Phase-factor-dependent symmetries and quantum phases in a three-level cavity QED system

Abstract

Unlike conventional two-level particles, three-level particles may support some unitary-invariant phase factors when they interact coherently with a single-mode quantized light field. To gain a better understanding of light-matter interaction, it is thus necessary to explore the phase-factor-dependent physics in such a system. In this report, we consider the collective interaction between degenerate V-type three-level particles and a single-mode quantized light field, whose different components are labeled by different phase factors. We mainly establish an important relation between the phase factors and the symmetry or symmetry-broken physics. Specifically, we find that the phase factors affect dramatically the system symmetry. When these symmetries are breaking separately, rich quantum phases emerge. Finally, we propose a possible scheme to experimentally probe the predicted physics of our model. Our work provides a way to explore phase-factor-induced nontrivial physics by introducing additional particle levels.

Symmetry and spontaneous symmetry breaking are central concepts in modern many-body physics¹,²,³, due to their natural and clear relations with quantum phase transitions⁴. It is the emergence of a new phase that breaks an intrinsic symmetry of the system. More importantly, different symmetry-broken phases usually exhibit different ground-state properties. As a fundamental model of many-body physics, the Dicke model describes the collective interaction between two-level particles (such as atoms, molecules, and superconducting qubits, etc.) and a single-mode quantized light field⁵. In general, this model possesses a discrete Z₂ symmetry. When increasing the collective coupling strength, this model exhibits a second-order quantum phase transition from a normal state to a superradiant state⁶,⁷,⁸,⁹,¹⁰,¹¹,¹²,¹³,¹⁴,¹⁵,¹⁶,¹⁷,¹⁸,¹⁹,²⁰,²¹,²², with the breaking of the discrete Z₂ symmetry (Here we intentionally use the wording “normal/superradiant state” instead of “normal/superradiant phase”, since the word “phase” in the latter may be confused with another nomenclature “phase difference” which we will mention below). In the Z₂-broken superradiant state, the ground state is doubly degenerate. In contrast, under the rotating-wave approximation, the Dicke model reduces to the Tavis-Cummings model²³, with a continuous U(1) symmetry. In its corresponding U(1)-broken superradiant state, an infinitely-degenerate ground state can be anticipated. The above important symmetry and symmetry-broken physics of the Dicke and Tavis-Cummings models have been explored experimentally²⁴,²⁵,²⁶,²⁷,²⁸,²⁹. Recently, based on this two-level Dicke model, novel transitions between different symmetries³⁰,³¹,³², especially from the discrete to the continuous³¹,³², have been revealed.

Notice that apart from its amplitude, the single-mode quantized light field ε_φ ≡ ae^iφ + a^†e^−iφ, where a and a^† are the corresponding annihilation and creation operators, has an important freedom of phase³³,³⁴,³⁵ φ. In this sense, the interaction Hamiltonian of the standard two-level Dicke model becomes a phase-factor-dependent form, i.e.,

where J_i (i = ±) are the collective spin ladder operators. However, this phase factor e^iφ can be removed by a simple unitary transformation . This means that the phase factor does not affect the system symmetries as well as the superradiance phase transitions³⁶. So, it is a trivial variable in the standard two-level Dicke model. However, things become quite different when an extra energy level is introduced. In fact, when a three-level particle couples with a single-mode quantized light field via the electric dipole interaction, a nontrivial phase factor of light field can emerge naturally (see the supplementary material for a simple analysis). When the light-matter coupling strength becomes sufficiently strong, the rotating-wave approximation, under which any nontrivial phase factors can be removed by a certain unitary transformation (see the supplementary material for detailed discussions), breaks down, and thus new physics induced by the phase factor of the single-mode light field can be expected. In this sense, compared with the two-level systems, the three-level cavity QED systems serve as an ideal platform for studying physical effects induced by the phase factor of the quantized light field. Although some authors have considered interaction between three-level particles and the single-mode quantized light field³⁷,³⁸,³⁹,⁴⁰,⁴¹, these previous works ignored the potential appearance of the phase factor e^iφ (or equivalently, they just set φ = 0). That is, the physical effects induced by the phase factor e^iφ are still unknown. To gain a better understanding of light-matter interaction, it is thus necessary to explore the phase-factor-dependent physics in such a system.

