1. Introduction
Bose–Einstein Condensation (BEC) of atomic gases has attracted a renewed theoretical and experimental interest in quantum many-body systems at extremely low temperatures (Pethick & Smith; 2002). This excitement stems from two favorable features: (1) by applying magnetic fields and lasers, most of the system parameters, such as the shape, dimensionality, internal states of the condensates, and even the strength of the interatomic interactions, are controllable; (2) due to the diluteness, the mean-field theory explains experiments quite well. In particular, the Gross–Pitaevskii (GP) equation demonstrates its validity as a basic equation for the condensate dynamics. The GP equation is the counterpart of the nonlinear Schrödinger (NLS) equation in nonlinear optics. Thus, a study based on nonlinear analysis is possible and important.
In nonlinear physics, a soliton is remarkable object not only for the fact that exact solutions can be obtained but also for its usefulness as a communications tool due to its robustness. In general, solitons are formed under the balance between nonlinearity and dispersion. For atomic condensates, the former is attributed to the interatomic interactions, while the latter comes from the kinetic energy. Either dark or bright solitons are allowable depending on the positive or negative sign of the interatomic coupling constants
The chapter is organized as follows. In Sec. 2, the mean field theory of condensate is briefly reviewed. Section 3 introduces an effective interatomic coupling in a q1D condensate. Using these results, we consider a spinor condensate in q1D regime in Sec. 4. Then, in Sec. 5, we show an integrable condition of the coupled nonlinear equations for spinor condensates in which the exact soliton solutions are derived. In Sec. 6 and 7, we analyze the spin properties of one-soliton and two-soliton, respectively. Finally we summarize our findings and remark some current progresses on this topic in Sec. 8.
2. Mean field theory
The dynamics of BEC wave function can be described by an effective mean-field equation known as the Gross-Pitaevskii (GP) equation. This is a classical nonlinear equation that takes into account the effects of interatomic interactions through an effective mean field.
In this section, we derive the GP equation for a single component condensate and discuss the theoretical background of it for later extension to a low dimensional case and a spinor case.
2.1. Hamiltonian
In order to derive the mean-field equation for atomic BECs, we start with the second quantized Hamiltonian. The Hamiltonian for the system of
where
In most of the experiments, the trap is well approximated by a harmonic oscillator potential,
Condensates are pancake-shape for
The atom-atom interaction
where
2.2. Bogoliubov theory
The mean-field theory for
where
This idea can be extended to non-uniform gases in trap potentials. If we introduce the r dependence of the condensate part, the field operator is expressed as
The scalar function Φ(r,
In the case of BEC, the number of the condensed atoms becomes macroscopic, i.e.,
In this sense, the “macroscopic” wave function Φ(r,
which obviously satisfies the symmetry under exchanges of two bosons.
Following the Bogoliubov prescription, we substitute (9) into (1) and retain
Equation (14) is called the Gross-Pitaevskii energy functional. The statistical and dynamical properties of the condensate are determined through a variation of
2.3. Gross-Pitaevskii equation
Even at the zero temperature, interactions may cause quantum correlation which gives rise to occupation in the excited states. The assumption that the quantum fluctuation part
where
we obtain the Gross-Pitaevskii (GP) equation:
This equation has been derived independently by Gross and Pitaevskii (Pethick & Smith; 2002) to deal with the superfluidity of 4He-II. The GP equation is a classical field equation for a scalar (complex) function Φ but contains ћ explicitly. In this sense, the description of the condensate in terms of Φ is a manifestation of the macroscopic de Broglie wave, where the corpuscular aspect of matter dose not play a role. Now the modulus and gradient of phase of Φ = |Φ|exp(i
where
3. Confinement induced resonance
In this section, we derive an effective one-dimensional (1D) Hamiltonian for bosons confined in an elongated trap. The interactions between atoms in the experiments are always three-dimensional (3D) even when the kinetic motion of the atoms in such a tight radial confinement is 1D like. Therefore, the trap-induced corrections to the strength of the atomic interactions should be taken into account properly.
