Trapped Bose-Einstein condensates with nonlinear coherent modes

V.I. Yukalov1,2, E.P. Yukalova3 and V.S. Bagnato2

1Bogolubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research, Dubna 141980, Russia

2Instituto de Fisica de São Carlos, Universidade de São Paulo,
CP 369, São Carlos 13560-970, São Paulo, Brazil

3Laboratory of Information Technologies,
Joint Institute for Nuclear Research, Dubna 141980, Russia

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

The review presents the methods of generation of nonlinear coherent
excitations in strongly nonequilibrium Bose-condensed systems of trapped atoms
and their properties. Non-ground-state Bose-Einstein condensates are
represented by nonlinear coherent modes. The principal difference of nonlinear
coherent modes from linear collective excitations is emphasized. Methods of
generating nonlinear modes and the properties of the latter are described.
Matter-wave interferometry with coherent modes is discussed, including such
effects as interference patterns, internal Josephson current, Rabi
oscillations, Ramsey fringes, harmonic generation, and parametric conversion.
Dynamic transition between mode-locked and mode-unlocked regimes is shown to
be analogous to a phase transition. Atomic squeezing and entanglement in a
lattice of condensed atomic clouds with coherent modes are considered.
Nonequilibrium states of trapped Bose-condensed systems, starting from weakly
nonequilibrium state, vortex state, vortex turbulence, droplet or grain
turbulence, and wave turbulence, are classified by means of effective Fresnel
and Mach numbers. The inverse Kibble-Zurek scenario is described. A method for
the formation of directed beams from atom lasers is reported.

Keywords: non-ground-state Bose-Einstein condensates, coherent modes, resonant
generation, resonant entanglement, atomic squeezing, quantum turbulence, atom
laser

Contents

1\. Introduction

2\. Condensate wave function

3\. Coherent modes

4\. Resonant generation

5\. Counterflow instability

6\. Energy levels

7\. Mode dynamics

8\. Mode locking

9\. Shape-conservation criterion

10\. Multiple mode generation

11\. Matter-wave interferometry

11.1. Interference patterns

11.2. Internal Josephson current

11.3. Rabi oscillations

11.4. Higher-order resonances

12\. Ramsey fringes

13\. Interaction modulation

14\. Strong interactions and noise

15\. Critical phenomena

16\. Atomic squeezing

17\. Cloud entanglement

17.1. Entangled states

17.2. Entanglement production

17.3. Resonant entanglement production

18\. Nonresonant mode generation

19\. Nonequilibrium states

19.1. Weak nonequilibrium

19.2. Vortex embryos

19.3. Vortex rings

19.4. Vortex lines

19.5. Vortex turbulence

19.6. Droplet state

19.7. Wave turbulence

20\. Classification of states

21\. Atom laser

22\. Conclusion

##  1 Introduction

Bose-Einstein condensation of trapped dilute atoms has been of high interest
in recent years [1, 2, 3] since 1995, triggering voluminous publications, as
can be inferred from review articles [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16]. Equilibrium states of Bose-condensed systems have been studied both
for atoms in traps and in optical lattices [17, 18, 19, 20, 21]. Elementary
excitations have been investigated, corresponding to small deviations from the
ground state, that is, to small density oscillations. These excitations are
usually described by considering linear deviations of density from the ground
state.

In 1997, the authors [22] suggested that it is feasible to generate in atomic
traps strongly nonlinear excitations corresponding to non-ground-state Bose-
Einstein condensates. Since the state of condensed atoms is equivalent to a
coherent state [23], these nonlinear excitations are called nonlinear coherent
modes. These modes enjoy several important properties allowing for the
realization of interesting effects. A trapped Bose condensate with nonlinear
coherent modes reminds an atom with several energy levels that can be
connected by resonant electromagnetic field. Therefore a condensate with
coherent modes allows for the occurrence of effects similar to those happening
for atoms, such as Rabi oscillations, appearance of interference patterns,
Josephson current, and Ramsey fringes. There also exist the effects of
harmonic generation and parametric conversion.

At the same time, a Bose-condensed atomic system is nonlinear due to
interactions between atoms, and it possesses highly nontrivial dynamics of
coherent modes. It demonstrates different regimes of motion. In a Bose-
condensed system, there is a mode-locked regime, when the modes are locked, so
that their fractional populations vary only in a part of the whole admissible
region. The other is mode-unlocked regime, when the mode populations vary in
the whole admissible interval. The dynamic transition between these regimes is
analogous to a phase transition in a statistical system. In a three-mode
system, there can occurs the regime of chaotic motion, which is not possible
in a two-mode system.

For Bose-condensed system of trapped atoms, it is possible to realize
collective squeezing effects. Simultaneous generation of coherent modes in
several connected traps, such as deep optical latices, allows for the creation
of entangled mesoscopic clouds and for the study of the effect of resonant
entanglement production.

Coherent atomic clouds released from a trap form what is called atom laser.
The difference of an atom laser from the photonic laser is that atoms fall
down due to gravity, but cannot be forwarded in the desired direction.
However, realizing special initial conditions one can trigger the beam
emanation in any chosen direction.

Nonequilibrium states can also be generated by strong nonresonant pumping
creating such coherent modes as quantum rings and vortices. Strong pumping
leads to the generation of vortex turbulence. Then there happens the formation
of droplets of Bose condensate floating among uncondensed atoms, and droplet
(or grain) turbulence develops. Even more strong pumping results in the
destruction of condensate and wave turbulence. The overall process of
consecutive generation of nonequilibrium states from an equilibrium condensate
through ring and vortex states, to vortex and droplet turbulence, and ending
with uncondensed state, has been named inverse Kibble-Zurek scenario. A
classification of nonequilibrium states is advanced, by means of relative
injected energy, effective temperature, Fresnel and Mach numbers.

Below, we use the system of units, where the Planck and Boltzmann constants
are set to one.

##  2 Condensate wave function

Atomic systems Bose-condensed in traps are usually rather rarified, such that
the effective interaction radius $r_{int}$ is much smaller than the mean
interparticle distance $a$, which can be written as

$\frac{r_{int}}{a}~{}\ll~{}1\;,\qquad\rho^{1/3}r_{int}~{}\ll~{}1\;,$ (1)

where $\rho\sim a^{-3}$ is the average density of particles. At the same time
the effective strength of particle interactions

$\gamma~{}\equiv~{}\rho^{1/3}a_{s}$ (2)

can be large, with $a_{s}$ being scattering length. In that sense, the
rarified system of trapped atoms is a gas cloud.

The energy Hamiltonian has the standard form

$\hat{H}~{}=~{}\int\psi^{\dagger}({\bf
r})\;\left(-\;\frac{\nabla^{2}}{2m}+U\right)\;\psi({\bf r})\;d{\bf
r}~{}+~{}\frac{1}{2}\int\psi^{\dagger}({\bf r})\;\psi^{\dagger}({\bf
r}^{\prime})\;\Phi({\bf r}-{\bf r}^{\prime})\;\psi({\bf
r}^{\prime})\;\psi({\bf r})\;d{\bf r}d{\bf r}^{\prime}\;,$ (3)

where $U=U({\bf r},t)$ is an external trapping potential. Short-range
interactions of rarified atoms can be represented by the effective interaction
potential

$\Phi({\bf r})~{}=~{}\Phi_{0}\delta({\bf
r})\qquad\left(\Phi_{0}~{}\equiv~{}4\pi\;\frac{a_{s}}{m}\right)\;.$ (4)

The Heisenberg equation of motion for the field operator reads as

$i\;\frac{\partial}{\partial t}\;\psi({\bf r},t)~{}=~{}H[\;\psi\;]\;\psi({\bf
r},t)\;,$ (5)

with the effective nonlinear Hamiltonian

$H[\;\psi\;]~{}\equiv~{}-\;\frac{\nabla^{2}}{2m}+U({\bf
r},t)+\int\psi^{\dagger}({\bf r}^{\prime})\;\Phi({\bf r}-{\bf
r}^{\prime})\;\psi({\bf r}^{\prime})\;d{\bf r}^{\prime}\;.$ (6)

Bose-Einstein condensation means the appearance of a coherent state. At
asymptotically low temperature, such that

$T~{}\ll~{}\rho\Phi_{0}\;,$ (7)

and very weak interactions, when

$\gamma~{}\ll~{}1\;,$ (8)

almost all atoms are Bose condensed, hence practically all the system is in a
coherent state.

Coherent states are the eigenstates of the annihilation field operator
$\psi({\bf r},t)$,

$\psi({\bf r},t)\;|\;\eta\;\rangle~{}=~{}\eta({\bf r},t)\;|\;\eta\;\rangle\;,$
(9)

with $\eta({\bf r},t)$ being called the coherent field or condensate wave
function. The coherent field is the vacuum state of a Bose-condensed system
[23]. Averaging Eq. (5) over the vacuum coherent state yields the equation for
the condensate function (coherent field)

$i\;\frac{\partial}{\partial t}\;\eta({\bf r},t)~{}=~{}H[\;\eta\;]\;\eta({\bf
r},t)\;,$ (10)

where the nonlinear Hamiltonian is

$H[\;\eta\;]~{}=~{}-\;\frac{\nabla^{2}}{2m}+U({\bf r},t)+\int\Phi({\bf r}-{\bf
r}^{\prime})\;|\;\eta({\bf r}^{\prime},t)\;|^{2}d{\bf r}^{\prime}\;.$ (11)

This equation, for an arbitrary interaction potential, was advanced by
Bogolubov in his book [24] that has been republished numerous times (see, e.g.
[25, 26, 27]) and studied in Refs. [28, 29, 30, 31, 32, 33, 34]. By its
mathematical structure, this is a nonlinear Schrödinger equation [35]. For
trapped atoms with the interaction potential (4), it reads as

$i\;\frac{\partial}{\partial t}\;\eta({\bf
r},t)~{}=~{}\left[\;-\;\frac{\nabla^{2}}{2m}+U({\bf
r},t)+\Phi_{0}\;|\;\eta({\bf r},t)\;|^{2}\;\right]\;\eta({\bf r},t)\;.$ (12)

The condensate function is normalized to the total number of atoms,

$\int|\;\eta({\bf r},t)\;|^{2}\;d{\bf r}~{}=~{}N\;.$

Recall that this equation assumes that all atoms are Bose-condensed and
temperature is zero, therefore it can provide a reasonable approximation only
for very low temperatures and asymptotically weak interactions, when
inequalities (7) and (8) are valid.

Since the number of atoms $N$ is conserved, it is possible to make the
replacement

$\eta({\bf r},t)~{}=~{}\sqrt{N}\;\varphi({\bf r},t)\;,$ (13)

so that the normalization condition becomes

$\int|\;\varphi({\bf r},t)\;|^{2}\;d{\bf r}~{}=~{}1\;.$

The external potential can be separated into a constant part of a trapping
potential and its temporal modulation,

$U({\bf r},t)~{}=~{}U({\bf r})+V({\bf r},t)\;.$ (14)

Thus we come to the equation

$i\;\frac{\partial}{\partial t}\;\varphi({\bf
r},t)~{}=~{}\left[\;\hat{H}[\;\varphi\;]+V({\bf r},t)\;\right]\;\varphi({\bf
r},t)\;,$ (15)

with a nonlinear Hamiltonian

$\hat{H}[\;\varphi\;]~{}=~{}-\;\frac{\nabla^{2}}{2m}+U({\bf
r})+N\Phi_{0}\;|\;\varphi({\bf r},t)\;|^{2}\;.$ (16)

The Cauchy problem (15), generally speaking, does not have a unique solution,
since it does not satisfy the conditions of the Cauchy-Kovalevskaya theorem
[36]. To find uniquely defined solutions, one has to concretize the sought
class of functions [37, 38]. We shall be looking for nonlinear solutions that
can be treated as analytical continuation for the solutions of the linear
Schrödinger equation.

##  3 Coherent modes

Let us consider the stationary solutions to Eq. (15). Substituting in (12) the
expression

$\eta({\bf r},t)~{}=~{}\eta({\bf r})e^{-iEt}$ (17)

gives the stationary eigenproblem

$\hat{H}[\;\varphi_{n}\;]\;\varphi_{n}({\bf r})~{}=~{}E_{n}\;\varphi_{n}({\bf
r})$ (18)

defining a spectrum $E_{n}$, labeled with a multi-index $n$, and
eigenfunctions $\varphi_{n}({\bf r})$. The eigenfunctions $\varphi_{n}({\bf
r})$ are called coherent modes [22, 39, 40, 41, 42].

Coherent modes are not compulsorily orthogonal, so that the scalar product

$(\varphi_{m},\;\varphi_{n})~{}=~{}\int\varphi_{m}^{*}({\bf
r})\;\varphi_{n}({\bf r})\;d{\bf r}\;,$

generally, is not a Kronecker delta. These modes do not form a resolution of
unity. However, the set $\\{\varphi_{n}({\bf r})\\}$ is total [43], and the
closed linear envelope over the set of the coherent modes

${\cal H}~{}=~{}{\rm span}_{n}\\{\;\varphi_{n}({\bf r})\;\\}$ (19)

composes a Hilbert space [42]. Then the solution to Eq. (15) can be
represented as the expansion

$\varphi({\bf r},t)~{}=~{}\sum_{n}b_{n}(t)\;\varphi_{n}({\bf r})\;.$ (20)

Approximate stationary solutions to Eq. (18) can be found in different ways.
At very small interaction parameters $\gamma$, it is possible to use simple
perturbation theory [44]. For arbitrary gas parameters $\gamma$, optimized
perturbation theory is applicable [45] or self-similar approximation theory
[46].

To find the temporal behaviour of $\varphi({\bf r},t)$, we need to specify the
explicit form of the modulation potential $V({\bf r},t)$. There are two ways
of generating nonequilibrium states in a system of trapped atoms, a resonant
and non-resonant pumping.

(i) Resonant generation. One way is by employing an external field oscillating
with a frequency $\omega$ that would be in resonance with a transition
frequency

$\omega_{21}~{}\equiv~{}E_{2}-E_{1}$ (21)

between the ground-state level, with the energy $E_{1}$, and an excited level,
with the energy $E_{2}$, so that the detuning be small,

$\left|\;\frac{\Delta\omega}{\omega_{21}}\;\right|~{}\ll~{}1\;,\qquad\Delta\omega~{}\equiv~{}\omega-\omega_{21}\;.$
(22)

An oscillating field can be produced by modulating the potential trapping the
atoms. More generally, it is possible to use several external fields with
their frequencies being in resonance with several transition frequencies
between different modes. Under resonant generation, the most important is to
preserve the resonance condition (22), while the modulation field can be
rather small.

(ii) Non-resonant generation. The other way is to use a nonresonant but
sufficiently strong modulation field. Then not a single or several selected
modes are excited but a bunch of modes is generated. If the generation process
lasts during the time much longer then the oscillation time, then the higher
modes decay into the most stable one that prevails.

##  4 Resonant generation

The modulation field can be chosen in the form

$V({\bf r},t)~{}=~{}V_{1}({\bf r})\cos(\omega t)+V_{2}({\bf r})\sin(\omega
t)\;,$ (23)

with a frequency $\omega$ satisfying condition (22). This situation is similar
to the resonance excitation of atoms in optics [47], because of which it is
called atom optics [48, 49].

The coefficient $b_{n}(t)$ in (20) can be written as

$b_{n}(t)~{}=~{}c_{n}(t)\exp(-iE_{n}t)\;,$ (24)

where $c_{n}(t)$ is a slow function of time, as compared to the quickly
oscillating exponential,

$\frac{1}{E_{n}}\;\left|\;\frac{dc_{n}(t)}{dt}\;\right|~{}\ll~{}1\;.$ (25)

This condition is analogous to the condition of slowly varying amplitude
approximation in optics [47, 50].

Then we substitute presentation (24) into equation (15) and use the averaging
techniques [51, 52] treating $c_{n}(t)$ as quasi-invariants [53]. In this way,
we need the notations for the transition amplitudes, due to atomic
interactions,

$\alpha_{mn}~{}\equiv~{}\Phi_{0}N\int|\;\varphi_{m}({\bf
r})\;|^{2}\left[\;2\;|\;\varphi_{n}({\bf r})\;|^{2}-|\;\varphi_{m}({\bf
r})\;|^{2}\;\right]\;d{\bf r}\;,$ (26)

and to the modulation field,

$\beta_{mn}~{}\equiv~{}\int\varphi_{m}^{*}({\bf r})\left[\;V_{1}({\bf
r})-iV_{2}({\bf r})\;\right]\;\varphi_{n}({\bf r})\;d{\bf r}\;.$ (27)

Note that (27) is an analog of the Rabi frequency.

In order that the averaging techniques be valid, one needs the inequalities

$\left|\;\frac{\alpha_{mn}}{\omega_{mn}}\;\right|~{}\ll~{}1\;,\qquad\left|\;\frac{\beta_{mn}}{\omega_{mn}}\;\right|~{}\ll~{}1\;,$
(28)

where $\omega_{mn}\equiv E_{m}-E_{n}$. Averaging over time, we use the
equality

$\lim_{\tau\rightarrow\infty}\;\frac{1}{\tau}\int_{0}^{\tau}\exp(i\omega_{mn}t)\;dt~{}=~{}\delta_{mn}\;.$

Thus wee come to the equations

$i\;\frac{dc_{n}}{dt}~{}=~{}\sum_{m(\neq
n)}\alpha_{mn}|\;c_{m}\;|^{2}c_{n}+\frac{1}{2}\;\delta_{n1}\beta_{12}c_{2}e^{i\Delta\omega
t}+\frac{1}{2}\;\delta_{n2}\beta_{12}^{*}c_{1}e^{-i\Delta\omega t}\;,$ (29)

with $c_{n}=c_{n}(t)$ satisfying the normalization condition

$\sum_{n}|\;c_{n}(t)\;|^{2}~{}=~{}1\;.$ (30)

From these equations, we find that all levels, except $n=1$ and $n=2$ are not
populated,

$c_{n}(t)~{}=~{}0\qquad(n=3,4,5,\ldots)\;,$ (31)

while for the levels $n=1,2$, we have the equations

$i\;\frac{dc_{1}}{dt}~{}=~{}\alpha_{12}|\;c_{2}\;|^{2}c_{1}+\frac{1}{2}\;\beta_{12}c_{2}e^{i\Delta\omega
t}\;,$
$i\;\frac{dc_{2}}{dt}~{}=~{}\alpha_{21}|\;c_{1}\;|^{2}c_{2}+\frac{1}{2}\;\beta_{12}^{*}c_{1}e^{-i\Delta\omega
t}\;.$ (32)

Thus, it is shown that in the case of resonance generation, with one
modulation field, only two modes are involved in dynamics. The properties of
the coherent modes, in addition to the literature cited above, also have been
studied theoretically in several articles [54, 55, 56, 57, 58, 59, 60]. A
nonlinear dipole mode in a two-component condensate was excited in experiment
[61].

##  5 Counterflow instability

The appearance of coherent modes is due to strong fluctuations of density
caused by an external alternating field. The density distribution inside the
trap becomes essentially nonuniform. Different parts of the trapped atomic
cloud quickly move with respect to each other, which leads to the occurrence
of a kind of counterflow instability [16, 62, 63, 64, 65, 66] arising when
different parts of a liquid system move with respect to each other. There
exist several types of hydrodynamic instabilities occurring in moving liquids
[67, 68]. For example, the Kelvin-Helmholtz instability occurs when there is a
velocity difference across the interface between two fluids. The Rayleigh-
Taylor instability is an instability of an interface between two moving fluids
of different densities which occurs when the lighter fluid is pushing the
heavier fluid. Counterflow instabilities appear in moving Bose systems [16,
62, 63, 64, 65, 66, 69, 70, 71, 72].

