Dynamical Transitions in Trapped Superfluids Excited by Alternating Fields

Abstract

The paper presents a survey of some dynamical transitions in nonequilibrium trapped Bose-condensed systems subject to the action of alternating fields. Nonequilibrium states of trapped systems can be implemented in two ways: resonant and nonresonant. Under resonant excitation, several coherent modes are generated by external alternating fields with the frequencies been tuned to resonance with some transition frequencies of the trapped system. A Bose system of trapped atoms with Bose–Einstein condensate can display two types of the Josephson effect, the standard one, when the system is separated into two or more parts in different locations, or the internal Josephson effect, when there are no any separation barriers but the system becomes nonuniform due to the coexistence of several coherent modes interacting one with another. The mathematics in both these cases is similar. We focus on the internal Josephson effect. Systems with nonlinear coherent modes demonstrate rich dynamics, including Rabi oscillations, the Josephson effect, and chaotic motion. Under the Josephson effect, there exist dynamic transitions that are similar to phase transitions in equilibrium systems. The bosonic Josephson effect is shown to be implementable not only for quite weakly interacting systems, but also in superfluids with not necessarily as weak interactions. Sufficiently strong nonresonant excitation can generate several types of nonequilibrium states comprising vortex germs, vortex rings, vortex lines, vortex turbulence, droplet turbulence, and wave turbulence. Nonequilibrium states are shown to be characterized and distinguished by effective temperature, effective Fresnel number, and dynamic scaling laws.

1. Introduction

In recent years, a high interest has been attacted to the study of dilute gases exhibiting, at low temperatures, Bose–Einstein condensation in traps [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] and in optical lattices [16,17,18,19]; see also consideration in the most recent monographs [20,21].

The present paper focuses on nonequilibrium Bose–Einstein condensates created by applying external alternating fields. The latter can be of two kinds: resonant and nonresonant. The resonant generation of nonequilibrium condensates excites the system by oscillating fields with the frequencies tuned in resonance with the transition frequencies corresponding to the chosen coherent modes. At the nonresonant excitation, nonequilibrium condensates are produced by sufficiently strong alternating fields with frequencies not neccessarily satisfying any resonant conditions.

When an external potential separates a Bose-condensed system into several clouds in different spatial locations, as in optical lattices or in a double well, there can arise the bosonic Josephson effect, similar to that arising in fermionic Josephson junctions [22,23,24,25]. When there exists a spatial separation of bosonic clouds, as in a double well or in a lattice, there exists the ordinary bosonic Josephson effect observed in trapped Bose gases [26,27,28] and in rather weakly linked reservoirs of superfluid helium [29,30]. A setup, carrying out an effective double-well potential, can also be created in a binary mixture, where two vortices of one component form two effective wells and the other component exhibits Josephson oscillations between the cores of the vortices [31].

The other setup is when there exist two interpenetrating populations, not separated by any barrier. Then, there can appear a current due to the different spatial shapes of the populations. The latter effect is called the internal Josephson effect [32] or quantum dynamical tunneling [33]. The mathematical formulation for both types of the Josephson effect are similar.

In the present paper, we consider the internal Josephson effect in trapped Bose systems. The coexistence of several different modes can be considered as follows. In equilibrium systems, Bose–Einstein condensation implies that the lowest energy level is occupied by a macroscopic number of atoms. By applying an external resonant field with the frequency $\omega$ in resonance with the transition frequency ${\omega }_{12}$ between two energy levels, a non-ground-state Bose condensate can be created, as is proposed in Refs. [34,35]. To keep the existence of non-ground-state condensates, the energy needs to be permanently pumped into the system, making the latter nonuniform due to the coexisting condensate modes of different shapes. The other possibility is to consider the coexistence of Bose condensates with different internal hyperfine states connected with each other by a resonant Rabi field, as is implemented in some experiments [36,37].

We consider the internal Josephson effect supported by the resonant generation of nonlinear coherent modes in a trapped Bose-condensed system. The modes are termed nonlinear, since they represent stationary states of the nonlinear Schrödinger equation characterizing a Bose–Einstein condensate confined in a trap. Then, the modes are called coherent as far as the condensate wave function describes the coherent part of a Bose-condensed system. General consideration of coherent states can be found in the monographs [38,39] and in the exhaustive review by Victor Dodonov [40]. Applications of coherent states in statistical and Bose-condensed systems are described in Refs. [13,41].

The nonresonant excitation of a trapped Bose–Einstein condensate creates a nonequilibrium system comprising, given the strength of the alternating field, different topological objects, such as vortex germs, vortex rings, and vortex lines, and it can generate highly nonequilibrium states, such as vortex turbulence, droplet turbulence, and wave turbulence.

The paper is organized as follows. Section 2 briefly recalls on the derivation of the coupled system of equations, necessary for the realization of the bosonic Josephson effect under relatively weak interactions and zero temperature. Then, Section 3 describes the dynamical transitions occurring in the system, and Section 4 generalizes the approach to the case of trapped atoms with relatively strong interactions. The generalization demonstrates the possibility of observing and studying the bosonic Josephson effect, as well as the related dynamical transitions, in a larger class of trapped superfluid systems. Section 5 describes some ramifications of the resonant method of coherent mode generation. In particular, the excitation of coherent modes by means of interaction modulation is discussed. The feasibility to generate several modes by applying several external oscillating fields is shown. As well, the feasibility of higher-order resonances, accompanied by harmonic generation and parametric conversion, is noted. Section 6 is devoted to nonresonant generation of qualitatively different nonequilibrium states of trapped atoms comprising topological coherent modes such as vortex germs, vortex rings, vortex lines, and coherent droplets. Different regimes of quantum vortex turbulence are shown to be distinguished by the types of dynamic scaling.

Throughout the paper, the system of units is employed where the Planck, ℏ, and Boltzmann, $k_B$, constants are set to one.

2. Resonant Generation

Let us consider a system of N trapped atoms interacting through the local potential

$$\mathrm{\Phi }\left(\mathbf{r}\right)={\mathrm{\Phi }}_0\delta \left(\mathbf{r}\right)\mathrm{with}{\mathrm{\Phi }}_0=4\pi a_s/m,$$

(1)

where $\mathbf{r}$ denotes the 3-dimensional radius-vector (position) of the atom, $a_s$ is a scattering length and m is atom mass. $\delta \left(\mathbf{r}\right)$ is the Dirak delta function. The second-quantized energy Hamiltonian, with the field operators $\psi$ (and their Hermitian conjugate ${\psi }^{†}$), reads

$$\begin{array}{ccc}\hfill \widehat{H}& =& \int {\psi }^{†}(\mathbf{r},t)\left[-{\displaystyle \frac{{\nabla }^2}{2m}}+U(\mathbf{r},t)\right]\psi (\mathbf{r},t)d\mathbf{r}\hfill \\ \hfill & +& {\displaystyle \frac{1}{2}}{\mathrm{\Phi }}_0\int {\psi }^{†}(\mathbf{r},t){\psi }^{†}(\mathbf{r},t)\psi (\mathbf{r},t)\psi (\mathbf{r},t)d\mathbf{r},\hfill \end{array}$$

(2)

where t denotes the time and the external potential

$$U(\mathbf{r},t)=U\left(\mathbf{r}\right)+V(\mathbf{r},t)$$

(3)

includes a trapping potential $U\left(\mathbf{r}\right)$ and a modulating potential $V(\mathbf{r},t)$ considered in the form

$$V(\mathbf{r},t)=V_1\left(\mathbf{r}\right)cos\left(\omega t\right)+V_2\left(\mathbf{r}\right)sin(\omega ,t).$$

(4)

At zero temperature and asymptotically weak interactions, all atoms can be assumed to be in the condensed state, that is, a coherent state. The latter is defined as the eigenstate of the destruction operator, according to the equation

$$\psi (\mathbf{r},t)|\eta \rangle =\eta (\mathbf{r},t\left)\right|\eta \rangle ,$$

(5)

where $|\eta \rangle$ is a coherent state and $\eta (\mathbf{r},t)$ is a coherent field representing the condensate wave function. Averaging the Heisenberg equation of motion over the coherent state results in the nonlinear Schrödinger equation for the condensate wave function

$$i{\displaystyle \frac{\partial }{\partial t}}\eta (\mathbf{r},t)=\widehat{H}\left[\eta \right]\eta (\mathbf{r},t),$$

(6)

with the nonlinear Hamiltonian

$$\widehat{H}\left[\eta \right]=-{\displaystyle \frac{{\nabla }^2}{2m}}+U(\mathbf{r},t)+{\mathrm{\Phi }}_0{\left|\eta (\mathbf{r},t)\right|}^2.$$

(7)

Equation (6) has been introduced by Nikolay Bogoliubov [42] (see also [43,44]) and later studied in a number of investigations starting with David Gross [45,46,47,48,49], Lev Pitaevskii [50], and Tai Tsun Wu [51].