This problem was first solved by Olshanii (Olshanii; 1998) within the pseudopotential approximation, yielding a new type of tuning mechanism for the scattering amplitude, now called
3.1. Model Hamiltonian
We start with the following model:
The trap potential is composed by an axially symmetric 2D harmonic potential of a frequency
Atomic motion for the
Interatomic interaction potential is represented by the Fermi-Huang pseudopotential:
where the coupling strength
The energy of atoms for both transverse and longitudinal motions is well below the transverse vibrational energy ћ
In the harmonic potential we can separate the center of mass and relative motion. Then we consider the Schrödinger equation for the relative motion,
where the reduced mass
From the above condition 4, we assume that the incident wave is factorized as
where
3.2. One-dimensional scattering amplitude
The asymptotic form of the scattering wave function is given by
where
To calculate the one-dimensional scattering amplitude we expand the solution,
where
to both side of the Schrödinger equation and taking the limit, in sequence,
Here we have used the normalization condition:
and the
We note that the regularization operator
the normalization condition of
Recall that due to the condition (24) the value inside the parentheses in eq. (31) is positive definite. Thus, the expression for the wave function along the
where the function Λ is defined as
the sum over
to the function Λ, and then, collecting
Here the zero-order term of the expansion has a form,
with
and
Substituting eq. (33) with eq. (36) into eq. (30), we get Ψreg in an explicit form.
We then write the final expression of the one-dimensional scattering amplitudes (25) as
with the 1D scattering length:
3.3. Effective one-dimensional coupling strength
The expression (40) is an exact result for the potential (21) with arbitrary strength of the transverse confinement
were the coupling strength:
Note that a simple average of the three-dimensional coupling
The resonance factor 1/[1 –
4. Spinor Bose–Einstein condensate
In this section, we extend the model of a single component condensate discussed in Sec. 2 to that of a multicomponent condensate with the spin degrees of freedom, which we call a
4.1. Hamiltonian
The hyperfine spin
Atoms in the
In order to discuss the properties of spinor Bose gases, we start with the following second quantized Hamiltonian,
where
Due to the Bose–Einstein statistics, the total spin
where
For
where
where a hat “ˆ” on f means an operator as projection. Solving these equations (52), (53) for
In this expression,
which are the magnitude of the density-density interaction and of the spin-spin interaction, respectively. Thus, the interaction Hamiltonian is rewritten as
where we may use the following expressions of spin-1 matrices f = (f
A construction of the interaction Hamiltonian for a general hyperfine spin
4.2. f = 1 spinor condensate in quasi 1D regime
From now on, we assume that the system is quasi-one dimensional: the trapping potential is suitably anisotropic such that the transverse spatial degrees of freedom (
As derived in Sec. 2, in the mean-field theory of the spinor BEC, the assembly of atoms in the
where the subscripts {1, 0,–1} denote the magnetic quantum numbers with the components subject to the hyperfine spin space. The normalization is imposed as
where
According to the discussion in Sec. 3, the effective 1D couplings
where
Thus, the Gross-Pitaevskii energy functional of this system is given by
with the particle number and spin densities, respectively, defined by
The coupling constants
The time-evolution of spinor condensate wave function Φ(
Substituting eq. (61) into eq. (64), we get a set of equations for the longitudinal wave functions of the spinor condensate:
5. Integrable model
To analyze the dynamical properties of the coupled system (65), we propose an integrable model as follows (Ieda et al.; 2004a,b). We consider the system with the coupling constants,
This situation corresponds to attractive mean-field interaction
The effective interactions between atoms in a BEC have been tuned with a Feshbach resonance (Pethick & Smith; 2002). In spinor BECs, however, we should extend this to alternative techniques such as an optically induced Feshbach resonance or a confinement induced resonance (Olshanii; 1998), which do not affect the rotational symmetry of the internal spin states. In the latter, the above condition is surely obtained by setting
in eq. (60) when
It is worth noting that the integrable property itself is independent of the sign of
where time and length are measured in units of
respectively, we rewrite eqs. (65) as follows, (we omit the arguments (
Now we find that these coupled equations (71) are equivalent to a 2×2 matrix version of nonlinear Schrödinger (NLS) equation:
with an identification,
Since the matrix NLS equation (72) is completely integrable (Tsuchida & Wadati; 1998), the integrability of the reduced equations (71) are proved automatically (Ieda et al.; 2004a). Remark that the general
5.1. Soliton solution
We summarize an explicit formula for the soliton solution of the 2 × 2 matrix version of NLS equation (72) with eq. (73) by considering a reduction of a general formula obtained in (Tsuchida & Wadati; 1998).
Under the vanishing boundary condition, one can apply the inverse scattering method (ISM) to the nonlinear time evolution equation (72) associated with the generalized Zakharov-Shabat eigenvalue problem:
Here Ψ1 and Ψ2 take their values in 2 × 2 matrices. The complex number
where the 2
Here we have introduced the following parameterizations:
The 2 × 2 matrices Π
must take the same form as
The same procedure can be performed for nonvanishing boundary conditions (Ieda et al.; 2007) which is relevant to formation of spinor dark solitons (Uchiyama et al.; 2006).
