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Interaction-induced
Lipkin-Meshkov-Glick model in a
Bose-Einstein condensate inside an
optical cavity
Gang Chen1,2 , J. -Q. Liang3 and Suotang Jia1
1 State
Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University,
Taiyuan 030006, P.R. China
2 Department of Physics, Shaoxing College of Arts and Sciences, Shaoxing 312000, P.R. China
3 Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, P.R. China
[email protected]
Abstract:
In this paper we present an experimentally feasible scheme
to simulate a generalized Lipkin-Meskov-Glick model in a Bose-Einstein
condensate coupled dispersively with an ultrahigh-finesse optical cavity.
This obtained Hamiltonian has a unique advantage in that all parameters
can be controlled independently by using Feshbach resonance technique, a
pump laser along cavity axis and an external driving laser. By the proper
choice of parameters, the macroscopic quantum coherent effect with a large
amplitude can be successfully achieved. Comparing with the exist schemes,
our proposal has a cleaner, perhaps significantly improved to observe this
whole coherent effect. Finally, we predict a novel interaction-induced
topological transition, which is an abrupt variation from π to zero of the
Berry phase.
© 2009 Optical Society of America
OCIS codes: (270.5580) Quantum electrodynamics; (020.1475) Bose-Einstein condenstate
References and links
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Received 27 Aug 2009; accepted 26 Sep 2009; published 15 Oct 2009
26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19682
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1.
Introduction
Interacting spin models have been regarded as the basic models to investigate the fundamental quantum phenomena in modern physics. Recently, they have become the important tools to
process quantum information and implement quantum computing as well as to explore the complex topological order that supports exotic anyonic excitations [1]. However, the observations
of these predicted quantum effects remain a huge experimental challenge since the parameters
are not accessible to control and the scaling to many qubits is not well achieved in physical
quantum systems. Due to the controllability of system parameters and long decoherence times,
the ultracold atoms has been regarded as a promising candidate for simulating interacting spin
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models [2, 3, 4].
The exactly-solvable Lipkin-Meshkov-Glick (LMG) model, which was originally introduced
in nuclear physics [5], has now become a basic model to describe magnetic properties of a
collective spin system with long-rang interactions. By changing the effective magnetic field, this
model has a rich phase diagram in both ground- and excited- states, independent of the systemsize [6, 7, 8, 9, 10, 11]. In quantum information it can be used to test the fundamental relation
between many-body entanglement and quantum phase transition [12, 13]. In this paper we
propose an experimentally-feasible scheme to realize a generalized LMG model when a twocomponent Bose-Einstein condensate (BEC) interacts dispersively with an ultrahigh-finesse
optical cavity. One of advantages in this device is that the strong coupling of a BEC to the
quantized field of an ultrahigh-finesse optical cavity has been achieved experimentally [14, 15].
This not only gives rise to a new regime of cavity quantum electradynamics, in which all atoms
occupying a single mode of a matter-wave field can couple identically to the photon induced by
the cavity mode, but also opens possibilities to a wealth of new phenomena that can be expected
in the cavity-mediated many-body physics of quantum gas [16, 17, 18, 19, 20, 21].
It will be shown that all parameters in our realized LMG model can be controlled independently by the s−wave scattering lengths via Feshbach resonance technique, a pump laser along
cavity axis and an external driving laser. By the proper choice of parameters, the macroscopic
quantum coherent effect with a large amplitude can be successfully achieved. Comparing with
the exist schemes, our proposal has a cleaner, perhaps significantly improved to observe this
whole coherent effect. More intriguingly, a novel interaction-induced topological transition,
which is an abrupt variation from π to zero of the Berry phase, is predicted. A topological transition in a quantum system is characterized by a topological invariant that takes on different
quantized values in different quantum phases [22].
The rest of this paper is organized as follows. Section 2 is devoted to introducing the LMG
model. Section 3 is devoted to presenting our proposed scheme for simulating a generalized
LMG model with controllable parameters. Section 4 is devoted to realizing a macroscopic quantum coherence effect with a large amplitude. Section 5 is devoted to predicting an interactioninduced topological transition. Conclusions and remarks are given in Section 6.
2.
The Lipkin-Meshkov-Glick model
In this section we briefly introduce the LMG model, which describes a set of N spins half
mutually interacting in the anisotropic x − y plane embedded in a perpendicular magnetic filed.
