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Spatial and temporal
structures in cavities with
oscillating boundaries
Nikolay N. Rosanov1,2,3 , George B. Sochilin1 ,
rsta.royalsocietypublishing.org
Vera D. Vinokurova1 and Nina V. Vysotina1
1 Theoretical Department, Vavilov State Optical Institute,
Review
Cite this article: Rosanov NN, Sochilin GB,
Vinokurova VD, Vysotina NV. 2014 Spatial and
temporal structures in cavities with oscillating
boundaries. Phil. Trans. R. Soc. A 372: 20140012.
http://dx.doi.org/10.1098/rsta.2014.0012
One contribution of 19 to a Theme Issue
‘Localized structures in dissipative media:
from optics to plant ecology’.
Subject Areas:
quantum physics
Keywords:
dynamical trap, dynamical billiard, quantum
particles, Bose–Einstein condensate,
cavity solitons
Author for correspondence:
Nikolay N. Rosanov
e-mail: [email protected]
St Petersburg 199053, Russia
2 Laser Optics, ITMO University, St Petersburg 197101, Russia
3 Laboratory of Atomic Radiospectroscopy, Ioffe Physical Technical
Institute, St Petersburg 194021, Russia
We review the general features of particles, waves and
solitons in dynamical cavities formed by oscillating
cavity mirrors. Considered are the dynamics of
classical particles in one-dimensional geometry of a
dynamical billiard, taking into account the non-elastic
collisions of particles with mirrors, the (quasi-energy)
states of a single quantum particle in a potential
well with periodically oscillating wells, and nonlinear
structures, including nonlinear Rabi oscillations,
cavity optical solitons and solitons of Bose–Einstein
condensates, in dynamical cavities or traps.
1. Introduction
To date, such essentially nonlinear phenomena as
solitons have been better studied in optics [1] owing to
the availability of high-power laser radiation and the
relative simplicity of nonlinear optical schemes. These
solitons are of two types: (i) conservative solitons [2],
which represent the balance between linear spreading
of wave packets and their nonlinear focusing, and
(ii) dissipative solitons, which result from the balance
between energy input and output in the region of
localization [3–6].
The main schemes that support spatial optical dissipative solitons—the cavity solitons—are shown in
figure 1a,b. In high-quality cavities with multiple
radiation trips, it is possible to achieve a high
concentration of radiation energy and, correspondingly,
high medium nonlinearity under resonance conditions.
2014 The Author(s) Published by the Royal Society. All rights reserved.
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(a)
(b)
(c)
2
M
M
pump
M
M
M
Figure 1. Three variants of excitation of two-mirror (M) cavity schemes. (a) Nonlinear interferometer driven by coherent holding
radiation (in) through a partially transparent mirror. The medium inside the cavity can be passive (without laser gain). (b) Laser
scheme with coherent or incoherent pump providing laser gain; mirrors can be non-transparent. (c) Scheme without coherent
holding radiation and pump; excitation is provided by oscillations of cavity mirrors that can be non-transparent. (Online version
in colour.)
For the scheme of a driven nonlinear interferometer (figure 1a), the medium inside the cavity can
be passive (without gain). Energy gain is due to external coherent holding radiation transmitted
by the partially transparent mirror. Resonances occur when the holding radiation frequency is
close to the cavity eigenfrequencies. In laser schemes (figure 1b), the external holding radiation
is not necessary, and energy supply is due to coherent or incoherent pumping resulting in
intracavity medium laser gain. The cavity mirrors can be non-transparent. More recently, owing
to progress in the formation of such macroscopic quantum objects as Bose–Einstein condensates
(BECs) [7], similar schemes have been studied for solitons of combined light–matter (polariton)
waves [8,9].
One difficulty for realization of cavity solitons of pure matter waves is the absence of effective
semitransparent mirrors. This can be resolved using cavities with oscillating mirrors, as in
figure 1c. In this case, the energy gain for the field inside the cavity is due to the kinetic energy
of the mirrors, which can be non-transparent. Similar schemes are known for electromagnetic
radiation [10,11], including current optomechanics [12,13], as well as for nonlinear acoustics [14].
The analysis of nonlinear structures in the scheme of such a dynamical billiard needs clarification
of the general nature of its nonlinear dynamics. At this point, it is useful to revisit the problem of
Fermi [15] with stochastic acceleration of particles colliding with moving bodies and that of Ulam
[16] with particles bouncing between one motionless and one oscillating wall.
