing the expression of the amplitude of the pump, signal, and idler tones along the transmission line. Subsequently, we will discuss two different quantum approaches for the description of the parametric amplifier dynamics. The first one, proposed by Grimsmo and Blais in 2017 [31], exploits a Hamiltonian based on continuous-mode operators to derive, in an interaction picture frame, a device’s output field. The second one instead, proposed by van der Reep in 2019 [32], derives a system of coupled mode equations for the creation and annihilation quantum operators starting from a Hamiltonian based on discrete-mode operators.

The theories presented in this chapter will be based on a series of simplifying assumptions, whose experimental realization could be difficult to be obtained. For instance, in a real device, the undepleted pump approximation is hardly respected along the entire extension of the device because, along the line, the pump tone transfers a non-negligible amount of energy to the signal and idler one. The depletion effects, resulting in a reduction of the gain and of the dynamics-range of the amplifier, have been studied both in a classical and quantum frame [28, 33].

In all of these models, a lossless electrical circuit composed by the repetition of an elementary cell, whose structure is shown in Figure 2, is taken into account. In order to standardize the notations, we assume that the Josephson junctions embedded in the transmission line are identical (i.e., they have the same critical current Ic) and that the current flowing through the n-th junction can be expressed through the nonlinear relation

Figure 2.

Electrical equivalent representation of a repetition of Josephson junctions embedded in a transmission line. The junctions are modeled as an LC resonant circuit. The length of the unitary cell length is represented by a.

IJ,n=IcsinφnE1

where φntis the phase difference across the junction of the macroscopic wave functions of the two superconductive electrodes. The relation between φnand the voltage-difference across the junction is given by the Faraday’s induction law

ΔVn=Vn+1Vn=Φ02πdφndt=Φ02πddtϕn+1ϕnE2

where Φ0=h/2eis the magnetic quantum flux (with h the Planck constant and e the elementary charge), whereas ϕntis the absolute phase in the n-th node of the circuit. The phase at the n-th node (ϕn) can be converted into a flux at the n-th node, and vice versa, through the relation Φn=Φ0/2πϕn.

Furthermore, we define CJthe capacitance associated to the n-th Josephson junction and LJ,nits inductance, defined as

LJ,n=ΔVndIJ,n/dt=Φ02π1Iccosφn=LJ0cosφnE3

where LJ0=Φ0/2πIcis the inductance of the Josephson junction for a phase difference φn=0.

The energy stored in the n-th Josephson junction can be expressed, using the definitions given in Eqs. (1) and (2), as

UJ,n=t0tVIdt=t0tΦ02πdφntdtIcsinφndt=IcΦ02π1cosφnt,E4

under the assumption that φnt0=0, and approximated through a first-order power expansion as

UJ,n=IcΦ02π1cosφn=12LJ0ΔΦn2124LJ02πΦ02ΔΦn4+OΔΦn6E5

where ΔΦn=Φn+1Φn.

Finally, we assume identical coupling capacitances C0between the transmission line and ground.

3.1. The classical theoretical model

In this subsection, we will present the main steps for the derivation of the classical model presented in [11, 24]. Under proper assumption, this model allows to determine analytically the amplitude of the signal’s and idler’s waves along the transmission line.

Expressing the current flowing through each branch of the circuit presented in Figure 2 in terms of absolute phases ϕnand imposing the current conservation in the n-th node (i.e., IJ,n1+ICJ,n1=IJ,n+ICJ,n+IC0,n), a differential equation for the absolute phase, in the case of a weak nonlinearity, can be obtained:

C0d2dt2ϕn=CJd2dt2ϕn+1+ϕn12ϕn1LJ0ϕn+1+ϕn12ϕn+Φ02π16Ic2LJ03ϕn+1ϕn3ϕnϕn13E6

where the last term derives from the first-order approximation of the nonlinear behavior of the Josephson’s inductance.

Assuming the length a of the elementary cell is much smaller than the wave lengths of the propagating waves λ(i.e., a/λ1), the discrete index n can be replaced by a continuous position x along the line (i.e., ϕntϕxt) and the phase differences can be expressed, at the second order approximation, as:

ϕn+1ϕnaϕx+12a22ϕx2E7
ϕnϕn1aϕx12a22ϕx2E8

In this way, it is possible to define a nonlinear differential equation for the continuous absolute phase ϕxt:

