Quantum switching at a mean-field instability of a Bose-Einstein condensate in an optical lattice

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by  V. S. Shchesnovich

Quantum switching at a mean-field instability of a Bose-Einstein condensate in an
optical lattice
V. S. Shchesnovich1 and V. V. Konotop2
1Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, Santo Andr´e, SP, 09210-170 Brazil,
2Centro de F´ısica Te´orica e Computacional, Universidade de Lisboa,
Complexo Interdisciplinar, Avenida Professor Gama Pinto 2,
Lisboa 1649-003, Portugal; Departamento de F´ısica, Faculdade de Ciˆencias,
Universidade de Lisboa, Campo Grande, Ed. C8, Piso 6, Lisboa 1749-016, Portugal
9 It is shown that bifurcations of the mean-field dynamics of a Bose-Einstein condensate can be
0 related with the quantum phase transitions of the original many-body system. As an example we
0 explore the intra-band tunneling in the two-dimensional optical lattice. Such a system allows for
2 easy control by the lattice depth as well as for macroscopic visualization of the phase transition.
The system manifests switching between two selftrapping states or from a selftrapping state to
n
a superposition of the macroscopically populated selftrapping states with the step-like variation
a
J of the control parameter about the bifurcation point. We have also observed the magnification
of the microscopic difference between the even and odd number of atoms to a macroscopically
0
distinguishable dynamics of the system.
1

PACSnumbers: 03.75.Lm;03.75.Nt
]
r
e
h
Introduction.- Since the very beginning of the quan- asymmetric states, characterized by population of only
t
o tummechanicsitsrelationtotheclassicaldynamicscon- one of the sites (the well knownphenomenonof selftrap-
t. stitutes one of the central questions of the theory. De- ping [10]). Now, exploring parallels between the semi-
a
pendence of the energy levels distribution on the type classicalapproachandthe mean-fieldapproximationone
m
of dynamics of the corresponding classical system [1], in can pose the natural question: what changes occur in a
-
general, and the quantum system response to variation manybody system when a control parameter crosses an
d
n of the bifurcation parameters controlling the qualitative instability (e.g. bifurcation) point of the limiting mean-
o changes of the classical behavior [2] are among the ma- field system?
c jor issues [3]. One of the main tools in studies of the Inthe presentLetter we givea partialanswershowing
[
quantum-classical correspondence is the WKB approxi- that one of the possible scenarios is the quantum phase

2 mation, where, loosely speaking, the Planck constant ~ transitionofthesecondtype,associatedwiththeswitch-
v is regarded as a small parameter. ing of the wave-function in the Fock space between the
7
“coherent” and “Bogoliubov” states possessing distinct
0 On the other hand, for a N-boson system the limit
3 N at a constant density, leading to the mean-field features. Consideringaflexible(time-dependent)control
0 app→rox∞imation, can also be understood as a semiclassi- parameter,we havealso founda strongsensitivity of the
. system to the parity of the total number of atoms N,
0 cal limit. This latter approach has received a great deal
1 of attention during the last decade [4], due its high rele- showing parity-dependent structure of the energy levels
8 and the macroscopically different dynamics for different
vancetothetheoryofBose-Einsteincondensates(BECs),
0 parity of N. Observation of the discussed phenomena is
many properties of whose dynamics are remarkably well
:
v described within the framework of the mean-field mod- feasible in the experimental setting available nowadays.
i Quantum and mean-field models.- We consider the
X els [5]. More recently,it was shown[6, 7] that the mean-
nonlinearity-induced intra-band tunneling of BEC be-
field description of a few-mode N-boson system can be
r
a recastinaformsimilartothe WKBapproximationfora tween the two high-symmetry X-points of a two-
dimensional square optical lattice (OL). The process is
discrete Schr¨odinger equation [8], emergent for the coef-
described by the two mode boson Hamiltonian (see [6]
ficients of the wavefunction expansion in the associated
for the details)
Fock space, where 1/N plays the role similar to that of
the Planck constant in the conventional WKB approxi-
1
mation. Hˆ = n2+n2+Λ 4n n +(b†b )2+(b†b )2 .