Below we review the following aspects of the problem. In §2, we present the nonlinear
dynamics of a classical particle in a one-dimensional dynamical billiard, revealing stationary, with
conserved particle kinetic energy, and quasi-chaotic regimes. Compared are the dynamics with
elastic and inelastic collisions of particles with the walls. In §3, studied are states, mainly quasienergetic ones, of a single quantum particle in a potential well with periodically oscillating walls.
Section 4 is devoted to nonlinear structures, including nonlinear Rabi oscillations and cavity
solitons (§4a) and ‘longitudinal’ (§4b) and ‘transverse’ solitons in dynamical traps. A general
conclusion is given in §5.
2. Classical particles in a dynamical billiard
Let us consider the one-dimensional motion of a classical particle in a ‘dynamical billiard’ or
cavity formed by two barriers: a motionless wall, coordinate z = 0, and an oscillating wall,
coordinate z = zw = L(t), where t is time (in the scheme of figure 1c, the left mirror is motionless
and the right mirror oscillates). For the sake of definiteness, we fix
zw (t) = L0 (1 + μ cos Ωt).
(2.1)
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M
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in
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and
2π 2 N2
4π 2 N2 μ
⎪
δvn + 1 −
δtn+1 = −
δtn .⎪
⎭
L0 (1 + μ)
1+μ
(2.2)
The eigenvalues λ of the corresponding transformation matrix are determined by the following
equation:
1−λ
2L0 μ
(2.3)
2
2
2
2
4π N μ = 0.
− 2π N
1
−
λ
−
L0 (1 + μ)
1+μ
Solutions to this quadratic equation are
a
λ1,2 = 1 − ±
2
a2
− a,
4
with a =
4π 2 N2 μ
.
1+μ
(2.4)
The condition of asymptotic stability |λ1,2 |2 < 1 cannot be fulfilled because λ1 λ2 = 1.
For μ < 0, the value a is negative, the eigenvalues are real and not coincident and the maximum
eigenvalue λ1 > 0. This means aperiodic instability.
.........................................................
⎫
⎪
⎪
⎬
δvn+1 = δvn + (2L0 μ)δtn
3
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The modulation depth μ is assumed to be small, μ2 1. Introducing dimensionless time Ωt → t,
one can set Ω = 1. This problem was considered first by Ulam [16] in connection with the Fermi
acceleration effect [15]; subsequent results are reviewed by Lichtenberg & Lieberman [17] and by
Loskutov [18]; a recent study of the Ulam problem with stochastic wall motion was performed by
Gelfreich et al. [19]. Note that Ulam [16] considered a sawtooth modulation of the wall’s position,
and the prevailing approximation in subsequent papers is neglecting shifts of the wall’s position
when calculating the time of collisions. These two assumptions do not allow one to describe
properly the stability of periodic and quasi-periodic regimes of particle reflections from the walls.
For the present review, it is important also to compare the dynamics of classical particles for elastic
and inelastic collisions with the walls [20] and that of solitons in similar schemes [21].
According to equation (2.1), the extrema of the oscillating wall’s position correspond to the
coordinates z0 = zw (0) = L0 (1 + μ) (maxima for μ > 0 and minima for μ < 0). First, let collisions
of the particle with the walls be elastic. Then, if the velocity of the particle colliding with an
oscillating wall at the moment t is v, the velocity of the reflected particle is v = −v + 2żw (t)
(the dot over a value denotes its temporal derivative). Correspondingly, depending on the
instantaneous wall velocity, the kinetic energy of a particle with v > 0 can increase (żw (t) < 0)
or decrease (żw (t) > 0) due to the collision, and the system considered is open. The dynamics of
the particle can be described by recurrence relations for times of collisions tn and corresponding
velocities vn , n = 1, 2, 3, . . . [20].
Fixed points of the system of governing equations correspond to regimes with conserved
particle kinetic energy. They are possible if all collisions are with an (instantaneously) motionless
wall (at time moments t = π N, N = 1, 2, . . .). For the first type of fixed points, the collisions occur
at time moments t = 2π N, N = 1, 2, . . . and the particle velocity is v0,N = z0 /π N = L0 (1 + μ)/π N.
This case includes variants of collisions at minimum (μ < 0) and maximum (μ > 0) cavity length.
The second type of fixed points corresponds to particles with velocity v0,M = L0 /π (M − 1/2),
M = 1, 2, . . ., reflecting alternately from the oscillating wall at its maximum and minimum
deviations. The degenerate case corresponds to motionless particles (v = 0) located in the interval
0 < z < L0 (1 − |μ|).