C02ϕt2a2LJ02ϕx2CJa24ϕx2t2=a42Ic2LJ032ϕx2ϕx2E9

In the case of a weakly nonlinear medium, the dispersion law can be derived from Eq. (9), considering the left-hand side being equal to zero and imposing a plane-wave solution ϕxteikxωt:

kω=ωLJ0C0a1LJ0CJω2E10

The solutions of Eq. (9) can be expressed, as shown by O’Brien et al. [11], in the form of a superposition of three waves (pump, signal, and idler) whose amplitudes are complex functions of the position along the line:

ϕxt=n=p,s,iReAnxeiknxωnt=12n=s,i,pAnxeiknxωnt+c.cE11

The case of a 4WM process with a degenerate pump can be taken into account by imposing the frequency matching condition 2ωp=ωs+ωi. Replacing this particular solution in Eq. (9) and assuming that, along the line, the amplitudes are slowly varying (i.e., 2An/x2knAn/xkn2An) and that As2and Ai2are negligible (i.e., As,iAp, strong pump approximation), we obtain a system of three coupled differential equations for the amplitudes Anxthat describe the energy exchange between the three waves along the line:

Apx=iϑpAp2Ap+2iXpApAsAieiΔkxE12
Asix=iϑsiAp2Asi+iXsiAp2AiseiΔkxE13

where Δk=2kpkskiis the chromatic dispersion. The term ϑpis responsible for the self-phase modulation of the pump tone, while ϑsiis responsible for the cross-phase modulation between the pump tone and the signal or idler, respectively. These terms can be expressed as

ϑp=a4kp516C0Ic2LJ03ωp2andϑsi=a4kp2ksi38C0Ic2LJ03ωsi2E14

while the coupling constants Xn, depending on the circuit parameters, are defined as

Xp=a4kp2kskikpΔk16C0Ic2LJ03ωp2andXsi=a4kp2kskiksi+Δk16C0Ic2LJ03ωsi2E15

Expressing the complex amplitudes Anxin a the co-rotating frame

Anx=An0eiϑnAp02xE16

it can be demonstrated that, working under the undepleted pump approximation Apx=Ap0Asix, the amplitude of the signal can be expressed as

Asx=As0coshg1xiΨ12g1sinhg1x+iXsiAp02g1Asi0sinhg1xeiΨ12xE17

where Ψ1=Δk+2ϑpϑsϑiAp02=Δk+ϑAp02is the total phase mismatch and g1is the exponential complex gain factor, defined as

g1=XsXiAp04Ψ122E18

The total gain of an amplifier, composed by the repetition of Nelementary cells, can then be expressed as GsaN=AsaN/As02.

3.2. Quantum Hamiltonian model based on continuous-mode operators

A standard method to treat quantum superconducting circuits is represented by the lumped element approach [34]. In this latter, the Hamiltonian of the quantum circuit is straightforwardly derived from its classical counterpart by promoting fields to operators and properly imposing commutating relations. In this view, one can proceed by deriving the Lagrangian of a TWJPA composed by the repetition of N unitary cells, under first nonlinear order approximation, as

L=n=0N1C02Φnt2+CJ2ΔΦnt2EJ01cos2πΦ0ΔΦnE19
n=0N1C02Φnt2+CJ2ΔΦnt212LJ0ΔΦn2124LJ02πΦ02ΔΦn4E20

where EJ0=IcΦ0/2π=IcLJ0. Under the assumption that a/λ1it is possible, as performed in the previous subsection, to replace the discrete index nwith a continuous position xalong the line (i.e., ΦntΦxt) and approximate, at the first order, ΔΦna∂Φxt/x. Furthermore, extending the system via two lossless semi-infinite transmission lines (characterized by a constant distributed capacitance c0and a constant distributed inductance l0), the Lagrangian can be expressed through a space integral extending from x=to x=+as

LΦ∂Φt=12cx∂Φt2+1ωJ2xlx2Φxt1lx∂Φx2+γx∂Φx4dxE21

where cxand lxare the distributed capacitance and inductance of the system, defined as

cx=c0x<0CJ/a0<x<zc0x>zandlx=l0x<0LJ0/a0<x<zl0x>zE22
ωJx=x<01/LJ0CJ0<x<zx>zandγx=0x<0a3EJ0/122π/Φ040<x<z0x>zE23

where zis the length of the TWJPA, ωJxis the junction’s plasma frequency, and γxis the term deriving from the nonlinearity of the junctions.

From Eq. (21), one can easily derive the Euler-Lagrange equation whose form, for 0<x<z, is equal to Eq. (9), giving the same dispersion relation (Eq. (10)) under analogous assumptions. Instead, outside the nonlinear region, the wave vector turns out to be kωx=c0l0ω2.

