PHYSICAL REVIEW A 80, 013610 共2009兲
Generalized purity and quantum phase transition for Bose-Einstein condensates
in a symmetric double well
T. F. Viscondi, K. Furuya, and M. C. de Oliveira
Instituto de Física “Gleb Wataghin,” Universidade Estadual de Campinas, 13083-970 Campinas, SP, Brazil
共Received 20 December 2008; published 16 July 2009兲
The generalized purity is employed for investigating the process of coherence loss and delocalization of the
Q function in the Bloch sphere of a two-mode Bose-Einstein condensate in a symmetrical double well with
cross collision. Quantum phase transition of the model is signaled by the generalized purity as a function of an
appropriate parameter of the Hamiltonian and the number of particles 共N兲. A power-law dependence of the
critical parameter with N is derived.
DOI: 10.1103/PhysRevA.80.013610
PACS number共s兲: 03.75.Lm, 03.67.Mn, 03.75.Gg, 64.70.Tg
II. GENERALIZED PURITY AND THE BECSDW MODEL
I. INTRODUCTION
Recently it has been pointed out that any bipartite and
multipartite entanglement measure can signal the presence of
a quantum phase transition 共QPT兲 关1–3兴 in many-particle
systems. Related to that, a subsystem-independent generalization of entanglement has been introduced based on coherent states and convex sets characterizing the unentangled
pure states as coherent states of a chosen Lie algebra 关4兴.
Such a notion of entanglement defined relative to a distinguished subspace of observables is pointed as particularly
useful for classifying multipartite entanglement and thus for
QPT characterization. The generalized purity of the state
relative to a certain distinguished subset of observables
forming a local Lie algebra is directly related to the MeyerWallach measure 关5兴, and whenever a specific subsystem can
be associated to the subset of observables the usual entanglement notion is recovered. On the other hand it has been
demonstrated that the model describing a two-mode BoseEinstein condensate in a symmetric double well 共BECSDW兲
关6兴, with cross-collisional terms 关7,8兴, presents interesting
dynamical regimes: The macroscopic self-trapping 共MST兲
and Josephson oscillations 共JO兲 of population 关6,8,9兴, both
experimentally observed 关10兴. Recently a discussion of the
transition from one regime to the other has been presented
from the point of view of critical phenomena, as a continuous QPT problem in terms of the usual subsystem entropy
关11–13兴. It is thus certainly important to investigate which
features of this transition are signaled by a subsystemindependent entanglement measure.
In this paper we apply the concept of generalized purity in
a BECSDW for a twofold purpose: 共i兲 To quantify the quality
of the semiclassical approach commonly employed in the
literature for the BECSDW; 共ii兲 To characterize the quantum
critical phenomena occurring in this model with a
subsystem-independent measure of quantum correlations 关4兴.
In Sec. II we introduce the model, define the generalized
purity of the su共2兲 algebra and analyze the quality of the
semiclassical results. Section III is devoted to the connection
between the classical bifurcation and the quantum phase
transition from JO to MST regime, and to the characterization of this critical phenomenon using the purity of the su共2兲
algebra. Conclusions are drawn in Sec. IV.