2N2 1 2 1 2 1 2 2 1
The mean-field equations of a system of interacting n h io(1)
bosons are nonlinear, hence, they naturally manifest where b and b† are the annihilation and creation oper-
j j
many common features of the nonlinear dynamics, in- ators of the two X-states, Λ (0 Λ 1) is the lat-
≤ ≤
cludingbifurcationsofthestationarysolutionscausedby tice parametereasilycontrollableby variationof the lat-
variationofthesystemparameters. Oneofthewellstud- tice depth (or period). The Schr¨odinger equation for
iedexamplesisaboson-Josephsonjunction[9],whichcan the BEC in a state Ψ reads ih∂ Ψ =Hˆ Ψ , where
τ
show either equally populated (symmetric) or strongly h = 2/N and τ =| (i2gρ/~)t, wit|h ig = |4πi~2a /m
s
2
and the atomic density ρ. The link with the semiclas- Now consider small deviations of Λ from the bifurca-
sical limit is evident for the Hamiltonian in the form tion point Λ . To this end, for a fixed N, one can use
c
(1): the Schr¨odinger equation written in the Fock ba- the basis consisting of the degenerate eigenstates of Hˆ :
0
(b†)k(b†)N−k (a†)m(a† )N−m
sis, k,N k = 1 2 0 , depends only on the E ,j = j 3−j 0 , j = 1,2, m = 0,...,N.
| − i k!(N k)!| i | m i m!(N m)! | i 2
− −
relative populations k/N and (N k)/N, while h serves The conditions for Vˆ to be treatedas a perturbationde-
p − p
as an effective “Planck constant”. pend onm as is seenfromthe diagonalmatrix elements:
Hamiltonian (1) represents a nonlinear version of the
well-known boson-Josephson model (see, e.g. [9, 11]), E ,j Vˆ E ,j = 1 + m 1 m . (4)
m m
where unlike in the previously studied models the states h | | i 4N 2N − N
(cid:16) (cid:17)
are coupled by the exchange of pairs of atoms. This is
At the lower levels (m N/2) the energy gaps between
a fairly common situation for systems with four-wave- ≪
thedegeneratesubspacesandtheperturbationbothscale
mixing, provided by the two-body interactions involving
as ∆E N−1, hence the condition of applicability is
fourbosons. The exchangeofthe bosonsbypairsresults ∼
Λ Λ 1 and the lower energy subspaces acquire
in the coupling of the states with the same parity of the | − c| ≪
simple shifts. At the upper energy levels (m N/2) the
populationandisreflectedinthedoubledegeneracyofall above energy gaps behave as ∆E N−2. Sin∼ce Vˆ 1
(N +1)/2 energy levels for odd N, due to the symmetry ∼ h i∼
inthiscase,theperturbationtheoryisapplicableonlyin
relation 2k,N 2k Ψ = N 2k,2k Ψ . For even N
h − | 1i h − | 2i anintervalofΛ ofthe sizeonthe orderofN−2. Thereis
the energy levels show quasi degeneracy (see below).
a dramatic transition in the energy levels, e.g. Fig. 1(b)
The mean-fieldlimit ofthe system(1)canbe formally
shows the exchange of the double degeneracy of the top
obtainedbyreplacingthebosonoperatorsb in(1)bythe
j levels for even N in this N−2-small interval of Λ. By
c-numbers b √Nxeiφ/4 and b N(1 x)e−iφ/4,
1 2 considering the phase of
→ → −
what gives the classical Hamiltonian [6]
p
N2 N 3
1 Em,j (b2†b1)2 Em,j = + + m(N m),(5)
=x(1 x)[2Λ 1+Λcosφ]+ , (2) h | | i − 4 4 2 −
H − − 2
it is easy to verify that the upper and lower eigenstates
where x = n1 /N is the population density and φ = correspond, respectively, to the mean-field stationary
h i
arg (b†)2b2 is the relative phase. possesses two sta- points P (φ=0) and P (φ=π).