The time-dependent wave function $\eta (\mathbf{r},t)$ can be expanded [52,53] over the stationary coherent modes ${\phi }_n\left(\mathbf{r}\right)$ satisfying the equation

$$H\left[{\phi }_n\right]{\phi }_n\left(\mathbf{r}\right)=E_n{\phi }_n\left(\mathbf{r}\right),$$

(8)

where

$$H\left[{\phi }_n\right]=-{\displaystyle \frac{{\nabla }^2}{2m}}+U\left(\mathbf{r}\right)+N{\mathrm{\Phi }}_0{\left|{\phi }_n\left(\mathbf{r}\right)\right|}^2,$$

N represents the number of atoms and n denotes the stationary states.

This expansion reads

$$\eta (\mathbf{r},t)=\sqrt{N}\sum _n{c}_n\left(t\right)e^{-i{E}_{n}t}{\phi }_n\left(\mathbf{r}\right),$$

(9)

where $E_n$ represents the stationary-state energy and $c_n$ are the expansion coefficients.

The coherent modes ${\phi }_n\left(\mathbf{r}\right)$ are not necessarily mutually orthogonal but are normalized:

$$\int {\phi }_n^{*}\left(\mathbf{r}\right){\phi }_n\left(\mathbf{r}\right)d\mathbf{r}=1.$$

(10)

Suppose that two coherent modes with energies $E_1<E_2$ to be generated with the transition frequency

$${\omega }_{21}\equiv E_2-E_1.$$

(11)

To this end, the frequency of the modulating field (4) has to be close to the transition frequency ${\omega }_{21}$, such that the quasi-resonance condition

$$\left|{\displaystyle \frac{\mathrm{\Delta }\omega }{\omega }}\right|\ll 1,\mathrm{\Delta }\omega \equiv \omega -{\omega }_{21},$$

(12)

to be valid.

Transitions between the energy levels are induced by two transition amplitudes, one being caused by atomic interactions

$${\alpha }_{mn}\equiv {\mathrm{\Phi }}_{0}N\int {\left|{\phi }_m\left(\mathbf{r}\right)\right|}^2\left(2|{\phi }_n{\left(\mathbf{r}\right)|}^2-{\left|{\phi }_m\left(\mathbf{r}\right)\right|}^2\right)d\mathbf{r},$$

(13)

and the other amplitude being due to the alternating field with the Rabi frequency:

$${\beta }_{mn}\equiv \int {\phi }_m^{*}\left(\mathbf{r}\right)\left(V_1\left(\mathbf{r}\right)-i{V}_2\left(\mathbf{r}\right)\right){\phi }_n\left(\mathbf{r}\right)d\mathbf{r}.$$

(14)

In order to preserve acceptable resonance and to avoid power broadening, the transition amplitudes (13) and (14) have to be smaller than the transition frequency:

$$\left|{\displaystyle \frac{{\alpha }_{12}}{{\omega }_{12}}}\right|\ll 1,\left|{\displaystyle \frac{{\alpha }_{21}}{{\omega }_{21}}}\right|\ll 1,\mathrm{and}\left|{\displaystyle \frac{{\beta }_{12}}{{\omega }_{12}}}\right|\ll 1.$$

(15)

If at the initial time only two modes are populated, then switching-on the resonant field does not populate other energy levels (leaving touched only the chosen two modes) which results in the evolution equations for the coefficients $c_n\left(t\right)$ of expansion (9):

$$i{\displaystyle \frac{d{c}_1}{dt}}={\alpha }_{12}{\left|c_2\right|}^2{c}_1+{\displaystyle \frac{1}{2}}{\beta }_{12}{c}_2{e}^{i\mathrm{\Delta }\omega ·t},$$

$$i{\displaystyle \frac{d{c}_2}{dt}}={\alpha }_{21}{\left|c_1\right|}^2{c}_2+{\displaystyle \frac{1}{2}}{\beta }_{12}^{*}{c}_{1}5e^{-i\mathrm{\Delta }\omega ·t},$$

(16)

with the normalization constraint

$$|c_1{|}^2+{|c_2|}^2=1.$$

(17)

It is convenient to introduce the population difference

$$s\equiv |c_2{|}^2-{\left|c_1\right|}^2.$$

(18)

Using the population difference (18), the coefficient functions $c_i$ can be expressed as

$$c_1=\sqrt{{\displaystyle \frac{1-s}{2}}}exp\left(i\left({\pi }_1+{\displaystyle \frac{\mathrm{\Delta }\omega }{2}}t\right)\right),$$

$$c_2=\sqrt{{\displaystyle \frac{1+s}{2}}}exp\left(i\left({\pi }_2-{\displaystyle \frac{\mathrm{\Delta }\omega }{2}}t\right)\right),$$

(19)

where ${\pi }_i={\pi }_i\left(t\right)$ represent real-valued phases.

Let us also introduce the average interaction amplitude (strength)

$$\alpha \equiv {\displaystyle \frac{1}{2}}({\alpha }_{12}+{\alpha }_{21}),$$

(20)

the notation for the Rabi frequency

$${\beta }_{12}\equiv \beta e^{i\nu }(\mathrm{with}\beta \equiv |{\beta }_{12}\left|\right),$$

(21)

where $\nu$ denotes a phase, and the effective detuning

$$\delta \equiv \mathrm{\Delta }\omega +{\displaystyle \frac{1}{2}}({\alpha }_{12}-{\alpha }_{21}).$$

(22)

The effective phase difference is then defined as

$$x\equiv {\pi }_2-{\pi }_1+\nu .$$

(23)

The population difference s (18) and the phase difference x (23) are the main quantities defining the dynamics of the generated populations. These two functions of time vary in the intervals

$$s\in [-1,1]\mathrm{and}x\in [0,2\pi )$$

(24)

and satisfy the equations

$${\displaystyle \frac{ds}{dt}}=-\beta \sqrt{1-s^2}sinx\mathrm{and}{\displaystyle \frac{dx}{dt}}=\alpha s+{\displaystyle \frac{\beta s}{\sqrt{1-s^2}}}cosx+\delta .$$

(25)

Equations (25) can be obtained from the Hamiltonian equations

$${\displaystyle \frac{ds}{dt}}=-{\displaystyle \frac{\partial \overline{H}}{\partial x}}\mathrm{and}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{\partial \overline{H}}{\partial s}}$$

(26)

with the Hamiltonian

$$\overline{H}={\displaystyle \frac{1}{2}}\alpha s^2-\beta \sqrt{1-s^2}cosx+\delta s.$$

(27)