The Hamiltonian of the system can be derived from the Lagrangian by taking into account πxt=δL/δ∂Φ/t, the canonical momentum of the flux Φxt:

HΦπ=π∂ΦtdxL=12cx∂Φ∂Φt2+1lx∂Φx2+1ωp2xlx2Φxt2dxγ20z∂Φx4dx=H0+H1E24

where the term H0represents the linear contributions to the energy of the system, while H1is the first-order nonlinear contribution.

This Hamiltonian can be converted to its quantum form promoting the field Φxtto the quantum operator Φ̂xt.

In direct analogy with Eq. (11), one can express the flux operator in terms of continuous-mode functions [34], such as Ĥ0is diagonal in the plane-waves unperturbed modes decomposition:

Φ̂xt=ν=L,R0lx4πkωxâνωei±kωxxωt+H.c.E25

where the subscript Rdenotes a progressive wave, while Ldenotes a regressive wave (i.e., ârepresents the annihilation operator of a right-moving field of frequency ω). In [35], it is demonstrated that by replacing the definition given in Eq. (25) into the linear Hamiltonian, Ĥ0takes the form

Ĥ0=ν=R,L0ωâνωâνωE26

(where the zero-point energy, which does not influence the dynamics of the amplifier, has been omitted).

Using the expansion of Φ̂xtintroduced above, under the hypothesis of a strong right-moving classical pump centered in ωp, and that the fields âνωare small except for frequencies closed to the pump frequency (i.e., replacing âνωwith âνω+bω, where bωis a complex valued function centered in ωp), the nonlinear Hamiltonian Ĥ1can be expressed under strong pump approximation, at the first order in bω, as the sum of three different contributions:

Ĥ1=ĤCPM+ĤSQ+HSPME27

In the expressions of these contributions, the fast rotating terms and the highly phase mismatched left-moving field have been neglected:

ĤCPM=2π0dωsdωidΩpdΩpkωskωiβΩpβΩpϒωsωiΩpΩpâRωsâRωi+H.cE28

describes the cross-phase modulation,

ĤSQ=4π0dωsdωidΩpdΩpkωskωiβΩpβΩpϒωsΩpωiΩpâRωsâRωi+H.c.E29

describes the broadband squeezing, and

HSPM=4π0dωsdωidΩpdΩpkωskωiβΩpβΩpϒωsωiΩpΩpbωsbωi+H.c.E30

describes the self-phase modulation. βΩis the dimensionless pump amplitude, proportional to the ratio between the pump current IJΩpand the critical current of the junctions Ic:

βΩp=IJΩp4IcE31

The function ϒω1ω2ω3ω4is the phase matching function, defined as

ϒω1ω2ω3ω4=0zeikω1xkω2x+kω3xkω4xdxE32

Assuming the nonlinear Hamiltonian Ĥ1as a perturbative term of the Hamiltonian Ĥ0for which the continuous modes are noninteracting, and assuming the initial time of the interaction t0=and the final time t1=+, it is possible to relate the input field of the system to the output one introducing the asymptotic output field

âout=ÛâÛ,E33

where Ûis the asymptotic unitary evolution operator (approximated to the first order in Ĥ1)

ÛÛ=eiK̂1whereK̂1=K̂CPM+K̂SQ+KSPME34

Working in the monochromatic degenerate pump limit bΩp=bΩpbΩpδΩpωpbp(where ωpis the pump frequency), the propagators take the form

K̂CPM=2βp2z0kωsâRωsâRωsdωsE35
K̂SQ=i2βp20kωskωi1ΔkeiΔkz1âRωsâRωidωs+H.c.E36

and

KSPM=zβp2kωpbpbpE37

where βωpβpand Δk=2kωpkωskωiare the chromatic dispersions.

Similar to the previous classical treatment, one can introduce the co-rotating framework by replacing the field operators with

âRωsz=a˜̂Rωsze2iβp2kωszandbpz=b˜pzeiβp2kωpzE38

In this framework, one can derive the following differential equation

a˜̂Rωszz=iddzK̂1a˜̂Rωs=iβp2kωskωieiΨ2ωsza˜̂RωiE39

and

b˜pzz=dKSPMdzb˜p=0E40

where Ψ2ωs=Δk+2βp2kωpkωskωiis the total phase mismatch. These latter are formally identical to Eqs. (12) and (13), up to a frequency-dependent normalization of the wave-amplitudes, under the undepleted pump approximation. Reference [11] derives an exact solution for Eq. (39), being

a˜̂Rωsoutz=coshg2ωsziΨ2ωs2g2ωssinhg2ωsza˜̂Rωs+iβp2kωskωig2ωssinhg2ωsza˜̂Rωi]eiΨ2ωs2zE41

where

g2ωs=βp22kωskωiΨ2ωs22E42

If a state moves inside a TWJPA of length z=aN, the power gain will be GωsaN=a˜̂RωsoutaNa˜̂RωsoutaN/a˜̂Rωsa˜̂Rωs.