Equation (72) is a completely integrable system whose initial value problems can be solved via, for example, the ISM (Tsuchida & Wadati; 1998) (Ieda et al.; 2007). The existence of the r-matrix for this system guarantees the existence of an infinite number of conservation laws which restrict the dynamics of the system in an essential way. Here we show explicit forms of some conserved quantities, i.e., total number, total spin (magnetization), total momentum and total energy.
Here tr{·} denotes the matrix trace and
6. Spin property of one-soliton solution
In this section, we discuss one-soliton solutions and classify them by their spin states. If we set
where
We have omitted the subscripts of the soliton number. Here and hereafter, the subscripts R and I denote real and imaginary parts, respectively. Throughout this section, we set
We use the term “amplitude” to indicate the peak(s) height of soliton’s envelope. Actual amplitude should be represented as
From a total spin conservation, one-soliton solution can be classified by the spin states. We shall show that the only two spin states are allowable,
Substituting eqs. (89)–(91) into eq. (83), we obtain the local spin density of the one-soliton solution:
We also give the explicit form of the number density:
To clarify the physical meaning of detΠ, we define here another important local density as
This quantity measures the formation of singlet pairs. Note that these “pairs” are distinguished from Cooper pairs of electrons or those of 3He owing to the different statistical properties of ingredient particles. Since Θ(
In the case of the one-soliton solution (89), the singlet pair density is proportional to the determinant of the polarization matrix Π,
This suggests that detΠ represents the magnitude of the singlet pairs. For the general
In what follows, we classify spin states of the one-soliton solution based on the values of detΠ.
6.1. Ferromagnetic state
Let detΠ = 0, then eq. (89) becomes a simple form:
Now all of
The total magnetization (82) becomes
with the modulus,
Next, we calculate the total momentum and the total energy of the ferromagnetic soliton. Substituting eq. (97) into eqs. (84), (86) and using detΠ = 0, we obtain
respectively. In infinite homogeneous 1D space as considered here, it can be shown that a single component GP equation for BEC with attractive interactions, i.e., the self-focusing NLS equation possesses the one-soliton solution that minimizes the total energy for fixed number of particles and total momentum. This remains true for the spinor GP equations (71). As we will see later, for given number of
6.2. Polar state
If detΠ ≠ 0, the local spin density has one node, i.e., f(
for each moment
Since each component of the local spin density is an odd function of
This implies that this type of soliton, on the average, belongs to the
To elaborate on this type of soliton, we further divide into two cases.
(i)
Under this constraint, we find the local spin (102) vanishes everywhere. Solitons in this state possess the symmetry of polar state locally. We, therefore, refer to only those solitons as polar solitons. Considering eq. (89) with the above condition, we recover a normal sech-type soliton solution:
Note that the amplitude of soliton is different from that of the ferromagnetic soliton, which leads to a relation between the total number and the spectral parameter as
The total momentum and the total energy are given by
respectively. The difference between ferromagnetic soliton energy and polar soliton energy with the same number of atoms
which is a natural consequence of the ferromagnetic interaction, i.e.,
(ii)
In this case, the local spin retains nonzero value, although the average spin amounts to be zero. The density profile (104) has the following structure. When
For a large value of
Hence, solitons of this type will be referred to as split solitons. The total number is the same as the case (i),
The total momentum and the total energy are the same values as those in the case (i):
This degeneracy is ascribed to the integrable condition for the coupling constants, i.e.,
which is a monotone decreasing function of
7. Two-soliton collision and spin dynamics
In this section, we analyze two-soliton collisions in the spinor model. The two-soliton solutions can be obtained by setting
For simplicity, we restrict the spectral parameters to regions:
Under the conditions, we calculate the asymptotic forms in the final state (
According to the classification of one-soliton solutions in the previous section, we choose the following three cases: i) Polar-polar solitons collision, ii) Polar-ferromagnetic solitons collision, iii) Ferromagnetic-ferromagnetic solitons collision. As we shall see later, the polar soliton dose not affect the polarization of the other solitons apart from the total phase factor. On the other hand, ferromagnetic solitons can ‘rotate’ their partners’ polarization, which allows for switching among the internal states.
7.1. Polar-polar solitons collision
We first deal with a collision between two polar solitons defined by
In the asymptotic regions, we can consider each soliton separately. Thus, the initial state is given by the sum of two polar solitons as
where the asymptotic form of soliton-
These can be proved by taking the limit
where
with
Equations (115) and (117) are the same form as polar one-soliton solution (105). Collisional effects appear only in the position shift (118) and the phase shifts (119). In Figs. 2, we show the polar-polar collision with
which are by themselves conserved through the collision. In this sense, the polar-polar collision is basically the same as that of the single-component NLS equation.
