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 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 system-size [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 two-component 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]

where σ_α(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 S_α=∑_iσⁱ_α with the total spin number S, Hamiltonian (1) can be written, apart from a constant v(1+γ)/2, as

Hamiltonian (2) commutes with S ² and the exp(iπS_z), and thus, possesses a parity (spin-flip) symmetry. Apart from a constant -2vγ S ², Hamiltonian (2) can be rewritten as H_LMG=-2v(1+γ)S ² _x+2vγS ² _z-2hS_z.

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 the interaction strength is increased above a critical value v_c=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 h_c=0 [6, 7].

 

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.

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3. Theoretical scheme and hamiltonian

Figure 1 shows our proposed scheme for simulating Hamiltonian (2). The BEC, consisting of N two-level ⁸⁷Rb ultracold atoms with a transition |F=1〉(|1〉)→|F′=2〉(|2〉) of the D ₂ 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 D ₂ line [14, 15]. Moreover, for the optical cavity with the length 178 µm and the TME₀₀ mode 25 µm, 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

Here, the Hamiltonian for the photon is given by H_ph=ωa ^† a (h̄=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

where Ψ_l(r) is the boson field operator with [Ψ_l(r),Ψ^† _l(r′)]=δ^l _rr′, V_l(r) is the single magnetic trapped potential with frequency ω_i(i=x,y, z), m_R and ω ₁₂ are the mass and the resonance frequency of the ultracold atoms, respectively, q_l=4πρ_l/m_R (q _1,2=4πρ _1,2/m_R) denotes the intraspecies (interspecies) interactions among the ultracold atoms with ρ_l(ρ _1,2) 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

with g̃ 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

and an external driving laser with Hamiltonian

where Ω_p and $\tilde{\Omega }$ _d are the magnitude of the pump laser and the driving laser with frequencies ω_p and ω_d, respectively.

In the two-mode approximation defined as Ψ₁(r)=c ₁ ϕ ₁(r) and Ψ₂(r)=c ₂ ϕ ₂(r), where c ₁ and c ₂ are the annihilation boson operators with [c ₁,c ^† ₁]=[c ₂,c ^† ₂]=1 [24], Hamiltonian (3) can be simplified as

where ${\omega }_l=\int d^3\mathbf{r}\left\{{\varphi }_l^{*}\left(\mathbf{r}\right)\left[-\genfrac{}{}{0.1ex}{}{{\nabla }^2}{2mR}+V_l\left(\mathbf{r}\right)\right]{\varphi }_l\left(\mathbf{r}\right),\right\}$, η_l=q_l∫d ³ r|ϕ_l(r)|⁴, χ=q _1,2∫d ³ r|ϕ ₁(r)|²|ϕ ₂(r)|², Ω_d=2$\tilde{\Omega }$ _d∫d ³ r ϕ ^* ₂(r)ϕ ₁(r), and λ=2g̃∫d ³ r ϕ ^* ₂(r)ϕ ₁(r). In the SU(2) Schwinger representation S_z=(c†₂ c ₂-c ^† ₁ c ₁)/2, S ₊=c ^† ₂ c ₁and S _-=c ^† ₁ c ₂ and rotating-wave approximation, Hamiltonian (8) becomes

where q=[χ-(η ₁ +η ₂)/2] and ω ₀=(N-1)(η ₂-η ₁)/2+ω ₁₂≃ω ₁₂. Finally, by using a time-dependent unitary transformation U(t)=exp[-i(ωp_a ^† a+S_z ω_d)t], Hamiltonian (9) can be transformed into a time-independent Hamiltonian

where Δ_p=ω-ω_p and Δ_a=ω ₁₂-ω_p-ω_d.

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 $U=\exp \left[\genfrac{}{}{0.1ex}{}{\lambda }{{\mathrm{\Delta }}_p}{S}_x\left(a^{\mathrm{†}}-a\right)\right]$ can be used to rewrite Hamiltonian (10) as H_L=UHU ^†. By means of Baker-Campbell-Haussdorf formula, a biaxial collective spin model can be obtained by

where p=λ ²/(ω-ω_p), Δ_a=ω ₁₂-ω_p-ω_d, and Ω=Ω_d-λΩ_p/(ω-ω_p). Hamiltonian (11) is our expected LMG model with a generalized version. As was shown previously, Hamiltonian (11) with Δ_a=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, Δ_a and Ω can be controlled independently. For example, since the parameter q is proportional to Δ_ρ=[ρ _1,2-(ρ ₁+ρ ₂)/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, ρ ₁, ρ ₂, and 2ρ ₁₂ 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 collision-induced interactions among the ultracold atoms. Two examples are illustrated as follows. For Δ_a=0, Hamiltonian (11) can be reduced to a standard LMG model

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 Ω_c=2N|p-q|. It has been shown that the quantum system is only microscopically excited in the normal phase (Ω>Ω_c), whereas it undergoes a macroscopic collective coherent excitation in the so-called deformed phase (Ω<Ω_c) [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 Ω_c=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 higher-order 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 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].

 

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).

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5. Topological transition

For Δ_a=Ω=0, Hamiltonian (11) can be reduced to a special collective spin model

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)|S_z|Ω(θ,φ)〉=Scosθ and Ω(θ,φ)|S_x|Ω(θ,φ)〉=Ssinθ cosφ. Thus, the semiclassical energy E(θ,φ)=〈Ω(θ,φ)|H _L2 |Ω(θ,φ)〉 can be obtained by

which has double-degenerate ground-states at cosφ=±1 and

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 Schrödinger 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 ground states |0〉 and |π〉, this transition-amplitude P can be formally written in the SU(2) spin-coherent-state representation as P=〈π|exp(-βH _L2)|0〉=∫𝓓{Ω}exp[-(S_E+S_WZ)] [32], where β is the imaginary time-period, S_E=lim_β→∞∫^β ₀ E(θ,φ)dτ with τ=it, and

 

Fig. 3. (Color online) The scaled ground-state Berry phase Γ/S and the ultracold atom population 〈S_z〉/S versus the controllable interaction constant q.

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S_E 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. S_WZ, which is only taken place in the spin-coherent-state path integral, is usually called the topologicalWess-Zumino action since the contribution of any path on the Bloch sphere S ₂, described by the parameters θ(τ) and φ(τ), to the Wess-Zumino action S_WZ 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

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 〈S_z〉=Scosθ 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

Figure 3 shows the scaled Berry phase and ground-state ultracold atom population as a function of q.

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.