In this report, we consider the collective interaction between degenerate V-type three-level particles and a single-mode light field, whose different components are labeled by different phases φ₁ and φ₂ [see the Hamiltonian (2) and Fig. 1 in the following]. Upon using this model, we mainly make a bridge between the phase difference, i.e., ϕ = φ₂ − φ₁, of the quantized light field and the system symmetry and symmetry broken physics. Specifically, when ϕ = π/2 or 3π/2, we find and symmetries as well as a nontrivial U(1) symmetry. If these symmetries are broken separately, three quantum phases, including an electric superradiant state, a magnetic superradiant state, and a U(1) electromagnetic superradiant state, emerge. When ϕ = 0 or π, we reveal a Z₂ symmetry and a trivial U^tr(1) symmetry. When ϕ ≠ 0, π/2, π, and 3π/2, only the Z₂ symmetry is found. If this Z₂ symmetry is broken, we predict a Z₂ electromagnetic superradiant state, in which both the electric and magnetic components of the quantized light field are collective excited and the ground state is doubly degenerate. Finally, we propose a possible scheme, in which the relative parameters can be tuned independently over a wide range, to probe the predicted physics of our model. Our work demonstrates that the additional particle level can highlight significant physics of the phase factor of the quadrature of the quantized light field, which can’t be captured in the two-level cavity QED systems.

Figure 1. Schematic picture of our considered system.

An ensemble of V-type three-level particles, in which and are degenerate, interacts with different phase-dependent components of a single-mode light field.

Results

Model and Hamiltonian

We consider N identical V-type three-level particles interacting with a single-mode quantized light field³⁷,³⁸, as sketched in Fig. 1. Each V-type particle consists of one ground state and two degenerate excited states and . Two transitions and are governed by different phase-factor-dependent components of the quantized light field, respectively. In the absence of the rotating-wave approximation, the total Hamiltonian reads³⁷

where ω is the frequency of the single-mode quantized light field, ω₀ is the transition frequency between the ground state and the two degenerate excited states and , λ_n (n = 1, 2) are the collective coupling strengths, φ_n and are the phases belonging to the quantized light field and the spin, respectively, and (i, j = 1, 2, 3) represent the collective spin operators. Two sets of spin operators {(A₃₃ − A_nn)/2, A_3n, A_n3}(n = 1, 2) construct the SU(2) angular momentum algebra, respectively, i.e., [A_3n, A_n3] = A₃₃ − A_nn, [(A₃₃ − A_nn)/2, A_3n] = A_3n, and [(A₃₃ − A_nn)/2, A_n3] = −A_n3.

Generally speaking, the parametric space of the spin-boson model is a direct product of several subspaces³⁵, i.e., , where J and X label the spin and bosonic subspaces, respectively. To illustrate clearly this feature in our model, we rewrite the Hamiltonian (2) as a compact form

where X(φ_m) =  and are the coordinates of the bosonic field and the spin in phase space³³,³⁴, respectively. Since and belong to two different spin subspaces and , they are independent, and can be removed by a unitary transformation . As a result, we set for simplicity. Whereas φ₁ and φ₂ belong to the same parametric space , and cannot be removed simultaneously. In fact, there exists a unitary-invariant phase difference ϕ = φ₂ − φ₁, and thus we may directly set φ₁ = 0 and ϕ = φ₂. We emphasize that the phase factor e^iϕ of the quantized light field is a unique feature of our model. If the counter-rotating wave terms are negelected³⁸,³⁹,⁴⁰,⁴¹ or two mode quantized light fields are considered⁴²,⁴³, the unitary-invariant phase factor e^iϕ disappears (see supplementary material for detailed discussions). As will be shown below, this phase factor e^iϕ plays an important role in determining symmetries and ground-state properties of the Hamiltonian (3).

Symmetries

It is straightforward to find that the Hamiltonian (3) is invariant when performing the following transformation:

which indicates that the Hamiltonian (3) has a Z₂ symmetry. In fact, when controlling ϕ as well as λ₁ and λ₂, the Hamiltonian (3) exhibits rich symmetries, as will be shown. For simplicity, we assume hereafter.

