Quantum phase transitions in the anti-Jaynes-Cummings triangle model

We carefully investigate the comprehensive impact of atom-cavity interaction and artificial magnetic fields on quantum phase transitions of anti-Jaynes-Cummings triangle model in the infinite frequency limit. We discover that ground states of the optical field can be a gapped normal phase (NP) or three kinds of gapless superradiant phases with infinite degeneracy. When the atom-cavity coupling is weak, the optical field is in a NP, which is a vacuum with no photons. Otherwise, it will stay at one of the superradiant phases: a normal superradiant phase without photon currents and another two chiral superradiant phases with opposite photon currents. The former only breaks the continuous U(1) symmetry and its gapless excitations are normal Goldstone modes. Nevertheless, the latter, mainly induced by an external synthetic gauge field, break both the continuous U(1) symmetry and chiral symmetry, thereby corresponding gapless excitations are chiral Goldstone modes. In addition, we also propose a detecting scheme to distinguish these superradiant phases.

Appearances of these quantum states are usually accompanied by quantum phase transitions (QPTs) or topological QPTs, during which an order parameter or a topological invariant will change [24][25][26][27].In Landau theory, spontaneous symmetry breaking and energy gap closing are the two typical characteristics in a continuous QPT [28], which results from the comprehensive interplay of intrinsic symmetries, particle interactions, external gauge field, etc.For example, for Dicke model implemented in cavity quantum electrodynamics (QEDs) or circuit QED, it was predicted to experience a superradiant phase transition that breaks the Z 2 symmetry if its atom-cavity interaction can exceed a threshold value [29][30][31][32]; synthetic gauge fields contribute to the emergence of Bose-Einstein condensates with non-zero momentum or angular momentum in ultra-cold atoms [33,34].The common point of these QPTs is that they generally happen in a many-body system under the thermodynamical limit [25].However, it was theoretically suggested that a QPT can also come out in few-body systems by taking the Rabi model as an example [35], which was observed in an experiment with a single trapped ion [36].Here the original thermodynamical limit is replaced by a frequency limit, that is, an infinite ratio of atomic transition frequency to cavity field frequency is required.Motivated by the easy manipulations, few-body systems turn out to be interesting and can be utilized to simulate many-body strongly-correlated systems [37,38].Recently, chiral photon currents are detected with a three-qubit loop in a synthetic magnetic field [39], as well as are predicted to exist in a Rabi triangle model by adjusting artificial gauge fields [40].Furthermore, the Rabi triangle model can be mapped into an effective magnetic model containing the XY exchange and Dzyaloshinskii Moriya interactions [41], which is useful for the study of topological phenomena in spin systems.
In this article, we explore the quantum ground states of an anti-Jaynes-Cummings (aJC) triangle model under the combined actions of atom-cavity interaction and artificial magnetic fields.Based on analytic approaches and numerical variation methods, we find out its quantum phase diagram in the infinite frequency limit.The cavity field is in a gapped normal phase (NP) possessing no photons when the atom-cavity coupling is weak; otherwise, it would rather remain at one of three superradiant phases breaking the continuous U(1) symmetry.We classify these novel states as normal superradiant phase (NSP) or chiral superradiant phase (CSP) depending on whether or not a photon current exists.Induced by an external synthetic gauge field, the CSPs have a non-zero photon current and also beak the chiral symmetry.All the three SPs are gapless with infinite degeneracy.Besides, a method is elaborated to successfully detect the NSP and CSPs in strong atom-cavity coupling.
This paper is organized as follows.In section 2, we introduce Hamiltonian of the aJC triangle model.We next derive effective Hamiltonians of the optical field in cases of both weak and strong atom-cavity interactions in section 3. Subsequently, we present the phase diagram and discuss NP and SPs as well as their spontaneous symmetry breaking in section 4. We also design a detecting scheme to discriminate these superradiant phases in section 5.At last, we conclude this article in section 6. Technical details can be found in the appendices.