1050-2947/2009/80共1兲/013610共5兲
The two-mode approximated BECSDW has been well
studied in the literature 共see, e.g., 关6,9兴兲 as a model presenting a nonlinear self-trapping phenomenon. More recently, the
importance of the cross-collisional terms for large number of
particles in the condensate 共N Ⰷ 1兲 has been noticed 关7兴 and
explored semiclassically with a time-dependent variational
principle based on coherent states 关8兴. For a fixed number of
condensed particles N, in order to explore the natural group
structure of the model, we conveniently adopt the Schwinger’s pseudospin operators defined in terms of the creation
†
, d⫾ on the approximated
and annihilation boson operators d⫾
localized states 兩u⫾典 关6,8兴: Jx ⬅ 共d−†d− − d+†d+兲 / 2, Jy ⬅ i共d−†d+
− d+†d−兲 / 2, and Jz ⬅ 共d+†d− + d−†d+兲 / 2, where J = N / 2. In that
form the two-mode BEC Hamiltonian writes as
冋
Ĥ = 2 2⌳共N − 1兲 +
册
⍀
Jz + 2共␬ − ␩兲J2x + 4␩Jz2 ,
2
共1兲
where ⍀ is the tunneling parameter and ␬ is the self-collision
parameter of the condensate which is much larger than the
so-called cross-collision terms ␩ = ␬⑀2 , ⌳ = ␬⑀3/2, with ⑀
= 具u+ 兩 u−典. ⍀⬘ ⬅ 2关2⌳共N − 1兲 + ⍀ / 2兴 is an effective tunneling
parameter dependent on N. The natural associated algebra of
the model is su共2兲. In this form Hamiltonian 共1兲 is a realization of the Lipkin-Meshkov-Glick model 关14兴, whose
ground-state entanglement has been investigated recently
关4,13兴. We should remark that this model with the crosscollision terms included describes more appropriately the experimental observation of an atom-number-dependent tunneling rate in an optical lattice realization 关7,8,10兴. Thus in
principle any conclusions derived here can be implemented
in the same experimental setup.
By means of a semiclassical method exploring SU共2兲 coherent states we demonstrated that for a sufficiently large N,
even a small amount of cross-collisional rate can change the
dynamical regime of oscillations of the condensate from
MST to JO 关8兴. The quantum and semiclassical dynamics of
the BECSDW are known to be qualitatively very similar
关6,8兴 for large number of particles 共N Ⰷ 1兲, except for the
presence of collapses and revivals in the quantum time evolutions for the mean values of 具Jx典共t兲. This breaking of
quantum-classical correspondence is due to quantum fluctuations, which drive the system state away from being coherent
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©2009 The American Physical Society
PHYSICAL REVIEW A 80, 013610 共2009兲
VISCONDI, FURUYA, AND DE OLIVEIRA
关15兴. When we treat the semiclassical dynamics employing a
time-dependent variational principle 关16兴, we restrict the
evolution of the state to a nonlinear subspace constituted
only by the coherent states of SU共2兲 关8兴. Such an evolution is
exact for an initially coherent state only in the classicalmacroscopic limit of N → ⬁ 关17兴. Also, in the time-dependent
variational principle approach we force the system quantum
representations to be always localized in the phase space.
From this point of view, the coherent states are the closest to
the classical ones, which are points in the classical phase
space, and converges to them in the limit N → ⬁. The delocalization of the quantum state is responsible for the quantitative disagreement between the two dynamics 关8兴. As a consequence, collapses and revivals of expectation values of
relevant observables occurring in the quantum description
are never recovered with the semiclassical approach.
For a more complete analysis of the quality of the semiclassical approximation, it becomes fundamental to have a
good quantitative measure of the “distance” of a given state
to the subspace of the coherent states. When the complete
dynamics of the system is restricted to a space that carries an
irreducible representation of SU共2兲 共here this condition is
satisfied due to the particle number conservation兲, there is a
simple measure 共obtained from the total uncertainty of the
algebra 关18兴兲 called generalized purity of the algebra Psu共2兲
关4,19兴, and defined as
Psu共2兲共兩␺典兲 =
1
兺 具␺兩Jk兩␺典2 .