7.2. Polar-ferromagnetic solitons collision
Under the condition (112), we set soliton 1 to be polar soliton and soliton 2 to be ferromagnetic soliton:
Then, the initial state is represented by eq. (114) with
The final state is given by eq. (116) with
Here we have defined
and also used eqs. (118), (119). Normalization of the new polarization matrix (125) turns out to be unity,
The determinant of it becomes
We can see clearly that the initial polar soliton breaks into a split type,
which means that the polar soliton keeps its shape against the collision and shows no mixing among the internal states except for the total phase shift. On the other hand, because of the total spin conservation, the ferromagnetic soliton always retains its polarization matrix and shows only the position and phase shifts similar to those of the polar-polar case.
In Fig. 3, we have density plots of a polar-ferromagnetic collision with the parameters shown in the caption. These pictures correspond to each component of the exact two-soliton solution for one collisional run. For simplicity, we choose the parameters to have |
invariant during this type of collision, the fraction of each component can vary not only in each soliton level but also in the total after the collision. This contrasts to an intensity coupled multicomponent NLS equation in which the total distribution among all components is invariant throughout soliton collisions while a switching phenomenon similar to Fig. 3 can be observed (Radhakrishnan et al.; 1997).
7.3. Ferromagnetic-ferromagnetic solitons collision
Finally, we discuss the collision between two ferromagnetic solitons,
The asymptotic forms are obtained for the initial state,
and for the final state,
Here we have defined
and, for (
which are shown to be normalized in unity,
Each polarization matrix Π
In this expression, the polarization matrices in the initial state Π
where, with (
This defines the collision property for the ferromagnetic-ferromagnetic soliton collision.
We can gain a better understanding of the collision between two ferromagnetic solitons by recasting it in terms of the spin dynamics. The total spin conservation restricts the motion of the spin of each soliton on a circumference around the total spin axis [Fig. 4(a)]. It will be interpreted as a spin precession around the total magnetization.
We calculate the magnetization for each soliton to investigate their collision. In the initial state, following eq. (99), we have the spin of soliton-
Thanks to the scattering property (137), the final state spins can be obtained through F1,2 by
where
The conserved total spin,
Considering spin rotation around the total spin FT, we can find ‘rotated spin’ as
where
with
The rotation angle
For the case that the magnitudes of the amplitude and velocity for each ferromagnetic soliton are, respectively, identical with each other, |
where (
The principal value should be taken for the arccosine function: 0 ≤ arccos
Setting
In Fig. 5–Fig. 7, we give examples of this type of collisions for different
8. Concluding remarks
The soliton properties in spinor Bose–Einstein condensates have been investigated. Considering two experimental achievements in atomic condensates, the matter-wave soliton and the spinor condensate, at the same time, we have predicted some new phenomena.
Based on the results provided in Sec. 2–4, in Sec. 5 we have introduced the new integrable model which describes the dynamics of the multicomponent matter-wave soliton. The key idea is finding the integrable condition of the original coupled nonlinear equations, i.e., the spinor GP equations derived in Sec. 4. The integrable condition expressed by the coupling constants, which is accessible via the confinement induced resonance explained in Sec. 3.
In Sec. 6, we classify the one-soliton solution. There exist two distinct spin states: ferromagnetic, |F
In Sec. 7, we have analyzed two-soliton solutions which rule collisional phenomena of the multiple solitons. Specifying the initial conditions, we have demonstrated two-soliton collisions in three characteristic cases: polar-polar, polar-ferromagnetic, ferromagnetic-ferromagnetic. In their collisions, the polar soliton is always “passive” which means that it does not rotate its partner’s polarization while the ferromagnetic soliton does. Thus, in the polar-ferromagnetic collision, one can use the polar soliton as a signal and ferromagnetic soliton as a switch to realize a coherent matter-wave switching device. Collision of two ferromagnetic solitons can be interpreted as the spin precession around the total spin. The rotation angle depends on the total spin, amplitude and velocity of the solitons. Only varying the velocity induces drastic change of the population shifts among the components.
Stability of spinor solitons has been investigated numerically and perturbatively (Li et al.; 2005) (Dabrowska-Wüster et al.; 2007) (Doktorov et al.; 2008). It is also interesting to pursue the soliton dynamics of spinor condensates under longitudinal harmonic trap (Zhang et al.; 2007). Recently, the integrability of the spinor GP equation has been studied in detail (Gerdjikov et al.; 2009). The behavior of spinor solitons shows a variety of nonlinear dynamics and it is worth exploring them experimentally.
Acknowledgments
This work was supported by Grant-in-Aid for Scientific Research No. 20740182 from MEXT.
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