The illustration of the appearance of counterflow instability in Bose-
condensed systems can be conveniently done by considering, first, a mixture of
two or more Bose systems. In the local-density picture, the mixture components
move with the corresponding velocities $v_{i}({\bf r,t})$. Collective
excitations in the mixture can be found by the standard methods, either by
linearizing the equations of motion or resorting to the random-phase
approximation [16, 63, 64, 66]. Assume that atoms interact through the local
potentials

$\Phi_{ij}({\bf r})~{}=~{}\Phi_{ij}\delta({\bf
r})\;,\qquad\Phi_{ij}~{}\equiv~{}\int\Phi_{ij}({\bf r})\;d{\bf r})\;,$ (33)

in which

$\Phi_{ij}~{}=~{}4\pi\;\frac{a_{ij}}{m_{ij}}\;,\qquad
m_{ij}~{}=~{}\frac{2m_{i}m_{j}}{m_{i}+m_{j}}\;,$ (34)

where $m_{i}$ and $a_{i}$ are masses and scattering lengths of the related
atoms.

Inside a trap, the motion of different components can be organized by
modulating the trapping potential, e.g., according to the law

$U_{i}({\bf
r},t)~{}=~{}\frac{1}{2}\;m_{i}\omega_{x}^{2}\;\left(x^{2}+y^{2}\right)+\lambda_{i}m_{i}^{2}\omega_{z}^{3}\;\left[\;z^{2}-b_{i}^{2}(t)\;\right]\;.$

In the long-wave limit, the collective spectra of each of the component, in
the local-density approximation, have the acoustic form with the sound
velocities defined by the relation

$s_{i}^{2}~{}=~{}\frac{\rho_{j}}{m_{i}}\;\Phi_{ii}\;,$ (35)

where $s_{i}=s_{i}({\bf r},t)$ and $\rho_{i}=\rho_{i}({\bf r},t)$. The mixture
of two components is stable provided the stability condition is true [63, 64,
66]:

$s_{12}^{2}~{}<~{}\left(s_{1}^{2}-v_{1}^{2}\right)\left(s_{2}^{2}-v_{2}^{2}\right)\;,$
(36)

where

$s_{12}^{2}~{}=~{}s_{12}^{2}({\bf r},t)~{}=~{}\sqrt{\frac{\rho_{1}({\bf
r},t)\rho_{2}({\bf r},t)}{m_{1}m_{2}}}\;\Phi_{12}.$ (37)

The stability condition can be rewritten as

$\frac{\Phi_{12}^{2}}{\Phi_{11}\Phi_{22}}~{}<~{}\left(1-\;\frac{m_{1}v_{1}^{2}}{\rho_{1}\Phi_{11}}\right)\left(1-\;\frac{m_{2}v_{2}^{2}}{\rho_{2}\Phi_{22}}\right)\;.$
(38)

Instead of considering a mixture of different components, it is admissible to
treat a nonequilibrium system as a mixture of different coherent modes, with
$m_{i}=m_{j}\equiv m$. In each mode, the atomic interactions are the same,
$\Phi_{ii}=\Phi_{jj}\equiv\Phi_{0}$. In order to distinguish the modes, one
needs to assume that at each spatial point, at each given time, there can
exist a single mode, so that $\rho_{i}({\bf r},t)\rho_{j}({\bf r},t)=0$ for
$i\neq j$. Then $s_{ij}=0$ if $i\neq j$. As a result, one gets the stability
condition

$0~{}<~{}\left(s_{1}^{2}-v_{1}^{2}\right)\left(s_{2}^{2}-v_{2}^{2}\right)\;.$
(39)

In that way, the system stability becomes lost as soon as one of the
velocities exceeds the related sound velocity.

##  6 Energy levels

The mode energy is defined by the eigenproblem (18). The energy levels can be
found using optimized perturbation theory (see reviews [73, 74]). For
instance, let us consider a cylindrical trap with the radial frequency
$\omega_{x}=\omega_{y}$ and the axial frequency $\omega_{z}$. The related
aspect ratio is denoted as

$\nu~{}\equiv~{}\frac{\omega_{z}}{\omega_{x}}~{}=~{}\frac{l_{x}^{2}}{l_{z}^{2}}\;.$
(40)

It is convenient to use the dimensionless cylindrical variables

$r~{}\equiv~{}\frac{\sqrt{r_{x}^{2}+r_{y}^{2}}}{l_{x}}\;,\qquad
z~{}\equiv~{}\frac{r_{z}}{l_{x}}\qquad\left(l_{x}\;\equiv\;\frac{1}{\sqrt{m\omega_{x}}}\right)$
(41)

and the dimensionless coupling parameter

$g~{}\equiv~{}4\pi\;\frac{a_{s}}{l_{x}}\;N\;.$ (42)

The nonlinear Hamiltonian (16), measured in units of the radial frequency, is

$\hat{H}~{}\equiv~{}\frac{\hat{H}[\;\varphi\;]}{\omega_{x}}~{}=~{}-\;\frac{\nabla^{2}}{2}+\frac{1}{2}\;\left(r^{2}+\nu^{2}z^{2}\right)+g|\;\psi\;|^{2}\;,$
(43)

where the dimensionless wave function is

$\psi(r,\varphi,z)~{}\equiv~{}l_{x}^{3/2}\varphi({\bf r})\;.$ (44)

In dimensionless units, the eigenproblem (18) reads as

$\hat{H}\psi_{nmj}(r,\varphi,z)~{}=~{}E_{nmj}\psi_{nmj}(r,\varphi,z)\;,$ (45)

where $n=0,1,2,\ldots$ is the radial quantum number, $m=0,\pm 1,\pm 2,\ldots$
is the azimuthal quantum number, and $j=0,1,2,\ldots$ is the axial quantum
number.

Then we use the Rayleigh-Schrödinger perturbation theory, starting with the
initial Hamiltonian

$\hat{H}_{0}~{}=~{}-\;\frac{\nabla^{2}}{2}+\frac{1}{2}\;\left(u^{2}r^{2}+v^{2}z^{2}\right)$
(46)

and calculating the spectrum in the orders $k=1,2,\ldots$,

$E_{nmj}^{(k)}~{}=~{}E_{nmj}^{(k)}(g,u,v)\;.$

The zero-order wavefunctions are

$\psi_{nmj}^{(0)}(r,\varphi,z)~{}=~{}\left[\;\frac{2n!u^{m+1}}{(n+|\;m\;|)!}\;\right]^{1/2}\;r^{|m|}\;\exp\left(-\;\frac{u}{2}\;r^{2}\right)\;\times$
$\times\;L_{n}^{|m|}(ur^{2})\;\frac{e^{im\varphi}}{\sqrt{2\pi}}\;\left(\frac{v}{\pi}\right)^{1/4}\;\frac{1}{\sqrt{2^{j}j!}}\;\exp\left(-\;\frac{v}{2}\;z^{2}\right)\;H_{j}(\sqrt{v}\;z)\;,$
(47)

where $L_{m}^{n}$ is a Laguerre polynomial and $H_{j}$ is a Hermit polynomial.
Here $u$ and $v$ are control parameters defined so that to control the
convergence of perturbation theory.

Calculating the energy $E_{nmj}^{(k)}$ in a $k$-th order of perturbation
theory, we define the control functions $u=u_{k}(g)$ and $v=v_{k}(g)$ by the
conditions

$\frac{\partial}{\partial
u}\;E_{nmj}^{(k)}~{}=~{}0\;,\qquad\frac{\partial}{\partial
v}\;E_{nmj}^{(k)}~{}=~{}0\;.$ (48)

We introduce the effective coupling

$\widetilde{g}~{}\equiv~{}2(2n+|\;m\;|+1)\;\sqrt{2j+1}\;I_{nmj}g\nu\;,$ (49)

in which

$I_{nmj}~{}\equiv~{}\frac{1}{u\sqrt{v}}\int|\;\psi_{nmj}(r,\varphi,z)\;|^{4}\;r\;drd\varphi
dz\;.$

For illustration, we may give the behavior of the energy spectrum obtained in
first-order optimized perturbation theory. Thus for weak coupling, we have

$E_{nmj}~{}\simeq~{}a_{0}+a_{1}\widetilde{g}\qquad(\widetilde{g}\rightarrow
0)\;,$ (50)

where

$a_{0}~{}=~{}(2n+|\;m\;|+1)+\frac{2j+1}{2}\;\nu\;,\qquad
a_{1}~{}=~{}\frac{1}{2(2n+|\;m\;|+1)\;\sqrt{(2j+1)\nu}}\;,$

and for strong coupling, we get

$E_{nmj}~{}\simeq~{}b_{0}\widetilde{g}^{2/5}+b_{1}\widetilde{g}^{-2/5}\qquad(\widetilde{g}\rightarrow\infty)\;,$
(51)

where

$b_{0}~{}=~{}\frac{5}{4}\;,\qquad
b_{1}~{}=~{}\frac{1}{2}\;(2n+|\;m\;|+1)^{2}+\frac{1}{4}\;(2j+1)^{2}\;\nu^{2}\;.$

Details can be found in [42]. The typical situation for trapped atoms is the
strong-coupling regime.

##  7 Mode dynamics

For studying the mode dynamics, it is convenient to introduce the population
difference

$s~{}\equiv~{}|\;c_{2}\;|^{2}-|\;c_{1}\;|^{2}$ (52)

varying in the interval $-1\leq s\leq 1$. Then the mode populations can be
written as

$|\;c_{1}\;|^{2}~{}=~{}\frac{1-s}{2}\;,\qquad|\;c_{2}\;|^{2}~{}=~{}\frac{1+s}{2}\;,$

which defines the presentation

$c_{1}~{}=~{}\sqrt{\frac{1-s}{2}}\;\exp\left\\{i\left(\zeta_{1}+\frac{\Delta\omega}{2}\;t\right)\right\\}\;,\qquad
c_{2}~{}=~{}\sqrt{\frac{1+s}{2}}\;\exp\left\\{i\left(\zeta_{2}-\;\frac{\Delta\omega}{2}\;t\right)\right\\}\;,$
(53)

where $\zeta_{i}=\zeta_{i}(t)$ are real-valued phases.

Also, we introduce the notations for the average interaction amplitude

$\alpha~{}\equiv~{}\frac{1}{2}\;(\alpha_{12}+\alpha_{21})\;,$ (54)

the transition amplitude

$\beta_{12}~{}=~{}\beta
e^{i\gamma}\;,\qquad\beta~{}\equiv~{}|\;\beta_{12}\;|\;,$ (55)

effective detuning

$\delta~{}\equiv~{}\Delta\omega+\frac{1}{2}\;(\alpha_{12}-\alpha_{21})\;,$
(56)

and the phase difference

$x~{}\equiv~{}\zeta_{2}-\zeta_{1}+\gamma\;.$ (57)

In terms of the above notations, equations (32) can be represented as

$\frac{ds}{dt}~{}=~{}-\beta\;\sqrt{1-s^{2}}\;\sin x\;,$
$\frac{dx}{dt}~{}=~{}\alpha s+\frac{\beta s}{\sqrt{1-s^{2}}}\;\cos
x+\delta\;.$ (58)

These equations can be rewritten in the Hamiltonian form

$\frac{ds}{dt}~{}=~{}-\;\frac{\partial H}{\partial
x}\;,\qquad\frac{dx}{dt}~{}=~{}\frac{\partial H}{\partial s}\;,$ (59)

with the Hamiltonian

$H~{}=~{}\frac{\alpha}{2}\;s^{2}\;-\;\beta\;\sqrt{1-s^{2}}\;\cos x+\delta
s\;.$ (60)

The population difference varies in the interval $-1\leq s\leq 1$ and the
phase, in the interval $0\leq x\leq 2\pi$.

We shall need the notations for the dimensionless Rabi frequency and detuning

$b~{}\equiv~{}\frac{\beta}{\alpha}\;,\qquad\varepsilon~{}\equiv~{}\frac{\delta}{\alpha}\;.$
(61)

The dimensionless detuning, by choosing $\Delta\omega$, can always be made
small, so that $\varepsilon\ll 1$.

In the case of $|b|>1$, Eqs. (58) possess three fixed points

$s_{1}^{*}~{}=~{}\frac{\varepsilon}{b}\;,\qquad x_{1}^{*}=0\;,$
$s_{2}^{*}~{}=~{}-\;\frac{\varepsilon}{b}\;,\qquad x_{2}^{*}=\pi\;,$
$s_{3}^{*}~{}=~{}\frac{\varepsilon}{b}\;,\qquad
x_{3}^{*}=2\pi\qquad(|\;b\;|~{}\geq~{}1)\;.$ (62)

The stability analysis shows that all these points are the centers.

When $0\leq b<1$, there are five fixed points, the three fixed points (62)
plus the points

$s_{4}^{*}~{}=~{}\sqrt{1-b^{2}}+\frac{b^{2}\varepsilon}{1-b^{2}}\;,\qquad
x_{4}^{*}=\pi\;,$
$s_{5}^{*}~{}=~{}-\;\sqrt{1-b^{2}}+\frac{b^{2}\varepsilon}{1-b^{2}}\;,\qquad
x_{5}^{*}=\pi\;\qquad(0~{}\leq b~{}<~{}1).$ (63)

The point $(s_{2}^{*},x_{2}^{*})$ is a saddle, and all other points are
centers.

For $-1<b\leq 0$, there occur seven fixed points, the three points, as in
(62), plus

$s_{4}^{*}~{}=~{}\sqrt{1-b^{2}}+\frac{b^{2}\varepsilon}{1-b^{2}}\;,\qquad
x_{4}^{*}=0\;,$
$s_{5}^{*}~{}=~{}-\;\sqrt{1-b^{2}}+\frac{b^{2}\varepsilon}{1-b^{2}}\;,\qquad
x_{5}^{*}=0\;,$ $s_{6}^{*}~{}=~{}s_{4}^{*}\;,\qquad x_{6}^{*}~{}=~{}2\pi\;,$
$s_{7}^{*}~{}=~{}s_{5}^{*}\;,\qquad
x_{7}^{*}~{}=~{}2\pi\qquad(-1~{}<~{}b~{}\leq 0)\;.$ (64)

The point $(s_{2}^{*},x_{2}^{*})$ remains a saddle, while all others are the
centers.

The trajectory, passing through a saddle point, is called the saddle
separatrix. This line separates the phase plane onto the regions of
qualitatively different regimes of motion. The separatrix equation is

$\frac{s^{2}}{2}\;-\;b\;\sqrt{1-s^{2}}\;\cos x+\varepsilon s~{}=~{}b\;.$ (65)

Here and in the equations above, the smallness of $\varepsilon\ll 1$ is used.
More details on the mode dynamics are given in Refs. [22, 42, 48, 49]. The
resonant excitation of the chosen mode can continue during the time of order
$\omega_{21}/\beta^{2}$, after which the effect of power broadening becomes
essential, when neighboring modes become involved in the process.

##  8 Mode locking

When the separatrix crosses the point of initial conditions
$\\{s_{0},x_{0}\\}$, the trajectory moves from one region of the phase space
to another region, which implies an abrupt qualitative change of dynamics. The
separatrix, crossing the point of initial conditions, defines the critical
line corresponding to the critical values of the parameters

$\frac{s_{0}^{2}}{2}\;-\;b_{c}\;\sqrt{1-s_{0}^{2}}\;\cos
x_{0}+\varepsilon_{c}s_{0}~{}=~{}b_{c}\;.$ (66)

Without the loss of generality, the initial condition for the phase can be
taken as $x_{0}=0$. Under the fixed initial conditions and detuning
$\varepsilon_{c}$, the critical pumping field reads as

$b_{c}~{}=~{}\frac{s_{0}^{2}+2\varepsilon_{c}s_{0}}{2(1+\sqrt{1-s_{0}^{2}})}\;.$
(67)

Crossing the critical line (66), the regime change happens abruptly, when $b$
moves between the regions with $b<b_{c}$ and $b>b_{c}$. When $b<b_{c}$, the
effect of mode locking exists, such that the variation of the population
difference is locked in a half of the allowed interval,

$-1~{}\leq~{}s~{}\leq~{}0\qquad(b<b_{c},\;s_{0}=-1)\;,$
$0~{}\leq~{}s~{}\leq~{}1\qquad(b<b_{c},\;s_{0}=1)\;,$ (68)

depending on initial conditions. And if $b>b_{c}$ then the period of the
population difference oscillations doubles, and the interval of variation
covers all allowed region,

$-1~{}\leq~{}s~{}\leq~{}1\qquad(b>b_{c})\;.$ (69)

Thus the value $b_{c}$ corresponds to a dynamic phase transition point. If the
initial condition for the population difference is $s_{0}=-1$, then the
critical point becomes

$b_{c}~{}=~{}\frac{1}{2}\;-\;\varepsilon_{c}\;.$ (70)

The dynamic transition between mode-locked regime and mode-unlocked regime is
illustrated in Fig. 1a (mode-locked regime), Fig. 1b (close to transition),
and Fig. 1c (mode-unlocked regime). Regular oscillations are presented in Fig.
1d.

Figure 1: The time dependence of the fractional populations $n_{1}(t)$ (solid
line) and $n_{2}(t)$ (dashed-dotted line) for $b=0.4999$ and (a) $\delta=0$;
(b) flattening of the fractional populations with oscillation period being
doubled for $\delta=0.0001$; (c) appearance of the upward cusps of $n_{1}(t)$
and downward cusps of $n_{2}(t)$ for $\delta=0.0001001$; (d) Regular
oscillations for $\delta=0.3$.

The qualitative change of the phase portrait is called dynamical phase
transition. In the vicinity of the dynamic critical line, there appear
critical phenomena that are analogous to thermodynamic phase transitions. This
can be shown by averaging the considered dynamical system over time [41, 75,
76].

The possibility of preserving resonance conditions is limited in time by a
value of order

$t_{res}~{}\sim~{}\frac{\omega_{21}}{\alpha^{2}+|\;\beta_{12}\;|^{2}}$ (71)

that can be called resonance time [42]. After this time, the effect of power
broadening comes into play, when other energy levels become involved, so that
good resonance cannot be anymore supported. This limiting resonance time, in
realistic trapping experiments, is of order of $10$ seconds, which is
comparable to the lifetime of atoms in a trap [6, 78, 79].

##  9 Shape-conservation criterion

In order to transfer trapped atoms into an excited coherent mode, it is
necessary to apply an alternating field. However not any, even resonant, field
can generate a coherent mode. Some types of fields, would merely move the
trapped atomic cloud without generating excited modes. The situation when the
cloud does not change its shape, hence when no coherent modes are generated,
despite the action of an alternating field, is described by the following
theorem [80].

Theorem. Let atoms be trapped in a potential $U({\bf r})$, so that their state
satisfies the trapping condition

$\lim_{r\rightarrow\infty}\;\varphi({\bf r},t)~{}=~{}0\;.$ (72)

And let these atoms, being at the initial moment of time in a real state
$\varphi_{0}({\bf r})=\varphi_{0}^{*}({\bf r})$, be subjected to the action of
an alternating field $V({\bf r},t)$. Then the solution to the nonlinear
Schrödinger equation (15) satisfies the shape-conservation condition

$|\;\varphi({\bf r},t)\;|~{}=~{}|\;\varphi({\bf r}-{\bf a}(t),0)\;|\;,$ (73)

where $a(t)$ is a function of time, if and only if the trapping potential is
harmonic,

$U({\bf r})~{}=~{}A_{0}+{\bf A}_{1}{\bf
r}+\sum_{\alpha\beta}A_{\alpha\beta}\;r^{\alpha}\;r^{\beta}\;,$ (74)

and the alternating field is linear with respect to the spatial variables,

$V({\bf r},t)=B_{0}(t)+{\bf B}_{1}(t)\cdot{\bf r}\;,$ (75)

with $B_{0}(t)$ and ${\bf B}_{1}(t)$ being arbitrary functions of time.

In order to efficiently excite nonlinear coherent modes, it is required that
at least one of the conditions (72) to (75) be invalid.

