Nonlinear phase-dynamics in a driven Bosonic Josephson junction
Erez Boukobza1, Michael G. Moore2, Doron Cohen3, and Amichay Vardi1,4
1Department of Chemistry, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel
2Department of Physics & Astronomy, Michigan State Univerity, East Lansing, Michigan 48824, USA
3Department of Physics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel
4ITAMP, Harvard-Smithsonian CFA, 60 Garden St., Cambridge, Massachusetts 02138, USA
We study the collective dynamics of a driven two mode Bose-Hubbard model in the Josephson
interaction regime. The classical phase-space is mixed, with chaotic and regular components, that
determine the dynamical nature of the fringe-visibility. For weak off-resonant drive, where the
chaotic component is small, the many-body dynamics corresponds to that of a Kapitza pendulum,
with the relative-phase ϕ between the condensates playing the role of the pendulum angle. Using a
master equation approach we show that the modulation of the inter-site potential barrier stabilizes
the ϕ = π ’inverted pendulum’ coherent state, and protects the fringe visibility.
The Josephson effect  is an unambiguous demonstra-
tion of macroscopic phase coherence between two cou-
pled Bose ensembles. The experimental realization of
BEC Josephson junctions [2–5] has led to observations of
Josephson oscillations [2, 4, 6–8] as well as macroscopic
self-trapping [4, 9] and the equivalents of the ac and dc
Josephson effect [5, 10], present also in the superconduc-
tor  or the superfluid Helium [13, 14] realizations.
Beyond these mean-field effects, BEC Josephson systems
allow for the observation of strong correlation phenom-
ena, such as the collapse and revival of the relative phase
between the two condensates [15–17] which was observed
with astounding precision in an optical lattice in Refs.
, in a double-BEC system in Ref. , and in a 1D
spinor BEC in Ref. .
The bosonic Josephson junction is often described by
a two-mode Bose-Hubbard Hamiltonian (BHH) [21, 22],
H = −KˆLx− EˆLz+ UˆL2
Here K, E, and U are coupling, bias, and interaction en-
ergies. The SU(2) generaratorsˆLx = (ˆ a†
ˆLy= (ˆ a†
fined in terms of the boson on-site annihilation and cre-
ation operators ˆ ai, ˆ a†
number n1+ n2 = N ≡ 2ℓ. Three distinct interaction
regimes are obtained, depending on the characteristic
strength of interaction u = UN/K [21, 23]. The quasi-
linear Rabi regime |u| < 1, the strong-coupling Josephson
regime 1 < |u| < N2, and the extremely quantum Fock
regime u > N2.
With adjustable parameters, the BHH can also real-
ize an atom interferometer, in which the bias, E, gener-
ates a phase-shift that can be measured by atom-number
counting. The interaction term allows for the creation
of non-classical input states, but also generates unde-
sired phase-diffusion noise. Atom interferometers based
on this model are of great current interest because they
can potentially resolve phase-shifts below the standard
quantum limit (SQL) of ∆φ ≥ 1/√N, and are limited
instead by the Heisenberg fundamental limit ∆φ ≥ 1/N.
In such a device, a highly correlated initial state would
1ˆ a2+ ˆ a†
1ˆ a2− ˆ a†
2ˆ a1)/(2i), andˆLz= (n1− n2)/2, are de-
i, with the conserved total particle
be prepared in the Josephson or Fock regime, but the
measurement would ideally be made in the Rabi regime.
While tunable, e.g. via Feshbach resonance, the interac-
tion parameter u will never be exactly zero. Thus un-
derstanding, and potentially harnessing, the dynamical
effects of the interaction-induced nonlinearity will play a
crucial role in designing such devices.