J2 k=x,y,z
共2兲
It should be noticed that in this last form it is easy to
relate the generalized purity with the concept of spin squeezing recently employed for experimental verification of entanglement for BEC trapped in a double or six-well potential
of an optical lattice 关20兴 共see references therein for a general
coverage of the spin squeezing concept and utilization兲. We
postpone entanglement analysis to the next section. We want
first to remark that the generalized purity is a good measure
of distance from a coherent state, among other reasons, because it is invariant under a transformation of the group
SU共2兲 on the state 兩␺典: Psu共2兲共兩␺典兲 = Psu共2兲共U兩␺典兲 , ∀ U
苸 SU共2兲. Thus, all states connected by a SU共2兲 transformation possess the same purity. However, the most interesting
property of Psu共2兲 共concerning the purpose of quantitatively
compare the correspondence between the semiclassical and
the exact quantum dynamics兲 is the fact that this measure
attains its maximum value at one, if and only if, 兩␺典 is the
coherent state closest to a classical one: 兩␪ , ␾典 = R共␪ , ␾兲兩J ,
−J典 = e−i␪共Jx sin ␾−Jy cos ␾兲兩J , −J典. As soon as the dynamics takes
the state away from the space consisting only of the states
兩␪ , ␾典, making the system more delocalized in the phase
space, Psu共2兲 decreases monotonically to zero. We remark
that Psu共2兲 only has such properties clearly defined for pure
states and only then it is a measure of existing quantum
correlations of the state in the classical phase space.
We choose as initial state 兩J , J典x ⬅ 兩␪ = ␲2 , ␾ = 0典, where N
= 100 particles are in the same well, with ⍀ = 1 and ␬ = 2N⍀ .
␬
For this set of parameters it is known that by setting ␩ = 100
the state is in the self-trapping region of the phase space,
FIG. 1. 共Color online兲 Time evolution of Psu共2兲 for MST and JO
dynamical regimes of the two-mode BEC model. The dashed curve
2⍀
␬
represents the MST regime with ␬ = N and ␩ = 100 , whereas the
␬
2⍀
solid curve represents the regime of JO with ␬ = N and ␩ = 10 . For
both cases the initial state of the system is the coherent state
␲
兩J , J典x ⬅ 兩␪ = 2 , ␾ = 0典, where N = 100 particles are initially in the
same well.
␬
whereas for ␩ = 10
it is outside the self-trapping region and
thus in the JO regime 关8兴. In Fig. 1 we plot Psu共2兲, where the
MST regime is in dashed line, and solid one for JO regime.
In the MST regime Psu共2兲 quickly drops down from 1 stabilizing close to 0.9 for ⍀t ⬇ 10, indicating that the dynamics
takes the state away from the subspace of the coherent states.
The region of almost constant purity coincides with the collapse region of population dynamics 关8兴. Note that close to
⍀t = 30 the purity shows small oscillations, and in the region
close to ⍀t = 60 the purity increases again until it practically
recovers the value 1. At this instant the quantum state gets
closer to a coherent one, being this responsible for the revival of the oscillations of the population dynamics 关8兴. In
the MST regime the large value of Psu共2兲 indicates a relative
good agreement between the quantum and semiclassical results. This agreement is due to the oscillation of the mean
value of the generator Jx in the MST since the purity depends
on the normalized square of such a mean value. The oscillations of 具Jx典 around a value close to J keeps also Psu共2兲 close
to its maximum possible value. In the JO regime the purity
also decays rapidly with the time as the system is driven
away from a coherent one, but the purity reaches much lower
values compared to the MST, never returning close to 1.
Hence, this regime presents lower quantitative agreement between the quantum and classical evolutions, when compared
to the MST regime. Close to ⍀t = 250 关inset of Fig. 1兴 the
state gets the closest possible to a coherent one, but Psu共2兲 is
still lower than 0.4. This result is expected because the classical orbit delocalizes much more on the Bloch sphere for
this regime, and this entails a correspondingly large delocalization of the semiclassical Q function on the sphere,
Q共␪ , ␾兲 = 具␪ , ␾兩␳兩␪ , ␾典, with ␳ = 兩␺典具␺兩, during its evolution
关8,21兴. The larger the region traveled by the classical trajectory in the phase space, the larger is the broadening of the
distribution and smaller the coherence left on the state.
Therefore, when the Hamiltonian has nonlinear terms in the
dynamical group generators, the semiclassical approximation
has better quantitative accordance with the exact quantum
results 共for finite N兲 when the classical orbits sweep smaller
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GENERALIZED PURITY AND QUANTUM PHASE…
“volume” in the phase space. Since the BECSDW model is
integrable, we cannot analyze how chaotic trajectories would
drive the system state away from a coherent one. However,
our present results indicate that the validity of the semiclassical method rapidly decreases for this type of less localized
trajectories.