To perform the linear stability analysis of the periodic regimes, let us introduce small
deviations of the velocity δvn = vn − v0,N and collision times δtn = tn − 2π N, n = 1, 2, . . ., from
corresponding unperturbed values. For the first type of fixed points, in the linear over δvn and
δtn approximation, one gets the following recurrence relations:
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(a)
(b)
0.3218
0.580
0.3216
0.575
0.3214
0.570
0.3212
0.565
4
v
0.3210
0.560
0
200
400
600
800
1000
(c)
(d )
0
100
200
300
400
0
400
800
n
1200
1600
2.0
0.20
1.6
0.16
1.2
0.12
v
0.08
0.8
0.04
0.4
0
0
0
400
800
1200
1600
2000
n
Figure 2. The dynamics of the velocity of the particle in a dynamical billiard; μ = 0.01 (a–c) and 0.3 (d). Regime close to the
periodic one of (a) the first (N = 1) and (b) the second (M = 1) type. Examples of chaotic variation of particle velocity: (c) initial
velocity v1 = 0.01 and (d) overcritical modulation depth.
For μ > 0, the value a > 0. If 0 < a < 4, then the radicand in equation (2.4) is negative, the two
eigenvalues are complex conjugates, and
|λ1,2 |2 = 1,
λ1,2 = e±iν
and
tan ν =
4a − a2
.
2−a
(2.5)
This means neutral stability. In this case, the linear analysis describes also long-term particle
dynamics if the initial deviations from the unperturbed values are small,
δvn = w cos(νn + ϕ).
(2.6)
Here the (small) amplitude w and phase ϕ are arbitrary.
For a > 4, both eigenvalues are real, and one of them λ1 < −1. Then we have oscillatory
instability. The stability boundary is determined by the condition acr = 4, therefore
μcr,N =
1
.
π 2 N2 − 1
Then μcr,1 = 0.113, μcr,2 = 0.026, μcr,3 = 0.011 and μcr,4 = 0.007.
(2.7)
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For quantum particles in a trap, essential are their wave nature and their energy spectrum
discreteness. If a classical particle can be motionless with an arbitrary position between oscillating
walls, a quantum particle cannot be at rest, and its wave function should oscillate because of
the walls’ oscillations. Additionally, owing to the wave features and corresponding ‘dispersion’
and ‘diffraction’, the wave packet of an initially localized quantum particle diffuses with time.
This diffusion can be compensated if we consider, not a single particle, but a large number of
interacting particles under critical temperature—the BEC [7]; however, the BEC is the subject
of §4. The diffusion and motion of atomic wave packets in a trap with oscillating walls were
studied by Steane et al. [23] and Saif et al. [24]. Here, we are interested in a different, purely
quantum, aspect connected with the discreteness of the quantum object’s energy spectrum. Below
we review the results of our paper [25] on quasi-energy states of a single quantum particle in a
dynamical trap.
The wave function ψ of a single quantum particle in a one-dimensional trap obeys the
Schrödinger equation,
ih̄
h̄2 ∂ 2 ψ
∂ψ
+ U(z, t)ψ,
=−
∂t
2mp ∂z2
(3.1)
with the coordinate z, time t, the reduced Planck constant h̄, the particle mass mp and the trap
potential U. For an infinite potential well with oscillating barriers, this equation is applied for
.........................................................
3. A single quantum particle in a dynamical trap
5
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The direct simulation of the dynamics of a particle colliding with oscillating and motionless
walls confirms these conclusions. For μ = 0.01, there are three stable fixed points of the first
type (N = 1, 2 and 3). Their neutral stability means that if deviations from the unperturbed
values are initially sufficiently small, then they remain small during the next evolution. This is
illustrated by figure 2a (N = 1) where one can see quasi-periodic variations of the particle velocity
v (also a dimensionless value, because we fix L0 = 1). The simulations agree very well with the
analytic expression (equation (2.6)). In figure 2b is presented the quasi-periodic dynamics near the
fixed point of the second type (M = 1). The modulation depth depends on the initial deviations.
However, if the initial deviations are fairly large, the dynamics becomes chaotic (figure 2c). This
figure shows also that, if the particle initial velocity is small, the mean value of the particle
energy increases with time. For overcritical values μ > μcr,1 , there are no stable fixed points,
and the dynamics is chaotic (figure 2d). Detailed characterization of deterministic chaos regimes
[22] needs special consideration. In the limit of large initial velocity of the particle, there are
quasi-periodic variations of the velocity near its initial value.
Now, let the collisions of the particles with the oscillating wall be not exactly elastic,
with the velocities of the incident v and reflected v particle connected by the relation
v − żw (t) = −q[v − żw (t)]. The value 1 − q2 is the measure of dissipation (inelasticity); the
previous purely elastic case corresponds to q = 1. In this case, there are also stable periodic
regimes, and now they are asymptotically stable (stable attractors, small initial deviations decay
with time) (figure 3a,b). For large initial deviations, chaotic dynamics takes place, as in the
previous case (figure 3c).