To model the nonlinear dynamics in the Josephson and
Rabi regimes, it is convenient to define the action-angle
variables ˆ a = eiϕi√ni, ˆ a†=√nie−iϕi. Using these def-
initions, the Hamiltonian (1) can be rewritten in terms
of the relative pair-number n = (n1−n2)/2 and relative-
phase eiϕ≡ eiϕ1e−iϕ2operators,
H = Un2− En −K
In the Josephson interaction regime, states initiated with
n ≪ N/2 remain confined to this small population im-
balance region, so that it is possible to use Josephson’s
HJosephson= Ec(n − nE)2− EJcos(ϕ) ,
with EC = U, EJ = KN/2, and nE = E/(2U). This
Hamiltonian matches that of a pendulum with ‘mass’
M = 1/(2U) and frequency ωJ ≈
pendulum angle is played by the relative phase ϕ between
the two condensate modes, while the relative population
imbalance, n, plays the role of angular momentum. One
consequence of this analogy is that a relative-phase of
ϕ = 0 is classically stable (ground state of the pendu-
lum), whereas a ϕ = π relative-phase (inverted pendu-
lum) is classically unstable.
The term “phase-diffusion” then describes the non-
linear effects generated by the anharmonic part of the
cos(ϕ) potential. The many-body manifestation of this
anharmonicity is the rapid loss of single-particle coher-
ence for a coherent state prepared with a π relative-
phase, as opposed to the slow phase-diffusion of phase-
locked condensates with ϕ = 0 [23, 24]. Several recent
works propose to control such phase-diffusion processes
√KUN. The role of
by means of external noise [25, 26] or modulation of the
Hamiltonian parameters to induce π flips of the relative
In this work, we build further on the pendulum analogy
to explore the effect of oscillatory driving on the collective
phase dynamics of the BHH. We consider two possible
time-dependent driving fields, “vertical” (v) and “hori-
zontal” (h), given by
Vv,h(t) = f(t)ˆ W = Dv,hsin(ωt + φ)ˆLx,y.(3)
Here ‘vertical’ and ‘horizontal’ are in reference to the
pendulum model. The classical phase-pendulum motion
is in the LxLy equatorial plane of the BHH, with the
ϕ = 0,π stationary points lying on the Lxaxis, making
it the ’gravitation’ direction of the pendulum. Hence, Vv
is equivalent to a vertical drive of the pendulum axis and
Vh corresponds to a horizontal drive. With respect to
the two-mode BHH, the first type of driving Vv(t) is a
modulation of the hopping frequency K, which may be
attained by changing the Barrier height, as illustrated
in Fig. 1(a); whereas Vh(t) may be induced by means
of shaking the double-well confining potential laterally,
as illustrated in Fig. 1(b), thus effectively introducing
the equivalent of an electromotive force in the oscillat-
ing frame.It is customary to define the dimension-
less frequency Ω ≡ ω/ωJ, and the dimensionless driving
strength q ≡
driving correspond to Ω ≫ 1 and Ω ≪ 1, respectively,
whereas q ≫ Ω and q ≪ Ω correspond to strong and
Within the 1D pendulum approximation, the angle
variable ϕ and the momentum n are canonical conjugate
variables. It is well known  that off-resonant, fast
driving is effectively equivalent to the additional static
‘pseudo-potential ’, Veff
trated in Fig. 1. It is possible to further refine this effec-
tive description, adding momentum dependent terms, as
described in . For sufficiently strong (q2> 2) verti-
cal drive, the effective term Veff
inverted pendulum (Fig. 1(c)), an effect known as the
Kapitza pendulum . By contrast, the effective term
can destabilize under the same conditions the ϕ = 0
pendulum-down ground state, and generate two new de-
generate quasi-stationary states (Fig. 1(d)).
Generally, the driven BHH has a mixed classical phase-
space structure similar to that of a kicked top , with
chaotic regimes bound by KAM trajectories, making the
bosonic Josephson junction a good system for studying
quantum chaos [31, 32] along with the kicked-rotor real-
ization by cold atoms in periodic optical lattice potentials
, ultracold atoms in atom-optics billiards , and the
recent realization of a quantum kicked top by the total
spin of single133Cs atoms . The unique features of
the driven BHH in this respect are: (i) It offers a novel
avenue of ’interaction-induced’ chaos, which should be
distinguished from the single-particle ’potential-induced’
?UN/K(D/ω) = D/(KΩ). Fast and slow
v,h= ±(1/4)q2Kℓsin2(ϕ), as illus-
can stabilize the ϕ = π
FIG. 1: (Color online) Schematics of a driven Bose-Josephson
junction:(a) The ’vertical’ driving obtained by time-
dependent modulation of the barrier height between the wells;
(b) The ’horizontal’ driving is via lateral shaking of the
double-well potential; (c) The potential term in the Joseph-
son Hamiltonian without driving (dash-dot), and with vertical
driving (solid) which includes the effective potential (dashed).