III. PHASE TRANSITION
Now we can take advantage of the many qualities of
Psu共2兲 to characterize the QPT 关3兴 in the BECSDW model.
The QPT is connected to a nonanalyticity of the energy of
the fundamental state of the system, when it is taken as a
function of some real continuous parameter of the Hamiltonian 关1兴 at zero temperature, in the macroscopic limit N
→ ⬁. Generally speaking, the energy of the ground state in a
finite system is an analytic function of any parameter of the
Hamiltonian and only shows nonanalyticity when N → ⬁.
Even though such a limit is not effectively taken here, we
can still observe the scaling of the system properties for increasing N and infer about the QPT.
The BECSDW suffers a sudden change in its dynamics
⍀
in the limit of no cross-collision terms and N
when ␬c = 2N
Ⰷ 1. With ⍀ , ␬ ⬎ 0, such a transition 共JO to MST兲 does not
occur due to a critical change in the ground state, but at the
largest energy state for the parameter value corresponding to
the bifurcation in the classical phase space, leading to the
appearance of a separatrix of motion. However, it can be
treated formally in the same way as a usual QPT since the
maximum energy state is just the fundamental state through
the transformation Ĥ → −Ĥ or, equivalently, ⍀ , ␬ → −⍀ , −␬.
Moreover, there is evidence that energy levels up to the energy of the corresponding separatrix orbit sense the quantum
phase transition 关22兴.
In Fig. 2, we have the Q functions for the largest energy
eigenstate of Ĥ for several values of ␬ and ␩. Figure 2共a兲
shows the coherent state 兩␪ = ␲ , ␾典 = 兩J , J典z, which is the
maximum energy state of the noninteracting case ␬ = ␩ = 0.
Increasing ␬, but still not considering the cross-collision
terms, the Q function broadens along the x axis, and consequently we expect the decreasing in Psu共2兲. At ␬ = ␬c 关Fig.
2共b兲兴 the distribution is greatly broadened but does not show
a bifurcation, namely, the formation of two peaks. This behavior is expected since for finite N the quantum transition
parameter ␬qc 共N兲 is slightly different from the value of transition ␬c of the classical limit. But for increased ␬, such as in
Fig. 2共c兲 for ␬ = ⍀N , two peaks emerge along the x axis as a
signature of the bifurcation. The two peaks move away with
increasing ␬, while the increase in the cross collision causes
␬
the opposite effect. For ␬ = 2N⍀ and ␩ = 10
, the two peaks of
the Q function become closer, as in Fig. 2共d兲.
Our results for the phase-space distribution of the maximum energy state are confirmed as we analyze the behavior
of Psu共2兲 as a function of the self-collision parameter and the
number of particles, as in Fig. 3, neglecting the cross collisions. Psu共2兲 initially decreases slowly with ␬⍀N , independent
of the value of N, corresponding to the region where the
distributions broadens along the x axis. However, close
FIG. 2. 共Color online兲 Q function for the largest energy state of
the Ĥ spectrum for several value of parameters ␬ and ␩. Considering the number of particles N = 100 and ⍀ = 1, we have the follow⍀
ing values for the collision rates: 共a兲 ␬ = ␩ = 0; 共b兲 ␬ = 2N , ␩ = 0; 共c兲
⍀
2⍀
␬
␬ = N , ␩ = 0, and 共d兲 ␬ = N , ␩ = 10 .
to ␬⍀N = 21 , Psu共2兲 begins to decrease quickly and, although
smoothly, the lowering of its value gets more steep as we
increase N. This behavior of Psu共2兲 suggests us a strong dependence between the derivative of Psu共2兲 with respect to ␬⍀N
and the number of particles. Since Psu共2兲 signals entanglement, it is clear that particle entanglement is considerably
higher in the MST phase than in the JO phase. We can understand this feature since in the MST the atomic interaction
is much stronger than in the JO regime, leading thus to a
stronger entanglement.