The results presented in this section indicate that classical particles in the dynamical billiard
with elastic collisions have dynamics intermediate between conservative and dissipative ones.
The neutral, in contrast to asymptotic, stability of fixed points is a feature of conservative systems.
On the other hand, the increase of average energy for evolution with small initial energy is
characteristic for dissipative systems. The dynamics of the particle resembles the dynamics of
conservative systems [22], though the particle energy is not conserved after collisions. This could
be explained by the fact that this energy can both increase and decrease due to collisions, and the
energy is conserved at the average over the period of wall position modulation. For non-elastic
collisions, we have asymptotic stability of periodic regimes, but chaotic dynamics takes place
again for large deviations of the initial values from the steady-state values.
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(a)
(b)
0.36
6
1.08
Dt/2p
1.00
0.32
0.96
0.30
0.92
0
400
800
n
1200
1600
400
0
800
n
1200
1600
(c) 0.04
0.03
v
0.02
0.01
0
0
2000
4000
n
6000
8000
Figure 3. Dynamics of a particle in a billiard with inelastic collisions, q = 0.99. Establishment of (a) velocity vn and (b) period
t = (tn − tn−1 )/(2π ) for the periodic regime with period 2π; μ = 0.02, v0 = 0.3. (c) Chaotic dynamics for μ = 0.001
and v0 = 0.02.
Lleft (t) < z < Lright (t), where U = 0, and the boundary conditions are
ψ(z = Lleft (t), t) = 0
and
ψ(z = Lright (t), t) = 0.
(3.2)
In the case of motionless walls (Lleft = 0, Lright = L0 = const., modulation depth μ = 0), solutions
to equations (3.1) and (3.2) are represented by the discrete energy spectrum
(0)
ψn (z, t) =
(0)
En =
2
exp
L0
(0)
En
(0)
t sin(kn z),
−i
h̄
h̄2 (0)2
π 2 h̄2 2
kn =
n ,
2mp
2mp L20
(0)
kn =
πn
L0
and
n = 1, 2, 3, . . . .
(3.3)
For periodic (harmonic) oscillations of barriers with the same period, T = 2π/Ω, for the left and
right barriers, there is a set of states with definite quasi-energy ε,
ψε (z, t) = uε (z, t)e−i(ε/h̄)t
and uε (z, t + T) = uε (z, t).
(3.4)
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0.34
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Periodic in time functions uε (x, t) can be decomposed into Fourier series,
7
(3.5)
l=−∞
Then functions χε,l (z) obey the ordinary differential equations
h̄2 d2 χε,l
= −(ε + h̄Ωl)χε,l ,
2mp dz2
(3.6)
with the boundary conditions (3.2). Owing to problem linearity, the general solution is given by a
linear superposition of partial solutions corresponding to states with different quasi-energies.
For small modulation depth, the quasi-energy states can be found by perturbation theory, and
the lowest-order solution is given by equations (3.3). Owing to the non-parabolic (rectangular)
(0)
shape of the trap potential, the energy spectrum is highly non-equidistant, En ∼ n2 . Therefore,
if the modulation frequency is close to the frequency of transition between the two levels n
and m,
(0)
(0)
h̄Ω = Em − En + h̄δΩ,
|δΩ|/Ω 1,
(3.7)
then only these two levels are subjected to the excitation due to modulation, and the scheme is
reduced to the two-level one [26,27]. The amplitudes of the resonance states an and am obey the
following equations:
dan
(0)
+ (−1)m−n μnmEm am = 0
dt
dam
(0)
+ (−1)m−n μnmEm an + h̄δΩam = 0.
ih̄
dt
ih̄
and
⎫
⎪
⎪
⎬
⎪
⎪
⎭
(3.8)
For pure quasi-energy states, the temporal dependence of the amplitudes is an,m ∼ e−i(δε/h̄)t .
Under the resonance conditions (equation (3.7)), there are splitting of quasi-energies and Rabi
oscillations with periodic exchange of the resonance levels’ populations. An additional feature
that is beyond the two-level approximation is the possibility of resonances of higher orders
corresponding to ‘multiphoton’ transitions. More detail can be found in [25,28].
4. The Bose–Einstein condensate in a dynamical trap
The BEC represents a macroscopic object that can be characterized by a single wave function ψ
obeying the nonlinear Gross–Pitaevskii equation [7]:
ih̄
h̄2
∂ψ
=−
ψ + U0 |ψ|2 ψ
∂t
2mp
and =
∂2
∂2
∂2
+
+
.