Anti-Jaynes-Cummings triangle model
We consider quantum ground states of the aJC triangle model that is composed of three identical optical cavities as illustrated in figure 1.Each cavity can store a two-level atom with transition frequency Ω and a single-mode optical field with frequency ω.Photons can interact with the atom in each cavity, and hop between adjacent cavities forming a closed loop.We suppose this atom-cavity interaction to be the aJC-type, which does not appear naturally in cavity QED.However, the aJC-type interaction can be brought about in trapped-ion systems by tuning laser frequencies close to the blue sideband [42][43][44].Hence the aJC triangle model can be accomplished through leveraging trapped ions [45].Hamiltonian of this setup can be described as where σ + n = (σ − n ) † = |e⟩ n ⟨g| and Pauli matrix σ z n = |e⟩ n ⟨e| − |g⟩ n ⟨g| describe operations on an atom in the nth cavity, a n and a † n denote the photonic annihilation and creation operators, g represents strength of the aJC-type interaction between itinerant photons and atoms, t stands for the hopping strength of photons and h is the reduced Planck constant.When photons tunnel between neighboring cavities, a phase θ = ´rn+1 rn A(r)dr is picked up due to existence of a (synthetic) magnetic field with vector potential A(r) [46,47].The artificial gauge field can be implemented by a Floquet modulation method [39,40].In the aJC-triangle model, total number of excitation is invariant because [N e , H] = 0.That is, Hamiltonian H is invariant under the global gauge transformation U(ϕ) = exp(iϕ N e ), which will give rise to a continuous U(1) symmetry.According to the Goldstone theorem, its spontaneous symmetry breaking would lead to the appearance of gapless Goldstone modes [48,49].

Effective Hamiltonian of the optical field
We investigate QPT in the frequency limit Ω/ω → ∞ and mainly consider low energy physics in case of hω ≫ t.Therefore, in this aJC triangle model, the atomic transition energy hΩ is dominated.The two-level atom in each cavity will stay at its ground state in weak atom-cavity coupling or fluctuate around it when the coupling is strong.In contrast, the optical field would exhibit novel quantum states under different situations.
For weak on-site coupling g, we first need to make a Schrieffer-Wolff transformation To second order of g, the transformed Hamiltonian H 1 = U † 1 HU 1 has following form Due to existence of the periodic boundary condition, we next make a Fourier transformation to the momentum space with p = 0, ± 2 3 π.Hamiltonian H np can now be rewritten as with ) + 2t cos(θ − p), from which we can realize that it is valid only when the on-site coupling is weak g < g 0 and tunneling t is small enough such that we can always have ξ np (p) ⩾ 0 for any p and θ ∈ (−π, π).In our numerical calculation, we set t = 0.1hω.In the original frame, its ground state can be presented as with a p |0⟩ a = 0. From equation (3), it can be known that this ground state |ψ⟩ np is dependent on the atom-field interaction g.Its corresponding energy is ).This state indicates that there exists no photon and all atoms are at their ground states.So we can name it as NP.As g increase across some value, instability occurs when we have ξ np (p) < 0. Thus QPT will take place when the on-site coupling g is strong enough.
For strong on-site coupling g, the optical field can not remain in a vacuum state any longer.We need to make a local translation to the cavity field To clarify clearly effects of the local translation α n , it can be helpful to apply a unitary transformation We will get a new translated Hamiltonian with phase θ n = θ + γ n+1 − γ n and a local atomic Hamiltonian It is needed to note that the continuous U(1) symmetry is manifested in freedom of selecting phase γ n : Hamiltonian H ′ 2 is invariant under a phase shift γ n → γ n + ϕ.From equation ( 12), we can realize that if there are photons in cavities, on one hand, they may induce an effective tunneling phase θ n ; on the other hand, they can perturb an atom in the nth cavity away from its ground state |g⟩ n .As to this two-level atom, its Hamiltonian H qn can be diagonalized as In its eigenstates bases, Hamiltonian H ′ 2 can be changed into (see equation (A.17)) As atomic energy hΩ n is predominant in Hamiltonian H ′ 2 , it is valid to assume that the atom in each cavity would keep occupying its ground state |−⟩ n .The first line in equation ( 14) can be eliminated by demanding that Under this situation, Hamiltonian H ′ 2 becomes which can be diagonalized by another Schrieffer-Wolff transformation U 2 = e −S2 with We mark the diagonalized Hamiltonian as H 2 and in the limit of Ω/ω → ∞, it becomes to second order of on-site coupling g (see equation (A.24)).Hamiltonian of the cavity field can be defined as with dimensionless parameters ), t ′ = t hω and undetermined energy In the Nambu basis , we can rewrite this effective Hamiltonian as Excitation spectra ξ 1,2,3 of Hamiltonian H sp can be obtained by a Bogoliubov transformation ζ to a new basis with ground state energy We can arrive at these excitation spectra by numerically diagonalizing ΣH sp with diagonal matrix Σ = diag{1, 1, 1, −1, −1, −1} [50].By exploiting numerical variation method, the ground states can be solved if the energy E sp is minimized and equation ( 15) is satisfied at the meantime.