We first consider the case of ϕ = π/2 or 3π/2, in which the Hamiltonian (3) becomes

where X¹ = X(0) and X² = X(ϕ) are two quadratures of the quantized light field, which are called electric and magnetic components of the quantized light field, respectively. These couplings X¹J₁ and X²J₂ support two different Z₂ symmetries and , which can be broken separately³¹,³²:

More importantly, in the case of λ₁ = λ₂ = λ, the Hamiltonian (5) reduces to . For this Hamiltonian H_λ, we find a conserved quantity

i.e., [C, H_λ] = 0. In terms of this conserved quantity, we have H_λ = e^iθCH_λe^−iθC (θ is an arbitrary real number), which implies that the Hamiltonian H_λ has a nontrivial U(1) symmetry (i.e., this U(1) symmetry can be broken by phase transitions), apart from the and symmetries. We present an intuitive description of the symmetric properties of the Hamiltonian (5) in Fig. 2(a).

Figure 2.

(a,b) Symmetric diagrams in the (λ₁, λ₂) plane and (c) symmetric properties for different ϕ. Parameters are chosen as (a) ϕ = π/2 or 3π/2, (b) ϕ = 0 or π, and (c) λ₁ = λ₂. In (a) and (b), the yellow points denote the critical point λ_c in Eq. (30). In (c), the big blue circle represents the Z₂ symmetry. At the green and red points, the U^tr(1) ⊗ Z₂ and U(1) ⊗ symmetries emerges, respectively.

When ϕ = 0 or π, the Hamiltonian (3) is a simple sum of two standard Dicke models and exhibits a trivial U^tr(1) symmetry, apart from the Z₂ symmetry. To demonstrate those, we introduce two orthogonal states for ϕ = 0, and for ϕ = π. Taking account of these orthogonal states, we rewrite the Hamiltonian (3) as

where with are the collective operators in the new basis, and is an effective coupling strength. The Hamiltonian (9) shows clearly that the state is completely decoupled from the system, and thus serves as a “dark state”. This dark state, which can be used to realize the coherent population trapping⁴⁴,⁴⁵, induces a trivial ground-state manifold. By introducing a unitary transformation , we find , which indicates that the Hamiltonian (3) has a new U^tr(1) symmetry. Because of the complete decoupling of the state, this U^tr(1) symmetry can not be broken and is thus trivial. In Fig. 2(b), we give an intuitive description of these different symmetries.

When ϕ ≠ 0, π/2, π, and 3π/2, the operators J₁ and J₂ are coupled to two nonorthogonal components of the quantized light field, respectively. In this case, only the Z₂ symmetry is found, and the predicted dark state is also absent.

From above discussions, it seems that the symmetries of the Hamiltonian (3) are sensitive to the phase difference ϕ, as shown in Fig. 2(c). The breaking of these symmetries are associated with rich quantum phases and their transitions, as will be discussed below (It should be noticed that the wording “symmetry breaking” refers to “spontaneous symmetry breaking” in this report, which is different from another nomenclature called “explicit symmetry breaking⁴⁶”).

Ground-state properties

To investigate quantum phases and their transitions, we need to consider ground-state properties of the Hamiltonian (2), which can be implemented by a generalized Holstein-Primakoff transformation⁴⁷ and a boson expansion method⁴⁸. In the case of three levels, we should apply the generalized Holstein-Primakoff transformation³⁷,³⁸,⁴²,⁴³, with a reference state called , to rewrite the operators A_ij as

where and b_k are the bosonic operators. For the Hamiltonian (2), we choose m = 3 to rewrite it as

To explore the ground-state properties of the Hamiltonian (11) in the thermodynamic limit, we redefine these bosonic operators as

where α = α₁ + iα₂, β = β₁ + iβ₂, and γ = γ₁ + iγ₂. These complex auxiliary parameters , , and are the ground-state expectation values of the operators a, b₁, and b₂, respectively. Substituting Eq. (12) into the Hamiltonian (11) and then using the boson expansion method⁴⁸, we obtain

where

with , is the scaled ground-state energy. Based on Eq. (14), the scaled populations

as well as the scaled mean-photon number

can be obtained by analyzing equilibrium equations ∂h₀/∂Y = 0 (Y = α_1,2, β_1,2, γ_1,2), i.e.,

where μ = α₂ sin ϕ + α₁ cos ϕ. In addition, in order to distinguish the excitations of different components of the quantized light field, two extra quantities, including the scaled electric component of the quantized light field

and the scaled magnetic component of the quantized light field

should be introduced.