h 2 1i H 1 2
tionary points describing equally populated X-states:
the classical energy maximum P = (x = 1,φ = 0) and 0.508
1 2 (a) 0.5011 (b)
minimum P =(x= 1,φ=π). P is dynamically stable
2 2 1
in the domain Λ > Λ = 1. For Λ < Λ it looses its
c 3 c 0.5035
stability, and another set of stationary points x=1 (S ) 0.50085
1
and x = 0 (S ) appears, which is a fairly general situa-
2
tionin nonlinearbosonmodels. The appearingsolutions Ek 0.499 0.5006
describe the symmetry breaking leading to selftrapping.
Energy levels near the critical point.- To describe the
0.4945 0.50035
spectrum of the Hamiltonian (1) in the vicinity of the
critical value Λ we rewrite Hˆ in terms of the operators
c 0.49
a1,2 =(b1∓ib2)/√2 0.32 0.33 Λ 0.34 0.35 0.50001.3326 0.3Λ33350.3340
Λ Λ+1 Λ
Hˆ =Hˆ0+ 1 Vˆ+ (Λ), (Λ)= + (3) FIG.1: (a)TheenergylevelsofHˆ forN =200and(b)ade-
− Λ E E 4 2N
(cid:18) c(cid:19) tailed picturein thevicinity of Λc. The classical energy lines
of the mean-field fixed points P1 and S2 are visibly formed.
2
where Hˆ = 2Λa†a a†a and Vˆ = 1 a†a +a†a . Thetopenergylevelsforsufficientlylarge |Λ−Λc|arequasi-
0 N2 1 1 2 2 4N2 1 2 2 1 degeneratewith theinter-leveldistancesindistinguishable on
At the critical point the energy spectrum(cid:16) is determin(cid:17)ed thescale of the figure(see the discussion in thetext below).
by Hˆ : E = 2Λcm(N m)+ (Λ ), where m is the
0 m N2 − E c
occupation number corresponding to the operator a†a .
1 1
The spectrumofHˆ is doublydegenerate(exceptforthe Spectrum in the limit N . Coherent states and
0
→ ∞
top level for even N) due to the symmetry m N m. selftrappingstates. -ForΛ−1 Λ−1 N−2 thequantum
→ − c − ≫
The ground state energy is (Λ ) = E = E , while statescorrespondingtoP canbeobtainedbyquantizing
min c 0 N 1
E
the top energy level has m = N/2 for even N and the local classical Hamiltonian (2), i.e. by expanding it
m=(N 1)/2 for odd N. Restricting ourselves to even withrespecttox 1/2andφandsettingφ= ih ∂ (see
± − − ∂x
number of bosons we get (Λ )= 1 + Λc. also Ref. [12]; on this way one looses the term of order
Emax c 2 2N
3
1/N in ). The “wave function” ψ(x) = √NC Fig.2bthedramaticdeformationofthetopenergyeigen-
max k
√N k,NE k ψ satisfies ≡ stateofHˆ (correspondingtotheS -P transition)about
2 1
h − | i
the critical Λ is shown. Finally, we note that for even
c
Λh2 ∂2 1 1 2 N the quasi double degeneracy of the energy levels for
+(3Λ 1) x ψ =Eψ. (6)
" 8 ∂x2 − 4 −(cid:18) − 2(cid:19) !# Λ−1−Λ−c1 ≫N−2 (c.f. Fig. 1(b))isduetotheexchange
symmetry between S and S resulting in equal energy
1 2
Eq. (6) is the negative mass quantum oscillator prob- levels of the Hamiltonians Hˆ and Hˆ .