Quantum phase transitions and phase diagram
It has been pointed out that QPTs would happen when the on-site interaction increases across some critical value in section 3.In the following we proceed to study effects of symmetries and gauge fields on the ground states.With analytic and numerical variational methods, we figure out a phase diagram for fixed and small enough tunneling t as shown in figure 2. When the on-site coupling g is weak, optical field prefers to stay in a NP, which actually is a vacuum; otherwise, its ground states evolve into superradiant phases with translation symmetry, that is, |α n | = α and γ n+1 − γ n = γ are not dependent on n.Three types of superradiant phases with infinite degeneracy are induced by the tunneling phase θ, which can make γ = 0 or γ = ± 2  3 π.If the on-site interaction g is weak, effective Hamiltonian of optical field now is H np (see equation ( 5) or ( 7)).In the limit Ω/ω → ∞, its ground state is a NP with rescaled energy and rescaled order parameter Phase transitions from NP to SPs occur when the energy gap ξ np (p)| min closes.From the varying characteristics of excitation spectra {ξ np (p)} shown in figure 3 , we can know that ξ np (p)| min = 0 can enable us to judge this boundary with f(θ, γ) = 1 + 2t ′ cos(θ + γ).By the aid of this equation, we can determine the boundaries between NP and SPs marked by a white-dashed line in the phase diagram (see figure 2).
If the on-site interaction g is strong enough, effective Hamiltonian of optical field now changes into H sp (see equation (19) or (22)).Its ground states can not be worked out at a glance.It is worth stressing that local translations {α n } should not only minimize the ground energy E sp but also satisfy the preconditions in equation (15).So we can find them out numerically by adapting variational methods, and learn that the translations {α n } should meet that Thus, effective Hamiltonian H sp of the optical field turns into which is translation invariant and also would be reasonable as to a strong and isotopic on-site interaction g.Using firstly a Fourier transformation to the momentum space and then a Bogoliubov transformation to a new basis b p with [b p1 , b † p2 ] = δ p1,p2 , Hamiltonian H sp can be diagonalized as (more details can be found in appendix B) with energy ] . Here we have defined whose eigenvalues are continuous and belong to (−∞, +∞).So the excitation spectra of H sp are Definition of single-particle energy ϵ(p) can be found in equation (B.5).Varying of the excitations along with on-site coupling g and tunneling phase θ are plotted in figure 3. Therefore, for the phase transitions from NP to SPs, the energy gap closes and does not reopen anymore.The NP is gapped and the SPs are gapless guaranteed by breaking of the continuous U(1) symmetry, which makes these SPs infinitely degenerate.In the original frame, its corresponding ground state is with b p |0⟩ p = 0 and x 0 |0⟩ x = 0 [51].From equations ( 9), ( 11), ( 17) and ( 26), we know that this ground state |ψ⟩ sp is determined by the atom-field interaction g, hopping strength t and phase θ.In the limit Ω/ω → ∞, its rescaled ground state energy is and its rescaled order parameter can be calculated as In figure 4, we illustrate that this order parameter will continuously increase from zero in NP to non-zero in SPs.Hence, phase transitions from NP to SPs are of second order, which can also be confirmed from discontinuous second-order derivative of the excitation energy with respect to on-site coupling strength g, i.e.Ω |⟨an⟩| in transition from NP to SP varying with on-site coupling g.Blue lines are its values in the limit of Ω/ω → ∞ and red lines stand for that obtained by numerical variation method.The three panels correspond to results with tunneling phase θ = −0.5π(a), 0.5π (b) and 0.9π (c) respectively.We set other parameters as ω = 1, Ω = 10 3 ω, t/h = 0.1ω and h = 1.
At the phase boundaries of NSP and CSP, their corresponding ground energies should be equal form which we can acquire that θ c = ± 2 3 π [see the black-solid lines in phase diagram].Similarly, phase boundary between the two CSPs can be ascertained by solving equation which leads to θ c = 0 (see the black-solid line in phase diagram).To discriminate these diverse SPs, it would be useful to define following current operator [39,40] Its rescaled mean value is which reveals that there is no photon current in the NSP, because γ = 0. We thus name it as normal supreradiant phase.It can be realized that this mean value ⟨I⟩ is also zero in the NP, because it is actually a vacuum state of the cavity field.Since 1 f 2 − g 4 0 g 4 > 0 in these SPs, direction of photon current ⟨I⟩ is dependent on the sign of sin(γ).When γ = 2  3 π, we know ⟨I⟩ < 0 and ω g 4 ] at the phase boundaries θ c = 0, − 2 3 π.Such direction-dependent (chiral) characteristic of the photon current is exhibited in figure 5. Accordingly, transitions between the SPs are all of first order.The NSP only breaks the continuous U(1) symmetry, so its corresponding gapless excitations are normal Goldstone modes.In CSPs, the photon currents are unidirectional (flowing clockwise or anti-clockwise), thus they also break the chiral symmetry and the gapless excitations are chiral Goldstone modes.