##  10 Multiple mode generation

It is feasible to generate not just a single coherent mode, but several of
them. For this purpose, one needs to invoke an alternating field with several
frequencies that are in resonance with the related transition frequencies.
Generally, a multimode alternating field can be written in the form

$V({\bf r},t)~{}=~{}\frac{1}{2}\sum_{j}\left[\;B_{j}({\bf
r})e^{i\omega_{j}t}+B_{j}^{*}({\bf r})e^{-i\omega_{j}t}\;\right]\;,$ (76)

with the modulation frequencies $\omega_{j}$ tuned to some transition
frequencies $\omega_{mn}$. For example, the excitation of two coherent modes
can be realized using one of three schemes that, similar to the optical
excitation schemes [50], can be called cascade, $V$-type and $\Lambda$-type.

In the cascade scheme, one couples the first (ground-state) level with the
second energy level and the second, with the third level by employing the
alternating-field frequencies in resonance with the related energy
differences:

$\omega_{1}~{}=~{}\omega_{21}\;,\qquad\omega_{2}~{}=~{}\omega_{32}\qquad(cascade).$
(77)

Here exact resonance conditions are defined. Generally, there can exist small
detunings from the resonance.

In the $V$-type scheme, two alternating-field frequencies induce resonance
with the transition frequencies, so that

$\omega_{1}~{}=~{}\omega_{21}\;,\qquad\omega_{2}~{}=~{}\omega_{31}\qquad(V-type).$
(78)

And in the $\Lambda$-type scheme, the coupling of the energy levels is
realized by means of the resonance

$\omega_{1}~{}=~{}\omega_{31}\;,\qquad\omega_{2}~{}=~{}\omega_{32}\qquad(\Lambda-
type).$ (79)

As in the case of two coherent modes, the solution to equation (15) is
represented as the expansion

$\varphi({\bf r},t)~{}=~{}\sum_{n}c_{n}(t)\;\varphi_{n}({\bf
r})\;e^{-iE_{n}t}\;.$ (80)

Then, following the same way as for two modes, we obtain the equations

$i\;\frac{dc_{1}(t)}{dt}~{}=~{}\left(\alpha_{12}|\;c_{2}\;|^{2}+\alpha_{13}|\;c_{3}\;|^{2}\right)\;c_{1}+f_{1}\;,$
$i\;\frac{dc_{2}(t)}{dt}~{}=~{}\left(\alpha_{21}|\;c_{1}\;|^{2}+\alpha_{23}|\;c_{3}\;|^{2}\right)\;c_{2}+f_{2}\;,$
$i\;\frac{dc_{3}(t)}{dt}~{}=~{}\left(\alpha_{31}|\;c_{1}\;|^{2}+\alpha_{32}|\;c_{2}\;|^{2}\right)\;c_{3}+f_{3}\;,$
(81)

in which the functions $f_{j}$ depend on the type of the generation scheme.
The related detunings $\Delta\omega_{mn}$ from the resonance are assumed to be
small

$\left|\;\frac{\Delta\omega_{mn}}{\omega_{mn}}\;\right|~{}\ll~{}1\;,\qquad(\Delta\omega_{mn}\equiv\omega_{j}-\omega_{mn})\;.$
(82)

For an example of the functions $f_{j}$, let us write down them for the
cascade generation:

$f_{1}~{}=~{}\frac{1}{2}\;\beta_{12}\;c_{2}\;e^{i\Delta\omega_{21}t}\;,$
$f_{2}~{}=~{}\frac{1}{2}\;\beta_{12}^{*}\;c_{1}\;e^{-i\Delta\omega_{21}t}+\frac{1}{2}\;\beta_{23}\;c_{3}\;e^{i\Delta\omega_{32}t}\;,$
$f_{3}~{}=~{}\frac{1}{2}\;\beta_{23}^{*}\;c_{2}\;e^{-i\Delta\omega_{32}t},$
(83)

where

$\Delta\omega_{21}~{}\equiv~{}\omega_{1}-\omega_{21}\;,\qquad\Delta\omega_{32}~{}\equiv~{}\omega_{2}-\omega_{32}\;.$

The number of modes that could be excited is defined by the available number
of states bound inside the trapping potential. For an infinitely rising
potential, there is infinite number of state. Of course, in real experiments,
trapping potentials are finite, hence they possess a finite number of states
[81].

Solutions in the case of three modes have been analyzed in Refs. [80, 82, 83]
using two ways, resorting to the averaging techniques [51] and by direct
numerical solution of the nonlinear Schrödinger equation (15). Both ways were
found to agree well with each other.

In the case of two coexisting nonlinear modes, the system of two complex
differential equations (32), taking account of the normalization condition
(30), has been reduced to two real differential equations (58). A two-
dimensional dynamical system cannot exhibit chaotic motion. In our case, the
fixed points where either centers or saddles.

For the case of three nonlinear modes, three complex equations (81) can be
reduced to the system of four real differential equations. For this purpose,
we introduce the real-valued phases $\zeta_{n}(t)$ by the representation

$c_{n}~{}\equiv~{}|\;c_{n}\;|\;\exp(i\zeta_{n})\;.$ (84)

We also use the notation

$\beta_{mn}~{}=~{}b_{mn}\;\exp(i\gamma_{mn})\;,\qquad
b_{mn}~{}\equiv~{}|\;\beta_{mn}\;|\;.$ (85)

Introduce the population differences

$s~{}\equiv~{}|\;c_{2}\;|^{2}-|\;c_{1}\;|^{2}\;,\qquad
p~{}\equiv~{}|\;c_{3}\;|^{2}-|\;c_{2}\;|^{2}\;,$ (86)

and the relative phases

$x~{}\equiv~{}\zeta_{2}-\zeta_{1}+\gamma_{12}+\Delta\omega_{21}t\;,\qquad
y~{}\equiv~{}\zeta_{3}-\zeta_{2}+\gamma_{23}+\Delta\omega_{32}t\;.$ (87)

Then the mode populations can be written as

$|\;c_{1}\;|^{2}~{}=~{}\frac{1}{3}\;(1-2s-p)\;,$
$|\;c_{2}\;|^{2}~{}=~{}\frac{1}{3}\;(1+s-p)\;,\qquad|\;c_{3}\;|^{2}~{}=~{}\frac{1}{3}\;(1+s+2p)\;.$
(88)

Using these expressions makes it possible to reduce the system of six real
equations (81) to the system of four equations. For instance, in the case of
the cascade generation, we get

$\frac{ds}{dt}~{}=~{}\frac{1}{3}\;\sqrt{1+s-p}\;(\;b_{23}\;\sqrt{1+s+2p}\;\sin
y-2b_{12}\;\sqrt{1-2s-p}\;\sin x\;)\;,$
$\frac{dp}{dt}~{}=~{}\frac{1}{3}\;\sqrt{1+s-p}\;(\;b_{12}\;\sqrt{1-2s-p}\;\sin
x-2b_{23}\;\sqrt{1+s+2p}\;\sin y\;)\;,$
$\frac{dx}{dt}~{}=~{}\alpha_{1}\;s+\delta_{1}\;p+\frac{3b_{12}s\cos
x}{2\sqrt{(1+s-p)(1-2s-p)}}\;-\;\frac{1}{2}\;b_{23}\;\sqrt{\frac{1+s+2p}{1+s-p}}\;\cos
y+\delta_{2}\;,$
$\frac{dy}{dt}~{}=~{}\alpha_{2}\;p+\delta_{3}\;s+\frac{3b_{23}p\cos
y}{2\sqrt{(1+s-p)(1+s+2p)}}+\frac{1}{2}\;b_{12}\;\sqrt{\frac{1-2s-p}{1+s-p}}\;\cos
x+\delta_{4}\;,$ (89)

where

$\alpha_{1}~{}\equiv~{}\frac{1}{3}\;(\alpha_{12}+\alpha_{13}+2\alpha_{21}-\alpha_{23})\;,\qquad\alpha_{2}~{}\equiv~{}\frac{1}{3}\;(\alpha_{32}+\alpha_{31}+2\alpha_{23}-\alpha_{21})\;,$
$\delta_{1}~{}\equiv~{}\frac{1}{3}\;(\alpha_{21}-\alpha_{12}+2\alpha_{13}-2\alpha_{23})\;,\qquad\delta_{2}~{}\equiv~{}\Delta\omega_{21}+\frac{1}{3}\;(\alpha_{12}-\alpha_{21}+\alpha_{13}-\alpha_{23})\;,$
$\delta_{3}~{}\equiv~{}\frac{1}{3}\;(\alpha_{23}-\alpha_{32}+2\alpha_{31}-2\alpha_{21})\;,\qquad\delta_{4}~{}\equiv~{}\Delta\omega_{32}+\frac{1}{3}\;(\alpha_{23}-\alpha_{32}+\alpha_{21}-\alpha_{31})\;.$

The four-dimensional dynamical system (89) demonstrates the effects similar to
those of the two-dimensional system (58), such as mode locking for small
pumping amplitudes $b_{ij}$, however with quasi-periodic motion, instead of
periodic Rabi oscillations. For large pumping amplitudes $b_{ij}$, there
appears chaotic motion. For example, in the case where $\alpha_{mn}=\alpha$,
$\beta_{mn}=\beta$, and $\Delta\omega_{mn}=0$, the chaotic motion arises when

$\left|\;\frac{\beta}{\alpha}\;\right|~{}\geq~{}0.639448\;.$

Note that here we consider cold bosonic atoms trapped in a potential that is
not periodic, although similar effects can be generated in nonequilibrium
optical latices as well [84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95].

##  11 Matter-wave interferometry

Composing a coherent system, Bose condensates enjoy many of the properties
typical of coherent optical systems and studied in atom interferometry [96].
In atoms, electrons can be excited into a higher individual energy level,
while in a condensate a large group of trapped Bose-condensed atoms can be
transferred into an excited collective energy level. The principal difference
in dealing with many atoms, instead of a single atom, is in the necessity of
taking account of interactions of condensed atoms.

###  11.1 Interference patterns

The existence of different coherent modes inside a trap leads to the
appearance of interference patterns [42]. The density of atoms in a trap is

$\rho({\bf r},t)~{}=~{}|\;\eta({\bf r},t)\;|^{2}~{}=~{}N\;|\;\varphi({\bf
r},t)\;|^{2}\;.$ (90)

With expression (20), this gives

$\rho({\bf
r},t)~{}=~{}N\left|\;\sum_{n}c_{n}(t)\;e^{-iE_{n}t}\;\varphi_{n}({\bf
r})\;\right|^{2}\;.$ (91)

The density of an $n$-th coherent mode is

$\rho_{n}({\bf r},t)~{}=~{}N\;|\;c_{n}(t)\;\varphi_{n}({\bf r})\;|^{2}\;.$
(92)

Therefore the interference pattern

$\rho_{int}({\bf r},t)~{}\equiv~{}\rho({\bf r},t)-\sum_{n}\rho_{n}({\bf r},t)$
(93)

reads as

$\rho_{int}({\bf r},t)~{}=~{}N\sum_{m\neq
n}c_{m}^{*}(t)\;c_{n}(t)\;\varphi_{m}^{*}({\bf r})\;\varphi_{n}({\bf
r})\;e^{i\omega_{mn}t}.$ (94)

Since the density of atoms can be observed, the interference pattern is also
observable.

###  11.2 Internal Josephson current

The appearance of several coherent modes inside a trap makes the trapped
atomic cloud essentially nonuniform, since different modes have different
topological shapes. Respectively, inside the trap there arise a density
current that is analogous to the current occurring between different potential
wells [97, 98]. The current appearing not because of external barriers, but as
a consequence of internally arising nonuniformities, is called internal
Josephson current [99, 100]. This effect is also named quantum dynamical
tunneling [101, 102].

The density current in a condensed system reads as

${\bf j}({\bf r},t)~{}\equiv~{}\frac{1}{m}\;{\rm Im}\;\eta^{*}({\bf
r},t)\;\vec{\nabla}\eta({\bf r},t)~{}=~{}\frac{N}{m}\;{\rm
Im}\;\varphi^{*}({\bf r},t)\;\vec{\nabla}\varphi({\bf r},t)\;.$ (95)

The current caused by a single mode is

${\bf j}_{n}({\bf r},t)~{}=~{}\frac{N}{m}\;{\rm
Im}\;|\;c_{n}(t)\;|^{2}\;\varphi_{n}^{*}({\bf
r},t)\;\vec{\nabla}\varphi_{n}({\bf r},t)\;.$ (96)

The difference between the total current in the system and the sum of density
currents of separate modes forms the internal Josephson current [42]

${\bf j}_{int}({\bf r},t)~{}=~{}\frac{N}{m}\;{\rm Im}\sum_{m\neq
n}c_{m}^{*}(t)\;c_{n}(t)\;e^{i\omega_{mn}t}\;\varphi_{m}({\bf
r},t)\;\vec{\nabla}\varphi_{n}({\bf r},t)\;.$ (97)

###  11.3 Rabi oscillations

Under the action of an external alternating field, the mode fractional
populations oscillate with time as in the case of Rabi oscillations [103].
Thus for two modes described by equations (32) the mode populations satisfy
the equations

$\frac{dn_{1}}{dt}~{}=~{}{\rm
Im}\;\left(\beta_{12}\;e^{it\Delta\omega}\;c_{1}^{*}c_{2}\right)\;,\qquad\frac{dn_{2}}{dt}~{}=~{}{\rm
Im}\;\left(\beta_{12}^{*}\;e^{-it\Delta\omega}\;c_{2}^{*}c_{1}\right)\;.$ (98)

When $|\beta_{12}/\alpha|\ll 1$, then the amplitudes $c_{n}$ can be treated as
fast functions, while $n_{1}$ and $n_{2}$, as slow. Then, employing the
averaging method, under the initial conditions

$c_{1}(0)~{}=~{}1\;,\qquad c_{2}(0)~{}=~{}0\;,$ (99)

we find [22, 42] the guiding centers for the populations as

$n_{1}~{}=~{}1\;-\;\frac{|\;\beta_{12}\;|^{2}}{\Omega^{2}}\;\sin^{2}\left(\frac{\Omega
t}{2}\right)\;,\qquad
n_{2}~{}=~{}\frac{|\;\beta_{12}\;|^{2}}{\Omega^{2}}\;\sin^{2}\left(\frac{\Omega
t}{2}\right)\;,$ (100)

with the effective Rabi frequency

$\Omega^{2}=[\;\Delta\omega+\alpha\;(n_{2}-n_{1})\;]^{2}+|\;\beta_{12}\;|^{2}\;,$
(101)

where, for simplicity, it is assumed that
$\alpha_{12}=\alpha_{21}\equiv\alpha$. The Rabi oscillations can be used for
studying the parameters of a two-level system.

###  11.4 Higher-order resonances

In addition to the resonance condition $\omega=\omega_{21}$, the generation of
coherent modes can also be realized at higher-order resonances, under the
effects of harmonic generation, employing a single external pumping field,
with

$n\omega~{}=~{}\omega_{21}\qquad(n=1,2,\ldots)\;,$

and at parametric conversion, when several fields $\omega_{j}$ are used, such
that

$\sum_{j}(\pm\omega_{j})~{}=~{}\omega_{21}\;.$

This can be shown [80, 82] by solving Eqs. (32) using the scale-separation
approach [104, 105].

##  12 Ramsey fringes

Ramsey [106] improved upon Rabi’s method by splitting one interaction zone
into two short interaction zones of time $\tau$, by applying two consecutive
pulses separated by a long time interval $T\gg\tau$. The idea is to study the
population $n_{2}$ of the excited atoms as a function of detuning or of
separation time. In the case of coherent modes, the population $n_{2}$
describes the fraction of atoms in an excited collective coherent mode.

For a two-mode situation, equations (32) cannot be solved exactly because of
the nonlinearity caused by atomic interactions. However, when
$|\beta_{12}/\alpha|\ll 1$, it is possible to use averaging techniques, since
then $c_{n}$ is classified as fast, while $n_{1}$ and $n_{2}$, as slow. Then
equations (32) can be solved keeping the populations as quasi-integrals of
motion. These equations can be rewritten [22] as

$\frac{d^{2}c_{1}}{dt^{2}}\;-\;i(\Delta\omega-\alpha)\;\frac{dc_{1}}{dt}+\left[\;\frac{|\;\beta_{12}\;|^{2}}{4}+\alpha_{12}n_{2}\;(\Delta\omega-\alpha_{21}n_{1})\;\right]\;c_{1}~{}=~{}0\;,$
$\frac{d^{2}c_{2}}{dt^{2}}+i(\Delta\omega+\alpha)\;\frac{dc_{2}}{dt}+\left[\;\frac{|\;\beta_{12}\;|^{2}}{4}-\alpha_{21}n_{1}\;(\Delta\omega+\alpha_{12}n_{2})\;\right]\;c_{2}~{}=~{}0\;.$
(102)

Introducing the notation for an effective detuning

$\Delta~{}\equiv~{}\Delta\omega+\alpha_{12}n_{2}-\alpha_{21}n_{1}$ (103)

and for an average interaction parameter

$\alpha~{}\equiv~{}\alpha_{12}n_{2}+\alpha_{21}n_{1}\;,$ (104)

we find the solutions of these equations

$c_{1}(t)~{}=~{}\left\\{\;c_{1}(0)\;\left[\;\cos\left(\frac{\Omega
t}{2}\right)-i\;\frac{\Delta}{\Omega}\;\sin\left(\frac{\Omega
t}{2}\right)\;\right]\;-\right.$
$\left.-\;i\;\frac{\beta_{12}}{\Omega}\;c_{2}(0)\sin\left(\frac{\Omega
t}{2}\right)\;\right\\}\;\exp\left\\{\;\frac{i}{2}\;(\Delta\omega-\alpha)\;t\;\right\\}\;,$
$c_{2}(t)~{}=~{}\left\\{\;c_{2}(0)\;\left[\;\cos\left(\frac{\Omega
t}{2}\right)-i\;\frac{\Delta}{\Omega}\;\sin\left(\frac{\Omega
t}{2}\right)\;\right]\;-\right.$
$-\;\left.i\;\frac{\beta_{12}^{*}}{\Omega}\;c_{1}(0)\sin\left(\frac{\Omega
t}{2}\right)\;\right\\}\;\exp\left\\{\;-\;\frac{i}{2}\;(\Delta\omega+\alpha)\;t\;\right\\}\;.$
(105)

If the first pulse acts during the time interval $0\leq t\leq\tau$ then at the
end of the pulse, starting from the initial conditions (99), the solutions are

$c_{1}(\tau)~{}=~{}\left[\;\cos\left(\frac{\Omega\tau}{2}\right)-i\;\frac{\Delta}{\Omega}\;\sin\left(\frac{\Omega\tau}{2}\right)\;\right]\;\exp\left\\{\;\frac{i}{2}\;(\Delta\omega-\alpha)\tau\;\right\\}\;,$
$c_{2}(\tau)~{}=~{}-i\;\frac{\beta_{12}^{*}}{\Omega}\;\sin\left(\frac{\Omega\tau}{2}\right)\;\exp\left\\{\;-\;\frac{i}{2}\;(\Delta\omega+\alpha)\tau\;\right\\}\;.$
(106)

Then, during the time interval $\tau\leq t\leq t+T$, the external pulses are
absent, which implies zero $\beta_{12}$. In that interval, the solutions have
the form (105), where the initial conditions $c_{n}(0)$ are replaced by
$c_{n}(\tau)$ from (106). Then, at time $\tau+T$, we have