The corresponding Hamiltonian is given by [5]
HLMG = −v ∑ (σxi σxj + γσyi σyj ) − h ∑ σzi ,
i< j
(1)
i
where σα (α = x, y, z) is the Pauli spin operator, v is the interacting energy, γ is the anisotropy
parameter, and h is the magnitude of magnetic field. By using the collective spin operator Sα =
∑i σαi with the total spin number S, Hamiltonian (1) can be written, apart from a constant
v(1 + γ )/2, as
(2)
HLMG = −2v(Sx2 + γ Sy2 ) − 2hSz ,
Hamiltonian (2) commutes with S2 and the exp(iπ Sz ), and thus, possesses a parity (spin-flip)
symmetry. Apart from a constant −2vγ S2 , Hamiltonian (2) can be rewritten as HLMG = −2v(1+
γ )Sx2 + 2vγ Sz2 − 2hSz .
It has been known that Hamiltonian (2) exhibits a second-order phase transition in the ferromagnetic regime (v > 0). For small interaction strength the system is in the normal phase,
where the ground state is unique and polarized in the direction of the magnetic field. When
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Pump laser
Detector
Driving laser
Fig. 1. (Color online) Schematic diagram for our considerations. The coherent dynamics of
the ultracold atoms and the cavity field can be driven by both applying a pump laser along
cavity axis and a classical driving laser.
the interaction strength is increased above a critical value vc = h, the system enters the broken
phase, in which the ground state becomes doubly degenerate. In the antiferromagnetic regime
(v < 0), Hamiltonian (2) has a first-order phase transition at a critical point hc = 0 [6, 7].
3.
Theoretical scheme and hamiltonian
Figure 1 shows our proposed scheme for simulating Hamiltonian (2). The BEC, consisting of N
two-level 87 Rb ultracold atoms with a transition |F = 1(|1) → |F = 2(|2) of the D2 line, is
created in a time-averaged, orbiting potential magnetic trap. The wavelength of such transition
is near 780 nm. In order to effectively strong coherent atom-photon dynamics, the wavelength
of empty cavity should be experimentally stabilized to the almost identical wavelength of the
D2 line [14, 15]. Moreover, for the optical cavity with the length 178 μ m and the TME00
mode 25 μ m, the maximum coupling strength between an ultracold atom and the cavity field is
g = 2π × 11.4 MHz [14, 15], which is larger than the cavity field decay rate κ = 2π × 1.3 MHz
and the ultracold atom dipole decay rate γ = 2π × 3.0 MHz. It means that the strong coupling
regime has been achieved.
When a pump laser along cavity axis and an external driving laser of the ultracold atom is
used to continuously control these coherent dynamics, the total time-dependent Hamiltonian
can be written as
(3)
H(t) = Hph + Hat + Hat−ph + Hpu + Hdr .
Here, the Hamiltonian for the photon is given by Hph = ω a† a (¯h = 1 hereafter), where a and
a† are the photon annihilation and creation operators with frequency ω . The ultracold atomic
Hamiltonian for the elastic two-body collisions with the δ -functional type potential is governed
by
Hat
=
d 3 r[ω12 Ψ†2 (r)Ψ2 (r) + q1,2 Ψ†1 (r)Ψ†2 (r)Ψ1 (r)Ψ2 (r)]
+
∑
l=1,2
d 3 r{Ψ†l (r)[−
(4)
∇2
ql
+Vl (r)]Ψl (r) + Ψ†l (r)Ψ†l (r)Ψl (r)Ψl (r)},
2mR
2
l , V (r) is the single magnetic
where Ψl (r) is the boson field operator with [Ψl (r), Ψ†l (r )] = δrr
l
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trapped potential with frequency ωi (i = x, y, z), mR and ω12 are the mass and the resonance
frequency of the ultracold atoms, respectively, ql = 4πρl /mR (q1,2 = 4πρ1,2 /mR ) denotes
the intraspecies (interspecies) interactions among the ultracold atoms with ρl (ρ1,2 ) being the
intraspecies (interspecies) s−wave scattering length. In experiment, these scattering lengths
can be controlled by the Feshbach resonance technique. The BEC-field interaction in the dipole
approximation via Hamiltonian
Hat−ph = g˜
d 3 rΨ†1 (r)Ψ2 (r)(a† + a) + H.c.