Detection of superradiant phases
As aJC-type interaction can be easily realized in trapped-ion systems, we consider detection of NSP and CSPs of the aJC triangle model in a trap-ion platform, where a phase transition from NP to SP in the Rabi model has been observed by detecting the spin population and average phonon number [36].To discriminate NSP  and CSPs in the phase diagram (see figure 2), the point is to measure phase γ of the complex translation α n , which can be probed by measuring spin of the 'n = 1' trapped ion.In the frequency limit Ω/ω → ∞, we know that ⟨σ z 1 ⟩ = −1 in NP and ⟨σ z 1 ⟩ = −Ω/ √ Ω 2 + g 2 α 2 ̸ = −1 in SPs as shown in figure 6.As a result, we can acquire the value of α by probing σ z 1 in an experiment.After observing a non-zero α, we can infer the phase γ by utilizing equation ( 26), then determine the SPs.
We can also verify the SPs by probing the blue sideband transitions of the 'n = 1' trapped ion, which consists of three steps (more details can be found in appendix C): (i) applying a short optical pumping pulse to pump internal state of the 'n = 1' trapped ion into |g⟩ 1 .This pulse should be set short enough so that its effect on the motional state (phonon state) population of the three trapped ions can be neglected.In the frequency limit Ω/ω → ∞, we can prepare the system into an initial state |ψ(0)⟩ (see equation (C.2)); (ii) a running wave light field is used to drive blue sideband transition of the 'n = 1' trapped ion for various time intervals t.Under the resonance condition of first blue sideband [52], its evolution is described by the interaction Hamiltonian with coupling frequency Γ determined by the external light field.At time t, wave function of the system evolves into |ψ( t)⟩ = e − i h H i t|ψ(0)⟩.
(iii) measuring the probability to find the 'n = 1' trapped ion in its ground state |g⟩ 1 , which can be defined as P g ( t) = ⟨ψ( t)|(|g⟩ 1 ⟨g|)|ψ( t)⟩.Through meticulous analysis, we can derive that where is the Hermitian polynomials and µ is defined in equation (B.6).Although there are many undetermined oscillation frequencies in the probability P g ( t), we can extract analytically two types of oscillation frequencies Γn and Γ.Just as expected, the frequencies Γn are closely related to populations of Fock states |n⟩ in the ground state |0⟩ x [52].The frequency Γ originates from the complex translation (α n ) and excitations (µ), which can give us information about γ and α.
Following the procedures provided above, we numerically calculate the spin evolution probability P g ( t) of the 'n = 1' trapped ion and its Fourier transformation P g (w) = ´Pg ( t)e −iwt d t as shown in figure 7. It is clear that oscillation of the spin probability P g ( t) turns out to be very intricate after evolution of long enough time in both CSPs (figures 7(a) and (c)) and NSP (figure 7(e)), which implies that abundant oscillation frequencies exist in the spin evolution probability P g ( t) (see figures 7(b), (d) and (f)).The subplots in figures 7(b), (d) and (f) explicitly display that a summit of the probability amplitude |P g (w)| arises at a frequency near the predicted value w = Γ = √ α 2 + 2 3 µ 2 Γ (the red-dash lines).Consequently, once this kind of summit is found in realistic evolution experiments, we can assert that the aJC triangle model is in a superradiant phase, and then identify the SP is NSP or CSP.It should be emphasized that the coupling frequency Γ should be set large enough such that the evolution time needed can be much less than the system's decoherence time.