S1 S2
lem with the frequency ω2 = 8 3 1 . The respec-
− Λ
tive descending energy levels re(cid:0)ad En(t(cid:1)op) = Emax + 100 (a) 0.338 (b)
1 Λ 1 hΛω n+ 1 .The eigenfunctions arelocal-
4 Λc − − 4 2
ize(cid:16)d in th(cid:17)e Fock s(cid:0)pace, e(cid:1).g. the n = 0 eigenfunction is 10−2 0.336
ψ0(x) = Cexp −2ωh x− 12 2 . In the original discrete g(|C|)k10−4 Λ 0.334
variable x = kh/N, t(cid:0)here ar(cid:1)eieven and odd eigenstates lo
C2k and C2k−1 related by the approximate symmetry 10−6 0.332
C C , hence the energy levels are quasi doubly
n n+1
≈
degenerate [c.f. Fig. 1(b)]. 10−8 0.33
The local approximation becomes invalid as 0 5 10 15 2k0 25 30 35 40 0 0.2 0.4k/N0.6 0.8 1
Λ−1 Λ−1 N−2 (the wave-function delocalizes).
c − ∼
The other set of the stationary points, S1,2, becomes FIG. 2: (Color online) (a) Convergence of the four upper
stable for Λ < Λc in the mean-field limit. In this case, eigenstates of the Hamiltonian (1) to the eigenstates of HˆS2
however, the phase φ is undefined. Let us first consider (shown by dots) for for N = 100 (solid lines) and N = 1000
the full quantum case, for example, the limit n N (dashed lines), for Λ=Λc−0.1. (b) The contour plot of the
1
h i ≪ state corresponding to the top energy level in the vicinity of
(i.e. the point S ). The resulting reduced Hamiltonian
2
can be either easily derived in the Fock basis or ob- Λc for N =200.
tained by formally setting b = N and retaining the
2
lowest-order terms in b and b†: In the mean-field description of the stationary point
1 1
S theassociatedHamiltonianisdefinedbyreplacingthe
2
Hˆ Hˆ = 1 + (2Λ−1)b†b + Λ [(b†)2+b2]. (7) boson operators in Eq. (7) by the c-numbers b1 =√Nα
≈ S2 2 N 1 1 2N 1 1 and b = √Nβ. Using α2 + β 2 = 1 and fixing the
2
| | | |
irrelevantcommonphasebysettingβ realwegetthedy-
Hamiltonian (7) can be diagonalized by the Bogoliubov
transformation c = cosh(θ)b sinh(θ)b†, where θ = namicalvariablesαandα∗ andtheclassicalHamiltonian
1 1
− intheform = 1+1(1 α2) 2(2Λ 1)α2+Λ[α2+
θ(Λ)>0 is determined fromtanh(2θ)=Λ/(1 2Λ). We HS2 2 2 −| | { − | |
− (α∗)2] , from which the stability of the point S (α=0)
get 2
}
for Λ<Λ follows.
c
Hˆ = Λ c†c+ Λtanhθ + 1. (8) Thus, the passage through the bifurcation point Λc of
S2 −Nsinh(2θ) 2N 2 the mean-field model, corresponds to the phase transition
in the quantum many-body system on an interval of the
Thus c†c gives the number of negative-energy quasi- control parameter scaling as N−2 and reflected in the de-
particles over the Bogoliubov (squeezed) vacuum solv-
formation of the spectrum and dramatic change of the
ing cvac =0. In the atom-number basis vac is
system wave-function in the Fock space. The described
| i | i
a superposition of the Fock states with C(vac) = change of the system is related to the change of the sym-
2k
tanhk(θ) (2k)!/(2kk!)C and C(vac) = 0, (C is a nor- metryoftheatomicdistribution, andthusitisthesecond
0 2k−1 0
malization constant). order phase transition.