In general, it is difficult to get a complete solution from the mean-field ground-state energy (14). In this report, however, we are able to analytically consider two specific cases discussed in the previous section, namely ϕ = π/2 and λ₁ = λ₂ = λ, to illustrate the crucial role of the phase factor e^iϕ in manipulating the ground state. Apart from their analytical solutions, another advantage of these two special choices is that they support typical symmetric properties of the system [see Fig. 2(a,c)], which signals the potential emergence of interesting symmetry-broken physics.

We first address the case of ϕ = π/2, in which the scaled ground-state energy in Eq. (14) turns into

After a straightforward calculation, solutions of Eqs (17)–(22), are given by

where  = 4λ² + ωω₀ and λ = λ₁ = λ₂ in Eq. (29).

By means of the stable condition (see Methods), we find that the solutions in Eqs (26)–(29), are stable in the following regions: (i) λ₁ < λ_c and λ₂ < λ_c, (ii) and , (iii) and , and (iv) , respectively, where

is a critical point.

Considering both above corresponding solutions and the order parameters defined in Eqs (15), (16), (23), and (24), we reveal the following four quantum phases:

• Normal state. When λ₁ < λ_c and λ₂ < λ_c, . This means that no collective excitations occurs. In this phase, the ground-state manifold in the parametric space is a single point [see Fig. 3(a)], and the Z₂ symmetry always exists.

• Magnetic superradiant state. When λ₂ > λ_c and λ₂ > λ₁, , , and . This means that the three-level particles in the state and the magnetic component of the quantized light field are excited simultaneously, and the system has a doubly-degenerate ground state along the direction of the magnetic component of the quantized light field (i.e., the α₂ axis) in the parametric space [see Fig. 3(b)]. In this quantum phase, the symmetry is broken.

• Electric superradiant state. When λ₁ > λ_c and λ₁ > λ₂, , , and This means that the three-level particles in the state and the electric component of the bosonic field are excited simultaneously, and the system has a doubly-degenerate ground state along the direction of the electric component of the quantized light field (i.e., the α₁ axis) in the parametric space [see Fig. 3(c)]. In this quantum phase, the symmetry is broken.

• U(1) electromagnetic superradiant state. When λ₁ = λ₂ = λ > λ_c, , , and . This means that the three-level particles in both the and states as well as two quadratures of the quantized light field are excited simultaneously. Notice that and , the ground-state manifold in the parametric space is thus a circular valley [see Fig. 3(d)], which signals the breaking of the nontrivial U(1) symmetry.

Figure 3. Energy surfaces h₀ in the (α₁, α₂) plane.

(a) λ₁/ω = 0.2 and λ₂/ω = 0.1, (b) λ₁/ω = 0.7 and λ₂/ω = 1.3, (c) λ₁/ω = 1.3 and λ₂/ω = 0.7, and (d) λ₁/ω = λ₂/ω = 1.3. The other parameters are chosen as ω₀/ω = 1 and ϕ = π/2.

In Fig. 4, we plot the corresponding phase diagram as a function of λ₁ and λ₂. In terms of the scaled ground-state energy h₀, we find that the transition from the normal state to the electric superradiant state or the magnetic superradiant state or the U(1) electromagnetic superradiant state is of second order. However, the transition from the electric superradiant state to the magnetic superradiant state is of first order. In addition, the results of ϕ = 3π/2 are the same as those of ϕ = π/2, and are thus not discussed here.

Figure 4. The ground-state phase diagram as a function of λ₁ and λ₂.