$c_{1}(\tau+T)~{}=~{}c_{1}(\tau)\;\left[\;\cos\left(\frac{\Omega(\tau+T)}{2}\right)-i\;\frac{\Delta}{\Omega}\;\sin\left(\frac{\Omega(\tau+T)}{2}\right)\;\right]\;\times$
$\times\;\exp\left\\{\;\frac{i}{2}\;(\Delta\omega-\alpha)(\tau+T)\;\right\\}\;,$
$c_{2}(\tau+T)~{}=~{}c_{2}(\tau)\;\left[\;\cos\left(\frac{\Omega(\tau+T)}{2}\right)-i\;\frac{\Delta}{\Omega}\;\sin\left(\frac{\Omega(\tau+T)}{2}\right)\;\right]\;\times$
$\times\;\exp\left\\{\;-\;\frac{i}{2}\;(\Delta\omega+\alpha)(\tau+T)\;\right\\}\;.$
(107)

Following this way, at the end of the second pulse we find the amplitude of
the excited mode

$c_{2}(2\tau+T)~{}=~{}-i\;\frac{\beta_{12}^{*}}{\Omega}\;\sin\left(\frac{\Omega\tau}{2}\right)\;\times$
$\times\;\left\\{\;\left[\;\cos\left(\frac{\Omega\tau}{2}\right)-i\;\frac{\Delta}{\Omega}\;\sin\left(\frac{\Omega\tau}{2}\right)\;\right]\;\exp\left[\;-i(\Delta\omega+\alpha_{12}n_{2})\;T\;\right]\right.\;+$
$+\;\left.\left[\;\cos\left(\frac{\Omega\tau}{2}\right)+i\;\frac{\Delta}{\Omega}\;\sin\left(\frac{\Omega\tau}{2}\right)\;\right]\;\exp(-i\alpha_{21}n_{1}\tau)\;\right\\}\;\exp\\{-i(\Delta\omega+\alpha)\tau\\}\;.$
(108)

Therefore, the related population difference $n_{2}=|c_{2}|^{2}$ reads as

$n_{2}(2\tau+T)~{}=~{}4\;\frac{|\;\beta_{12}\;|^{2}}{\Omega^{2}}\;\sin^{2}\left(\frac{\Omega\tau}{2}\right)\;\times$
$\times\;\left[\;\cos\left(\frac{\Omega\tau}{2}\right)\;\cos\left(\frac{\Delta
T}{2}\right)+\frac{\Delta}{\Omega}\;\sin\left(\frac{\Omega\tau}{2}\right)\;\sin\left(\frac{\Delta
T}{2}\right)\;\right]^{2}.$ (109)

Usually, one works with the so-called $\pi/2$ pulses, when $\Omega\tau=\pi/2$.
Then

$n_{2}(2\tau+T)~{}=~{}\frac{|\;\beta_{12}\;|^{2}}{\Omega^{2}}\;\left[\;\cos\left(\frac{\Omega\tau}{2}\right)+\frac{\Delta}{\Omega}\;\sin\left(\frac{\Delta
T}{2}\right)\;\right]^{2}\;.$ (110)

The excited mode population is studied as a function of the detuning
$\Delta\omega$. The corresponding oscillating function $n_{2}$, demonstrating
Ramsey fringes, is shown in Refs. [107, 108]. The Ramsey fringes, caused by
coherent modes, are similar to Ramsey fringes of binary Bose condensate
mixtures of two different components [109] or of mixtures with two different
internal states [110], or of mixtures consisting of atoms and molecules [111].

##  13 Interaction modulation

The atomic scattering length can be varied in the presence of an external
magnetic field $B$ due to Feshbach resonance [112]. The magnetic field can be
varied in time, $B=B(t)$. Then the related scattering length

$a_{s}(B)~{}=~{}a_{s}\;\left(1\;-\;\frac{\Delta B}{B-B_{res}}\right)\;,$ (111)

where $a_{s}$ is the scattering length far outside of the resonance field
$B_{res}$ and $\Delta B$ is the resonance width, also becomes time dependent,
as well as the interaction strength

$\Phi(t)~{}=~{}4\pi\;\frac{a_{s}(B)}{m}\;,\qquad B~{}=~{}B(t)\;.$ (112)

If the magnetic field $B(t)$ is modulated around a field $B_{0}$, so that

$B(t)~{}=~{}B_{0}+b(t)\;,$ (113)

with a small modulation amplitude $|b(t)/B_{0}|\ll 1$ oscillating as

$b(t)~{}=~{}b_{1}\cos(\omega t)+b_{2}\sin(\omega t)\;,$ (114)

then the effective interaction strength acquires the form

$\Phi(t)~{}=~{}\Phi_{0}+\Phi_{1}\cos(\omega t)+\Phi_{2}\sin(\omega t)\;,$
(115)

where

$\Phi_{0}~{}=~{}\frac{4\pi}{m}\;a_{s}\left(1\;-\;\frac{\Delta
B}{B_{0}-B_{res}}\right)\;,$ $\Phi_{1}~{}=~{}\frac{4\pi a_{s}b_{1}\Delta
B}{m(B_{0}-B_{res})^{2}}\;,\qquad\Phi_{2}~{}=~{}\frac{4\pi a_{s}b_{2}\Delta
B}{m(B_{0}-B_{res})^{2}}\;.$

Resonance modulation of the effective interaction can generate coherent modes,
similarly to the trap modulation, as has been mentioned in [113] and studied
in [15, 114, 115].

Thus we are looking for the solution of the nonlinear Schrödinger equation

$i\;\frac{\partial}{\partial t}\;\varphi({\bf
r},t)~{}=~{}\hat{H}(t)\;\varphi({\bf r},t)\;,$ (116)

with the effective time-dependent Hamiltonian

$\hat{H}(t)~{}=~{}-\;\frac{\nabla^{2}}{2m}+U({\bf
r})+N\;\Phi(t)\;|\;\varphi({\bf r},t)\;|^{2}\;.$ (117)

Substituting here the mode expansion

$\varphi({\bf r},t)~{}=~{}\sum_{n}c_{n}(t)\;e^{-iE_{n}t}\;\varphi_{n}({\bf
r})\;,$ (118)

keeping in mind the two-mode case, following the same procedure as above, and
using the notation

$\gamma_{1}~{}\equiv~{}(\Phi_{1}-i\Phi_{2})\;N\int\varphi_{1}^{*}({\bf
r})|\;\varphi_{1}({\bf r},t)\;|^{2}\varphi_{2}({\bf r})\;d{\bf r}\;,$
$\gamma_{2}~{}\equiv~{}(\Phi_{1}-i\Phi_{2})\;N\int\varphi_{1}^{*}({\bf
r})|\;\varphi_{2}({\bf r},t)\;|^{2}\varphi_{2}({\bf r})\;d{\bf r}\;,$ (119)

we obtain the equations for the mode amplitudes

$i\;\frac{dc_{1}}{dt}~{}=~{}\alpha_{12}|\;c_{2}\;|^{2}c_{1}+\left(\gamma_{1}\;|\;c_{1}\;|^{2}+\frac{1}{2}\;\gamma_{2}\;|\;c_{2}\;|^{2}\right)\;c_{2}e^{i\Delta\omega
t}+\frac{1}{2}\;\gamma_{1}^{*}\;c_{2}^{*}\;c_{1}^{2}\;e^{-i\Delta\omega t}\;,$
$i\;\frac{dc_{2}}{dt}~{}=~{}\alpha_{21}|\;c_{1}\;|^{2}c_{2}+\left(\gamma_{2}^{*}\;|\;c_{2}\;|^{2}+\frac{1}{2}\;\gamma_{1}^{*}\;|\;c_{1}\;|^{2}\right)\;c_{1}e^{-i\Delta\omega
t}+\frac{1}{2}\;\gamma_{2}\;c_{1}^{*}\;c_{2}^{2}\;e^{i\Delta\omega t}\;.$
(120)

Generally, it is possible to combine the mode generation by means of trap
modulation as well as by scattering length modulation. This can be used for
generating multiple coherent modes or for enhancing the resonant generation of
a special mode. Then, employing the same procedures as above, we derive [115]
the equations

$i\;\frac{dc_{1}}{dt}~{}=~{}\alpha_{12}|\;c_{2}\;|^{2}c_{1}+\frac{1}{2}\;\left(2\gamma_{1}\;|\;c_{1}\;|^{2}+\gamma_{2}\;|\;c_{2}\;|^{2}+\beta_{12}\right)\;c_{2}\;e^{i\Delta\omega
t}+\frac{1}{2}\;\gamma_{1}^{*}\;c_{2}^{*}\;c_{1}^{2}\;e^{-i\Delta\omega t}\;,$
$i\;\frac{dc_{2}}{dt}~{}=~{}\alpha_{21}|\;c_{1}\;|^{2}c_{2}+\frac{1}{2}\;\left(2\gamma_{2}^{*}\;|\;c_{2}\;|^{2}+\gamma_{1}^{*}\;|\;c_{1}\;|^{2}+\beta_{12}^{*}\right)\;c_{1}\;e^{-i\Delta\omega
t}+\frac{1}{2}\;\gamma_{2}\;c_{1}^{*}\;c_{2}^{2}\;e^{i\Delta\omega t}\;.$
(121)

Equations (120) and (121) have the structure similar to Eqs. (32), hence they
can produce the same effects involving coherent modes [15, 114, 115]. The
generation of coherent modes can be effectively done by both methods, by means
of external potential modulation or by modulating atomic interactions [15,
114, 115, 116, 117].

##  14 Strong interactions and noise

In the above sections, generation of coherent modes is treated in the regime
of asymptotically weak interactions and in the absence of noise corresponding
to temperature. The condensate wave equation (15) has been obtained assuming
that all atoms are condensed. This has allowed us, for deriving the condensate
wave-function equation, to resort to the coherent approximation, resulting in
the nonlinear Schrödinger equation (15). Of course, at finite, and moreover
under strong interactions, the condensate can be essentially depleted. In
addition, there can exist intrinsic noise imitating temperature effects.

Thus, three questions arise. First, what is the wave-function equation for
finite and strong atomic interactions? Second, is it feasible to generate
coherent modes with that condensate equation? And, third, what is the
influence of intrinsic noise on the mode dynamics? This section provides
answers to these questions. Below, we follow the self-consistent theory of
Bose-condensed systems [15, 16, 23, 118, 119, 120, 121].

When in a Bose system, described by the Hamiltonian (3), there occurs Bose-
Einstein condensation, the global gauge symmetry becomes broken. The symmetry
breaking is the necessary and sufficient condition for Bose condensation [12,
15]. The explicit and convenient way of symmetry breaking is by means of the
Bogolubov shift

$\psi({\bf r},t)~{}=~{}\eta({\bf r},t)+\psi_{1}({\bf r},t)\;,$ (122)

where $\eta({\bf r},t)$ is the condensate wave function and $\psi_{1}({\bf
r},t)$ is the field operator of uncondensed atoms. The field variables of
condensed and uncondensed particles are mutually orthogonal,

$\int\eta^{*}({\bf r},t)\;\psi_{1}({\bf r},t)\;d{\bf r}~{}=~{}0\;.$ (123)

The condensate wave function is normalized to the number of condensed atoms

$N_{0}~{}=~{}\int|\;\eta({\bf r},t)\;|^{2}\;d{\bf r}\;,$ (124)

while the number of uncondensed atoms is

$N_{1}~{}=~{}\int\langle\;\psi_{1}^{\dagger}({\bf r},t)\;\psi_{1}({\bf
r},t)\;\rangle\;d{\bf r}\;.$ (125)

The total number of atoms

$N~{}=~{}N_{0}+N_{1}$ (126)

is fixed.

In order that the quantum numbers, such as spin or momentum, be conserved,
requires to impose the conservation condition

$\langle\;\psi_{1}({\bf r},t)\;\rangle~{}=~{}0\;.$ (127)

The general form of the conservation condition is written as

$\langle\;\hat{\Lambda}\;\rangle~{}=~{}0\;,$ (128)

with the condition operator

$\hat{\Lambda}~{}=~{}\int\left[\;\lambda({\bf r},t)\;\psi_{1}^{\dagger}({\bf
r},t)+\lambda^{*}({\bf r},t)\;\psi_{1}({\bf r},t)\;\right]\;d{\bf r}\;,$ (129)

in which $\lambda({\bf r},t)$ is a Lagrange multiplier.

Taking into account the normalization conditions (124) and (125), as well as
the conservation condition (128), leads to the grand Hamiltonian

$H~{}=~{}\hat{H}-\mu_{0}N_{0}-\mu_{1}\hat{N}_{1}-\hat{\Lambda}\;,$ (130)

where

$\hat{N}_{1}~{}=~{}\int\psi_{1}^{\dagger}({\bf r},t)\;\psi_{1}({\bf
r},t)\;d{\bf r}\;.$ (131)

The evolution equations for condensed and uncondensed atoms are

$i\;\frac{\partial}{\partial t}\;\eta({\bf
r},t)~{}=~{}\left\langle\;\frac{\delta H}{\delta\eta^{*}({\bf
r},t)}\;\right\rangle$ (132)

and, respectively,

$i\;\frac{\partial}{\partial t}\;\psi_{1}({\bf r},t)~{}=~{}\frac{\delta
H}{\delta\psi_{1}^{\dagger}({\bf r},t)}\;.$ (133)

The conservation condition (128) requires that the grand Hamiltonian must have
no linear in $\psi_{1}$ terms, which is achieved by setting the Lagrange
multiplier

$\lambda({\bf r},t)~{}=~{}\left[\;-\;\frac{\nabla^{2}}{2m}+U({\bf
r},t)+\frac{1}{2}\int\Phi({\bf r}-{\bf r}^{\prime})\;|\;\eta({\bf
r}^{\prime},t)\;|^{2}\;d{\bf r}^{\prime}\;\right]\;\eta({\bf r},t)\;.$ (134)

To proceed further, let us introduce several notations, such as the normal
density matrix

$\rho_{1}({\bf r},{\bf
r}^{\prime},t)~{}\equiv~{}\langle\;\psi_{1}^{\dagger}({\bf
r}^{\prime},t)\;\psi_{1}({\bf r},t)\;\rangle\;,$ (135)

the so-called anomalous average

$\sigma_{1}({\bf r},{\bf r}^{\prime},t)~{}\equiv~{}\langle\;\psi_{1}({\bf
r}^{\prime},t)\;\psi_{1}({\bf r},t)\;\rangle\;,$ (136)

the density of condensed atoms

$\rho_{0}({\bf r},t)~{}\equiv~{}|\;\eta({\bf r},t)\;|^{2}\;,$ (137)

the density of uncondensed atoms

$\rho_{1}({\bf r},t)~{}\equiv~{}\langle\;\psi_{1}^{\dagger}({\bf
r},t)\;\psi_{1}({\bf r},t)\;\rangle~{}=~{}\rho_{1}({\bf r},{\bf r},t)\;,$
(138)

the total density of atoms

$\rho({\bf r},t)~{}=~{}\rho_{0}({\bf r},t)+\rho_{1}({\bf r},t)\;,$ (139)

the diagonal anomalous average

$\sigma_{1}({\bf r},t)~{}\equiv~{}\langle\;\psi_{1}({\bf r},t)\;\psi_{1}({\bf
r},t)\;\rangle~{}=~{}\sigma_{1}({\bf r},{\bf r},t)\;,$ (140)

and the triple anomalous average

$\xi({\bf r},{\bf r}^{\prime},t)~{}\equiv~{}\langle\;\psi_{1}^{\dagger}({\bf
r}^{\prime},t)\;\psi_{1}({\bf r}^{\prime},t)\;\psi_{1}({\bf r},t)\;\rangle\;.$
(141)

With these notations, the equation for the condensate wave function reads as

$i\;\frac{\partial}{\partial t}\;\eta({\bf
r},t)~{}=~{}\left[-\;\frac{\nabla^{2}}{2m}+U({\bf
r},t)-\mu_{0}\;\right]\;\eta({\bf r},t)\;+$ $+\;\int\Phi({\bf r}-{\bf
r}^{\prime})\;\left[\;\rho({\bf r}^{\prime},t)\;\eta({\bf r},t)+\rho_{1}({\bf
r},{\bf r}^{\prime},t)\;\eta({\bf r}^{\prime},t)+\sigma({\bf r},{\bf
r}^{\prime},t)\;\eta^{*}({\bf r}^{\prime},t)+\xi({\bf r},{\bf
r}^{\prime},t)\;\right]\;d{\bf r}^{\prime}\;.$ (142)

Note that this is an exact equation valid for any integrable interaction
potential. For the contact potential (4), the equation takes the form

$i\;\frac{\partial}{\partial t}\;\eta({\bf
r},t)~{}=~{}\left[-\;\frac{\nabla^{2}}{2m}+U({\bf
r},t)-\mu_{0}\;\right]\;\eta({\bf r},t)\;+$ $+\;\Phi_{0}\\{\;[\;\rho_{0}({\bf
r},t)+2\rho_{1}({\bf r},t)\;]\;\eta({\bf r},t)+\sigma_{1}({\bf
r},t)\;\eta^{*}({\bf r},t)+\xi({\bf r},t)\;\\}\;,$ (143)

in which

$\xi({\bf r},t)~{}\equiv~{}\xi({\bf r},{\bf r},t)\;.$ (144)

The solution to this equation can be sought in the form

$\eta({\bf r},t)~{}=~{}\sqrt{N_{0}(t)}\;\sum_{n}c_{n}(t)\;\varphi_{n}({\bf
r})\;\exp\\{-i[\;E_{n}+\varepsilon_{n}(t)-\mu_{0}\;]\;t\\}\;.$ (145)

If the number of condensed atoms changes slowly in time, so that

$\left|\;\frac{1}{E_{n}}\;\frac{dN_{0}(t)}{dt}\;\right|~{}\ll~{}1$ (146)

and the phase $\varepsilon_{n}(t)$ is also a slow function of time,

$\left|\;\frac{1}{E_{n}}\;\frac{d\varepsilon_{n}(t)}{dt}\;\right|~{}\ll~{}1\;,$
(147)

then, in the case of two coherent modes, we obtain [15, 119] the same
equations (121). However, the phase $\varepsilon_{n}(t)$ is a slow function
when $\rho_{1}$, $\sigma_{1}$, and $\xi$ are small as compared to $\rho_{0}$.

In this way, when interactions between atoms are strong and temperature is
high, the phase of the condensate wave function can very rather fast. In order
to understand how fast phase fluctuations influence the dynamics of the
condensate wave function, it is possible to consider the equations

$\frac{ds}{dt}~{}=~{}-\beta\;\sqrt{1-s^{2}}\;\sin x\;,$
$\frac{dx}{dt}~{}=~{}\alpha\;s+\frac{\beta s}{\sqrt{1-s^{2}}}\;\cos
x+\delta+\sigma\;dW_{t}\;,$ (148)

in which phase fluctuations are modeled by random noise, where $W_{t}$ is the
standard Wiener process and $\sigma$ is the standard noise deviation. These
equations are to be compared with equations (58) containing no random phase
fluctuations. If the strength of the noise is not too large, its influence
leads to the distortion of Rabi oscillations, but does not change the overall
qualitative picture [122, 123], as is shown in Fig. 2 for the mode-locked
(subcritical) regime) and Fig. 3 for the mode-unlocked (supercritical regime).

Figure 2: Subcritical regime with $b=0.25<b_{c}=0.282$. (a) Population
imbalance s(t) as functions of dimensionless time, for $\sigma=0.5$ (solid
line) and $\sigma=0$ (dashed-dotted line). (b) Phase difference x(t) as
functions of dimensionless time, for $\sigma=0.5$ (solid line) and $\sigma=0$
(dashed-dotted line).

Figure 3: Supercritical regime with $b=0.5>b_{c}=0.282$. (a) Population
imbalance s(t) as functions of dimensionless time, for $\sigma=0.5$ (solid
line) and $\sigma=0$ (dashed-dotted line). (b) Phase difference x(t) as
functions of dimensionless time, for $\sigma=0.5$ (solid line) and $\sigma=0$
(dashed-dotted line).