3.3. Quantum Hamiltonian model based on discrete-mode operators

An alternative approach for the derivation of the quantum dynamics of a TWJPA is the one proposed in [32]. In this model, the quantum Hamiltonian for a 4WM parametric amplifier is expressed as the integral, along an arbitrary quantization length lq, of the linear energy density stored in each element of the circuit. The energy stored per unit length in a Josephson junction can be derived from Eq. (5) by dividing each term by the elementary cell length aand replacing ΔΦnwith its continuous counterpart. Instead, the energy stored per unit length in a capacitance C can be alternatively expressed in terms of flux difference ΔΦor stored charge Q as

UC=1at0tVIdt=1at0tdΔΦdtCddtdΔΦdtdt=12CaΔΦt2ΔΦt2E43
=1at0tQCdQdtdt=12a1CQ2tE44

under the assumption that ΔΦt0=0and Qt0=0.

Therefore, the quantum Hamiltonian of the system can be expressed, with an approximation to the first nonlinear order, as

Ĥ=lqUJ+UCJ+UC0dxlq12aLJ0ΔΦ̂124aLJ02πΦ02ΔΦ̂3+12CJa2ΔΦ̂t2ΔΦ̂+12a1C0Q̂C02dxE45

where Q̂and Φ̂are quantum operators. The former can be expressed, as suggested in [34] and adapted for discrete-mode operators in [36], as

Q̂C0=nC0aV̂C0,n=nC0aωna2C0lqâneiknxωnt+H.c.E46

(here kn=knand ωn=ωn).

Before defining the flux operator, it is necessary to define an effective inductance Leffof the transmission line (modeled, as shown in Figure 2, as a parallel of the nonlinear Josephson inductance LJand the capacitance CJ):

1jωnLeff=1jωnLJ+jωnCJhenceLeff=LJ1ω2LJCJLJΛnE47

Using the telegrapher’s equation [37], the discrete-mode current operator, under slowly varying amplitude approximation (ân/x0), can be derived from the discrete-mode voltage operator as:

ÎLeff=nsgnnωna2Lefflqâneiknxωnt+H.c.E48

Therefore, the flux operator can be expressed as

ΔΦ̂=LJaÎLeffwhereLJΔΦ=ΔΦIJ=2πΦ0ΔΦIc2πΦ0sin2πΦ0ΔΦ=LJ02πΦ0ΔΦsin2πΦ0ΔΦE49

The recursive relation deriving from Eq. (49) can be solved iteratively. Exploiting a power series expansion of the sine function and considering just the first order of interaction, it results that

ΔΦ̂=n[1+Λn122πΦ0ΔΦ̂02+O2πΦ0ΔΦ̂04ΔΦ̂n0E50

where ΔΦ̂0is the zero-order approximation of the flux quantum operator

ΔΦ̂0=nΔΦ̂n0=nknaωnωna2C0lqâneiknxωnt+H.c.E51

Substituting Eqs. (46) and (50) in Eq. (45), and limiting the expression to the first nonlinear order, the Hamiltonian for a 4WM amplifiers turns up

Ĥ=nωnânân+12+n,m,l,ki2a96LJ0Ic2lq2ΔknmlkeiΔωnmlkteiΔknmlklq1×{14LJ0ΛnCJωk2a˜̂+H.c.n×m×l×k+4LJ0ΛnCJ[2ωia˜̂+H.c.n×m×a˜̂+H.c.l×k+a˜̂+H.cn×mωia˜̂+H.c.l×k]}E52

where a˜̂sgnnΛnωnân, Δknmlk±kn±km±kl±kk, Δωnmlk±ωn±ωm±ωl±ωk(a ± sign refers to a corresponding annihilation (creation) operator) and the subscript i×jindicates a multiplication (i.e., Λωi×j=ΛiωiΛjωj).