Introducing the dimensionless quantities for the Rabi frequency (21) (pumping parameter) and effective detuning (22)

$$b\equiv \beta /\alpha \mathrm{and}ϵ=\delta /\alpha ,$$

(28)

respectively, and measuring time in units of $1/\alpha$ reduce the effective Hamiltonian (27) to

$$H(s,x)\equiv {\displaystyle \frac{\overline{H}}{\alpha }}={\displaystyle \frac{1}{2}}{s}^2-b\sqrt{1-s^2}cosx+ϵs,$$

(29)

and the dynamical equations (26) to

$${\displaystyle \frac{ds}{dt}}=-b\sqrt{1-s^2}sinx\mathrm{and}{\displaystyle \frac{dx}{dt}}=s+{\displaystyle \frac{bs}{\sqrt{1-s^2}}}cosx+ϵ.$$

(30)

3. Dynamical Transitions

Under a ‘dynamical transition’ one understands the qualitative change of the phase portrait formed by the set of fixed points [54,55]. Dynamic and stationary solutions of the evolution equations, similar to Equation (30), have already been studied in Refs. [34,35,52,53,56,57,58,59,60,61,62,63,64,65] and are summarized in the reviews [12,14]. Here, we follow the analysis of the papers [34,35]. Throughout, a small detuning parameter (see Equation (28)) is considered: $\left|ϵ\right|\ll 1$.

Given the pumping parameter b (see Equation (28)), characterizing the dimensionless Rabi frequency (as been normalized by the interaction strength: $b=\beta /\alpha$), there may exist several dynamic regimes. In the case when the Rabi frequency is larger than the interaction strength, there are two fixed points:

$$s_1^{*}={\displaystyle \frac{ϵ}{b}},x_1^{*}=0;$$

and

$$s_2^{*}=-{\displaystyle \frac{ϵ}{b}},x_2^{*}=\pi ;(\mathrm{with}{b}^2\ge 1),$$

(31)

both being the centers. This regime is close to the regime of classical Rabi oscillations.

When the Rabi frequency is smaller than the interaction strength, with the interactions being repulsive, then there are four fixed points:

$$s_1^{*}={\displaystyle \frac{ϵ}{b}},x_1^{*}=0;$$

$$s_2^{*}=-{\displaystyle \frac{ϵ}{b}},x_2^{*}=\pi ;$$

$$s_3^{*}=\sqrt{1-b^2}+{\displaystyle \frac{b^2ϵ}{1-b^2}},x_3^{*}=\pi ;$$

and

$$s_4^{*}=-\sqrt{1-b^2}+{\displaystyle \frac{b^2ϵ}{1-b^2}},x_4^{*}=\pi ;(\mathrm{with}0<b<1).$$

(32)

The fixed point $\{s_2^{*},x_2^{*}\}$ is a saddle point, while all other points are centers.

If the interactions are attractive, then $b<0$, and instead of the fixed points (32), one has

$$s_1^{*}={\displaystyle \frac{ϵ}{b}},x_1^{*}=0;$$

$$s_2^{*}=-{\displaystyle \frac{ϵ}{b}},x_2^{*}=\pi ;$$

$$s_5^{*}=\sqrt{1-b^2}+{\displaystyle \frac{b^2ϵ}{1-b^2}},x_5^{*}=0;$$

and

$$s_6^{*}=-\sqrt{1-b^2}+{\displaystyle \frac{b^2ϵ}{1-b^2}},x_6^{*}=0;(\mathrm{with}-1<b<0).$$

(33)

As one can see, the dynamics is similar to that of the case (32), except that the effective phase differences $x_5^{*}$ and $x_6^{*}$ are shifted by $\pi$. This occurs due to the symmetry of the evolution equations (30) with respect to the replacement $b\to -b$, $x\to x-\pi$, and $ϵ\to -ϵ$. Therefore, it is admissible to limit the consideration by the positive $b>0$. Let us recall that $b=0$ denotes the absence of pumping and no mode generation, with s keepng a constant value.

In the case of a pure resonance, when $\omega ={\omega }_{21}$, hence $ϵ=0$, the dynamical transition, happening at $b=1$, is the transition from the set of two stable centers

$$s_1^{*}=0x_1^{*}=0\left(\mathrm{center}\right);$$

and

$$s_2^{*}=0x_2^{*}=\pi \left(\mathrm{center}\right);(\mathrm{with}{b}^2\ge 1),$$

(34)

to the set of three centers and a saddle point

$$s_1^{*}=0,x_1^{*}=0\left(\mathrm{center}\right);$$

$$s_2^{*}=0,x_2^{*}=\pi \left(\mathrm{saddle}\right);$$

$$s_3^{*}=\sqrt{1-b^2},x_3^{*}=\pi \left(\mathrm{center}\right);$$

and

$$s_4^{*}=-\sqrt{1-b^2},x_4^{*}=\pi \left(\mathrm{center}\right);(\mathrm{with}0<b<1).$$

(35)

The regime with $b<1$, the case (35), when the Rabi frequency is smaller than the interaction strength, is sometimes associated with the Josephson dynamics. The change in the dynamic regime at $b=1$ is named a ‘pitchfork bifurcation’ [54]. The variation in the fixed point $s^{*}$ as a function of b is shown in Figure 1.

Since $b=\beta /\alpha$ is the ratio of the Rabi frequency $\beta$ to the interaction strength $\alpha$, the regime with $b>1$ can be called the ‘regime of strong pumping’ or ‘the regime of weak interactions’, while the regime with $b<1$ is then the ‘regime of weak pumping’ or the ‘regime of strong interactions’. Sometimes, one names the regime of $b>1$ as the regime of Rabi dynamics and one of $b<1$ the regime of Josephson dynamics, although for all $b>0$ the approximate solutions for the mode populations can be represented as [34]

$$n_1=1-{\displaystyle \frac{{\left|\beta \right|}^2}{{\mathrm{\Omega }}^2}}{sin}^2{\displaystyle \frac{\mathrm{\Omega }t}{2}},n_2={\displaystyle \frac{{\left|\beta \right|}^2}{{\mathrm{\Omega }}^2}}{sin}^2{\displaystyle \frac{\mathrm{\Omega }t}{2}}$$

(36)

with the effective Rabi frequency $\mathrm{\Omega }$ defined as

$${\mathrm{\Omega }}^2={(\alpha (n_1-n_2)-\mathrm{\Delta }\omega )}^2+{\left|\beta \right|}^2,$$

(37)

where the notations

$$n_1\equiv |c_1{|}^2\mathrm{and}{n}_2\equiv {\left|c_2\right|}^2$$

(38)

are used and the initial condition

$$c_1\left(0\right)=1\mathrm{and}{c}_2\left(0\right)=0$$

(39)

is set. Thus, the dynamics is always of the effective Rabi type, to keep in mind the effective Rabi frequency (37). However, the latter depends on the relation between the standard Rabi frequency $\beta$ (21) and the interaction strength $\alpha$ (20) and is a function of time. For the convenience of nomenclature, one can also term the regime of $b>1$ the Rabi regime and one of $b<1$ the Josephson regime.

In addition to the dynamical transition between the Rabi and Josephson dynamics, there is also another nonstandard dynamical transition related to the effect of saddle separatrix crossing [14,34,35]. In the Josephson regime, the fixed point $\{s_2^{*},x_2^{*}\}=\{0,\pi \}$ is a saddle (see Equation (35)). The trajectory traversing the saddle is called the saddle separatrix as soon as this trajectory separates the basins of attraction for different fixed points. The separatrix satisfies the condition (see Equation (29))

$$H(s,x)=H(s_2^{*},x_2^{*}),$$

(40)

which yields the separatrix equation

$${\displaystyle \frac{1}{2}}{s}^2-b\sqrt{1-s^2}cosx+ϵs=b.$$

(41)

The separatrix (41) is shown in Figure 2 for the case of the resonance with the detuning parameter $ϵ=0$ and in Figure 3 for $ϵ=0.1$.