##  15 Critical phenomena

The qualitative change of dynamics between the mode-locked regime and mode-
unlocked regime reminds a phase transition in a statistical system. This
similarity can be emphasized by considering an effective stationary system
obtained by averaging the original system over time [41, 75, 76, 77].

For simplicity, we set $\alpha_{12}=\alpha_{21}=\alpha$. In the case of two
coherent modes, the evolution equations (32) can be derived from the
Hamiltonian equations

$i\;\frac{dc_{1}}{dt}~{}=~{}\frac{\partial H}{\partial c_{1}^{*}}\;,\qquad
i\;\frac{dc_{2}}{dt}~{}=~{}\frac{\partial H}{\partial c_{2}^{*}}\;,$ (149)

with the effective Hamiltonian

$H~{}=~{}\alpha\;|\;c_{1}\;c_{2}\;|^{2}+\frac{1}{2}\;\left(\beta\;c_{1}^{*}\;c_{2}\;e^{i\Delta\omega
t}+\beta^{*}\;c_{2}^{*}\;c_{1}\;e^{-i\Delta\omega t}\right)\;,$ (150)

in which $\beta\equiv\beta_{12}$. Resorting to the averaging techniques yields
solution (105).

Averaging Hamiltonian (150) over time, with using the normalization condition

$n_{1}+n_{2}~{}=~{}1\;,$ (151)

gives the effective energy

$E~{}=~{}\frac{\alpha|\;\beta\;|^{2}}{2\overline{\Omega}^{2}}\;\left(\frac{|\;\beta\;|^{2}}{2\overline{\Omega}^{2}}+\varepsilon\right)\;,$
(152)

with $\varepsilon\equiv\delta/\alpha$, the average Rabi frequency given by the
expression

$\overline{\Omega}^{2}~{}=~{}\alpha^{2}\;\left(\varepsilon+\overline{n}_{2}-\overline{n}_{1}\right)^{2}+|\;\beta\;|^{2}\;,$
(153)

and the average fractions

$\overline{n}_{1}~{}=~{}1-\overline{n}_{2}\;,\qquad\overline{n}_{2}~{}=~{}\frac{|\;\beta\;|^{2}}{2\overline{\Omega}^{2}}\;.$
(154)

The order parameter can be defined as the average population difference

$\eta~{}\equiv~{}\overline{n}_{1}-\overline{n}_{2}~{}=~{}1\;-\;\frac{|\;\beta\;|^{2}}{\overline{\Omega}^{2}}\;.$
(155)

Also, let us introduce the system capacity to store the energy pumped into the
system

$C_{\beta}~{}\equiv~{}\frac{\partial E}{\partial|\;\beta\;|}$ (156)

and the susceptibility with respect to detuning

$\chi_{\varepsilon}~{}\equiv~{}\frac{\partial\eta}{\partial|\;\varepsilon\;|}\;.$
(157)

The abrupt change of dynamics happens on the critical line
$b_{c}+\varepsilon=0.5$, where $b\equiv\beta/\alpha$. In the vicinity of the
dynamic phase transition, the system characteristics strongly depend on the
variable

$\tau~{}\equiv~{}\frac{|\;b-b_{c}\;|}{b_{c}}\;.$ (158)

When $\tau\rightarrow 0$, we observe the critical behavior of the order
parameter

$\eta-\eta_{c}~{}\simeq~{}\frac{1}{\sqrt{2}}\;(1-2\varepsilon)\;\tau^{1/2}\qquad(\eta_{c}~{}=~{}0.5+\varepsilon)\;,$
(159)

effective system capacity and susceptibility:

$C_{\beta}~{}\simeq~{}\frac{1}{4\sqrt{2}}\;\tau^{-1/2}\;,\qquad\chi_{\varepsilon}~{}\simeq~{}\frac{1}{\sqrt{2}}\;\tau^{-1/2}\;.$
(160)

The latter diverge as at phase transitions of second order.

The critical exponents of the order parameter, capacity, and susceptibility
satisfy the scaling relation

$\frac{1}{2}+1+\frac{1}{2}~{}=~{}2$ (161)

typical of equilibrium second-order phase transitions.

##  16 Atomic squeezing

Bose-Einstein condensation is accompanied by global gauge symmetry breaking,
when the arising condensate is characterized by a wave function. Rigorous
description of the Bose-condensate assumes the thermodynamic limit. In a
finite volume, strictly speaking, there is no Bose-condensate, but there
exists a quasi-condensate that is described by field operators. Hence, a
finite cloud of trapped atoms should be characterized by operators instead of
non-operator functions. In order to consider a finite quasi-condensate, one
has to replace the amplitudes $c_{n}$ by Bose operators ${\hat{c}}_{n}$.

When one is interested in the case of two modes, one can make the canonical
transformation to quasi-spin operators

$\hat{c}_{1}^{\dagger}\;\hat{c}_{1}~{}=~{}\frac{1}{2}\;-\;S_{z}\;,\qquad\hat{c}_{2}^{\dagger}\;\hat{c}_{2}~{}=~{}\frac{1}{2}\;+\;S_{z}\;,$
$\hat{c}_{1}^{\dagger}\;\hat{c}_{2}~{}=~{}S_{x}-iS_{y}\;,\qquad\hat{c}_{2}^{\dagger}\;\hat{c}_{1}~{}=~{}S_{x}+iS_{y}\;,$
(162)

where

$S_{\alpha}~{}\equiv\sum_{j=1}^{N}S_{j}^{\alpha}\qquad(\alpha=x,y,z)\;.$

The evolution equations can be derived [42] using the standard Heisenberg
equations of motion with the Hamiltonian

$H~{}=~{}-\;\frac{\alpha}{N}\;S_{z}^{2}+\frac{\beta}{2}\;(S^{-}+S^{+})-S_{z}\;\delta\;,$
(163)

in which $S_{j}^{\alpha}$ are half-spin operators and the ladder operators are

$S^{\pm}~{}\equiv~{}S_{x}\pm iS_{y}\;.$

Averaging these equations, with employing the mean-field approximation, gives
equations equivalent to Eqs. (58).

Then it is possible to study the problem of squeezing in trapped atomic
systems, similarly to the squeezing of light in optical systems [50]. In
general, atomic squeezing can be achieved for both bosons as well as for
fermions. Squeezing for Bose-Einstein condensates was considered for two-
component mixtures [124], for atoms with two internal states [125], and for
atoms in linked mesoscopic traps formed by an optical lattice [126], which is
equivalent to a multicomponent mixture. Atomic squeezing can be used for
atomic spectroscopy, creation of atomic clocks [127], and for atom
interferometers [128].

For any two operators, $\hat{A}$ and $\hat{B}$, there exists the Heisenberg
uncertainty relation

${\rm var}(\hat{A})\;{\rm
var}(\hat{B})~{}\geq~{}\frac{1}{4}\;\left|\;\langle\;[\;\hat{A},\;\hat{B}\;]\;\rangle\;\right|^{2}\;,$
(164)

where the variance is

${\rm
var}(\hat{A})~{}\equiv~{}\langle\;\hat{A}^{+}\;\hat{A}\;\rangle-|\;\langle\;\hat{A}\;\rangle\;|^{2}\;.$

One says that the operators $\hat{A}$ and $\hat{B}$ are not squeezed with
respect to each other, when

$\displaystyle\begin{array}[]{l}{\rm
var}(\hat{A})~{}\geq~{}\frac{1}{2}\;|\;\langle\;[\;\hat{A},\;\hat{B}\;]\;\rangle\;|\\\
{\rm
var}(\hat{B})~{}\geq~{}\frac{1}{2}\;|\;\langle\;[\;\hat{A},\;\hat{B}\;]\;\rangle\;|\end{array}~{}~{}~{}(not\;squeezed)\;.$
(167)

And one says that $\hat{A}$ is squeezed with respect to $\hat{B}$, if

${\rm
var}(\hat{A})~{}<~{}\frac{1}{2}\;\left|\;\langle\;[\;\hat{A},\;\hat{B}\;]\;\rangle\;\right|\qquad(squeezed)\;.$
(168)

The measure of squeezing is quantified by the squeezing factor

$Q_{AB}~{}\equiv~{}\frac{2{\rm
var}(\hat{A})}{|\;\langle\;[\;\hat{A},\;\hat{B}\;]\;\rangle\;|}\;.$ (169)

For two operators, squeezing happens when the squeezing factor is smaller than
unity,

$\displaystyle\begin{array}[]{ll}Q_{AB}~{}\geq~{}1{}{}&~{}~{}(no\;squeezing)\;,\\\
Q_{AB}~{}<~{}1{}{}&~{}~{}(squeezing)\;.\end{array}$ (172)

The occurrence of squeezing of $\hat{A}$ with respect to $\hat{B}$ implies
that the observable corresponding to $\hat{A}$ can be measured more precisely
than that one related to $\hat{B}$.

In our case, the operator $S_{z}$ corresponds to the population difference,
while $S^{\pm}$, to the transition dipoles, since

$\langle\;S_{z}\;\rangle~{}=~{}\frac{N}{2}\;s\;,\qquad|\;\langle\;S^{\pm}\;\rangle\;|~{}=~{}\frac{N}{2}\;|\;\langle\;\hat{c}_{1}^{\dagger}\;\hat{c}_{2}\;\rangle\;|\;.$
(173)

For the related operators, the Heisenberg uncertainty relation is

${\rm var}(S_{z})\;{\rm
var}(S^{\pm})~{}\geq~{}\frac{1}{4}|\;\langle\;S^{\pm}\;\rangle\;|^{2}\;.$
(174)

Squeezing of $S_{z}$ with respect to $S^{\pm}$ is measured by the squeezing
factor

$Q~{}=~{}\frac{2{\rm var}(S_{z})}{|\;\langle\;S^{\pm}\;\rangle\;|}\;,$ (175)

which leads to

$Q~{}=~{}\sqrt{1-s^{2}}\;.$ (176)

In the mode-locked regime, $s^{2}\leq 1$, and $Q<1$ for all times. In the
mode-unlocked regime $s^{2}$ for the most of time is smaller than one,
touching one only at some moments of time. Hence almost always $S_{z}$ is
squeezed with respect to $S^{\pm}$. This means that the population difference,
hence the mode populations, can be measured with a better accuracy than the
quantities associated with the transition dipoles, such as density current.

Similarly, it is possible to consider the effect of squeezing for several
coupled modes. Thus, for three modes, it is straightforward to introduce the
quasi-spin operators

$J_{+}~{}\equiv~{}\sqrt{2}\;(a_{2}^{\dagger}\;a_{1}+a_{3}^{\dagger}\;a_{2})\;,\qquad
J_{-}~{}\equiv~{}\sqrt{2}\;(a_{1}^{\dagger}\;a_{2}+a_{2}^{\dagger}\;a_{3})\;,\qquad
J_{z}~{}\equiv~{}a_{3}^{\dagger}\;a_{3}-a_{1}^{\dagger}\;a_{1}\;,$

satisfying the commutation relations

$[\;J_{+},\;J_{-}\;]~{}=~{}2J_{z}\;,\qquad[\;J_{z},\;J_{\pm}\;]~{}=~{}\pm
J_{\pm}\;.$

Then the squeezing factor is defined in the same way as for two modes.

##  17 Cloud entanglement

It is necessary to distinguish two different notions, entanglement of states
and entanglement production. Entanglement of a state concerns the structure of
a statistical operator, while entanglement production by an operator describes
the ability of an operator, by acting on functions of a Hilbert space, to
create entangled functions.

###  17.1 Entangled states

When one talks about entanglement, one has, first of all, to specify what are
the objects whose entanglement is considered. In quantum physics it is
customary to consider entanglement of particles. However in the case of Bose-
Einstein condensate, it is possible to study entanglement between large atomic
clouds. Such coherent atomic clouds can be organized in optical lattices with
sufficiently deep wells at lattice sites. Each lattice well can house
condensate clouds containing between several atoms to $10^{4}$ atoms [129,
130]. Shaking the optical lattice by means of magnetic fields or lasers, it is
possible to excite coherent modes simultaneously in all lattice sites.

Suppose Bose-condensed atoms are loaded into a deep lattice with $N_{L}$
sites, enumerated by the index $j=1,2,\ldots,N_{L}$, where at each lattice
site there is a cloud of $N$ atoms, with $N\gg 1$. Assume that there can be
excited $M$ coherent modes enumerated by the index $m=1,2,\ldots,M$. A
coherent mode $m$ at a lattice site $j$, which is characterized by a coherent
field $\eta_{jm}({\bf r})$, in the Fock space is represented [131] by a column

$|\;\eta_{jm}\;\rangle~{}=~{}\left[\;\frac{e^{-N/2}}{\sqrt{n!}}\;\prod_{i=1}^{n}\eta_{jm}({\bf
r}_{i})\;\right]\;,$ (177)

with the rows enumerated by $n=0,1,2,\ldots$. These modes form a basis in the
space

${\cal H}_{j}~{}=~{}{\rm span}_{m}\\{\;|\;\eta_{jm}\;\rangle\;\\}\;.$ (178)

The overall system of $N_{L}$ clouds is described by the Hilbert space

${\cal H}~{}=~{}\bigotimes_{j=1}^{N_{L}}{\cal H}_{j}~{}=~{}{\rm
span}_{m}\left\\{\bigotimes_{j=1}^{N_{L}}|\;\eta_{jm}\;\rangle\;\right\\}\;.$
(179)

The coherent wave state of the whole system in the Fock space is

$|\;\eta(t)\;\rangle~{}=~{}\sum_{m=1}^{M}c_{m}(t)\bigotimes_{j=1}^{N_{L}}|\;\eta_{jm}\;\rangle\;.$
(180)

The related statistical operator, or statistical state, is

$\hat{\rho}(t)~{}=~{}|\;\eta(t)\;\rangle\langle\;\eta(t)\;|\;.$ (181)

Notice that the wave state (180) is a multimode multicat entangled state.

The structure of the state (181),

$\hat{\rho}(t)~{}=~{}\sum_{m=1}^{M}\sum_{n=1}^{M}c_{m}(t)\;c_{n}^{*}(t)\bigotimes_{j=1}^{N_{L}}|\;\eta_{jm}\;\rangle\langle\;\eta_{jn}\;|\;,$
(182)

shows that this is an entangled state, with respect to clouds located at the
lattice wells [132], provided that there exist at least two nontrivial modes,
when $M\geq 2$. The state (181) is entangled, since it is not separable [133],
hence cannot be represented in the form

$\hat{\rho}_{sep}~{}=~{}\sum_{k}\lambda_{k}\bigotimes_{j=1}^{N_{L}}\hat{\rho}_{jk}\qquad\left(\lambda_{k}\geq
0,\;\sum_{k}\lambda_{k}=1\right)\;,$ (183)

where $\hat{\rho}_{jk}$ are statistical operators in the spaces (178).

###  17.2 Entanglement production

Entanglement production by an operator characterizes the ability of this
operator of entangling the functions of the Hilbert space it acts on. An
operator is called entangling, if there exists at least one separable pure
state such that it becomes entangled under the action of the operator.

Suppose an operator $\hat{A}$ acting on a Hilbert space
$\mathcal{H}=\otimes_{j}\mathcal{H}_{j}$ is a trace-class operator with a
finite nonzero trace,

$0~{}<~{}|\;{\rm Tr}\hat{A}\;|~{}<~{}\infty\;.$ (184)

Generally, an operator, acting on a state, pertaining to $\mathcal{H}$,
produces an entangled state, even when this operator is separable, being
defined similarly to (183) as

$\hat{A}_{sep}~{}=~{}\sum_{k}\lambda_{k}\bigotimes_{j}\hat{A}_{jk}\qquad\left(\lambda_{k}\geq
0,\;\sum_{k}\lambda_{k}=1\right)\;.$ (185)

Even when it acts on a product state, it generates an entangled state

$\hat{A}_{sep}\bigotimes|\;\varphi_{j}\;\rangle~{}=~{}\sum_{k}\lambda_{k}\bigotimes_{j}\hat{A}_{jk}\;|\;\varphi_{j}\;\rangle\;,$
(186)

if at least two $\lambda_{k}$ differ from zero.

The sole operator that does not entangle product states is the operator of the
factor form

$\hat{A}^{\otimes}~{}=~{}\bigotimes_{j}\hat{A}_{j}\;,$ (187)

which is clear from its action

$\hat{A}^{\otimes}\bigotimes_{j}|\;\varphi_{j}\;\rangle~{}=~{}\bigotimes_{j}\hat{A}_{j}\;|\;\varphi_{j}\;\rangle.$
(188)

The measure of entanglement production has been introduced [134, 135, 136] by
comparing the action of $\hat{A}$ with that of its nonentangling counterpart

$\hat{A}^{\otimes}~{}\equiv~{}\frac{\bigotimes_{j}\hat{A}_{j}}{({\rm Tr}_{\cal
H}\hat{A})^{N_{L}-1}}\;,$ (189)

where

$\hat{A}_{j}~{}\equiv~{}{\rm Tr}_{{\cal H}/{\cal H}_{j}}\;\hat{A}$ (190)

and the normalization holds:

${\rm Tr}_{{\cal H}}\;\hat{A}^{\otimes}~{}=~{}{\rm Tr}_{{\cal H}}\;\hat{A}\;.$
(191)

The measure of entanglement production is defined [134, 135] as

$\varepsilon(\hat{A})~{}\equiv~{}\log\;\frac{||\;\hat{A}\;||}{||\;\hat{A}^{\otimes}\;||}\;,$
(192)

where any logarithm base can be accepted. This measure is semi-positive, zero
for nonentangling operators, continuous, additive, and invariant under local
unitary operations [134].

The fact that the only nontrivial operators preserving separability are the
operators having the form of tensor products of local operators is proved in
Refs. [137, 138, 139, 140, 141, 142]. The operator preserving separability is
called nonentangling [143]. The operator transforming at least one non-
entangled state into an entangled state is termed entangling [144].

###  17.3 Resonant entanglement production

Existence of entanglement in a statistical operator implies the occurrence of
quantum correlation between subsystems. Then the statistical average of the
operator of an observable cannot be factorized into partial averages. This
kind of factorization, reminding a mean-field approximation, happens only when
both, the operator of an observable, as well as the statistical operator are
non-entangling, so that

${\rm Tr}_{\cal H}\;\hat{\rho}\bigotimes_{j}\hat{A}_{j}~{}=~{}\prod_{j}{\rm
Tr}_{{\cal
H}_{j}}\;\hat{\rho}_{j}\;\hat{A}_{j}\qquad(\hat{\rho}=\otimes_{j}\hat{\rho}_{j})\;,$
(193)

which can be represented as

$\langle\;\bigotimes_{j}A_{j}\;\rangle~{}=~{}\prod_{j}\langle\;\hat{A}_{j}\;\rangle\;.$
(194)

The statistical operator (182) for a system of atomic clouds, with resonantly
generated coherent modes, is entangled, so that the considered above
factorization cannot happen. The statistical operator (182) is also
entangling. Thus its action on the non-entangled coherent basis state of space
(179) yields an entangled state

$\hat{\rho}(t)\bigotimes_{j=1}^{N_{L}}|\;\eta_{jm}\;\rangle~{}=~{}c_{m}^{*}(t)\sum_{n}c_{n}(t)\bigotimes_{j=1}^{N_{L}}|\;\eta_{jn}\;\rangle\;.$
(195)

The entangling power of this statistical operator is quantified [145] by the
measure of entanglement production

$\varepsilon(\hat{\rho}(t))~{}=~{}\log\;\frac{||\;\hat{\rho}(t)\;||}{||\;\hat{\rho}^{\otimes}(t)\;||}\;,$
(196)

in which

$\hat{\rho}^{\otimes}~{}=~{}\bigotimes_{j=1}^{N_{L}}\hat{\rho}_{j}(t)\;,\qquad\hat{\rho}_{j}(t)~{}\equiv~{}{\rm
Tr}_{{\cal H}/{\cal H}_{j}}\;\hat{\rho}(t)\;.$ (197)

Using the standard operator norm gives the entanglement measure produced by
the statistical operator (182) as

$\varepsilon(\hat{\rho}(t))~{}=~{}(1-N_{L})\;\log\;\sup_{m}\;|\;c_{m}(t)\;|^{2}\;.$
(198)

Creation of entanglement in a lattice of atomic clouds with resonantly
generated coherent modes can be called resonant entanglement production. The
behavior of measure (198) has been studied in Refs. [145, 146]. A number of
other examples of entanglement production are considered in Refs. [147, 148,
149].