(5)
with g˜ being the BEC-cavity coupling strength [23]. The coherent dynamics of the ultracold
atoms and the cavity field is driven by both applying a pump laser with Hamiltonian
Hpu (t) = Ω p [a† exp(−iω pt) + a exp(iω pt)],
(6)
and an external driving laser with Hamiltonian
˜d
Hdr (t) = Ω
d 3 r[Ψ†2 (r)Ψ1 (r) exp(−iωd t)] + H.c.,
(7)
˜ d are the magnitude of the pump laser and the driving laser with frequencies
where Ω p and Ω
ω p and ωd , respectively.
In the two-mode approximation defined as Ψ1 (r) = c1 φ1 (r) and Ψ2 (r) = c2 φ2 (r), where c1
and c2 are the annihilation boson operators with [c1 , c†1 ] = [c2 , c†2 ] = 1 [24], Hamiltonian (3) can
be simplified as
H(t) = ω a† a +
∑ (ωl c†l cl + ω12 c†2 c2 +
l=1,2
ηl † †
λ
cl cl cl cl ) + χ c†1 c1 c†2 c2 + (c†1 c2 + c†2 c1 )(a† +(8)
a)
2
2
Ωd
†
c1 exp(−iωd t) + c+
+ [c+
1 c exp(iωd t)] + Ω p [a exp(−iω p t) + a exp(iω p t)]
2 2
3
∇2
d r{φl∗ (r)[− 2m
+ Vl (r)]φl (r),
ηl = ql d 3 r |φl (r)|4 ,
χ =
R
3
3 ∗
3 ∗
2
2
˜
q1,2 d r |φ1 (r)| |φ2 (r)| , Ωd = 2Ωd d rφ2 (r)φ1 (r), and λ = 2g˜ d rφ2 (r)φ1 (r). In
where
ωl
=
the SU(2) Schwinger representation Sz = (c†2 c2 − c†1 c1 )/2, S+ = c†2 c1 and S− = c†1 c2 and
rotating-wave approximation, Hamiltonian (8) becomes
H(t) = ω a† a + λ (S− a† + S+ a) + ω0 Sz − qSz2 +
Ωd
[S+ exp(−iωd t) + S− exp(iωd t)] (9)
2
+Ω p [a† exp(−iω pt) + a exp(iω pt)],
where q = [χ − (η1 + η2 )/2] and ω0 = (N − 1)(η2 − η1 )/2 + ω12 ω12 . Finally, by using a
time-dependent unitary transformation U(t) = exp[−i(ω p a† a + Sz ωd )t], Hamiltonian (9) can
be transformed into a time-independent Hamiltonian
H = Δ p a† a + λ (S− a† + S+ a) + Δa Sz − qSz2 + Ωd Sx + Ω p (a† + a),
(10)
where Δ p = ω − ω p and Δa = ω12 − ω p − ωd .
In the dispersive regime Δ p λ , the maximum of the scaled mean intracavity photon number
is far away from the critical intracavity photon number. As a result, the photon is virtually
excited. Therefore, a time-dependent unitary transformation U = exp[ Δλp Sx (a† − a)] can be
used to rewrite Hamiltonian (10) as HL = UHU † . By means of Baker-Campbell-Haussdorf
formula, a biaxial collective spin model can be obtained by
HL = −pSx2 − qSz2 + Δa Sz + ΩSx
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(11)
Received 27 Aug 2009; accepted 26 Sep 2009; published 15 Oct 2009
26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19686
where p = λ 2 /(ω − ω p ), Δa = ω12 − ω p − ωd , and Ω = Ωd − λ Ω p /(ω − ω p ). Hamiltonian (11)
is our expected LMG model with a generalized version. As was shown previously, Hamiltonian
(11) with Δa = 0 has been achieved theoretically in the trapped ions [25] and the four-level
alkali-metal atoms [26]. Although promising, these approaches are difficult since all trapped
ions (atoms) can not couple identically to the laser fields (cavity modes) , which is, however,
the main advantage of the ultracold atoms in BEC.