In general, the separatrix (41) consists of two parts, the upper (positive, $s>0$) and the lower (negaitve, $s<0$) parts. If the initial point ($s_0,x_0)$ happens to be above the upper separatrix, the trajectory is always locked from below by the upper separatrix. If, however, the initial point is below the lower separatrix, the trajectory is always locked from above by the lower separatrix. This is called the effect of mode locking [14,34,35]. The critical line on the parametric plane $\{b,ϵ\}$, where the effect of the transition between the ‘mode-locked’ and ‘mode-unlocked’ regimes occurs, is the separatrix line crossing the initial point ($s_0,x_0$) of the trajectory, such that

$$H(s_2^{*},x_2^{*})=H(s_0,x_0),$$

(42)

which yields the critical-line equation

$${\displaystyle \frac{1}{2}}{s}_0^2-b_c\sqrt{1-s_0^2}cos{x}_0+{ϵ}_c{s}_0=b_c$$

(43)

with the critical pumping parameter $b_c$ then via $s_0$ and $x_0$:

$$b_c={\displaystyle \frac{s_0^2+2{ϵ}_c{s}_0}{2(1+\sqrt{1-s_0^2}cos{x}_0)}}.$$

(44)

For $b<b_c$, given the chosen initial condition, the mode locking implies that the trajectory of the population difference s is locked in the lower or upper half of the interval $[-1,1]$, so that, for example, either

$$-1\le s<0\mathrm{at}b<b_c\mathrm{and}{s}_0=-1,$$

(45)

or

$$0\le s<1\mathrm{at}b<b_c\mathrm{and}{s}_0=1.$$

(46)

However, for $b>b_c$, the population difference oscillates in the entire interval $[-1,1]$; that is, the mode is not locked.

For instance, under the initial condition $s_0=\pm 1$, the critical line (44) simplifies to

$$b_c={\displaystyle \frac{1}{2}}\pm {ϵ}_c.$$

(47)

Beyond the critical line, the dynamics changes from the mode-locked to mode-unlocked regimes and the oscillation period doubles. The critical dynamics in the vicinity of the critical line as been studied in Refs. [14,34,35].

Summarizing, the dynamics of the coherent mode generation, under the initial condition with $s_0=\pm 1$, contains the following regimes:

$$b=0\left(\mathrm{at}\mathrm{equilibrium}\right),$$

$$0<b<b_c\left(\mathrm{mode}\text{-}\mathrm{locked}\mathrm{Josephson}\mathrm{regime}\right),$$

$$b=b_c\left(\mathrm{critical}\mathrm{dynamics}\right),$$

$$b_c<b<1\left(\mathrm{mode}\text{-}\mathrm{unlocked}\mathrm{Josephson}\mathrm{regime}\right),$$

$$b=1(\mathrm{pitchfork}\mathrm{bifurcation}),$$

$$b>1\left(\mathrm{Rabi}\mathrm{regime}\right).$$

(48)

As the pumping parameter b increases, the system passes from the mode-locked regime to the mode-unlocked regime. Crossed the critical line, with increasing b, the system dynamics changes dramatically. In particular, the oscillation amplitude and the oscillation period approximately double [14].

The dynamic transition on the critical line is similar to a phase transition in an equilibrium statistical system [34,35,61,66]. To show this, one needs to define an effective stationary energy. To this end, let us notice that the evolution Equations (16) can be represented in the Hamiltonian form

$$i{\displaystyle \frac{d{c}_1}{dt}}={\displaystyle \frac{\partial H_{\mathrm{eff}}}{\partial c_1^{*}}}\mathrm{and}i{\displaystyle \frac{d{c}_2}{dt}}={\displaystyle \frac{\partial H_{\mathrm{eff}}}{\partial c_2^{*}}}$$

(49)

with the effective Hamiltonian

$$H_{\mathrm{eff}}=\alpha n_1{n}_2+{\displaystyle \frac{1}{2}}\left(\beta e^{i\mathrm{\Delta }\omega ·t}{c}_1^{*}{c}_2+{\beta }^{*}{e}^{-i\mathrm{\Delta }\omega ·t}{c}_2^{*}{c}_1\right).$$

(50)

Equations (16), in the Josephson regime, i.e., when $b<1$, can be solved resorting to the averaging techniques [67] and the scale separation approach [68]. Then, the functional variables $n_i$ are treated as slow, as compared to the fast varying variables $c_i$. This solution and the averaging of the effective Hamiltonian (50) over time yield the effective energy

$$E_{\mathrm{eff}}\equiv {\displaystyle \frac{\alpha b^2}{2u^2}}\left({\displaystyle \frac{b^2}{2u^2}}+\delta \right),$$

(51)

where the bar denotes the time averaging and

$$u^2={\displaystyle \frac{1}{2}}\left(1+\sqrt{1-4b^2}\right).$$

Then the order parameter can be introduced as the difference of the time-averaged mode populations (36):

$$M\equiv {\overline{n}}_1-{\overline{n}}_2=1-b^2/u^2.$$

(52)

The pumping capacity, describing the capacity of the system to store the energy pumped in, can be then defined as

$$C_{\beta }\equiv \partial E_{\mathrm{eff}}/\partial \beta .$$

(53)

The dependence of the order parameter (52) on the effective detuning (22) is characterized by the detuning susceptibility

$${\chi }_{\delta }\equiv \left|{\displaystyle \frac{\partial M}{\partial \delta }}\right|.$$

(54)

In the vicinity of the critical pumping $b_c$ (47), the characteristics (52)–(54) introduced then describe the critical behavior with respect to the diminishing variable

$$\tau \equiv |b-b_c|,$$

(55)

namely,

$$\eta \simeq {\displaystyle \frac{1}{\sqrt{2}}}(1-2\delta ){\tau }^{1/2}\mathrm{for}\tau \to 0$$

(56)

for the order parameter (52),

$$C_{\beta }\simeq {\displaystyle \frac{1}{4\sqrt{2}}}\mathrm{for}\tau \to 0$$

(57)

for the pumping capacity (53), and

$${\chi }_{\delta }\simeq {\displaystyle \frac{1}{\sqrt{2}}}\mathrm{for}\tau \to 0$$

(58)

for the the detuning susceptibility (54).

The related critical exponents satisfy the same sum rule as in the case of equilibrium statistical systems: $0.5+2\times 0.5+0.5=2$ [20].

4.  Strong Bose Particles Interactions

The dynamics studied above assumes that the system is gaseous, being composed of significantly weakly interacting particles, such that these interactions are as weak that all particles in the system are Bose-condensed and the interactions do not disturb the evolution of the condensed part of the system significantly. In order to study the influence of interactions in a more general case, one has, foremost, to address the general form of the equation for the condensate. In this Section, we analyze the possibility of realizing the Josephson effect in a Bose-condensed system of particles with sufficiently strong interactions.