Circles denote stable stationary points whereas ’x’ denotes in-
stabilities; (d) The same for horizontal driving. It should be
noted that ’vertical’ and ’horizontal’ refer to the motion of
the pendulum axis in the Kapitza analogy, which incidentally
match the direction of potential modulation
chaos that had been highlighted in past experiments with
cold atoms; (ii) The pertinent dynamical variables are
relative-phase and relative-number, leading to nonlinear
and possibly chaotic phase dynamics, which may be mon-
itored via fringe-visibility measurements in interference
Figure 2 shows representative results for near-resonant
(Ω ≈ 1) horizontal driving. Stroboscopic Poincare plots
of the classical (mean-field) evolution at drive-period in-
tervals are shown on the left for varying drive strength,
q, demonstrating the growth of the stochastic component
to form a chaotic ’sea’ surrounding regular ’islands’ of
non-chaotic motion. On the right, we represent the full
many-body BHH evolution via the Husimi Q function
Q(n,ϕ) = |?n,ϕ|ψ(t)?|2, which provides visualization for
the expansion of the time-dependent many-particle state
|ψ(t)? in the spin coherent states |n,ϕ? basis. For weak
driving the initial preparation |0,0? lies within a regu-
lar region of phase-space and retains its coherence. In
contrast, for larger values of q the initial coherent state
spreads quickly throughout the chaotic sea, resulting in
a highly correlated many-body state, as manifest in the
dynamics of fringe visibility g(1)
Similar results, with a slightly different classical phase-
space structure are obtained for vertical driving.
For an off-resonant weak driving with Ω ≫ 1 and q ≪
Ω, the chaotic component of phase-space becomes hard
to resolve. In this case we obtain the many-body mani-
festation of the Kapitza pendulum physics , where a
FIG. 2: (Color online) Mean-field stroboscopic phase-space
plots of classical trajectories (left), and a representative
N=100 many-body quantum Hussimi distribution (right), for
the dynamics that is generated by the BHH with Vh(t) driv-
ing, assuming |0,0? coherent preparation. The parameters
are u=30, and Ω=1. The strength of the driving is q=0.1
(top), and q=0.5 (middle), and q=1.0 (bottom). The evolu-
tion of the fringe-visibility is plotted in the lower panel for
q=0.1 (bold solid, green) and q=1.0 (solid, blue). The circle
indicates the time of the Hussimi plot.
periodic vertical drive of the pendulum axis stabilizes the
ϕ = π inverted pendulum and a horizontal drive desta-
bilizes its ground ϕ = 0 state.
In generalizing the Kapitza pendulum physics into the
context of the driven bosonic Josephson junction, we
have to deal with two modifications: (i) The phase-
space of the full BHH is spherical and not canoni-
cal, as opposed to the truncated, cylindrical Joseph-
son phase-space; (ii) Realistically f(t) may include
a noisy component whose effect on the effective po-
tential should be determined. We therefore re-derive
the Kapitza physics for the full BHH, using a mas-
ter equation approach rather than by the standard
timescale separation methodology. The quantum state
of the system is represented by the probability matrix
ρ, satisfying dρ/dt = i[H,ρ] where H = H+f(t)W with
f(t) = sin(ωt + φ). The small parameter is the driving
period δt = 2π/ω for harmonic drive, or the correlation
time if f(t) is noisy. Using a standard iterative procedure
the difference ρ(t + δt) − ρ(t) can be expressed to 1st or-
der as an integral over [H+f(t′)W,ρ]. The 2nd order
adds a double integral over [H+f(t′)W,[H+f(t′′)W,ρ]],
andthe3rd order adds
tains a noisy component, as in the standard master equa-
tion treatment, we obtain after integration over the sec-
ond order contribution a diffusion term [W,[W,ρ]]. In
the familiar classical Focker-Planck context, with W = x,
this terms takes the form ∂2ρ(x,p)/∂p2. However, for
a strictly periodic noiseless driving the time integration
over a period vanishes, and evaluation to 3rd order is
required. Integrating the 3rd order contribution over a
period we get terms that can be packed as [[W,[W,H]],ρ].