##  18 Nonresonant mode generation

As is stressed above, coherent modes can be excited by modulating either the
trapping potential or the scattering length. There are also two ways of
generation, resonant and nonresonant. In the case of resonant generation, the
frequency of the alternating field has to be in resonance with the transition
frequency corresponding to the transition between the ground-state energy
level and the energy of the mode to be generated. Then the amplitude of
modulation by an external field does not need to be very large.

However the strict occurrence of the resonance is not compulsory for mode
generation due to the effect of power broadening, when modes can be generated
if the modulation frequency is not in resonance with the transition frequency,
but the alternating field amplitude is sufficiently strong to generate higher-
order coherent modes. Depending on the strength of the alternating field,
those modes are excited for which the energy pumped by the field is sufficient
for their creation.

If the aim would be to generate quantum vortices, then it could be done by
means of a rotating laser beam. However, if one wishes to generate different
kinds of modes, then there is no reason of imposing anisotropy on the system,
so that the alternating potential should be more or less isotropic.

Thus in experiments [150, 151] the harmonic trapping potential

$U({\bf
r})~{}=~{}\frac{m}{2}\;\omega_{x}^{2}\;(x^{2}+y^{2})+\frac{m}{2}\;\omega_{z}^{2}\;z^{2}\;,$
(199)

with the trapping frequencies $\omega_{x}=2\pi\times 210$ Hz and
$\omega_{z}=2\pi\times 23$ Hz, is subject to the alternating field with the
frequency $\omega=2\pi\times 200$ Hz and the modulating amplitude
$0.2\omega_{x}$. The modulated potential has the form

$U({\bf
r},t)~{}=~{}\frac{m}{2}\;\Omega_{x}^{2}(t)\;(x-x_{0})^{2}+\frac{m}{2}\;\Omega_{y}^{2}(t)\;(y^{\prime}-y_{0}^{\prime})^{2}+\frac{m}{2}\;\Omega_{z}^{2}(t)\;(z^{\prime}-z_{0}^{\prime})^{2}\;,$
(200)

where the oscillation centers are given by the equation

$\displaystyle\left[\;\begin{array}[]{l}y^{\prime}-y_{0}^{\prime}\\\
z^{\prime}-z_{0}^{\prime}\end{array}\right]~{}=~{}\left[\;\begin{array}[]{rr}\sin\vartheta_{0}{}&~{}\cos\vartheta_{0}\\\
\cos\vartheta_{0}{}&~{}-\sin\vartheta_{0}\end{array}\right]\;\left[\;\begin{array}[]{l}y-y_{0}\\\
z-z_{0}\end{array}\right]$ (207)

and the oscillation frequencies are

$\Omega_{\alpha}(t)~{}=~{}\omega_{\alpha}\;\delta_{\alpha}\;[\;1-\cos(\omega
t)\;]\;.$

By increasing the strength of the alternating potential and the modulation
time, it is possible to pump into the trap large amounts of energy. The
energy, per atom, injected into the trap during the period of time $t$ can be
written as

$E_{inj}~{}=~{}\frac{1}{N}\int_{0}^{t}\left|\;\frac{\partial H(t)}{\partial
t}\;\right|\;dt\;,$ (208)

where

$H(t)~{}=~{}\int\eta^{*}({\bf r},t)\;\left[\;-\;\frac{\nabla^{2}}{2m}+U({\bf
r},t)\;\right]\;\eta({\bf r},t)\;d{\bf
r}+\frac{1}{2}\;\Phi_{0}\int|\;\eta({\bf r},t)\;|^{4}\;d{\bf r}\;.$ (209)

This type of modulation imposes no anisotropy on the system.

The overall procedure of generating coherent modes and different
nonequilibrium states, by employing the above alternating potential, has been
studied in detail by realizing experimental observations, providing
theoretical description, and accomplishing numerical calculations [152, 153,
154, 155, 156, 157, 158, 159, 160, 161, 162].

The same setup has been used in numerical simulations as well as in
experiments with 87Rb [150, 151]. At the initial moment of time, practically
all $N=1.5\times 10^{5}$ atoms, having mass $m=1.445\times 10^{-22}$ g and
scattering length $a_{s}=0.577\times 10^{-6}$ cm, are Bose-condensed in a
cylindrical harmonic trap with a radial frequency of $\omega_{x}=2\pi\times
210$ Hz and an axial frequency of $\omega_{z}=2\pi\times 23$ Hz. Hence the
aspect ratio is $\nu=\omega_{z}/\omega_{x}=0.11$. In experiments, the trapped
atomic cloud had a radius of $R=4\times 10^{-4}$ cm and a length of $L=6\times
10^{-3}$ cm. In the Thomas-Fermi approximation, the cloud radius and length
are

$R_{TF}~{}=~{}1.036\;l_{x}(g\nu)^{1/5}\;,\qquad
L_{TF}~{}=~{}\frac{2.072}{\nu}\;l_{x}(g\nu)^{1/5}\;,$

while the oscillator lengths are $l_{x}=0.74\times 10^{-4}$ cm and
$l_{z}=2.25\times 10^{-4}$ cm. The central density is $\rho=2.821\times
10^{14}$ cm-3. The coherence length, or healing length, is

$l_{coh}~{}=~{}\frac{\hbar}{mc}~{}=~{}\frac{1}{\sqrt{4\pi\rho
a_{s}}}~{}=~{}1.37\;\frac{l_{x}}{(g\nu)^{1/5}}\;,$

which makes $l_{coh}=2.18\times 10^{-5}$ cm. The effective coupling parameter
(42) is $g=1.96\times 10^{4}$.

##  19 Nonequilibrium states

In the process of generation of nonlinear modes by a strong nonresonant
modulation field, a whole sequence of nonequilibrium states has been observed
in experiments [152, 153, 154, 155] and modeled in computer simulations [156,
157, 158, 159, 160, 161, 162]. Initially, elementary collective modes populate
the system, followed by the production of vortices and solitons, which quickly
begin to react, generating a cloud with the presence of large density
fluctuations. The appearance of strong density fluctuations in the atomic
cloud has been observed by subtracting from the current density distribution
the averaged density, thus leaving only the strong fluctuations, as is shown
in Fig. 4. When we vary the amplitude for short excitation times, where
vortices can be seen, we notice that the formation of visible vortices grows
with the amplitude and with the excitation time. For sufficiently large
amplitudes and excitation times, the evolution of the momentum distribution,
projected onto the observation plane, evolves towards the occupation of the
largest moments. The presence of just a few vortices does not seem to produce
large variations in the density distribution, when compared with the
excitation-free case. Passing through the turbulence threshold, the density
distribution becomes displaced to the side of atoms with greater moments, and
there appear the regions of momenta with a power law dependence in the
distribution of moments, shown in Fig. 5. This is a signature of the presence
of a direct cascade of particles, which is a characteristic feature of
turbulence.

Figure 4: Aspects of the excited cloud during free expansion and the
distribution of the fluctuations obtained by subtracting the averaged profile.

Figure 5: Evolution of the projected momentum distribution measured by time of
flight.

In more detail, the studied sequence of nonequilibrium states, observed in
experiments [152, 153, 154, 155] and modeled in computer simulations [156,
157, 158, 159, 160, 161, 162] is discussed below.

###  19.1 Weak nonequilibrium

At the beginning of the process of imposing an alternating potential, when the
injected energy $E_{inj}$ is yet insufficient for the creation of coherent
modes, being less than twice the energy $E_{emb}$ required for creating at
least a couple of embryos, or germs, of vortex rings,

$0~{}<~{}E_{inj}~{}<~{}2E_{emb}\;,$ (210)

the trapped atomic cloud is only slightly perturbed with small density
fluctuations. Vortex embryos appear by pairs, since each of them possesses
vorticity $\pm 1$, while the vorticity of the whole system is zero, which
requires that each vortex has to be compensated by an antivortex. The energy
of creating a vortex embryo, $E_{emb}$, is smaller then that sufficient for
generating a vortex ring, $E_{ring}$. For estimates, it is possible to accept
that $E_{emb}\sim E_{ring}/2$.

Let us bring attention of the reader that the injected energy (208) is defined
as the energy per particle. Therefore to compare it with the energies of
coherent modes, the corresponding energies below are also defined as
characteristic reduced energies, that is the energies per particle.

###  19.2 Vortex embryos

When the injected energy reaches the level sufficient for generating a pair of
vortex embryos, these objects arise being just pieces of broken vortex rings.
This happens when the injected energy is in the interval

$2E_{emb}~{}<~{}E_{inj}~{}<~{}2E_{ring}\;.$ (211)

The embryos can be of different sizes. They have vorticity $\pm 1$. If the
external pumping is switched off after the germs are created, they survive for
about $0.2$ s, when they do not move, exhibiting only small oscillations. The
local equilibration time is

$t_{loc}~{}=~{}\frac{m}{\hbar\rho
a_{s}}~{}=~{}\frac{87.8}{(g\nu)^{2/5}\omega_{x}}\;,$

which is $t_{loc}\approx 10^{-3}$ s. Since the embryos lifetime of $0.2$ s is
much larger than the local equilibration time, the embryos are metastable
objects. Typical vortex embryos are shown in Fig. 6 obtained from numerical
simulation.

Figure 6: Typical example of vortex germs. The time of the trap modulation is
$t=5.3$ ms.

###  19.3 Vortex rings

The following pumping leads to the appearance of well formed rings of
different radii $R_{ring}$, as in Fig. 7 representing numerical results. A
vortex ring is a circular line of zero density, with a winding number $\pm 1$
around each of the line elements [163, 164, 165, 166, 167, 168]. The rings
with opposite vorticity appear by pairs. These ring modes exist in the
interval of energies

$2E_{ring}~{}<~{}E_{inj}~{}<~{}2E_{vort}\;,$ (212)

where $E_{vort}$ is the energy of a vortex line. The reduced energy of a ring
is

$E_{ring}~{}=~{}\frac{2\pi^{2}}{N}\;\rho\;R_{ring}\;l_{x}^{2}\;\ln\left(2.25\;\frac{R_{ring}}{l_{coh}}\right)\;\omega_{x}\;.$
(213)

Here $R_{ring}$ is the ring radius, $l_{coh}$ is the coherence length, and
$\rho$ is the density of the system at the location of the ring, if the ring
would be absent. In the trap, the ring oscillates [169, 170] with the period

$T_{ring}~{}=~{}\frac{4\pi
R}{l_{coh}\omega_{z}\sqrt{\ln(R/l_{coh})}}~{}=~{}\frac{4\pi(g\nu)^{2/5}}{\omega_{z}\sqrt{\ln(0.5g\nu)}}\;,$
(214)

which, for the accepted parameters, makes $T_{ring}\approx 0.7$ s. If the
pumping is switched off, the ring lifetime is about $0.1$ s, which is much
longer than the local equilibration time $t_{loc}\approx 10^{-3}$ s. This
shows that the rings are metastable objects. In the trap, the rings look as
practically immovable. A pair of rings, found in numerical simulation, is
shown in Fig. 7.

Figure 7: Typical example of vortex rings. The time of the trap modulation is
$t=11.5$ ms.

###  19.4 Vortex lines

The continuing pumping leads to the generation of pairs of vortex lines in the
energy interval

$2E_{vort}~{}<~{}E_{inj}~{}<~{}E_{turb}\;,$ (215)

in which $E_{turb}$ is the pumped energy after which vortex turbulence
develops. Vortices can have different lengths $L_{vort}$. The reduced energy
of a vortex of length $L_{vort}$, in the Thomas-Fermi approximation [171],
reads as

$E_{vort}~{}=~{}\frac{2\pi}{3N}\;\rho\;L_{vort}\;l_{x}^{2}\;\ln\left(0.95\;\frac{R}{l_{coh}}\right)\;\omega_{x}\;.$
(216)

The vortex length $L_{vort}$ is approximately about the cloud length $L$. Then
the reduced vortex energy is $E_{vort}=0.296\omega_{x}$.

The vortices move randomly due to the Magnus force caused by the difference in
pressure between the opposite sides of a vortex line. The vortex lifetime,
after switching off the pumping, is about $0.2$ s. Numerically found vortex
lines are shown in Fig. 8.

Figure 8: Typical example of vortex lines directed along the $z$ axis. The
time of the trap modulation is $t=17$ ms.

###  19.5 Vortex turbulence

Injecting into the trap more energy creates more and more vortices and anti-
vortices, as well as distorts their straight lines and forces them to incline
in different directions. The used modulating potential does not impose a
rotation axis, because of which the increasing number of vortices does not
lead to the creation of a vortex lattice, but results in the formation of a
random vortex tangle corresponding to the definition of Vinen quantum
turbulence [172, 173, 174, 175, 176, 177]. Being released from the trap, the
atomic cloud expands isotropically, which is typical of Vinen turbulence [152,
153, 154, 178].

Vortex turbulence exists in the energy interval

$E_{turb}~{}<~{}E_{inj}~{}<~{}E_{drop}\;,$ (217)

where $E_{drop}$ is the upper boundary of the vortex turbulence.

One distinguishes random vortex turbulence, or Vinen turbulence, as opposed to
the correlated vortex turbulence, or Kolmogorov turbulence [176]. In a Vinen
vortex tangle, the intervortex distance is much larger than the coherence
length. Therefore, the summary energy of vortices in the random Vinen tangle
can be approximated by the product of the vortex number and a single vortex
energy. Thus the vortex turbulence energy threshold

$E_{turb}~{}=~{}E_{vort}\;N_{vort}$ (218)

is reached after the injected energy creates the critical number of vortices
$N_{vort}$ that is so large that the mean distance between the outer parts of
the vortices becomes equal to their diameter $2\xi$, so that no vortex can be
inserted inbetween without touching the neighbors. This implies that in the
cross-section of the trap the number $N_{vort}$ is such that the equality

$N_{vort}\pi(4\xi)^{2}~{}=~{}\pi R^{2}$

holds true. Hence the critical number of vortices is

$N_{vort}~{}=~{}\left(\frac{R}{4\xi}\right)^{2}\;.$ (219)

For the experiments with 87 Rb, the atomic cloud radius is $R\approx 4\times
10^{-4}$ cm and the coherence length is $\xi\approx 2\times 10^{-5}$ cm. Thus,
the critical vortex number is $N_{vort}\approx 25$, which is in perfect
agreement with both experiments and numerical simulations.

The overwhelming majority of vortices usually have the lowest circulation
number $\pm 1$. In the case where there would arise vortex lines with
different circulation, it is possible to represent the system as a mixture of
fluids containing vortex lines with different vorticity [179].

The column integrated radial momentum distribution obeys a power law
$n_{r}(k)\propto k^{-\gamma}$ in the range $\pi/\xi<k<2\pi/\xi$, which is a
key quantitative expectation for an isotropic turbulent cascade [180]. This
power low has been observed [181, 182] for turbulent atomic clouds of trapped
atoms, with the slope characterized by $\gamma\approx 2$.

It is interesting to note that there exists a similar effect in laser media,
which is called optical turbulence or turbulent photon filamentation. In high
Fresnel lasers and nonlinear media there appear bright filaments randomly
distributed in the sample cross-section and not correlated with each other.
The theory of the photon turbulence is developed in Refs. [183, 184, 185].
This effect has been observed in many experiments, as can be inferred from
review articles [186, 187, 188, 189, 190, 191, 192, 193, 194].

###  19.6 Droplet state

Starting from an equilibrium state of a practically completely Bose-condensed
system, and injecting energy into the trap by applying an alternating field,
we produce more and more excited condensate. In weak nonequilibrium, the whole
atomic cloud is yet Bose-condensed. The appearing vortex excitations, such as
vortex embryos, vortex rings, and vortex lines can be treated as germs, or
nuclei, of uncondensed phase. The following pumping of energy into the trap
little by little destroys the condensate. When the injected energy reaches the
threshold value $E_{drop}$, vortex turbulence transforms into a new state
composed of coherent Bose-condensed droplets immersed into an incoherent
surrounding.

The state of coherent droplets is analogous to heterophase states where
different thermodynamic phases are intermixed on mesoscopic scale [195, 196,
197]. This state exists in the energy interval

$E_{drop}~{}<~{}E_{inj}~{}<~{}E_{c}\;,$ (220)

where $E_{c}$ is the critical amount of energy after which Bose-Einstein
condensate is completely destroyed and the lower energy threshold can be
estimated as

$E_{drop}~{}=~{}E_{vort}\;N^{*}_{vort}\;.$ (221)

Here $N_{vort}^{*}$ is the maximal number of vortices that do not yet
essentially interact with each other, so that their centers are separated by
the distance $d_{int}$, which defines the number of vortices

$N_{vort}^{*}~{}=~{}\left(\frac{R}{d_{int}}\right)^{2}\;.$ (222)

With $R\approx 4\times 10^{-4}$ cm and $d_{int}\approx 0.7\times 10^{-4}$ cm,
we have $N_{vort}^{*}\approx 30$, which is in good agreement with experiments.
Since $E_{vort}=0.296\omega_{x}$, the energy, after which the droplet state
develops is $E_{drop}\approx 8.88\omega_{x}$.

The droplet state develops because the number of vortices becomes so large and
their interactions so strong that vortices destroy each other. Numerical
simulations show that the grains have the sizes close to the coherence length
$\xi\approx 10^{-5}$ cm. The phase inside a grain is constant, hence the
grains represent coherent droplets. After switching off the pumping by an
alternating potential, each grain survives during the lifetime $10^{-2}$ s,
which is much longer than the local equilibration time of $10^{-4}$ s. This
tells us that, the droplets are metastable objects. The density inside a
droplet is up to $100$ times larger than that of their incoherent surrounding.
The number of atoms in a droplet is around $40$. The coherent droplets, whose
number is about $400$, are randomly distributed inside the trap.

Figure 9 shows experimentally measured density distributions for different
states: vortex state with several vortices (a); vortex turbulence (b); and
droplet state (c).

Figure 9: Density distributions for different states, observed in experiments:
(a) vortex state with several vortices; (b) vortex turbulence; (c) droplet
state.

The droplet state can be called heterophase droplet mixture, droplet
turbulence, or grain turbulence. It also reminds fog. In numerical
simulations, considering the opposite process of equilibrium Bose-condensate
formation from an uncondensed gas, this regime is termed strong turbulence
[180, 198]. Since the location of droplets in space is random, after averaging
over spatial configurations, the system looks as a mesoscopic mixture of two
phases with different density [199, 200]. It is possible to show that the
averaging over phase configurations is similar to the averaging over time
[201].

###  19.7 Wave turbulence

After the energy injected into the trap reaches the value $E_{c}=k_{B}T_{c}$,
with $T_{c}$ being the Bose-Einstein condensation temperature, the condensed
state cannot exist. Then there develops a state modeling the turbulent normal
state. This state corresponds to what is termed wave turbulence or weak
turbulence. These waves are not so big, with the density only about $3$ times
larger than their surrounding. The typical size of a wave is of order
$10^{-4}$ cm. The phase inside it is random, which is in line with the absence
of Bose-Einstein condensate. However, strictly speaking, the transition
between the droplet state and wave turbulence is a gradual crossover. The
crossover line, conditionally separating the regimes of droplet state and wave
turbulence, can be located at the point, where the system coherence,
associated with droplets, is destroyed in half of the system. For 87Rb atoms,
the critical temperature is $T_{c}=2.76\times 10^{-7}$ K, hence
$E_{c}=27.4\omega_{x}$.