In these abbreviations, NP denotes the normal state, ESP denotes the electric superradiant state, MSP denotes the magnetic superradiant state, and U(1) EMSP denotes the U(1) electromagnetic superradiant state (Red solid line). The other parameters are chosen as ϕ = π/2 and ω = ω₀.

We now address the other case of λ₁ = λ₂ = λ, in which the ground-state energy in Eq. (14) turns into

Following the previous procedure, the stable solutions of Eqs (26)–(29), should be divided into two cases, including (i) ϕ ≠ π/2 and 3π/2, and (ii) ϕ = π/2 or 3π/2.

When ϕ ≠ π/2 and 3π/2, we have

for λ < λ_c(ϕ), and

for , where

and

is a phase-dependent critical point. In the case of ϕ = π/2 or 3π/2, these stable solutions become

for λ < λ_c(ϕ), and

for

The stable solutions in Eqs (32)–(38), govern the interesting phase-dependent ground-state properties. In Fig. 5, we plot four order parameters, 〈a^†a〉/N, , , and 〈A₁₁〉/N, as functions of λ and ϕ. This figure shows that when varying ϕ from 0 to 2π, these four order parameters as well as the critical point exhibit a periodic behavior, which is a manifestation of the competition between the electric and magnetic components of the quantized light field. In particular, when ϕ ≠ π/2 and 3π/2 with λ > λ_c(ϕ), 〈a^†a〉/N = |α|² ≠ 0, and the superradiant state occurs. In this case, all of the nonzero parameters α_1,2, β_1,2 and γ_1,2 have two feasible values, which means the breaking of the Z₂ symmetry. When ϕ ≠ 0 and π in the Z₂-broken superradiant state, both two quadratures of the quantized light field are excited [see Fig. 5(d,f)]. This indicates that a new phase, called the Z₂ electromagnetic superradiant state, is predicted. A particularly interesting case is ϕ = π/2 or 3π/2 for λ > λ_c(ϕ), in which the scaled electric component (magnetic component) of the quantized light field, i.e., (), shows a nonanalytic behavior. As has discussed previously, under such a condition, the two quadratures of the quantized light field can acquire any available values continuously [see also Fig. 3(d)]. This is a definite signature of the breaking of the continuous U(1) symmetry and the U(1) electromagnetic superradiant phase thus emerges.

Figure 5. Four order parameters as functions of λ and ϕ.

(a,b) 〈a^†a〉/N, (c,d) , (e,f) , (g,h) 〈A₁₁〉/N. The result of 〈A₂₂〉/N is the same as that of 〈A₁₁〉/N, and is thus not plotted here. The other parameter is chosen as ω = ω₀.

In Fig. 6, we plot the corresponding phase diagram as a function of λ and ϕ. In terms of the scaled ground-state energy h₀, we find that the transition from the normal state to the Z₂ electromagnetic superradiant state is of second order. However, the transition from the Z₂ electromagnetic superradiant state to the U(1) electromagnetic superradiant state is of first order.

Figure 6. The ground-state phase diagram as a function of λ and ϕ.

In these abbreviations, NP denotes the normal state, ESP denotes the electric superradiant state (Blue solid line), Z₂ EMSP denotes the Z₂ electromagnetic superradiant state, and U(1) EMSP denotes the U(1) electromagnetic superradiant state (Red solid line). The other parameter is chosen as ω = ω₀.

In the standard Dicke model, there is only one component of the quantized light field governing the system properties⁸. However, in our consideration, because of the existence of a finite phase difference, both two quadratures of the quantized light field contribute to the properties of the Hamiltonian (3). In fact, as shown in Fig. 7, the phase factor dramatically modulates the ground-state distributions of the quantized light field in its phase space, which give rise to a possibility of simultaneous excitations of and .

Figure 7. Schematic diagram of the ground-state distributions of the light field in phase space.

If ϕ ≠ π/2 and 3π/2, and have two possible values, which is an intuitive manifestation of the breaking of the Z₂ symmetry. However, if ϕ = π/2 or 3π/2, and can be located at any point of a fixed circle. This, in contrast, signals the breaking of the U(1) symmetry. The other parameters are chosen as ω = ω₀ = λ.