∂x2
∂y2
∂z2
(4.1)
The boundary conditions to equation (4.1) have the form of equations (3.2) in our case.
This equation is valid for weakly non-ideal atomic gases at sufficiently low temperature. The
nonlinearity parameter U0 depends on the external magnetic field and can be either positive or
negative. A similar mean-field equation for exciton or polariton condensates in semiconductors
is known as the Keldysh equation [29].
(a) Two-level scheme and nonlinear Rabi oscillations
For the resonance conditions (equation (3.7)), and neglecting transverse effects, the consideration
can also be reduced to the two-level scheme [25,28]. In this case, the governing equations for the
.........................................................
al χε,l (z)e−ilΩt .
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20140012
uε (z, t) =
∞
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(a)
(d)
A
B
0
1.0
C
0
0.6
F
4
8
12
16
–0.4
20
(f)
4
15
10
F
5
t
1
2
t
3
4
5
F
0
4.0
1.0
0.9
0.8
3
1.0
0.2
0
t
0.6
X
1
B
0.4
1.0
An
0
An 0.6
–0.2
0
D
E
0.8
0.2
An
0.4
(e)
0.4
0.8
(c)
X
1
.........................................................
b
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(b)
b
c
c D
C
8
A
3.5
F
An 0.8
3.0
0.7
t
0.4
0
2
4
t
6
8
2
10
0.6
0
1
2
t
3
4
5
2.5
Figure 4. (a) Phase plane for zero detuning. The circle A, dotted (blue) curve, with the centre 0 and radius 1 is divided into
two cells by separatrix D, broken (red) curve. Each of the two cells contains a fixed point, B in the right cell and C in the
left cell. Through any point inside each of the cells passes one trajectory, a closed curve wrapped around the corresponding
fixed point. Solid (black) curves with arrows b and c are examples of these trajectories; arrows show the direction of time
evolution. (b, c) Temporal dependence of amplitude An (t), solid (red) curve, and phase difference Φ(t), dotted (blue) curve, for
trajectories b and c in figure 4a. (d–f ) The same as in figure 4(a–c) for the case of non-zero frequency detuning. (Online version
in colour.)
resonance states’ amplitudes have the following form:
dan
(0)
+ (−1)m−n μnmE1 am − U0 34 |an |2 + |am |2 an = 0
dt
(0)
dam
ih̄ dt + (−1)m−n μnmE1 an + h̄δΩ − U0 34 |am |2 + |an |2 am = 0.
ih̄
and
⎫
⎪
⎬
⎪
⎭
(4.2)
In the linear case, U0 = 0, they coincide with equations (3.8). It is possible to solve equations (4.2)
analytically [28]. The main results are illustrated by figure 4a–c (exact resonance) and d–f (nonzero frequency detuning). The real amplitudes of the resonance states, An and Am , are periodic
functions of time, with the ‘Rabi period’ depending on the initial conditions (cf. figure 4b with 4c,
and also figure 4e with 4f ). As for the phase difference of the resonance states’ amplitudes Φ, it
can be either a periodic (figure 4b,c,f ) or a monotonic (figure 4e) function of time.
It is convenient to analyse the solutions to equations (4.2) with the help of the phase plane
of this system where the solutions are represented by closed lines (An , Φ) and to treat An as the
polar radius and Φ as the polar angle of the phase plane. For normalization, the amplitudes of the
resonance levels are connected by the relation A2n + A2m = 1; therefore, the state is characterized
fully by the values An and Φ, neglecting an inessential constant shift of the phase. Trajectories
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In this section, we follow [21]; we do not use here the resonance approximation. In onedimensional geometry, equation (4.1) takes the form
ih̄
h̄2 ∂ 2 ψ
∂ψ
=−
+ U0 |ψ|2 ψ.
∂t
2mp ∂z2
(4.3)
For U0 = 0, it coincides with the Schrödinger equation (equation (3.1)). Mathematically,
equation (4.3) is the well-known nonlinear Schrödinger equation [2]. In infinite space, without
a trap, known are solutions to equation (4.3) describing modulational instability, cnoidal waves,
bright and dark solitons, and oscillating localized structures—breathers. For a dynamical trap
with finite length (equations (3.2)), it is possible to find quasi-energy states, as in §3 [25]; however,
owing to the problem nonlinearity, the superposition principle is not applicable in this case.