Conclusion
In this paper, we have studied QPTs of the aJC triangle model in the infinite frequency limit by analytical and numerical variation methods.The on-site atom-cavity coupling gives rise to a continuous phase transition from a gapped NP to a gapless superradiant phase breaking the continuous U(1) symmetry.Meantime, the external artificial gauge field can lead to first-order phase transitions between NSP and CSPs.These SPs are of infinite degeneracy.There exist two types of directed photon currents in the CSPs, which therefore also break the chiral symmetry.Combined interaction of the atom-cavity coupling and artificial gauge fields can yield novel quantum states in the aJC triangle model.Distinguishing of NSP and CSPs can be fulfilled by utilizing the suggested detection scheme that is intended to seek out the phase γ and special oscillation frequency Γ of the spin evolution probability P g ( t).Our study may advance the search for exotic quantum states in few-body systems and their applications in quantum simulation or quantum metrology.
in the frequency limit Ω/ω → ∞.Here we have defined a finite parameter g 0 = 2 √ Ωω.Thus, to second order of g, this Hamiltonian becomes As transition frequency Ω of the two-level atom is dominated, atoms should stay at their ground states Then, physical behaviors of the optical field are governed by Hamiltonian H np = ⟨0|H 1 |0⟩ σ , whose concrete form becomes At strong on-site coupling g, the optical field can no longer remain in a vacuum state.We need to make a local translation to the cavity field with α n = |α n |e iγn , and get a translated Hamiltonian H ′ 2 = D † HD.Then this new Hamiltonian H ′ 2 reads To clarify effects of the local translation α n , it can be helpful to apply a unitary transformation Because it can be identified that we will get a new translated Hamiltonian H ′ 2 = U † H ′ U and know that with eigenstates and sin( In this eigenstates bases, we can have that And Hamiltonian H ′ 2 can be changed into As atomic energy hΩ n is predominant in Hamiltonian H ′ 2 , it is valid to assume that the atom in each cavity would keep in its ground state |−⟩ n .The first line in equation (A.17) can be eliminated by demanding that which is equal to with n = 1, 2, 3.Under this situation, Hamiltonian H ′ 2 becomes which can be diagonalized by another Schrieffer-Wolff transformation U 2 = e −S2 with We mark the transformed Hamiltonian using H 2 and expand it in this way In the limit of Ω/ω → ∞, we can find that and With the aid of these two equations, Hamiltonian H 2 becomes to second order of on-site coupling g.So Hamiltonian of the cavity field can be defined as

Appendix B. Diagonalizing effective Hamiltonian H sp
In superradiant phases, effective Hamiltonian of optical field is H sp (see equation (19) or ( 22)).We find that the translations {α n } should meet with f(θ, γ) = 1 + 2t ′ cos(θ + γ).Thus, effective Hamiltonian H sp will turn into Because it is translation invariant, we can make a Fourier transformation to the momentum space and will discover that with single-particle energy Using a Bogoliubov transformation It should be noted that because ϵ(0) = λhω, we can not use the same way to diagonalize H(0) [53], whose concrete form is The ground energy E sp can be calculated as Secondly, applying a running wave light field to drive the 'n = 1' trapped ion for various time intervals t.
Under the resonance condition of first blue sideband [52], its evolution is described by following interaction Hamiltonian

Figure 1 .
Figure 1.Schematic diagram of the aJC triangle model.It consists of three uniform optical cavities with photon frequency ω.A two-level atom resides in each cavity and can interact with the intracavity photons.Transition frequency of the two-level atom is Ω.Photons can tunnel between adjacent cavities forming a closed loop.The complex tunneling strength is labeled as te ±iθ .