For the condensate state, a system with global gauge symmetry breaking has to be considered [5,9,12,13,69,70]. In general, to consider a Bose system with broken gauge symmetry, it is sufficient to employ the Bogoliubov shift for the field operator

$$\psi (\mathbf{r},t)=\eta (\mathbf{r},t)+{\psi }_1(\mathbf{r},t),$$

(59)

where $\eta$ is the condensate wave function and ${\psi }_1$ is an operator of uncondensed particles. $\psi$ and $\eta$ are mutually orthogonal:

$$\int {\eta }^{*}(\mathbf{r},t)\psi (\mathbf{r},t)d\mathbf{r}=0.$$

(60)

Then, the so-called grand Hamiltonian has the form

$$H=\widehat{H}-{\mu }_0{N}_0-{\mu }_1{\widehat{N}}_1-\widehat{\mathrm{\Lambda }}$$

(61)

where $\widehat{H}$ is the energy Hamiltonian (2), ${\mu }_0$ is a condensate chemical potential ensuring the normalization

$$N_0=\int {\left|\eta (\mathbf{r},t)\right|}^{2}d\mathbf{r},$$

(62)

to the number $N_0$ of condensed particles, ${\mu }_1$ is the chemical potential of uncondensed particles preserving the normalization

$$N_1=\langle {\widehat{N}}_1\rangle ,{\widehat{N}}_1=\int {\psi }_1^{†}(\mathbf{r},t){\psi }_1(\mathbf{r},t)d\mathbf{r},$$

(63)

to the number $N_1$ of uncondensed particles, the anglar brackets denote the statistical averaging, and the operator $\widehat{\mathrm{\Lambda }}$ defined as

$$\widehat{\mathrm{\Lambda }}=\int \left(\lambda (\mathbf{r},t){\psi }_1^{†}(\mathbf{r},t)+{\lambda }^{*}(\mathbf{r},t){\psi }_1(\mathbf{r},t)\right),$$

(64)

where the asterisk denote the complex conjugate and $\lambda$ denotes the Lagrange multipliers, ensures the quantum-number conservation condition

$$\langle \widehat{\mathrm{\Lambda }}\rangle =0.$$

(65)

To satisfy the condition (65), the Lagrange multipliers $\lambda$ in Equation (64) are chosen in a way to cancel the terms linear in the operators ${\psi }_1$ in the Hamiltonian (61).

The condensate wave function $\eta$ is defined by the equation

$$i{\displaystyle \frac{\partial }{\partial t}}\eta (\mathbf{r},t)=〈{\displaystyle \frac{\delta H}{\delta {\eta }^{*}(\mathbf{r},t)}}〉$$

(66)

of the variation $\delta H$ of the Hamiltonian (61) relative to the variation $\delta {\eta }^{*}$, which leads to the condensate equation

$$i{\displaystyle \frac{\partial }{\partial t}}\eta (\mathbf{r},t)=\left(-{\displaystyle \frac{{\nabla }^2}{2m}}+U-{\mu }_0\right)\eta (\mathbf{r},t)$$

$$+\int \mathrm{\Phi }(\mathbf{r}-{\mathbf{r}}^{\prime })\left(\rho ({\mathbf{r}}^{\prime },t)\eta (\mathbf{r},t)+{\rho }_1(\mathbf{r},{\mathbf{r}}^{\prime },t)\eta ({\mathbf{r}}^{\prime },t)+{\sigma }_1(\mathbf{r},{\mathbf{r}}^{\prime },t){\eta }^{*}({\mathbf{r}}^{\prime },t)+{\xi }_1(\mathbf{r},{\mathbf{r}}^{\prime },t)\right)d{\mathbf{r}}^{\prime },$$

(67)

where

$$\rho (\mathbf{r},t)={\rho }_0(\mathbf{r},t)+{\rho }_1(\mathbf{r},t)$$

(68)

represents the particle density,

$${\rho }_0(\mathbf{r},t)\equiv {\left|\eta (\mathbf{r},t)\right|}^2$$

(69)

represents the density of condensed particles,

$${\rho }_1(\mathbf{r},t)\equiv {\rho }_1(\mathbf{r},\mathbf{r},t)=\langle {\psi }_1^{†}(\mathbf{r},t){\psi }_1(\mathbf{r},t)\rangle$$

(70)

represents the density of uncondensed particles,

$${\rho }_1(\mathbf{r},{\mathbf{r}}^{\prime },t)=\langle {\psi }_1^{†}({\mathbf{r}}^{\prime },t){\psi }_1(\mathbf{r},t)\rangle$$

(71)

represents the single-particle density matrix,

$${\sigma }_1(\mathbf{r},t)\equiv {\sigma }_1(\mathbf{r},\mathbf{r},t)=\langle {\psi }_1(\mathbf{r},t){\psi }_1(\mathbf{r},t)\rangle ,$$

(72)

represents the amplitude of pairing particles (or anomalous average), with

$${\sigma }_1(\mathbf{r},{\mathbf{r}}^{\prime },t)=\langle {\psi }_1({\mathbf{r}}^{\prime },t){\psi }_1(\mathbf{r},t)\rangle ,$$

(73)

and

$${\xi }_1(\mathbf{r},{\mathbf{r}}^{\prime },t)=\langle {\psi }_1^{†}({\mathbf{r}}^{\prime },t){\psi }_1({\mathbf{r}}^{\prime },t){\psi }_1(\mathbf{r},t)\rangle .$$

(74)

represents the triple anomalous average.

For the local interaction potential (1), the condensate Equation (67) reads

$$i{\displaystyle \frac{\partial }{\partial t}}\eta (\mathbf{r},t)=\left(-{\displaystyle \frac{{\nabla }^2}{2m}}+U-{\mu }_0\right)\eta (\mathbf{r},t)$$

$$+{\mathrm{\Phi }}_0\left(\left({\rho }_0(\mathbf{r},t)+2{\rho }_1(\mathbf{r},t)\right)\eta (\mathbf{r},t)+{\sigma }_1(\mathbf{r},t){\eta }^{*}(\mathbf{r},t)+{\xi }_1(\mathbf{r},t)\right),$$

(75)

where

$${\xi }_1(\mathbf{r},t)\equiv {\xi }_1(\mathbf{r},\mathbf{r},t).$$

(76)

If the Hartree–Fock–Bogoliubov approximation is used, the expression (76) turns to zero, while is finite in general.

In equilibrium, the functions in Equation (75) are time-independent. That is, introducing the supplementary Hamiltonian $H_{\sup }\left[\eta \right]$ acting on the condensate function according to the definition

$$H_{\sup }\left[\eta \right]\eta \left(\mathbf{r}\right)=\left(-{\displaystyle \frac{{\nabla }^2}{2m}}+U\left(\mathbf{r}\right)\right)\eta \left(\mathbf{r}\right)$$

$$+{\mathrm{\Phi }}_0\left(\left({\left|\eta \left(\mathbf{r}\right)\right|}^2+2{\rho }_1\left(\mathbf{r}\right)\right)\eta \left(\mathbf{r}\right)+{\sigma }_1\left(\mathbf{r}\right){\eta }^{*}\left(\mathbf{r}\right)+{\xi }_1\left(\mathbf{r}\right)\right),$$

(77)

Equation (67), in the absence of the external perturbation (4), reduces to the equilibrium eigenvalue form

$$H_{\sup }\left[\eta \right]\eta \left(\mathbf{r}\right)={\mu }_0\eta \left(\mathbf{r}\right).$$

(78)

In general, the eigenvalue equation of the type of Equation (78) leads to a set of stationary solutions of the equation

$$H_{\sup }\left[{\eta }_n\right]{\eta }_n\left(\mathbf{r}\right)=E_n{\eta }_n\left(\mathbf{r}\right),$$

(79)

with the lowest energy level corresponding to the chemical potential

$${\mu }_0=\underset{n}{min}{E}_n.$$

(80)

To compare the problem with the zero-temperature considered in the previous Sections, it is convenient to pass from the functions ${\eta }_n$, normalized to $N_0$, to the functions ${\phi }_n$, normalized to one, by the substitution (cf. Equation (9))

$${\eta }_n\left(\mathbf{r}\right)=\sqrt{N_0}{\phi }_n\left(\mathbf{r}\right),$$

(81)

satisfying the normalizations

$$\int |{\eta }_n{\left(\mathbf{r}\right)|}^{2}d\mathbf{r}=N_0,\int {\left|{\phi }_n\left(\mathbf{r}\right)\right|}^{2}d\mathbf{r}=1.$$

(82)