When a moving bright soliton collides with an ideal motionless mirror, its kinetic energy does
not change. In fact, one can replace this problem by the collision of a soliton with its antiphase
mirror image, the summed field at the mirror location being zero; this problem is solved by the
inverse scattering transform method [2]. If the mirror moves with some constant velocity, the
problem is reduced to the previous one using Galilean transformation symmetry; depending
on the sign of the mirror velocity, the soliton can be accelerated or decelerated. For periodic
oscillations of mirrors, increase or decrease of soliton kinetic energy depends periodically on the
oscillation phase at the moment of collision, as for point classical particles (we consider below the
case of narrow solitons with dimensionless width w 1 and oscillations with small frequency
Ω w−2 ). Additionally, as well as for classical particles, sufficiently slow solitons can collide
with the same mirror several times repeatedly before they move to the cavity centre. Results of
.........................................................
(b) ‘Longitudinal’ solitons
9
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pass through each point of the phase plane inside the circle with radius An = 1. At An = 1 (only
the nth level is occupied) and An = 0 (only the mth level is occupied), there are singularities of
corresponding equations.
Fixed points of the phase plane can be found when equalizing to zero the derivatives in
equations (4.2). They correspond to the quasi-energy states. In the case of exact resonance, one
can see from figure 4a that the phase plane is divided by separatrix D into two cells. In each
cell, trajectories are closed lines disposed concentrically around the corresponding fixed point:
B in the right cell and C in the left cell. The trajectories correspond to periodic oscillations with
time of both amplitude A0 (t) and phase difference Φ(t), as shown in figure 4b,c. The period of the
oscillations (the Rabi period) depends on the initial conditions.
For non-zero detuning, the phase plane has a more complicated structure. In figure 4d–f , shown
are results for fairly large detuning δω = 1.5. Now separatrix D does not include the coordinate
origin 0, and separatrix E appears that passes through 0. The temporal dependence of the phase
difference, Φ(t), can be of two types, depending on the initial conditions: (i) periodic, as in the
previous case, and (ii) monotonic, which can be decomposed into a sum of a periodic function
and a component linear in time. The phase plane is divided into three cells (figure 4d). The left
cell—a ‘half-moon’—is bounded by the left semicircle and the separatrix D. It is of the same type
as in the previous case, i.e. it consists of closed trajectories wrapping around the fixed point C;
for trajectories in this cell, both amplitude An and phase difference Φ vary periodically with time
(the first type of trajectories). The same are features of trajectories inside the second separatrix E.
However, in the cell bounded by separatrices D and E and the right semicircle C, trajectories wrap
on the beginning of coordinates 0 and correspond to periodic temporal variation of amplitude
An and monotonic variation of phase difference Φ (the second type of trajectories). These two
types of trajectories are illustrated in figure 4e,f . They are similar to trajectories of a classical
pendulum with angle periodic variation for small initial velocities and monotonic variation
for large velocities. Beyond the two-level approximation, the direct numerical solution of the
one-dimensional Gross–Pitaevskii equation (equation (4.1)) reveals some additional phenomena
studied in [28].
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0.2
10
2
0
3
–0.1
–0.2
0
0.4
0.8
1.2
1.6
2.0
dt
Figure 5. Dependence of the difference of velocities δV = Vi − |Vr | on mirror oscillation phase δt; Vi,r are velocities of the
incident and reflected soliton, μ = 0.1, Vi = 1 (curve 1, black), Vi = 0.4 (curve 2, red) and 0.04 (curve 3, blue). (Online version
in colour.)
0.52
3
0.51
2
z 0.50
1
0.49
0.48
0
100
200
300
400
500
t
Figure 6. Temporal dependence of the bright soliton position for its central initial position and initial velocity V = 0 (curve 1,
red), V = 0.0001 (curve 2, blue) and V = 0.0008 (curve 3, black). (Online version in colour.)
simulations of soliton collisions with a single oscillating mirror are illustrated in figure 5. For
large initial velocity, Vi = 1, the dependence is very close to sinusoidal. With decrease of the
initial velocity, this dependence deforms. The dip near δt = 0.5 for Vi = 0.04 is because, for these
conditions, the soliton collides with the mirror not a single time, but twice before it moves away
from the mirror.
Now let us consider the case of a two-mirror cavity. Note that classical particles can be at rest
in any place inside the cavity not available for oscillating mirrors, and for any non-zero initial
velocity they travel along the whole cavity. Opposite are features of bright solitons: even with
motionless mirrors, a soliton’s position necessarily oscillates, if it is not disposed in the cavity
centre with zero velocity; generally, a soliton with small initial velocity oscillates in the vicinity
of the cavity centre (figure 6). This is due to the interaction of the soliton’s tails with the mirrors
even when the soliton width is much less than the cavity length.