p
The validity condition of the approximation (7), Inourcasethisscenariocorrespondstolossofstability
given by nˆ ,∆n N, can be rewritten in the form oftheselftrappingsolutionsS1 andS2 andappearanceof
tanh−2(2θh)≫i 1+N≪−2,whatisthesameasΛ−1−Λ−c1 ≫ the stable stationary point P1. In the quantum descrip-
N−2. Inthiscase,theeigenstatesof(7)arewell-localized tionthis happensby asetofavoidedcrossingsofthe top
in the atom-number Fock space, i.e. the coefficients C energy levels (and splitting of the quasi-degenerate en-
2k
decay fast enough. The condition for this excludes the ergy levels for even N) as the parameter Λ sweeps the
same small interval as in the perturbation theory, hence small interval on the order of N−2 about the critical
the transition between the coherent states and the self- valueΛc (seeFig.1). Forlowerenergylevelsthe avoided
trapping (Bogoliubov) states occurs on the interval of crossingsappearalongthetwostraightlinesapproximat-
Λ of order of N−2. The convergence of the eigenstates ing the classical energies of the two involved stationary
of Hˆ to that of the full Hamiltonian (1) turns out points: (P )= 1 + (3Λ−1) (for Λ < Λ ) and = 1
S2 H 1 2 4 c Emax 2
to be remarkably fast as it is shown in Fig. 2(a). In (Λ>Λ ), see Fig. 1.
c
4
Dynamics of the phase transition.- Let us see how the To estimate the physical time scale, t t τ =
ph
≡
quantumphasetransitionshowsupinthesystemdynam- md2ℓ⊥ τ, we assume that a condensate of 87Rb atoms
icswhenΛistime-dependent. TheselftrappingstatesS 8π~asNpc
1 is loaded in a square lattice with the mean density of
and S , eigenstates of the Hamiltonian (1), correspond
2 =20atomspercite. Ifthelatticeconstantd=2µm
pc
to occupation of just one of the X-points. Such an ini- N
and the oscillator length of the tight transverse trap (to
tialconditioncanbeexperimentallycreatedbyswitching
assurethe two-dimensionalapproximation)ℓ =0.1µm,
⊥
on a moving lattice with Λ < Λ (see e.g. [7]). As the
c thent 0.2msandthetimenecessaryforthecreation
ph
lattice parameter Λ(τ) passes the critical value from be- ∼
ofthemacroscopicsuperpositionofFig. 3(c),(d)isabout
low, the selftrapping states are replaced by the coherent
20 ms.
states with comparable average occupations of the two
X-points.
A more intriguing dynamics is observed when Λ(τ) is Conclusion.- We have shown that behind the mean-
asmoothstep-likefunctionbetweenΛ andΛ suchthat field instability in the intra-band tunneling of BEC in
1 2
Λ < Λ < Λ . In this case, the system dynamics and an optical lattice is a quantum phase transition be-
1 c 2
the emerging states dramatically depend also on parity tween macroscopically distinct states, giving a macro-
of the number of atoms. For fixed Λ the system be- scopicmagnificationofthemicroscopicquantumfeatures
1,2
havior crucially depends on the time that Λ(τ) spends ofthesystem. Aspectaculardemonstrationofthisisthe
above Λ . More specifically, one can identify two dis- dynamic formation of the superposition of macroscopi-
c
tinct scenarios, which can be described as a switching cally distinct states, which, besides being responsible for
dynamics between the selftrapping states at the two X- the difference between the mean-field and quantum dy-
points, Fig. 3(a),(b) or dynamic creation of the super- namics (see also recent Ref. [13]), shows also an anoma-
position of macroscopically distinct states, well approx- lous dependence on parity of BEC atoms reflecting dis-
imated by (C k,N k +C N k,k ) with a tinct energy level structure for even and odd number of
k N−k
k<km | − i | − i atoms.
small k /NP, Fig. 3(c),(d) (where k /N 0.2). In the
m m
≈
case of macroscopic superposition the dynamics shows
The work of VSS was supported by the FAPESP of
anomalous dependence on parity of N, i.e. showing the
Brazil.
samebehaviorforlargeN ofthe sameparitybutmacro-
scopically distinct behavior for N and N +1, Fig. 3(c).
Note that the mean-field dynamics is close to the quan-
tum one in the switching case, Fig. 3(a), while it is dra-
matically different in the superposition case, Fig. 3(c).
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1
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