Hamiltonian (11) has a unique advantage in that the parameters p, q, Δa and Ω can
be controlled independently. For example, since the parameter q is proportional to Δρ =
[ρ1,2 − (ρ1 + ρ2 )/2], it can range continuously from the positive to the negative values due to
the competition among the s−wave scattering lengths. Far from the resonance, ρ1 , ρ2 , and 2ρ12
are approximately equal, and the collision-induced interactions among the ultracold atoms are
therefore suppressed. However, near the resonance nonzero s-wave scattering length Δρ leads
to a significant nonlinear interaction energy with a small value because Δρ can be changed by a
few 10% from its background value [27]. For typical experimental parameters with B = 9.131
G and N = 60 the parameter q can be evaluated as 2π × 4.6 Hz [28]. The cavity-assisted interactions can be driven by frequency of the pump laser, and especially, vary continuously from
the antiferromagnetic (p < 0) to the ferromagnetic (p > 0) cases. Also, the effective Rabi frequency Ω and the detuning Δa of the ultracold atoms depend on both the pump laser and the
driving laser.
4.
Macroscopic quantum coherent effect with a large amplitude
By properly choosing some parameters we can realize some interesting collective spin models
with rich phase diagrams, which are strongly determined by the cavity-assisted or the collisioninduced interactions among the ultracold atoms. Two examples are illustrated as follows. For
Δa = 0, Hamiltonian (11) can be reduced to a standard LMG model
HL1 = −pSx2 − qSz2 + ΩSx .
(12)
This Hamiltonian can exhibit a typical second-order phase transition from the normal phase to
the deformed phase via the effective Rabi frequency Ω in the ferromagnetic regime (p > 0). By
using the mean-field approximation [9], the critical point can be evaluated as Ωc = 2N |p − q|.
It has been shown that the quantum system is only microscopically excited in the normal phase
(Ω > Ωc ), whereas it undergoes a macroscopic collective coherent excitation in the so-called
deformed phase (Ω < Ωc ) [6, 7]. Experimentally, this quantum phase transition can be detected
by using the transmission spectroscopy with a weak probe laser since in our proposal the photon
of the cavity mode is virtually excited.
However, in the antiferromagnetic regime (p < 0) Hamiltonian (12) has a first-order phase
transition at the critical point Ωc = 0. Moreover, for finite-number ultracold atoms, it possesses
a macroscopic quantum coherent effect that the energy gap is periodically driven by the effective Rabi frequency Ω shown in Fig. 2. This phenomenon arises from the quantum phase
interference induced by the gauge potential, which is generated by the effective Rabi frequency
Ω along the hard (the highest energy) anisotropy direction [29]. The macroscopic quantum
coherent effect was partly observed experimentally in molecular magnetic of Fe8 due to higherorder anisotropies [30], and very recently, predicted in a BEC inside a double-well potential
[31]. It should be noticed that in their models the value for |p| / |q|, which determines the magnitude of the energy gap, is very small (for example, |p| / |q| ∼ 10−3 in Fe8 ). So, it is very
difficult to observe the fully macroscopic quantum coherent effect when the high-order effects
exist [30]. However, here |p| / |q| can arrive at a larger value of ∼ 103 . Therefore, we argue
that our proposal has a cleaner, perhaps significantly improved to observe this coherent effect.
Experimentally, we suggest a dynamic approach used in Fe8 to detect the energy gap. When the
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Fig. 2. (Color online) The energy gap of Hamiltonian (12) versus the effective Rabi frequency Ω. The parameters are given by q = 0.1 Hz and N = 20 with p = −10q (Red-solid
line) and p = −12q (Blue-dashed line).
effective Rabi frequency Ω is varied through the tunneling resonance, a coherent superposition
state is generated. By measuring the probability in each state this energy gap can be identified
by using the famous Landau-Zener formula [30].
5.
Topological transition
For Δa = Ω = 0, Hamiltonian (11) can be reduced to a special collective spin model
HL2 = −pSx2 − qSz2
(13)
In the SU(2) spin-coherent-state representation |Ω(θ , ϕ ) with the north pole gauge,
the expectation values for spin angular momenta are given by Ω(θ , ϕ )| Sz |Ω(θ , ϕ ) =
S cos θ and Ω(θ , ϕ )| Sx |Ω(θ , ϕ ) = S sin θ cos ϕ . Thus, the semiclassical energy E(θ , ϕ ) =
Ω(θ , ϕ )| HL2 |Ω(θ , ϕ ) can be obtained by
E(θ , ϕ ) = −pS2 sin2 θ cos2 ϕ − qS2 cos2 θ ,
which has double-degenerate ground-states at cos ϕ = ±1 and
0, q < p
cos θ =
.