By defining the supplementary Hamiltonian $H_{\sup }\left[{\phi }_n\right]$ by its action

$$H_{\sup }\left[{\phi }_n\right]{\phi }_n\left(\mathbf{r}\right)=\left({\displaystyle \frac{{\nabla }^2}{2m}}+U\left(\mathbf{r}\right)\right){\phi }_n\left(\mathbf{r}\right)$$

$$+{\mathrm{\Phi }}_0\left(\left(N_0{\left|{\phi }_n\left(\mathbf{r}\right)\right|}^2+2{\rho }_1^{\left(n\right)}\left(\mathbf{r}\right)\right){\phi }_n\left(\mathbf{r}\right)+{\sigma }_1^{\left(n\right)}\left(\mathbf{r}\right){\phi }_n^{*}\left(\mathbf{r}\right)+{\displaystyle \frac{{\xi }_1^{\left(n\right)}\left(\mathbf{r}\right)}{\sqrt{N_0}}}\right),$$

(83)

one obtains the eigenvalue equation

$$H_{\sup }\left[{\phi }_n\right]{\phi }_n\left(\mathbf{r}\right)=E_n{\phi }_n\left(\mathbf{r}\right)$$

(84)

for the coherent modes ${\phi }_n$. Here, the functions ${\rho }_1^{\left(n\right)}$, ${\sigma }_1^{\left(n\right)}$, and ${\xi }_1^{\left(n\right)}$ are the solutions to the equations where the role of the condensate function is played by the mode (81); cf. Equations (70)–(74).

The condensate wave function can be represented as the expansion over the coherent modes:

$$\eta (\mathbf{r},t)=\sqrt{N_0}\sum _n{B}_n\left(t\right){\phi }_n\left(\mathbf{r}\right)e^{-i{\omega }_{n}t},$$

(85)

where $B_n\left(t\right)$ is a slow varying function as compared to the fast oscillating exponential and

$${\omega }_n\equiv E_n-{\mu }_0.$$

(86)

Assuming that the external modulating field (4) is in resonance with a chosen mode with the frequency, for exmaple, ${\omega }_2$, so that the resonance condition (12) is valid, one finds that now

$$\mathrm{\Delta }\omega =\omega -{\omega }_{21},E_1\equiv \underset{n}{min}{E}_n={\mu }_0,$$

$$\mathrm{and}{\omega }_{21}={\omega }_2-{\omega }_1=E_2-E_1={\omega }_2.$$

(87)

Let us assume that at the initial moment of time the system is in its equilibrium state so that only the ground-state coherent mode, with the energy level $E_1$, is occupied. Substituting expansion (85) into Equation (75), one has to keep in mind the case of slowly varying in time ${\rho }_1$ and ${\sigma }_1$. Multiplying Equation (75) from the left by ${\phi }_n^{*}\left(\mathbf{r}\right)exp\left(i{\omega }_2\right)t$, averaging over time and then integrating over the spatial variable $\mathbf{r}$, one arrives at the equations

$$i{\displaystyle \frac{d{B}_1}{dt}}={\alpha }_{12}{\left|B_2\right|}^2{B}_1+{\displaystyle \frac{1}{2}}{\beta }_{12}{B}_2{e}^{i\mathrm{\Delta }\omega ·t}+{\gamma }_1{B}_1,$$

$$i{\displaystyle \frac{d{B}_2}{dt}}={\alpha }_{21}{\left|B_1\right|}^2{B}_2+{\displaystyle \frac{1}{2}}{\beta }_{12}^{*}{B}_1{e}^{-i\mathrm{\Delta }\omega ·t}+{\gamma }_2{B}_2,$$

(88)

where $B_n=B_n\left(t\right)$ and

$${\gamma }_n={\mathrm{\Phi }}_0\int {\phi }_n^{*}\left(\mathbf{r}\right)\left(2\left({\rho }_1(\mathbf{r},t)-{\rho }_1^{\left(n\right)}\left(\mathbf{r}\right)\right){\phi }_n\left(\mathbf{r}\right)-{\sigma }_1^{\left(n\right)}\left(\mathbf{r}\right){\phi }_n^{*}\left(\mathbf{r}\right)-{\displaystyle \frac{{\xi }_1^{\left(n\right)}\left(\mathbf{r}\right)}{\sqrt{N_0}}}\right)d\mathbf{r}.$$

(89)

Recall that ${\omega }_1=E_1-{\mu }_0=0$, when $E_1$ corresponds to the lowest energy level.

Employing the representation

$$B_n=C_n{e}^{-i{\gamma }_{n}t}$$

(90)

reduces the system (88) to

$$i{\displaystyle \frac{d{C}_1}{dt}}={\alpha }_{12}{\left|C_2\right|}^2{C}_1+{\displaystyle \frac{1}{2}}{\beta }_{12}{C}_2{e}^{i{\mathrm{\Delta }}_{12}t},$$

$$i{\displaystyle \frac{d{C}_2}{dt}}={\alpha }_{21}{\left|C_1\right|}^2{C}_2+{\displaystyle \frac{1}{2}}{\beta }_{12}^{*}{C}_1{e}^{-i{\mathrm{\Delta }}_{12}t},$$

(91)

with

$${\mathrm{\Delta }}_{12}\equiv \mathrm{\Delta }\omega +{\gamma }_1-{\gamma }_2.$$

(92)

In this way, one arrives at the system (91) that is similar to the system (16), while differ in the following. The energy levels $E_n$ in the eigenvalue Equation (84) are defined by an essentially more complicated expression (83), and the detuning ${\mathrm{\Delta }}_{12}$ in the Equations (91) now includes the terms ${\gamma }_n$, depending on the interaction strength, as compared to the detuning $\mathrm{\Delta }\omega$ in the Equations (16) independent of the strength of interactions. The larger detuning may destroy the resonance conditions, thus making impossible the resonant generation of the upper coherent modes. Nevertheless, since the terms ${\gamma }_1$ and ${\gamma }_2$ enter in the combination (${\gamma }_1-{\gamma }_2$), the $\gamma$s can compensate one another, making their difference essentially smaller than each ${\gamma }_n$ value is. That is, it is not entirely impossible the coherent modes to be generated, even in a system with rather strong interactions, provided the compensation effect is present.

The treated case of relatively strong interactions reduces to the case of weaker interactions given the interaction strength is so small that the characteristics of non-condensed particles ${\rho }_1$ (70), ${\sigma }_1$ (72), and ${\xi }_1$ (74) tend to zero. Then, ${\gamma }_n$ also tends to zero. Hence, $C_n$ turns to $B_n$ and the system (88) for $B_n$ reduces to the system (16) for $c_n$.

5. Generalizations and Extensions

The resonant generation of coherent modes, described so far, is performed via the trapping potential modulation with the frequency in resonance with the transition frequency between two coherent modes. In such a process, there occur several dynamical regimes and intriguing dynamical transitions between these regimes. In addition, there a number of other nontrivial dynamical effects may happen, some of which are flashed below.