Simulations show that, in traps with oscillating barriers, the solitons survive even after a large
number of reflections from barriers. Similar to the classical case, there are stable periodic and
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1
0.1
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(a)
11
100
200
300
400
(b)
0.8
0.4
V 0
–0.4
–0.8
0
50
100
300
350
400
t
Figure 7. (a) Periodic and (b) chaotic temporal dependence of dimensionless velocity V on dimensionless time t of a bright
soliton bouncing between two oscillating mirrors. The velocity is constant when the soliton is sufficiently far from the mirrors.
quasi-periodic regimes of soliton reflections from oscillating mirrors (figure 7a). The velocity
modulation depth depends on the soliton initial velocity, as well as for elastic reflections of
classical particles. It is interesting that, for small initial velocity, the soliton dynamics is chaotic
(figure 7b), as well as for a classical particle (see figure 2c,d).
(c) ‘Transverse’ solitons
Generalization of the resonance approach governing equations (equations (4.2)) to the (2 + 1)Dgeometry (two transverse coordinates and time) gives [30]
⎫
⎪
h̄2
∂
(0)
⎪
m−n
2
2
3
⎪
⊥ an + (−1)
μnmE1 am − U0 4 |an | + |am | an = 0
ih̄ +
⎪
⎪
∂t 2mp
⎪
⎬
(4.4)
⎪
⎪
⎪
h̄2
∂
⎪
(0)
m−n
2
2
3
⎪
am = 0.⎪
⊥ am + (−1)
μnmE1 an + h̄δΩ − U0 4 |am | + |an |
and
ih̄ +
⎭
∂t 2mp
Here, ⊥ = ∂ 2 /∂x2 + ∂ 2 /∂y2 is the transverse Laplacian, and x and y are the transverse
coordinates. According to equations (4.4), the total number of particles is conserved for localized
structures,
dr⊥ (|an |2 + |am |2 ) = const.
(4.5)
Next, equations (4.4) have the Galilean symmetry: if functions An,m (x, y, t) give a solution to
equations (4.4), then there is a family of solutions with an arbitrary transverse velocity V,
V2
V
an,m = exp i x − i t An,m (x − Vt, y, t).
(4.6)
2
4
Evidently, equations (4.4) are invariant to a phase shift of both amplitudes, an,m → an,m eiδΦ,
δΦ = const., and to shifts of transverse coordinates, (x, y) → (x + δx, y + δy). For exact resonance,
δΩ = 0, equations (4.4) are also invariant to the replacement n → m.
In the case of exact resonance, there are solutions of equations (4.4) with equal populations of
the two resonance levels, am = ±an ≡ a. Then, after replacement a = b exp[±i(−1)m−n t] and using
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0
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0.6
0.3
V 0
–0.3
–0.6
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(a)
80
(b)
(c)
2
20
max( an 2)
x 40
0.6
0.4
2
0.2
0
0.4
0.2
0
0
100
200
1
2
0
300
0
100
t
200
300
0
100
t
t
200
Figure 8. Dynamics of the collision of two vector solitons. Soliton 1 (curves 1, red) is initially a motionless in-phase soliton, and
soliton 2 (curves 2, blue) is initially an antiphase soliton moving with velocity V2 (t = 0) = 0.1. (a) Temporal dependences of
soliton centre positions. The solitons do not overlap during the collision, and after approaching the minimum distance x =
5.3, they move away from each other with velocities V1 = 0.145 and V2 = −0.071, correspondingly. (b) Temporal dependences
of total population for solitons 1 and 2. (c) Temporal dependences of the lower resonance level population for solitons 1 and 2.
(Online version in colour.)
a n 2, a m 2, B
0.4
t=0
t = 80
1
1, 2
1, 2
1, 2
2
0
3
3
–0.4
a n 2, a m 2, B
0.4
t = 90
1
2
1
2
0
1
t = 100
2
2
1
3
3
–0.4
a n 2, a m 2, B
0.4
t = 110
t = 150
2
2
1, 2
1, 2
1
0
1
3
3
–0.4
30
40
50
x
60
30
40
50
60
x
Figure 9. Transverse profiles of the populations of the lower resonance level (curves 1, black), upper level (curves 2, red) and
value B (curves 3, dotted) at the time moments indicated, V = 0.1. (Online version in colour.)
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1
60
12
0.6
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max( an 2 + am 2)
0.8
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50
1
0.6
0.4
2
0.2
0
50
t
1
0.2
2
0
0
30
0.4
0
50
t
0
50
t
Figure 10. (a–c) The same as in figure 8 for initial velocity of antiphase soliton 2, V2 (t = 0) = 1. (Online version in colour.)
dimensionless values, equations (4.4) are reduced to the standard (two-dimensional) nonlinear
Schrödinger equation
∂b
(4.7)
i + ⊥ b − 7ν|b|2 b = 0.