Neglecting the constant zero-point energy and assuming a strong degenerate classical pump (as shown in [36])

âpiωpC0lq2aApE53

it is possible to approximate the Hamiltonian in Eq. (52) to the second order in âs,ias

ĤCPn=s,iωn+ξnAp2ânânχAp2âsâi+H.c.E54

where

ξn=kp2a2Λnωn32Ic2LJ0243δpn1+23ΛpΛn+ΛnΛp2E55

represents the quantum self-phase modulation (when n=p) and the quantum cross-phase modulation (when n=s,i), whereas the coupling constant χis defined as

χ=kp2a2ΛsωsΛiωi16Ic2LJ021+LJ0CJ6ωpωs2Λp+5Λs3Λi+ωpωi2Λp3Λs+5Λi+ωsωi4Λp2Λs2Λi])E56

Starting from the Hamiltonian ĤCP, it is possible to calculate the Heisenberg equation of motion for the classical pump amplitude and for the quantum operators âsand âi, obtaining the coupled mode equations:

Apt=iωp+2ξpAp2Ap+2iχApâsâiE57
âsit=iωsi+ξsiAp2âsi+Ap2âisE58

In [32], the hypothesis under which the classical coupled mode equations Eqs. (12) and (13) can be obtained from Eqs. (57) and (58) is described in detail.

Moving to a co-rotating frame (âsiâsieiξsiAp02z), the Hamiltonian (Eq. (54)) can be expressed as

ĤrotCP=χAp2âsâieiΨ3't+H.c.E59

where Ψ3=4ξpξsξiAp02. In this frame, introducing the undepleted pump assumption (Eq. (58)) turns into

âsit=Ap02âiseiΨ3'tE60

whose solutions are

âsit=âsi0coshg3't+iΨ3'2g3'sinhg3't+Ap02g3'âis0sinhg3'tE61

where the exponential complex gain factor is defined as

g3=χ2Ap04Ψ322E62

If a state spends a time tin the amplifier, the gain can be expressed as GsQt=âstâst/âs0âs0.

To make the results of this last treatment, in which the operators are expressed as a function of the time, comparable with the previous ones, in which the operators are expressed as a function of the space coordinate, we need to take into account the phase velocity of the tones. It turns out that:

Ψ3=Δk+4ξpωpkpξsωsksξiωikiandg3=χ2Ap04ωpkp2Ψ322E63

where Δk=2kpkskiis the chromatic dispersion.

3.4. Models comparison

The three exponential complex gain factors (gi) and the three total phase mismatches (Ψi) derived in these models are analytically different but numerically similar, as shown in Figure 3 (where the two insets report the differences between the quantum predictions and the classical ones).

Figure 3.

Comparison of total phase mismatches ( Ψ i ) and exponential complex gain factors ( g i ) predicted by the three treatments presented in Section 3. For these calculations, typical constructive parameters have been assumed: a = 50 μm , I c = 5 μA , C J = 300 fF , and C 0 = 35 fF , in such a way that the characteristic impedance is Z ≈ 50 Ω . Furthermore ω p / 2 π = 6 GHz and I p = Ic / 2 . In the insets, the differences between the quantum predictions and the classical ones are presented.

It is important to observe that in Eqs. (17), (41), and (61), the last term is always equal to zero in the case of a zero initial idler amplitude. In such a case, and under the hypothesis of a perfect phase matching (Ψi=0), giis real and the amplification gain increases exponentially with the line length; whereas, in the case of a nonzero phase mismatch, giis imaginary and the gain increases quadratically [11].

4. Conclusions

In the present chapter, we have presented the state-of-the-art of the experimental evidences in the field of Josephson junctions-based traveling-wave metamaterials through a historical review in Section 2. Moreover, in Section 3, we have reported three different theoretical approaches for the prediction of a TJWPA dynamics, in the particular case of a 4WM process. Assuming similar simplifying hypothesis, like the use of a classical undepleted degenerate pump, the presence of slowly varying fields along the transmission line and approximating the nonlinearities of the system up to the first order, a similar expression for the signal amplitude (or field annihilation/creation operators in the case of quantum theories) expressed in a co-rotating frame, is derived in the three treatments. Although the results of the quantum theories are similar to the classical ones, the description of the system dynamics with a quantum theory grants the possibility to evaluate photon-number distributions, squeezing effects and averages, standard deviations or higher-order moments of the measurements operators, taking into account the commutation relations between operators explicitly. For instance, detailed calculations of the output state of a TWJPA in the case of a single-photon input state and in the case of a coherent input state are presented in [32].

Acknowledgments

The author would like to thank Luca Callegaro for the stimulating discussion. This work was partially funded by the Joint Research Project PARAWAVE of the European Metrology Programme for Innovation and Research (EMPIR). This project has received funding from the EMPIR program co-financed by the Participating States and from the European Unions’ Horizon 2020 research and innovation program.

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Luca Fasolo, Angelo Greco and Emanuele Enrico (October 10th 2019). Superconducting Josephson-Based Metamaterials for Quantum-Limited Parametric Amplification: A Review [Online First], IntechOpen, DOI: 10.5772/intechopen.89305. Available from:

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