Hence, the effective static potential is,
a triple integral
If f(t) con-
Other terms also exist (producing the tilt of the islands in
Figs. 3c, 4c), however they depend on the driving phase
φ, and vanish if the stroboscopic sampling is averaged. In
the standard canonical case withˆ W = W(ˆ x) this expres-
sion gives the familiar Kapitza result [W′(x)]2/(4Mω2)
as in . For the BHH non-canonical spherical phase-
space, with W ∝ Lxor W ∝ Ly, it is straightforward to
verify that Eq.(4) generates the expected Kapitza terms
x, as well as additional terms that slightly re-
normalize the bare values of U and K.
The predicted Kaptiza physics effects are confirmed
numerically in Fig. 3 and Fig. 4 for the vertical Vv(t)
and horizontal Vh(t) drive, respectively. Comparison of
the stroboscopic Poincare plots for the undriven (a) and
driven (c) BHH, clearly shows the stabilization of the
|0,π? coherent state by the verticalˆLxdriving (Fig. 3),
and the destabilization of the |0,0? preparation by the
horizontalˆLydriving (Fig. 4). These classical effect are
mirrored in the evolution of the quantum Husimi func-
tion, thereby affecting the many-body fringe-visibility
dynamics, leading to the protection of coherence by Vv(t)
driving for ϕ = π coherent preparation, and to its de-
struction by Vh(t) for ϕ = 0.
To conclude, the driven BHH, currently attainable in a
number of experimental setups, presents a wealth of non-
linear phase-dynamics effects. Strong, resonant driving
fields result in large chaotic phase-space regions, open-
ing the way for the generation of exotic highly correlated
quantum states . The properties of such states, as
well as their manifestation in interference experiments,
FIG. 3: (Color online) Quantum Kapitza pendulum with ver-
tical driving, and |0,π? coherent preparation. The panels are
arranged as in Fig. 1. The parameters are u=100, Ω=30, and
N=100. The strength of the driving is q=0 (a,b, lower panel
dashed blue) and q=3 (c,d, lower panel solid red). Circles
denote the time at which the Husimi distribution in (c,d) is
plotted. The classical stabilization of the inverted pendulum
results in a protected single-particle coherence of the initial
and the more conventional tunneling effects between reg-
ular islands, are novel manifestations of semiclassical
physics. For weak and fast off-resonant drive we have ob-
tained the many-body equivalents of the Kapitza pendu-
lum effects, with the relative-phase between the conden-
sates acting as the pendulum angle. Such effects could be
readily observed in interference experiments and utilized
to protect fringe-visibility. We note that noise-protected
coherence was also studied in Ref. , yet with a rather
different quantum-Zeno underlying physics.
We thank Saar Rahav  for usefull communication.
This work was supported by the Israel Science Founda-
tion (Grant 582/07), by grant nos. 2006021, 2008141
from the United States-Israel Binational Science Foun-
dation (BSF), and by the National Science Foundation
through a grant for the Institute for Theoretical Atomic,
Molecular, and Optical Physics at Harvard University
and Smithsonian Astrophysical Observatory.
 B. D. Josephson, Phys. Lett. 1, 251 (1962).
 F. S. Cataliotti et al., Science 293, 843 (2001).
FIG. 4: (Color online) Quantum Kapitza pendulum with hor-
izontal driving, and |0,0? coherent preparation. The panels
are arranged as in Fig. 3, with the same parameters, except
u = 30. One observes the splitting of the ground state. Due
to this destabilization the fringe visibility of the preparation
 T. Anker et al., Phys. Rev. Lett. 94, 020403 (2005).
 M. Albiez et al., Phys. Rev. Lett. 95, 010402 (2005).
 S. Levy et al.,Nature 449, 579 (2007).
 J. Javanainen, Phys. Rev. Lett. 57, 3164 (1986).
 F. Dalfovo, L. Pitaevskii, and S. Stringari, Phys. Rev. A
54, 4213 (1996).