1, q > p
(14)
(15)
We also denote these two degenerate orientations of the giant-spin as |0 (ϕ = 0) and |1 (ϕ =
π ), respectively.
In general, quantum tunneling between these two degenerate ground states prevails. As a consequence, the degeneracy is removed and two low-lying eigenstates with a small tunnel splitting d are formed by symmetric and antisymmetric superpositions of the macroscopic
quantum
√
states (the so-called Schr¨odinger cat states), such that |ψ± = (|0 ± |π )/ 2. In order to evaluate this tunnel splitting d, it is necessary to consider the imaginary-time transition-amplitude P
since the tunnel splitting is inversely proportional to the transition-amplitude. In the degenerate
written in the SU(2) spinground states |0 and |π , this transition-amplitude P can be formally
coherent-state representation as P = π | exp(−β HL2 ) |0 = D{Ω} exp[−(SE + SW Z )] [32],
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Fig. 3. (Color online) The scaled ground-state Berry phase Γ/S and the ultracold atom
population Sz /S versus the controllable interaction constant q.
where β is the imaginary time-period, SE = limβ →∞
SW Z =
θ (τ ),ϕ (τ )
θ (0),ϕ (0)
β
0
E(θ , ϕ )d τ with τ = it, and
iS(1 − cos θ )ϕ˙ d τ .
(16)
SE is the Euclidean action evaluated along the instanton trajectory, which is, as a matter of fact,
the tunneling path between the degenerate ground-states. Indeed, instanton may be visualized
as a pseudo-particle moving between degenerate vacua under the barrier region and has nonzero
topological charge but zero energy. SW Z , which is only taken place in the spin-coherent-state
path integral, is usually called the topological Wess-Zumino action since the contribution of any
path on the Bloch sphere S2 , described by the parameters θ (τ ) and ϕ (τ ), to the Wess-Zumino
action SW Z is equal to iS times the area swept out between the path and the north pole [33]. For
closed paths, this is identical to the Berry phase [34].
For Hamiltonian (13) with the degenerate condition in Eq. (15), the scaled Berry phase (mod
n, where n is the winding number, counting the number of times that the path wraps around the
north pole) can be immediately evaluated by
π, q < p
Γ/S = π (1 − cos θ ) =
.
(17)
0, q > p
Equation (17) reveals a novel interaction-induced topological transition for both integer and
half-integer spins, which is an abrupt variation from π to zero of the Berry phase. In terms of
Sz = S cos θ and Eq.(17), we can suggest in experiment to measure the ultracold atom population by using the transmission spectroscopy with a weak probe laser to detect this topological
transition [14]. By means of Eq. (15), the scaled ground-state ultracold atom population can be
obtained easily by
0, q < p
Sz /S = cos θ =
.
(18)
1, q > p
Figure 3 shows the scaled Berry phase and ground-state ultracold atom population as a function
of q.
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6.
Conclusions and remarks
In this paper, we have simulated a controllable LMG model based on recent experimental development about the BEC and the cavity. We have also obtained some interesting collective
spin Hamiltonians with rich phase diagrams, which are strongly determined by the nonlinear
quadratic terms arising from the cavity-assisted and the collision-induced interactions among
the ultracold atoms. Especially, we have predicted a novel interaction-induced topological transition, which is an abrupt variation from π to zero of the Berry phase, and a macroscopic quantum coherent effect with a large amplitude. Compared with the existed schemes, our proposal
has a cleaner, perhaps significantly improved to observe this coherent effect.
Acknowledgements
We thank H. B. Xue for helpful discussions and suggestions. Gang Chen and J. -Q. Liang
thank the supports of the Natural Science Foundation of China under Grant Nos.10704049
and 10775091, respectively. Suotang Jia thanks the supports of the 973 Program under Grant
No.2006CB921603 and the Natural Science Foundation of China under Grant Nos.10574084
and 60678003.
#115749 - $15.00 USD
(C) 2009 OSA
Received 27 Aug 2009; accepted 26 Sep 2009; published 15 Oct 2009
26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19690