##  20 Classification of states

The described nonequilibrium states can be distinguished by the intervals of
injected energy, keeping in mind the relations $E_{emb}\approx 0.5E_{ring}$
and $E_{ring}\approx\nu E_{vort}$, with $\nu=0.11$. Then the characteristic
energies are:

$E_{emb}~{}=~{}0.016\omega_{x}\;,\qquad E_{ring}~{}=~{}0.033\omega_{x}\;,$
$E_{vort}~{}=~{}0.296\omega_{x}\;,\qquad E_{turb}~{}=~{}7.4\omega_{x}\;,$
$E_{drop}~{}=~{}8.88\omega_{x}\;,\qquad E_{c}~{}=~{}27.4\omega_{x}\;.$ (223)

Introducing the dimensionless injected energy $E\equiv E_{inj}/\omega_{x}$, it
is possible to list the related energy intervals as follows:

$\displaystyle\begin{array}[]{rlllll}0&<&E&<&0.032{}{}{}&~{}~{}~{}(weak\;nonequilibrium)\;,\\\
0.032&<&E&<&0.066{}{}{}&~{}~{}~{}(vortex\;embryos)\;,\\\
0.066&<&E&<&0.592{}{}{}&~{}~{}~{}(vortex\;rings)\;,\\\
0.592&<&E&<&7.4{}{}{}&~{}~{}~{}(vortex\;lines)\;,\\\
7.4&<&E&<&8.88{}{}{}&~{}~{}~{}(vortex\;turbulence)\;,\\\
8.88&<&E&<&27.4{}{}{}&~{}~{}~{}(coherent\;droplets)\;,\\\
&&E&>&27.4{}{}{}&~{}~{}~{}(wave\;turbulence)\;.\end{array}$ (231)

When a system, prepared in a strongly nonequilibrium symmetric state,
equilibrates to a state with broken symmetry, there appear topological
defects, such as vortices, protodomains, cells, grains, strings, and
monopoles. This is called the Kibble-Zurek mechanism [202, 203, 204, 205]. In
our experiments, the opposite procedure of generating nonequilibrium states
with topological modes is realized, which can be called inverse Kibble-Zurek
scenario [158].

The nonequilibrium states can also be characterized [161, 162] by effective
temperature, Fresnel, and Mach numbers. The effective temperature is given by
the relation

$T_{eff}~{}\equiv~{}\frac{2}{3}\;[\;E_{kin}(t)-E_{kin}(0)\;]\;,$ (232)

where $E_{kin}(t)$ is the current kinetic energy per particle and $E_{kin}(0)$
is the initial kinetic energy of the system at equilibrium. The Fresnel number
can be defined as

$F~{}\equiv~{}\frac{\pi R^{2}}{\lambda
L}\qquad\left(\lambda\equiv\sqrt{\frac{2\pi}{mT_{eff}}}\right)\;,$ (233)

with $\lambda$ being thermal wavelength. And the Mach number is written as

$M~{}\equiv~{}\frac{v}{c}~{}=~{}\sqrt{\frac{3T_{eff}}{mc^{2}}}\;,$ (234)

where $v$ is the particle velocity corresponding to the kinetic energy and $c$
is the sound velocity in the system. The classification of the nonequilibrium
states of trapped ${}^{87}Rb$ is summarized in Table 1. The values of
effective temperature, Fresnel number and Mach number correspond to the lower
boundary of the related states, where these states appear.

Table 1: Nonequilibrium states of a trapped Bose-Einstein condensate, characterized by effective temperature (in units of $\omega_{x}$), Fresnel number, and mach number. The figures correspond to the lower boundary of the related states. Nonequilibrium states | Effective temperature | Fresnel number | Mach number
---|---|---|---
wave turbulence | 23.5 | 1.01 | 1.07
coherent droplets | 8.56 | 0.61 | 0.66
vortex turbulence | 5.54 | 0.49 | 0.53
vortex lines | 2.26 | 0.31 | 0.25
vortex rings | 1.21 | 0.23 | 0.22
vortex embryos | 0.29 | 0.11 | 0.08
weak nonequilibrium |  |  |

##  21 Atom laser

If trapped atoms could be ejected from the trap, this would remind the action
of a laser emitting photons. Bose-Einstein condensate is a coherent system,
hence emitting Bose-condensed atoms would be similar to emitting coherent
light. Thus, one compares with the optical laser the simple process of Bose
atoms falling out of the trap under the influence of gravity [206, 207, 208,
209, 210]. The direction of an atomic beam can be regulated by means of Raman
scattering [211]. A mechanism allowing for guiding atomic beams in any
required direction has been proposed and analyzed in Refs. [212, 213, 214,
215].

Let us consider a system of atoms, with spin operators $S_{i}$, with
interaction potential $\Phi_{ij}$, in a magnetic field ${\bf B}_{i}$, and
subject to the Earth gravity force ${\bf g}$. The Hamiltonian is

$H~{}=~{}\sum_{i=1}^{N}\left(-\;\frac{\nabla_{i}^{2}}{2m}\;-\;\mu_{S}\;{\bf
S}_{i}\cdot{\bf B}_{i}-m\;{\bf g}\cdot{\bf
r}_{i}\right)+\frac{1}{2}\sum_{i\neq j}\Phi_{ij}\;.$ (235)

Here $\mu_{S}$ is the atomic magnetic moment that can be positive or negative.
Since magnetic moments of atoms are often mainly due to electrons, for
concreteness, we assume that $\mu_{S}<0$, although the results do not
principally depend on the sign of the magnetic moment.

Denoting the quantum-mechanical average by angular brackets, it is
straightforward to derive the evolution equations for the mean position
vector,

$m\;\frac{d^{2}}{dt}\;\langle\;{\bf
r}_{i}\;\rangle~{}=~{}\mu_{S}\langle\;\vec{\nabla}_{i}({\bf S}_{i}\cdot{\bf
B}_{i})\;\rangle-\sum_{j(\neq
i)}^{N}\langle\;\vec{\nabla}_{i}\;\Phi_{ij}\;\rangle+m\;{\bf g}\;,$ (236)

and the average spin,

$\frac{d}{dt}\;\langle\;{\bf S}_{i}\;\rangle~{}=~{}\mu_{S}\;\langle\;{\bf
S}_{i}\times{\bf B}_{i}\;\rangle\;.$ (237)

Being interested in the motion of an atomic cloud as a whole, we introduce the
center-of-mass vector

${\bf R}~{}\equiv~{}\frac{1}{N}\sum_{i=1}^{N}\langle\;{\bf r}_{i}\;\rangle\;,$
(238)

the average spin

${\bf S}~{}\equiv~{}\;\frac{1}{N}\sum_{i=1}^{N}\langle\;{\bf
S}_{i}\;\rangle\;,$ (239)

and the collision force

${\bf I}~{}\equiv~{}-\;\frac{1}{N}\sum_{i\neq
j}^{N}\langle\;\vec{\nabla}_{i}\;\Phi_{ij}\;\rangle$ (240)

due to atomic interactions.

Assume that the magnetic field is slowly varying in space, so that

${\bf B}_{i}~{}\equiv{\bf B}({\bf r}_{i})~{}\cong~{}{\bf B}({\bf
R})~{}\equiv~{}{\bf B}.$ (241)

Then the approximate equalities are valid:

$\frac{1}{N}\sum_{i=1}^{N}\langle\;\vec{\nabla_{i}}({\bf S}_{i}\cdot{\bf
B}_{i})\;\rangle~{}\cong~{}\vec{\nabla}({\bf S}\cdot{\bf B})\;,$
$\frac{1}{N}\sum_{i=1}^{N}\langle\;{\bf S}_{i}\times{\bf
B}_{i}\;\rangle~{}\cong~{}{\bf S}\times{\bf B}\;.$

Therefore, we have the equations of motion for the center of mass

$m\;\frac{d^{2}{\bf R}}{dt^{2}}~{}=~{}\mu_{S}\vec{\nabla}({\bf S}\cdot{\bf
B})+{\bf I}+m{\bf g}$ (242)

and for the average spin

$\frac{d{\bf S}}{dt}~{}=~{}\mu_{S}\;{\bf S}\times{\bf B}\;.$ (243)

The magnetic field of the trap can be taken in the typical of magnetic traps
form [216, 217, 218], as the sum

${\bf B}~{}=~{}{\bf B}_{1}+{\bf B}_{2}$ (244)

of a quadrupole field

${\bf B}_{1}~{}=~{}B_{1}^{\prime}\;(R_{x}{\bf e}_{x}+R_{y}{\bf e}_{y}-\lambda
R_{z}{\bf e}_{z})\qquad(B_{1}^{\prime}>0)$ (245)

and a transverse field

${\bf B}_{2}~{}=~{}B_{2}(h_{x}{\bf e}_{x}+h_{y}{\bf e}_{e})\qquad(B_{2}>0)\;,$
(246)

where

$h_{x}^{2}+h_{y}^{2}~{}=~{}1\qquad(B_{2}=|\;{\bf B}_{2}\;|)\;.$

There exist two types of quadrupole traps, dynamic quadrupole traps, with the
transverse field depending on time,

$h_{x}~{}=~{}\cos(\omega t)\;,\qquad h_{y}~{}=~{}\sin(\omega t)\;,$ (247)

and static quadrupole traps, with a time-independent transverse field,

$h_{x}~{}=~{}const\;,\qquad h_{y}~{}=~{}const\;.$ (248)

Also, there can exist an additional so-called cooling field serving for
removing fast atoms from the center of the trap. However this cooling field
does not play role in describing the mechanism of creating directed atomic
beams [213].

The method we are suggesting does not depend on whether a dynamic or static
trap is accepted [219, 220]. For concreteness, below we illustrate the results
for the dynamic quadrupole trap.

The spatial motion of the center of mass is characterized by the frequency
$\omega_{1}$ defined by the equation

$\omega_{1}^{2}~{}=~{}-\;\frac{\mu_{S}(B_{1}^{\prime})^{2}}{mB_{2}}~{}>~{}0\;,$
(249)

while the spin motion is characterized by the Zeeman frequency

$\omega_{2}~{}=~{}-\mu_{S}B_{2}~{}>~{}0\;.$ (250)

We stress that these frequencies are positive, keeping in mind that
$\mu_{S}<0$.

The spin motion is faster than the spatial atomic motion, which implies that
$\omega_{2}\gg\omega_{1}$. The rotation of the transverse field is faster than
the atomic motion, $\omega\gg\omega_{1}$, but slower than the spin motion in
order not to induce transitions between the Zeeman energy levels, hence
$\omega\ll\omega_{2}$. Thus there are the relations

$\omega_{1}~{}\ll~{}\omega~{}\ll~{}\omega_{2}\;.$ (251)

The size of the atomic cloud is defined by the effective radius

$R_{0}~{}\equiv~{}\frac{B_{2}}{B_{1}^{\prime}}\;.$ (252)

This radius can be used for defining dimensionless spatial variables
describing the location of the center of mass,

${\bf r}~{}\equiv~{}\frac{{\bf R}}{R_{0}}~{}=~{}\\{x,y,z\\}\;.$ (253)

Let us introduce the collision rate $\gamma$ by the equality

$\gamma\vec{\xi}~{}\equiv~{}\frac{{\bf I}}{mR_{0}}\;.$ (254)

The cloud of trapped atoms is rarified, so that the collision rate is small,
in the sense that

$\gamma~{}\ll~{}\omega_{1}\;,$ (255)

and $\xi$ can be approximated by a random variable centered at zero because of
the random directions of gradients in the collision force (240).

The shape of the trap can be described by a shape factor, which can be defined
by the function

$f({\bf
r})~{}=~{}\exp\left\\{-\;\frac{x^{2}+y^{2}}{(R/R_{0})^{2}}\;-\;\frac{z^{2}}{(L/R_{0})^{2}}\right\\}\;,$
(256)

where the trap radius $R\geq R_{0}$ and the trap length $L\geq R_{0}$, or by a
step-like function

$f({\bf
r})~{}=~{}1-\Theta\left(x^{2}+y^{2}\;-\;\frac{R^{2}}{R_{0}^{2}}\right)\;\Theta\left(|\;z\;|-\;\frac{L}{2R_{0}}\right)\;.$
(257)

As we have checked, the concrete choice of the shape factor does not change
the overall picture and does not influence the described mechanism [219, 220].

Thus, the center-of-mass spatial motion is given by the equation

$\frac{d^{2}{\bf r}}{dt^{2}}~{}=~{}\left(\;{\bf
F}+\gamma\vec{\xi}\;\right)\;f+\frac{{\bf g}}{R_{0}}\;,$ (258)

with $f=f({\bf r})$ and the force

${\bf F}~{}=~{}-\omega_{1}\;(S_{x}{\bf e}_{x}+S_{y}{\bf e}_{y}-\lambda
S_{z}{\bf e}_{z})\;.$ (259)

The spin motion, characterized by equation (243), can be represented as

$\frac{d{\bf S}}{dt}~{}=~{}\omega_{2}\hat{A}{\bf S}\;,$ (260)

with an antisymmetric matrix $\hat{A}=[A_{\alpha\beta}]$ having the elements

$A_{\alpha\alpha}~{}=~{}0\;,\qquad
A_{\alpha\beta}~{}=~{}-A_{\beta\alpha}\;,\qquad A_{21}=\lambda\;z\;,$
$A_{23}~{}=~{}x+h_{x}\;,\qquad A_{31}~{}=~{}y+h_{y}\;.$ (261)

According to condition (251), the transverse field (247), rotating with the
frequency $\omega$, is fast, as compared to the atomic motion characterized by
the frequency $\omega_{1}$. Then, it is possible to resort to the averaging
method [221], following which, we find the solution to the fast variable, that
is the spin ${\bf S}$, keeping the slow variables, that is the atomic
coordinates ${\bf r}$, fixed, and substitute the found fast solution to the
equation for the slow variable (258), with averaging the right-hand side of
this equation over the random variable $\vec{\xi}$ and over time (see details
in Refs. [222, 223, 224]). Thus we come to the equation

$\frac{d^{2}{\bf r}}{dt^{2}}~{}=~{}{\bf F}_{eff}\;f+\frac{{\bf g}}{R_{0}}\;,$
(262)

where $f=f({\bf r})$ and the effective force is

${\bf
F}_{eff}~{}=~{}-\;\frac{\omega_{1}^{2}[\;(1+x)S_{0}^{x}+yS_{0}^{y}-\lambda
zS_{0}^{z}\;](x{\bf e}_{x}+y{\bf e}_{y}+2\lambda^{2}z{\bf
e}_{z})}{2[\;(1+2x+x^{2}+y^{2}+\lambda^{2}z^{2})(1+x^{2}+y^{2}+\lambda^{2}z^{2})\;]^{1/2}}\;.$
(263)

Here $S_{0}^{\alpha}$ is the initial spin polarization.

The standard initial spin polarization is $S_{0}^{x}=S>0$ and
$S_{0}^{y}=S_{0}^{z}=0$. This guarantees atomic confinement for $r<1$, with
the attractive effective force

${\bf F}_{eff}~{}\cong~{}-\;\frac{1}{2}\;\omega_{1}^{2}\;S\;(x{\bf
e}_{x}+y{\bf e}_{y}+2\lambda^{2}z{\bf e}_{z})\;,$

leading to the effective potential well

$U_{eff}~{}\cong~{}\frac{1}{4}\;\omega_{1}^{2}\;S\;(x^{2}+y^{2}+2\lambda^{2}z^{2})\;.$

On the contrary, we suggest to arrange the initial spin polarization under
conditions

$S_{0}^{x}~{}=~{}S_{0}^{y}~{}=~{}0\;,\qquad S_{0}^{z}~{}=~{}S\;,$ (264)

which results in the effective force

${\bf F}_{eff}~{}=~{}\frac{\omega_{1}^{2}\lambda Sz(x{\bf e}_{x}+y{\bf
e}_{y}+2\lambda^{2}z{\bf
e}_{z})}{2[\;(1+2x+x^{2}+y^{2}+\lambda^{2}z^{2})(1+x^{2}+y^{2}+\lambda^{2}z^{2})\;]^{1/2}}\;.$
(265)

If the spin polarization has been aligned with the axis $x$, then the desired
initial spin polarization (264) along the axis $z$ can be prepared [225] by
the action of an oscillating $\pi/2$ pulse with the frequency in resonance
with $\omega_{2}$, turning the spin from the axis $x$ to the axis $z$.

Let us denote a dimensionless gravity acceleration

$\vec{\delta}~{}\equiv~{}\frac{{\bf g}}{R_{0}\lambda
S\omega_{1}^{2}}~{}=~{}\frac{m{\bf g}}{|\;\lambda
S\mu_{s}B_{1}^{\prime}\;|}\;.$ (266)

Then equations (262) acquire the form

$\frac{d^{2}x}{dt^{2}}~{}=~{}\frac{1}{2}\;z\;x\;f+\delta_{x}\;,$
$\frac{d^{2}y}{dt^{2}}~{}=~{}\frac{1}{2}\;z\;y\;f+\delta_{y}\;,\qquad\frac{d^{2}z}{dt^{2}}~{}=~{}\lambda^{2}\;z^{2}\;f+\delta_{z}\;.$
(267)

The detailed analysis of these equations at small $x$, $y$, and $z$, as well
as numerical solution for arbitrary spatial coordinates, demonstrate the
effect of atomic semiconfinement [212, 213, 214, 215, 219, 220, 222, 223,
224], when atoms are confined from one side of the trap, but are expelled from
another side. In the present setup, atoms are ejected in the direction of the
trap axis, with $z>0$. The typical trajectories of the emitted atomic beams
are shown in Fig. 10 for the trap axis $z$ inclined by $45$ degrees to the
horizon, so that $\delta_{z}=-\delta_{x}$; the initial spatial location of the
atomic cloud center of mass is close to the trap center and the initial
dimensionless velocities are close to zero, being varied in the interval
$-0.1\leq v_{0}^{\alpha}<0.1$. The aspect ratio $\nu=<x>^{2}/<z>^{2}$ is
rather small, being $10^{-4}$, which means that the atomic beam is stretched
in the $z$ direction about $100$ times larger then in the $x$ direction, that
is the beam is very well collimated. The collimation is not essentially
influenced by thermal fluctuations, provided the temperature is less than the
critical temperature

$T_{c}~{}\approx~{}\frac{\hbar\omega_{1}^{3}}{k_{B}\gamma^{2}}\;,$

which is close to the Bose condensation temperature. After leaving the trap,
the trajectories become curved due to gravity. The velocity of emitted atoms
depends on the setup parameters. For realistic parameters, the velocity can
reach $100$ cm/s.

Figure 10: Typical trajectories for the trap parameters $L=40$, $R=1$,
$\lambda=10$, the gravity components $\delta_{x}=0.03$,
$\delta_{z}=-\delta_{x}$, $\delta_{y}=0$, and different velocities
$-0.1<\dot{r}<0.1$: (a) for the time interval $0\leq t\leq 400$; (b) beginning
of the process;

##  22 Conclusion

The problem of generating non-ground-state Bose-Einstein condensates is
discussed. Such non-ground-state Bose condensates are represented by nonlinear
coherent modes of trapped atoms. Coherent modes are the solutions to the
stationary nonlinear Schrödinger equation. The solution to the time-dependent
nonlinear Schrödinger equation can be expanded over the coherent modes, which
leads to the equations for the amplitudes of the fractional mode populations.

The generation of the coherent modes can be realized by alternating external
fields, when either an external trapping potential or atomic interactions are
modulated. There exist two ways of generation, resonant generation and
nonresonant generation. In the resonant generation, the perturbing alternating
field does not need to be strong, but the alternating frequency has to be in
resonance with the transition frequency corresponding to the difference
between the energy levels of the ground and excited modes. In the nonresonant
generation, the perturbing field has to be sufficiently strong, however there
is no need to tune the frequencies to resonant conditions.