∂t
A wider class of solitons described by equations (4.4) was studied in [30].
In one-dimensional geometry, well known are bright, sech-type solitons and their collisions
[2]. However, even in the resonance case (equation (4.7)), we deal not with scalar but with vector
solitons, because both in-phase, am = an , and antiphase, am = −an , solitons are described by this
equation. Below we present results of computer simulation of these solitons’ collisions [31]. More
exactly, we will consider here only the case of exact resonance and collisions of a soliton moving
with velocity V with an initially motionless in-phase soliton; the colliding solitons have different
widths and, correspondingly, different maximum amplitudes.
The collision scenario depends strongly on the (relative) velocity of approach of the solitons.
Below, two limiting cases are considered with fairly small (figures 8 and 9, V = 0.1) and large
(figures 10 and 11, V = 1) velocities.
For small velocities (V = 0.1), the solitons initially approach, reaching a minimum distance
x = 5.3, and then move away (figure 8a). It is possible to say that the second soliton is reflected
from the first soliton pushing it. As figure 8b shows, the total population of the resonance levels
of the two solitons changes after the collision, increasing for the first soliton and decreasing
for the second soliton. The populations of the first (antiphase) soliton separate levels begin to
oscillate; it transforms into a long-living breather or oscillon (figure 8c). These oscillations are not
so pronounced for the second (in-phase) soliton (figure 8c). Temporal evolution of the profiles
of the population, both lower (|an |2 ) and upper (|am |2 ), is illustrated in figure 9. To distinguish
between in-phase and antiphase solitons, we present here also profiles of value B = −|an − am |2 /4;
for in-phase solitons B = 0.
For large velocity, V = 1, the collision scenario is different (figures 10 and 11). First, the initially
moving soliton traverses the motionless soliton practically without change of its velocity; the
initially motionless soliton moves in the same direction with small velocity (figure 10a). Second,
after collision, both solitons begin to oscillate, transforming to breathers (figure 10c).
5. Conclusion
The results presented confirm the efficiency of excitation of various nonlinear structures inside
dynamical billiards—cavities or traps with oscillating mirrors (barriers). In such schemes, the
power supply is due to kinetic energy of the mirrors, which can be non-transparent.
A single classical particle in a one-dimensional dynamical billiard with periodic modulation of
the barriers’ position has, in a certain range of parameters, one or a number of stable regimes with
conserved kinetic energy. With increase of modulation depth, these regimes become unstable, and
the particle dynamics becomes chaotic.
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2
x
13
0.6
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70
(c)
0.8
max( an 2)
(b)
90
max( an 2 + am 2)
(a)
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t=0
t = 10
14
0.4
2
1, 2
1,2
1
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0.2
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an 2, am 2, B
0.6
1, 2
1
0
2
3
3
–0.2
–0.4
an 2, am 2, B
0.6
t = 11
t = 12
0.4
1
2
0.2
2
1
0
3
–0.2
3
–0.4
an 2, am 2, B
0.6
t = 13
t = 14
0.4
1
2
2
0.2
1
0
3
3
–0.2
–0.4
an 2, am 2, B
0.6
t = 15
t = 20
1
0.4
1
2
0.2
2
0
2
3
3
–0.2
1
–0.4
30
40
50
60
30
x
40
50
60
x
Figure 11. The same as in figure 9 for velocity V = 1. (Online version in colour.)
The billiard serves as a trap for single quantum particles, and simultaneously its oscillations
excite the particle to higher energy levels. For periodic oscillations, there is a discrete set of particle
quasi-energies. When the oscillation frequency is close to a frequency of transition between quasienergy levels, resonance occurs, with strong Rabi oscillations of resonance levels population.
An atomic BEC under the resonance conditions has two types of dynamics: (i) with periodic
variation of populations and resonance states phase difference, and (ii) with periodic variation
only of populations and monotonic variation of the phase difference. The period of Rabi
oscillations depends strongly on the initial distribution of populations. Neglecting the transverse
distribution, ‘longitudinal’ Schrödinger-like solitons exist in the dynamical trap, and their
dynamics can be regular or chaotic, as in the case of classical particles. For transversely distributed
schemes, various types of solitons exist, including vector spatial solitons. Their collisions can
change soliton type, transforming, for example, a stationary soliton to a breather.
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Sciences ‘Fundamental problems of nonlinear dynamics in mathematical and physical sciences’ and by the
Government of the Russian Federation, grant no. 074-U01. The current stage of our research is supported by
the Russian Scientific Foundation, grant no. 14-12-00894.
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