The peaking of the derivative of Psu共2兲 共or any other conceivable measure of entanglement兲 with ␬⍀N signals a QPT
FIG. 3. 共Color online兲 Psu共2兲 for the largest energy eigenstate as
a function of normalized self-collision parameter and the number of
particles, with cross-collision rate ␩ = 0.
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PHYSICAL REVIEW A 80, 013610 共2009兲
VISCONDI, FURUYA, AND DE OLIVEIRA
the overall behavior can be inferred from the discussion
above 共see 关7,8兴 for a more complete discussion on the effect
of the cross-collision terms兲.
IV. CONCLUSION
FIG. 4. 共Color online兲 Derivative of Psu共2兲 of the eigenstate
␬N
of largest energy eigenvalue with respect to ⍀ for various values of
the total number of particles N. The dashed vertical line represents
␬ cN 1
the position at the value of the classical transition point ⍀ = 2 . All
the curves for finite systems cross exactly at this critical value.
关1,2兴. In Fig. 4 we show dPsu共2兲 / d共␬N / ⍀兲 for several N. For
increasing N we see the minimum derivative value shifting
to the left 共closer to the classical critical value 0.5兲 as
it becomes more pronounced. We define the value of ␬ at
the minimum as the critical value of the quantum dynamical
transition ␬qc 共N兲. It is clear that the value of ␬qc 共N兲 is brought
⍀
closer to the classical transition value ␬c = 2N
for increasing
N, but we still need to characterize how this approximation
N 共 ␬ q− ␬ 兲
happens. The values of ln关 ⍀c c 兴 from the curves in Fig. 4
suggest a power-law between 共␬qc − ␬c兲 and N. A linear
interpolation of the data points gives ␬qc − ␬c
= ⍀e0,31⫾0,05N−1,657⫾0,009. It is clear that ␬qc → ␬c as N → ⬁,
evidencing the signaling of the transition by Psu共2兲 in contrast
to the findings of 关4兴 for the ground state. This is so because
in the N → ⬁ limit, after the transition, the ground state is
twofold degenerated. However, in our finite N inference,
there is no degeneracy in the maximum energy level and
Psu共2兲 is perfectly suited for indicating the QPT occurring at
␬c, even though nothing can be said about its order 关2兴. The
effect of the cross-collisional term ␩ is merely to shift the
transition point ␬c and, despite of not being considered here,
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In conclusion we considered the generalized purity, Psu共2兲,
to analyze the dynamics of coherence loss of initially coherent state for the BECSDW model. In the MST regime Psu共2兲
remains high without significant departure from a coherentstate description. For the JO regime, on the other hand, we
have seen that the decay of Psu共2兲 was in a similar time scale
to the MST one, but the system state is strongly driven away
from a coherent one and no considerable recurrence to any
coherent state is observed. Since the coherent state represents
the closest to the classical state, the value of the generalized
purity enabled us to quantify the quality of the semiclassical
approximation at each time in both regimes. Moreover, we
have employed Psu共2兲 as a tool for characterizing a QPT in
the same model. We have shown for finite number of particles the bifurcation of the Q function of the largest energy
state as the self-collision parameter ␬ becomes larger than a
critical value ␬qc 共N兲. By increasing the number of particles N,
Psu共2兲 has shown a more and more steep behavior near the
critical value of ␬⍀N ; moreover, its value has approached the
␬N
known classical value ⍀c = 21 as N → ⬁. Finally we have
shown that ␬qc − ␬c is consistent with a power law in N.
Therefore, Psu共2兲 is an excellent measure of the dynamical
loss of coherence, distinguishing the two types of population
dynamics and indicating the QPT. When a state suffers fundamental changes resulting from the QPT, its generalized
purity must follow its behavior because it has all the information about its coherence and the degree of localization
over the phase space.
ACKNOWLEDGMENTS
We acknowledge J. Vidal for bringing several aspects of
the LMG model to our knowledge and for his comments and
L. Viola for a careful reading and suggestions. Our work is
supported by FAPESP and CNPq.
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