5.1. Modulation Through Interactions

The other way of generating the coherent modes is by modulating particle interactions, which can be accomplished by modulating an external magnetic field employed in Feshbach resonance [71]. An external magnetic field $B=B\left(t\right)$ influences the effective scattering length

$$a_s\left(B\right)=a_s\left(1-{\displaystyle \frac{\mathrm{\Delta }B}{B-B_{\mathrm{res}}}}\right),$$

(93)

where $a_s$ denotes the scattering length far outside of the resonance field $B_{\mathrm{res}}$ and $\mathrm{\Delta }B$ is the resonance width. Then, the interaction potential reads

$${\mathrm{\Phi }}_s\left(t\right)=4\pi a_s\left(B\right)/m.$$

(94)

If the magnetic field oscillates around $B_0$ as

$$B\left(t\right)=B_0+b\left(t\right),$$

(95)

with a relatively small amplitude

$$b\left(t\right)=b_{1}cos\left(\omega t\right)+b_{2}sin\left(\omega t\right),$$

(96)

then the effective potential interaction reads

$$\mathrm{\Phi }\left(t\right)\cong {\mathrm{\Phi }}_0+{\mathrm{\Phi }}_{1}cos\left(\omega t\right)+{\mathrm{\Phi }}_{2}sin\left(\omega t\right),$$

(97)

with

$${\mathrm{\Phi }}_0={\displaystyle \frac{4\pi }{m}}{a}_s\left(1-{\displaystyle \frac{\mathrm{\Delta }B}{B_0-B_{\mathrm{res}}}}\right)$$

$${\mathrm{\Phi }}_1={\displaystyle \frac{4\pi a_s{b}_1\mathrm{\Delta }B}{m{(B_0-B_{\mathrm{res}})}^2}},\mathrm{and}{\mathrm{\Phi }}_2={\displaystyle \frac{4\pi a_s{b}_2\mathrm{\Delta }B}{m{(B_0-B_{\mathrm{res}})}^2}},$$

with $B_0$ a fixed magnetic field.

The generation of coherent modes by the interaction modulation is similar to their generation by the trap modulation [12,14,53,72,73].

5.2. Multi-Mode Generation

In general, it is possible to generate not merely a single coherent mode but several modes by applying several alternating fields, for instance, by applying the multi-frequency field

$$V(\mathbf{r},t)={\displaystyle \frac{1}{2}}\sum _j\left[B_j\left(\mathbf{r}\right)e^{i{\omega }_{j}t}+B_j^{*}\left(\mathbf{r}\right)e^{-i{\omega }_{j}t}\right],$$

(98)

where j counts the modes and the frequencies are in resonance with the required transition frequencies ${\omega }_{mn}$. For example, two upper modes can be generated by using two alternating fields with the frequencies ${\omega }_1$ and ${\omega }_2$. Then, similarly to optical schemes [39], three coherent modes can coexist when different types of mode generation are used. In the cascade generation, one uses the resonance conditions

$${\omega }_1={\omega }_{21},{\omega }_2={\omega }_{32}\left(\mathrm{cascade}\mathrm{scheme}\right),$$

(99)

in the V-type scheme, the conditions

$${\omega }_1={\omega }_{21},{\omega }_2={\omega }_{31}\left(V\text{-}\mathrm{type}\mathrm{scheme}\right),$$

(100)

and in the $\mathrm{\Lambda }$-type scheme, the resonance conditions

$${\omega }_1={\omega }_{31},{\omega }_2={\omega }_{32}\left(\mathrm{\Lambda }\text{-}\mathrm{type}\mathrm{scheme}\right)$$

(101)

are used.

By varying the system parameters, various dynamic regimes can be performed, exhibiting quasi-periodic oscillations [14,74]. Contrary to the two-mode case, three or more coexisting modes can develop chaotic motion, when the strength of a generating field becomes sufficiently large, such that

$$\left|{\beta }_{mn}/{\alpha }_{mn}\right|\ge 0.639448.$$

(102)

5.3. Higher-Order Resonances

Except for the standard resonance, with $\omega ={\omega }_{21}$, there also appear higher-order resonances occurring under the effects of harmonic generation when

$$l\omega ={\omega }_{21},l=1,2,\dots ,$$

(103)

and under parametric conversion when

$$\sum _l(\pm {\omega }_l)={\omega }_{21}.$$

(104)

These effects require a relatively strong generating field [14,74].

6. Nonresonant Excitation

The generation of coherent modes, considered in the previous Sections, requires the use of resonance, or quasi-resonance, conditions. Then, there the following questions rise naturally. First, how long can the required resonance conditions be supported, not being spoiled by the effect of power broadening? Second, what is the influence of external noise on the dynamics of coherent modes? Third, can the coherent modes be generated without resorting to resonances, but merely applying a sufficiently strong external modulating field?

6.1. Power Broadening

The existence of the effect of power broadening does pose a limit to the ability of supporting the generation of coherent modes, even in the case of pure resonance, since, in addition to resonant transitions, there always occur nonresonant transitions, although their probability is quite small; however, the effect of the transitions accumulates with time. The interval of time, when, even under well-defined resonance between two coherent modes, the generation of these modes can be realized, but after which no the resonant generation is possible any longer, has been obtained to be [35]

$$t_{mn}={\displaystyle \frac{{\alpha }_{mn}^2+{\omega }_{mn}^2}{{\beta }_{mn}^2{\omega }_{mn}}}.$$

(105)

For typical traps, the time (105) is of the order of 10–100 s, which is considerably long being comparable to the typical lifetime of atoms in a trap [2,3]. Recently, the lifetime of trapped atoms has been shown to allow for an extension of up to 50 min [75].

The existence of external noise, surely, introduces irregularity in the dynamics of coherent modes, but if the noise is not exceptionally strong, it does not essentially disturb the overall dynamical picture [76].

A nonresonant alternating field can also generate coherent modes, provided the energy pumped into the system becomes sufficient for this mode generation. The energy per particle, injected into the system during the time period t, reads

$$E_{\mathrm{inj}}={\displaystyle \frac{1}{N}}{\int }_0^t\left|{\displaystyle \frac{\partial \langle \widehat{H}\rangle }{\partial t}}\right|dt.$$

(106)

A mode n can be generated when the injected energy surpasses the n-th mode energy: $E_{\mathrm{inj}}\ge E_n$.

6.2. Nonequilibrium-State Characteristics

Different nonequilibrium states, comprising different modes, can be classified [14,77] by the effective temperature

$$T_{\mathrm{eff}}\equiv {\displaystyle \frac{2}{3}}\left(E_{\mathrm{kin}}\left(t\right)-E_{\mathrm{kin}}\left(0\right)\right)$$

(107)

expressed through the difference of kinetic energies at time t and at the initial state, or by the effective Fresnel number

$$F_{\mathrm{eff}}\equiv {\displaystyle \frac{\pi R^2}{{\lambda }_{\mathrm{eff}}L}},\mathrm{where}{\lambda }_{\mathrm{eff}}\equiv \sqrt{{\displaystyle \frac{2\pi }{m{T}_{\mathrm{eff}}}}}$$

(108)

and R and L represent the radius and length of the trap.

Qualitatively different nonequilibrium states, displaying the appearance of different coherent modes, were studied in the experiments with trapped ⁸⁷Rb atoms [78,79,80] and with computer simulations [81,82], both being quite in agreement one with another. The observed sequence of nonequilibrium states is listed in

Table 1. Nonequilibrium states of a trapped Bose–Einstein condensate, characterized by the effective temperature $T_{\mathrm{eff}}$ (107) and effective Fresnel number $F_{\mathrm{eff}}$ (108). The effective temperature $T_{\mathrm{eff}}$ is given in units of the transverse trap frequency ${\omega }_x=2\pi \times 210$ Hz as obtained in measurements [78,79,80] and via simulations [81,82].

State	${\mathit{T}}_{\mathbf{eff}}$	${\mathit{F}}_{\mathbf{eff}}$
Weak nonequilibrium	0	0
Vortex germs	0.29	0.11
Vortex rings	1.21	0.23
Vortex lines	2.26	0.31
Vortex turbulence	5.54	0.49
Droplet turbulence	8.56	0.61
Wave turbulence	23.5	1.01

, where the numbers correspond to the lower threshold for the appearance of the related states.

By increasing the amount of energy injected into the trap, the system passes through several dynamical regimes with quite distinct properties. The subsequence of the regimes is as follows.

(i)

Weak nonequilibrium. At the beginning of the pumping procedure, there are no topological coherent modes, but there occur only elementary excitations describing density fluctuations.