In the behaviour of coherent modes, there are many similarities with optical
phenomena, such as mode locking, interference patterns, Josephson current,
Rabi oscillations, Ramsey fringes, and the occurrence of higher-order
resonances related to harmonic generation and parametric conversion. Dynamic
transition between mode-locked and mode unlocked regimes is similar to a
critical phase transition in a statistical system.

Creating coherent modes in a system of coupled traps or in a deep optical
lattice allows for the realization of mesoscopic entanglement of atomic clouds
and entanglement production. With coherent modes, one can observe atomic
squeezing

Employing nonresonant generation it is possible to form strongly
nonequilibrium states of Bose condensates with different topological modes
including vortex embryos, vortex rings, vortex lines, and condensate droplets.
Such nonequilibrium states as vortex turbulence, droplet turbulence, and wave
turbulence can be produced. The nonequilibrium states can be classified by
means of effective temperature, Fresnel and Mach numbers. The overall
procedure of perturbing an initially equilibrium condensate to nonequilibrium
states with topological defects, such as vortex embryos, vortex rings, vortex
lines, and condensate droplets is called inverse Kibble-Zurek scenario.

The possibility of producing well collimated atomic beams from atom lasers,
which can be guided in any desired direction is described.

This research did not receive any specific grant from funding agencies in the
public, commercial, or not-for-profit sectors.

## References

  * [1] Anderson M H, Ensher J R, Matthews M R, Wieman C E and Cornell E A 1995 Science 269 198
  * [2] Bradley C C, Sackett C A, Tollett J J and Hulet R G 1995 Phys. Rev. Lett. 75 1687
  * [3] Davis K B, Mewes M O, Andrews M R, van Drutten N J, Durfee D S, Kurn D M and Ketterle W 1995 Phys. Rev. Lett. 75 3969
  * [4] Parkins A S and Walls D F 1998 Phys. Rep. 303 1
  * [5] Dalfovo F, Giorgini S, Pitaevskii L P and Stringari S 1999 Rev. Mod. Phys. 71 463
  * [6] Courteille P W, Bagnato V S and Yukalov V I 2001 Laser Phys. 11 659
  * [7] Andersen J O 2004 Rev. Mod. Phys. 76 599
  * [8] Yukalov V I 2004 Laser Phys. Lett. 1 435
  * [9] Bongs K and Sengstock K 2004 Rep. Prog. Phys. 67 907
  * [10] Yukalov V I and Girardeau M D 2005 Laser Phys. Lett. 2 375
  * [11] Posazhennikova A 2006 Rev. Mod. Phys. 78 1111
  * [12] Yukalov V I 2007 Laser Phys. Lett. 4 632
  * [13] Proukakis N P and Jackson B 2008 J. Phys. B 41 203002
  * [14] Yurovsky V A, Olshanii M, Weiss D S 2008 Adv. At. Mol. Opt. Phys. 55 61
  * [15] Yukalov V I 2011 Phys. Part. Nucl. 42 460
  * [16] Yukalov V I 2016 Laser Phys. 26 062001
  * [17] Morsch O and Oberthaler M 2006 Rev. Mod. Phys. 78 179
  * [18] Moseley C, Fialko O and Ziegler K 2008 Ann. Phys. (Berlin) 17 561
  * [19] Yukalov V I 2009 Laser Phys. 19 1
  * [20] Krutitsky K V 2016 Phys. Rep. 607 1
  * [21] Yukalov V I 2018 Laser Phys. 28 053001
  * [22] Yukalov V I, Yukalova E P and Bagnato V S 1997 Phys. Rev. A 56 4845
  * [23] Yukalov V I 2006 Laser Phys. 16 511
  * [24] Bogolubov N N 1949 Lectures on Quantum Statistics (Kiev: Ryadyanska Shkola)
  * [25] Bogolubov, N N 1967 Lectures on Quantum Statistics (New York: Gordon and Breach) Vol. 1
  * [26] Bogolubov, N N 1970 Lectures on Quantum Statistics (New York: Gordon and Breach) Vol. 2
  * [27] Bogolubov N N 2015 Quantum Statistical Mechanics (Singapore: World Scientific)
  * [28] Gross E P 1957 Phys. Rev. 106 161
  * [29] Gross E P 1958 Ann. Phys. (N.Y.) 4 57
  * [30] Gross E P 1960 Ann. Phys. (N.Y. 9 292
  * [31] Gross E P 1961 Nuovo Cimento 20 454
  * [32] Wu T T 1961 J. Math. Phys. 2 105
  * [33] Pitaevskii L P 1961 J. Exp. Theor. Phys. 13 451
  * [34] Gross E P 1963 J. Math. Phys. 4 195
  * [35] Malomed B 2005 Nonlinear Schrödinger equations, in Encyclopedia of Nonlinear Science ed Scott A (New York: Routledge) p. 639
  * [36] Vinogradov I M 1988 ed Encyclopedia of Mathematics (Dordrecht: Kluwer) Vol. 2 p. 67
  * [37] Calogero F and Degasperis A 1982 Spectral Transform and Solitons (Amsterdam: North-Holland)
  * [38] Faddeev L D and Takhtajan L A 1987 Hamiltonian Methods in the Theory of Solitons (Berlin: Springer)
  * [39] Yukalov V I, Yukalova E P and Bagnato V S 2000 Laser Phys. 10 26
  * [40] Bagnato V S, Yukalova E P and Yukalov V I 2000 in Bose-Einstein Condensates and Atom Lasers ed Martelucci S, Chester A, Aspect A and Inguscio M (New York: Kluwer) p. 201
  * [41] Yukalov V I, Yukalova E P and Bagnato V S 2001 Laser Phys. 11 455
  * [42] Yukalov V I, Yukalova E P and Bagnato V S 2002 Phys. Rev. A 66 043602
  * [43] Klauder J R and Skagerstam B S 1985 Coherent States (Singapore: World Scientific)
  * [44] Serov V V, Derbov V L, Vinitsky S I and Yukalov V I 2002 Proc SPIE 4706 124
  * [45] Yukalov V I, Yukalova E P and Bagnato V S 2002 Laser Phys. 12 1325
  * [46] Yukalov V I, Yukalova E P and Bagnato V S 2002 Phys. Rev. A 66 025602
  * [47] Allen L and Eberly J H 1975 Optical Resonance and Two-Level Atoms New York: Wiley 1975.
  * [48] Yukalov V I, Yukalova E P and Bagnato V S 2003 Laser Phys. 13, 551
  * [49] Yukalov V I, Yukalova E P and Bagnato V S 2003 Laser Phys. 13, 861
  * [50] Mandel L and Wolf E 1995 Optical Coherence and Quantum Optics (Cambridge: Cambridge University Press)
  * [51] Bogolubov N N and Mitropolsky Y A 1961 Asymptotic Methods in the Theory of Nonlinear Oscillations (New York: Gordon and Breach)
  * [52] Murdock J A 1991 Perturbations (New York: Wiley)
  * [53] Yukalov V I and Yukalova E P 2002 J. Phys. A 35 8603
  * [54] Marzlin, K P and Zhang W 1998 Phys. Rev. A 57 3801
  * [55] Marzlin, K P and Zhang W 1998 Phys. Rev. A 57 4761
  * [56] Ostrovskaya E A, Kivshar, Y S, Lisak M, Hall B, Cattani F and Anderson, D 2000 Phys. Rev. A 61 031601
  * [57] Feder D L, Pindzola M S, Collins L A, Schneider B I and Clark C W 2000 Phys. Rev. A 62 053606
  * [58] Kivshar Y S, Alexander T J and Turitsyn S K Phys. Lett. A 278 225
  * [59] Damski B, Karkuszewski Z P, Sacha K, Zakrewski J 2001 Phys. Rev. A 65 013604
  * [60] D’Agosta R, Malomed B A and Precilla C 2002 Laser Phys. 12 37
  * [61] Williams J, Walser R, Cooper J, Cornell E A and Holland M 2000 Phys. Rev. A 61 033612
  * [62] Nepomnyashchy Y A 1974 Theor. Math. Phys. 20 904
  * [63] Yukalov V I 1980 Acta Phys. Pol. A 57 295
  * [64] Yukalov V I and Yukalova E P 2004 Laser Phys. Lett. 1 50
  * [65] Abad M, Recati A, Stringari S and Frédéric C 2015 Eur. Phys. J. D 69 126
  * [66] Yukalov V I 2022 Laser Phys. Lett. 19 103001
  * [67] Chandrasekhar S 1961 Hydrodynamics and Hydromagnetic Stability (Oxford: Clarendon Press)
  * [68] Kul H J 1991 Phys. Rep. 206 197
  * [69] Sasaki K, Suzuki N, Akamatsu D and Saito H 2009 Phys. Rev. A 80 063611
  * [70] Takeuchi H, Suzuki N, Kasamatsu K, Saito H and Tsubota M 2010 Phys. Rev. B 81 094517
  * [71] Ishino S, Tsubota M and Takeuchi H 2011 Phys. Rev.A 83 063602
  * [72] Fujimoto K and Tsubota M 2012 Phys. Rev. A 85 033642
  * [73] Yukalov V I 2019 Phys. Part. Nucl. 50 141
  * [74] Yukalov V I and Yukalova E P 2021 Physics 3 829
  * [75] Yukalov V I, Yukalova E P and Bagnato V S 2001 Proc. SPIE 4243 150
  * [76] Yukalov V I, Yukalova E P and Bagnato V S 2002 Laser Phys. 12 231
  * [77] Ramos E R, Sanz L, Yukalov V I and Bagnato V S 2007 Phys. Rev. A 76 033608
  * [78] Barrett M D, Sauer J A and Chapman M S 2001 Phys. Rev. Lett. 87 010404
  * [79] Ryu C, Andersen M F, Clade P, Natarajan V, Helmerson K and Phillips W D 2007 Phys. Rev. Lett. 99 260401
  * [80] Yukalov V I, Marzlin K P and Yukalova E P 2004 Phys. Rev. A 69 023620
  * [81] Simon B 1970 Helv. Phys. Acta 43 607
  * [82] Yukalov V I, Marzlin K P and Yukalova E P 2004 Laser Phys. 14 565
  * [83] Yukalov V I, Marzlin K P, Yukalova E P and Bagnato V S 2005 AIP Conf. Proc. 770 218
  * [84] Jauregui R, Poli N, Roati G and Modugno G 2001 Phys. Rev. A 64 033403
  * [85] Kumakura M, Shirahata Y, Takasu Y, Takahashi Y and Yabuzaki T 2003 Phys. Rev. A 68 021401
  * [86] Poli N, Brecha R J, Roati G and Modugno G 2002 Phys. Rev. A 65 021401
  * [87] Marzlin K P and Yukalov V I 2005 Eur. Phys. J. D 33 253
  * [88] Alberti A, Ferrari G, Ivanov V V, Chiofalo M L and Tino G M 2010 New J. Phys. 12 065037
  * [89] Parker C V, Ha L C and Chin C 2013 Nature Phys. 9 769
  * [90] Zheng W and Zhai H 2014 Phys. Rev. A 89 061603
  * [91] Zhang S L and Zhou Q 2014 Phys. Rev. A 90 051601
  * [92] Niu L, Hu D, Jin S, Dong X, Chen X and Zhou X 2015 Opt. Express 23 10064
  * [93] Hudomal A, Regnault N and Vasic I 2019 Phys. Rev. A 100 053624
  * [94] Boulier T, Maslek J, Bukov M, Bracamontes C, Magnan E, Lellouch S, Demler E, Goldman N and Porto J 2019 Phys. Rev. X 2019 9 011047
  * [95] Zheng N, Liu W, Wu J, Li Y, Sovkov V and Ma J 2022 Photonics 9 442
  * [96] Cronin A D, Schmiedmayer J and Pritchard, D E 2009 Rev. Mod. Phys. 81 1051
  * [97] Milburn G J, Corney J, Wright E M and Walls D F 1997 Phys. Rev. A 55 4318
  * [98] Raghavan S, Smerzi A, Fantoni S and Shenoy S R, 1999 Phys. Rev. A 59 620
  * [99] Leggett A J 1975 Rev. Mod. Phys. 47 331
  * [100] Ohberg P and Stenholm S 1999 Phys. Rev. A 59 3890
  * [101] Davis M J and Heller E J 1981 J. Chem. Phys. 75 246
  * [102] Heller E J and Davis M J 1981 J. Phys. Chem. 85 307
  * [103] Rabi I 1937 Phys. Rep. 51 652
  * [104] Yukalov V I 1998 Phys. At. Nucl. 61 1882
  * [105] Yukalov V I and Yukalova E P 2000 Phys. Part. Nucl. 31 561
  * [106] Ramsey N F 1990 Rev. Mod. Phys. 62 541
  * [107] Ramos E R F, Sanz L, Yukalov V I and Bagnato V S 2007 Phys. Lett. A 365 126
  * [108] Ramos E R F, Sanz L, Yukalov V I and Bagnato V S 2007 Nucl. Phys. A 790 776
  * [109] Eschmann A, Ballagh R J and Caradoc-Davies B M 1999 J. Opt. B: Quantum Semiclass. Opt. 1 383
  * [110] Döring D, Debs J E, Robins N P, Figl C, Altin P A and Close J D 2009 Opt. Express 17 20661
  * [111] Kokkelmans S J and Holland M J 2002 Phys. Rev. Lett. 89 180401
  * [112] Timmermans E, Tommasini P, Hussein M and Kerman A 1999 Phys. Rep. 315 199
  * [113] Yukalov V I and Yukalova E P 2005 J. Low Temp. Phys. 138 657
  * [114] Ramos E R, Henn E A, Seman J A, Caracanhas M A, Magalhães K M, Helmerson K, Yukalov V I and Bagnato V S 2008 Phys. Rev. A 78 063412
  * [115] Yukalov V I and Bagnato V S 2009 Laser Phys. Lett. 6 399
  * [116] Nguyen J H V, Tsatsos M C, Luo D, Lode A U J, Telles G D, Bagnato V S and Hulet R G, 2019 Phys. Rev. X 9 011052
  * [117] Shukuno N, Sano Y and Tsubota M 2023 J. Phys. Soc. Jap. 92 064602
  * [118] Yukalov V I 2005 Phys. Rev. E 72 066119
  * [119] Yualov V I 2006 Laser Phys. Lett. 3 406
  * [120] Yukalov V I 2006 Phys. Lett. A 359 712
  * [121] Yukalov V I 2008 Ann. Phys. (N.Y.) 323 461
  * [122] Yukalov V I, Yukalova E P and Sornette D 2013 Laser Phys. Lett. 10 115502
  * [123] Yukalov V I, Yukalova E P and Sornette D 2014 J. Phys. Conf. Ser. 497 012034
  * [124] Poulsen U V and Mølmer K 2001 Phys. Rev. A 64 013616
  * [125] Sørensen A, Duan L M, Cirac J I and Zoller P 2001 Nature 409 63
  * [126] Orzel C, Tuchman A K, Fenselau M L, Yasuda M and Kasevich M A 2001 Science 291 2387
  * [127] Wineland D J, Bollinger J J, Itano W M and Heinsen D J 1994 Phys. Rev. A 50 67
  * [128] Bouyer P and Kasevich M A 1997 Phys. Rev. A 56 1083
  * [129] Hadzibabic Z, Stock S, Battelier B, Bretin V and Dalibard J 2004 Phys. Rev. Lett. 93 180403
  * [130] Cennini G, Geckeler C, Ritt G and Weitz M 2005 Phys. Rev. A 72 051601
  * [131] Yukalov V I 2013 Laser Phys. 23 062001
  * [132] Yukalov V I and Yukalova E P 2006 Laser Phys. 16 354
  * [133] Keyl M 2002 Phys. Rep. 369 431
  * [134] Yukalov V I 2003 Phys. Rev. Lett. 90 167905
  * [135] Yukalov V I 2003 Phys. Rev. A 68 022109
  * [136] Yukalov V I 2003 Mod. Phys. Lett. B 17 95
  * [137] Marcus M and Moyls B N 1959 Pacif. J. Math. 9 1215
  * [138] Westwick R 1967 Pacif. J. Math. 23 613
  * [139] Beasley L 1988 Linear Algebra Appl. 107 161
  * [140] Alfsen E and Shultz F 2010 J. Math. Phys. 51 052201
  * [141] Johnston N 2011 Linear Multilinear Algebra 59 1171
  * [142] Friedland S, Li C K, Poon Y T and Sze N S 2011 J. Math. Phys. 52 042203
  * [143] Gohberg J and Goldberg S 1987 J. Math. Anal. Appl. 125 124
  * [144] Fan H Y 2001 Mod. Phys. Lett. B 15 1475
  * [145] Yukalov V I and Yukalova E P 2006 Phys. Rev. A 73 022335
  * [146] Yukalov V I and Yukalova E P 2008 J. Phys. Conf. Ser. 104 012003
  * [147] Yukalov V I and Sornette D 2010 Phys. At. Nucl. 73 559
  * [148] Yukalov V I, Yukalova E P and Yurovsky V A 2019 Laser Phys. 29 065502
  * [149] Yukalov V I 2020 Entropy 22 565
  * [150] Henn E A L, Seman J A, Ramos E R F, Caracanhas M, Castilho P, Olimpio E P, Roati G, Magalhães D V, Magalhães K M F and Bagnato V S 2009 Phys. Rev. A 79 043618
  * [151] Seman J A, Henn E A L, Haque M, Shiozaki R F, Ramos E R F, Caracanhas M, Castilho P, Castelo Branco C, Tavares P E S, Poveda-Cuevas F J, Roati G, Magalhães K M F and Bagnato V S 2010 Phys. Rev. A 82 033616
  * [152] Henn E A L, Seman J A, Roati G, Magalhães K M F and Bagnato V S 2009 Phys. Rev. Lett. 103 045301
  * [153] Shiozaki R F, Telles G D, Yukalov V I and Bagnato V S 2011 Laser Phys. Lett. 8 393
  * [154] Seman J A, Henn E A L, Shiozaki R F, Roati G, Poveda-Cuevas F J, Magalhães K M F, Yukalov V I, Tsubota M, Kobayashi M, Kasamatsu K and Bagnato V S 2011 Laser Phys. Lett. 8 691
  * [155] Bagnato V S and Yukalov V I 2013 Prog. Opt. Sci. Photon. 1 377
  * [156] Yukalov V I, Novikov A N and Bagnato V S 2014 Laser Phys. Lett. 11 095501
  * [157] Yukalov V I, Novikov A N and Bagnato V S 2015 J. Low Temp. Phys. 180 53
  * [158] Yukalov V I, Novikov A N and Bagnato V S 2015 Phys. Lett. A 379 1366
  * [159] Novikov A N, Yukalov V I and Bagnato V S 2015 J Phys. Conf. Ser. 594 012040
  * [160] Yukalov V I, Novikov A N, Yukalova E P and Bagnato V S 2016 J. Phys. Conf. Ser. 691 012019
  * [161] Yukalov V I , Novikov A N and Bagnato V S 2018 Laser Phys. Lett. 15 065501
  * [162] Yukalov V I, Novikov A N, Yukalova E P and Bagnato V S 2020 J. Phys. Conf. Ser. 1508 012006
  * [163] Iordanskii S V 1965 J. Exp. Theor. Phys. 21 467
  * [164] Amit D and Gross E P 1966 Phys. Rev. 145 130
  * [165] Roberts P H and Grant J 1971 J. Phys. A 4 55
  * [166] Jones C A and Roberts P H 1982 J. Phys. A 15 2599
  * [167] Barenghi C F and Donnelly R J 2009 Fluid Dyn. Res. 41 051401
  * [168] Belyaev S T 2016 J. Exp. Theor. Phys. 149 1107