 I. Zapata, F. Sols, and A. J. Leggett, Phys. Rev. A 57,
 A. Smerzi et al., Phys. Rev. Lett. 79, 4950 (1997)
 S. Giovanazzi, A. Smerzi, and S. Fantoni, Phys. Rev.
Lett. 84, 4521 (2000).
 G. J. Milburn et al.Phys. Rev. A 55, 4318 (1997).
 K. K. Likharev, Rev. Mod. Phys. 51, 101 (1979).
 S. V. Pereverzev et al., Nature 388, 449 (1997).
 K. Sukhatme et al., Nature 411, 280 (2001).
 A. J. Leggett and F. Sols, Found. Phys. 21, 353 (1998).
 E. M. Wright, D. F. Walls and J. C. Garrison Phys. Rev.
Lett. 77, 2158 (1996).
 J. Javanainen and M. Wilkens, Phys. Rev. Lett. 78, 4675
(1997); Phys. Rev. Lett. 81, 1345 (1998).
 M. Greiner et al., Nature 419, 51 (2002).
 G.-B. Jo et al., Phys. Rev. Lett. 98, 030407 (2007).
 A. Widera et al., Phys. Rev. Lett. 100, 140401 (2008).
 Y. Makhlin, G. Sch¨ on, and A. Shnirman, Rev. Mod.
Phys. 73, 357 (2001); R. Gati and M. K. Oberthaler,
J. Phys. B 40, R61 (2007).
5 Download full-text
 Gh-S. Paraoanu et al., J. Phys. B: At. Mol. Opt. Phys.
34, 4689 (2001); A. J. Leggett, Rev. Mod. Phys. 73, 307
 E. Boukobza et al., Phys. Rev. Lett. 102, 180403 (2009).
 A. Vardi and J. R. Anglin, Phys. Rev. Lett. 86, 568
 Y. Khodorkovsky, G. Kurizki, and A. Vardi, Phys. Rev.
Lett. 100, 220403 (2008); Y. Khodorkovsky, G. Kurizki,
and A. Vardi, Phys. Rev. A 80, 023609 (2009).
 D. Witthaut, F. Trimborn, and S. Wimberger, Phys. Rev.
Lett. 101, 200402 (2008); D. Witthaut, F. Trimborn, and
S. Wimberger, Phys. Rev. A 79, 033621 (2009).
 N. Bar-Gill et al., Phys. Rev. A 80, 053613 (2009).
 Collected Papers of P. L. Kapitza, editted by D. ter Haar
(Pergamon, Oxford, 1965); P. L. Kapitza, Zh. Eksp. Teor.
Fiz. 21, 588 (1951); L. D. Landau and E. M. Lifshitz,
Mechanics (Pergamon, Oxford, 1976); T. P. Grozdanov
and M. J. Rakovi´ c, Phys. Rev. A 38, 1739 (1988).
 S. Rahav, I. Gilary, and S. Fishman, Phys. Rev. Lett. 91,
 F. Haake, M. Kus, and R. Scharf, Z. Phys. B 65, 381
 S. Ghose, P. M. Alsing, and I. H. Deautch, Phys. Rev. E
64, 056119 (2001).
 C. Weiss and N. Teichmann, Phys. Rev. Lett. 100,
 D. A. Steck, W. H. Oskay, and M. G. Raizen, Science
293, 274 (2001); W. K. Hensinger et al., Nature 412 52
 V. Milner et al., Phys. Rev. Lett 86, 1514 (2001); N.
Friedman et al., Phys. Rev. Lett. 86, 1518 (2001); A.
Kaplan et al., Phys. Rev. Lett. 87, 274101 (2001); M. F.
Andersen et al., Phys. Rev. Lett. 97, 104102 (2006).
 S. Chaundry et al., Nature 461, 768 (2009).