(ii)

Vortex germs. Then, when the injected energy is not yet sufficient for the generation of the whole vortex rings, there arise vortex germs reminding one of broken pieces of vortex rings.

(iii)

Vortex rings. With the increasing injected energy, the whole rings appear in pairs, having the typical ring properties [83,84,85,86,87,88,89].

(iv)

Vortex lines. At the next stage, the pairs of vortex lines appear [90]. The vortices arise in pairs, since no rotation is imposed on the system, so that the total vorticity has to be zero.

(v)

Vortex turbulence. Upon generating a large number of vortices, the regime of quantum vortex turbulence develops. Due to the absence of any imposed anisotropy, the vortices form a random tangle characteristic of the Vinen turbulence [91,92,93,94,95,96], as opposed to the Kolmogorov turbulence of correlated vortex lines [95]. Several specific features confirm the existence of quantum vortex turbulence. Thus, when released from the trap, the atomic cloud expands isotropically, which is typical of Vinen turbulence [78,79,80]. The radial momentum distribution, obtained by averaging in the axial direction, exhibits a specific power law typical of an isotropic turbulent cascade [97,98,99]. The system relaxation from the vortex turbulent state displays a characteristic universal scaling [100].

(vi)

Droplet turbulence. Further increasing the amount of the injected energy by a longer pumping or by rising the amplitude of the alternating field transforms the system into an ensemble of coherent droplets floating in a sea of uncondensed cloud. The density of the coherent droplets is around 100 times larger than that of their incoherent surroundings. Each droplet consists of about 40 atoms. The lifetime of a droplet is of about of $0.01$ s. This state can be called droplet turbulence, or grain turbulence [77,81,82].

(vii)

Wave turbulence. When coherence in the system is completely destroyed, the system enters the regime of wave turbulence, which is the regime of weakly nonlinear dispersive waves [101]. Actually, the transformation of the droplet turbulence into wave turbulence is not a sharp transition but a gradual crossover. The transition point is conditionally accepted as the point where the number of coherent droplets diminishes by half.

When a nonequilibrium system relaxes to its equilibrium state, passing from a state with a symmetry to the state where the symmetry is broken, the system passes though the stage with the appearing topological defects, such as grains, cells, vortices, strings, and others. This is called the Kibble–Zurek mechanism [102,103,104,105]. In the experiments and computer modeling [78,79,80,81,82], we follow the reverse way of transforming an equilibrium system with broken global gauge symmetry to a nonequilibrium gauge-symmetric system, passing through the stages of arising topological defects, such as vortex germs, vortex rings, vortex lines, and coherent droplets. That is, this reverse way can be named the inverse Kibble–Zurek scenario [82].

6.3. Dynamic Scaling

Nonequilibrium regimes can be distinguished and characterized by scaling laws. It is known that quite some dynamical systems exhibit a kind of self-similarity in their evolution. This was noticed by Fereydoon Family and Tamás Vicsek [106,107] in the process of diffusion-limited aggregation of clusters in two dimensions. The Family–Vicsek dynamic scaling describes the behavior of a probability distribution $f(x,t)$ of a variable x at different instants of time t, such that

$$f(x,t)={\left({\displaystyle \frac{t}{t_0}}\right)}^{{\alpha }^{\prime }}F\left(x{\left({\displaystyle \frac{t}{t_0}}\right)}^{{\beta }^{\prime }},t_0\right){\left({\displaystyle \frac{x}{x_0}}\right)}^{\gamma },$$

(109)

where $F(x,t)$ is a universal function, $x_0$ and $t_0$ are fixed reference values, and ${\alpha }^{\prime }$, ${\beta }^{\prime }$, and $\gamma$ are universal scaling exponents. Numerous cases of nonequilibrium dynamics display scaling laws, for instance, polymer degradation [108], kinetics of aggregation [109,110,111,112,113], complex networks [114], growth models [115,116], fractional Poisson processes [117], and other dynamical processes [118]. Scaling laws and universal critical exponents appear in the theory of nonthermal fixed points when the system is far from equilibrium [119,120,121], which distinguishes the scaling-law regime from the quasi-stationary stage of prethermalization [122,123,124,125]. Cold trapped Bose gas serves as a highly convenient object for studying quantum turbulence [78,79,80,81,82,94,95,96,98,99,100] and its relaxation [100,126,127,128,129].

Distinct stages in the relaxation dynamics of a harmonically trapped three-dimensional Bose–Einstein condensate of ⁸⁷Rb, driven to a turbulent state by an external oscillating field, are analyzed in Ref. [100]. The angular-averaged two-dimensional momentum distribution $n(k,t)$ is measured, for small momenta $k\to 0$, in the time-of-flight experiment [130]. The universal dynamical scaling in the time evolution of the momentum distribution is observed:

$$n(k,t)={\left({\displaystyle \frac{t}{t_0}}\right)}^{{\alpha }^{\prime }}n\left(k{\left({\displaystyle \frac{t}{t_0}}\right)}^{{\beta }^{\prime }},t_0\right),$$

(110)

where $t_0$ is an arbitrary reference time within the temporal window where the scaling is observed. The universal exponents are

$${\alpha }^{\prime }=-0.5\mathrm{and}{\beta }^{\prime }=-0.25.$$

This universal scaling (110) corresponds to a direct energy cascade from the low-momentum to the high-momentum states, when the condensate becomes depleted.

Then, after a prethermalization stage, there appears an inverse energy cascade from the high-momentum to the low-momentum states, which implies the repopulation of the condensate. At the condensate revival stage, the dynamic scaling has the form

$$n(k,t)={\left({\displaystyle \frac{t_b-t}{t_b-t_0}}\right)}^{\lambda }n\left(k{\left({\displaystyle \frac{t_b-t}{t_b-t_0}}\right)}^{\mu },t_0\right),$$

(111)

with $t_b$ being the time of condensate density divergence and the universal exponents

$$\lambda =-1.5\mathrm{and}\mu =-0.9,$$

showing that the condensate fraction sharply increases, which is called [131,132] the condensate blowup.

7. Conclusions

Dynamic transitions between different nonequilibrium states of trapped Bose–Einstein condensates, subject to the action of alternating fields, are surveyed. The applied external fields can be of two types, resonant and nonresonant. A resonant field implies that its frequency is tuned close to resonance with some transition frequency of the trapped system. The transition frequency is the difference between two chosen energy levels of the trapped system. Several external alternating fields with different frequencies can also be used. Resonant fields do not need to be strong enough. More important is the presence of resonance conditions.

Resonant fields generate nonlinear coherent modes in trapped condensates. Depending on the ratio between the amplitude of the alternating field and the interaction strength of atoms, there can appear several dynamic states, including the mode-locked Josephson regime, critical dynamics, mode-unlocked Josephson regime, pitchfork bifurcation, and Rabi regime. The dynamic transition, occurring on the critical line in the effect of separatrix crossing, reminds one of a phase transition in equilibrium statistical systems. These dynamical transitions can be implemented in quite weakly interacting trapped Bose gases. We show that the generation of coherent modes and the related dynamic transitions can, actaully, be implemented in quite strongly interacting superfluids as well, although it is a far more complicated task.

Employing several alternating fields, it is possible to generate several coherent modes and realize higher-order resonance phenomena, such as harmonic generation and parametric conversion.

The other way of generating nonequilibrium states in trapped Bose condensates is through the use of sufficiently strong nonresonant fields. Then, one can produce a sequence of nonequilibrium states containing vortex germs, vortex rings, and vortex lines, as well as generate different turbulent regimes, such as vortex turbulence, droplet turbulence, and wave turbulence. Nonequilibrium states of superfluids can be characterized by effective temperature and effective Fresnel number. Different stages of nonequilibrium systems can be distinguished by the existence of specific dynamic scaling.