Possible experimental realization

The lack of experimentally-tunable parameters in the conventional three-level atoms prevents the direct observation of above predicted phenomena. Here we first propose a generalized balanced Raman channels¹¹ to simulate this Hamiltonian, and then give a possible experimental implementation, based on the current experimental techniques of ultracold atoms in high-Q cavities²⁴,²⁷,⁴⁹. This scheme has a distinct advantage that the corresponding parameters in the realized Hamiltonian can be independently tuned over a wide range.

As shown in Fig. 8, an ensemble of seven-level atoms is coupled with two pairs of Raman lasers and a single photon mode (i.e., quantized light field). Each atom has three ground states and , and four excited states , , , and . The photon mode mediates the transitions , , , and (red solid lines), with coupling strengths , and , respectively. While two pairs of Raman lasers govern the other transitions , , , and (blue dashed lines), with Rabi frequencies , , , and , respectively. , , , and denote the detunings of the Raman lasers.

Figure 8. Atomic energy levels and their transitions.

The total dynamics in Fig. 8 is governed by the following Hamiltonian:

where

In the Hamiltonians (40)–(42), , , , , , and are the atomic frequencies, , , , and are the frequencies (initial phases) of two pairs of incident Raman lasers, φ is the phase of photon mode, z_j is the location of the jth atom in the laser beams, which support the wave numbers , , , , and k (note that ), and phases , , , and are acquired through three tunable optical lengths L₁, L₂, and L₃.

In the interaction picture with respect to the free Hamiltonian , the Hamiltonian (39) is transformed as

where , , , ω_A = ω_cav − ω_s, , , , and .

In the large-detuning limit, i.e., , where γ and κ are the atomic excited states’ linewidth and photon loss rate, respectively, we make an adiabatic approximation to eliminate all excited states of the Hamiltonian (43)¹¹,⁵⁰,⁵¹, and then obtain an effective Hamiltonian

where , , , , , , , , , and , with (α = r₁, r₂, s₂).

When the parameters are chosen as , , , , , and , the Hamiltonian (44) reduces to

where

The Hamiltonian (45) is our required Hamiltonian. In this Hamiltonian, all parameters can be tuned independently. For example, λ₁ and λ₂ can be driven by the Rabi frequencies or detunings of Raman lasers¹¹,²⁷, and φ₁ and φ₂ can be individually tuned by adjusting the optical lengths L₁, L₂, and L₃ or the wave-number difference δk_α (α = r₁, r₂, s₂) of Raman lasers⁵².

We now specify the implementation in an actual experimental setup. We consider an ensemble of ultracold atoms, loaded in a high-Q cavity, interacts simultaneously with a quantized cavity field and two pairs of Raman lasers. As shown in Fig. 9(a), a guided magnetic field B is applied along z direction to fix a quantized axis and split the Zeeman sublevels of the atomic ensemble, which confirms the distinct Raman channels. These two pairs of Raman lasers are right- and left-handed circularly polarized, respectively, and are assumed to co-propagate along the direction of the magnetic field. Moreover, the cavity field is linearly polarized along the y axis, which is perpendicular to the magnetic field. The detailed transitions are chosen as the D₂ line of ⁸⁷Rb atom, in which the three stable ground states and are chosen as some specific hyperfine sublevels of 5S_1/2, such as , , and , respectively, whereas the excited energy levels are assumed to be 5 and 5P_3/2 [see Fig. 9(b)].

Figure 9.

(a) The proposed experimental setup and (b) possible atomic excitation scheme based on the D₂ line of ⁸⁷Rb atom.

Based on above energy levels and their transitions⁵³, together with the current experimental conditions²⁷, the atom-photon coupling strengths can reach  MHz and  MHz, respectively. The atom number is set as²⁴,²⁷ . A proper choice of (), ranging from 0 to 0.04, is reasonable for the adiabatic condition in deriving Eq. (44). The practical parameters of the line width of cavity and atom are (κ/2π, γ/2π) = (0.07, 3.0) MHz²⁷. Note that due to the far-detuning coupling, the spontaneous emission rate of the atomic excited states can be suppressed strongly by a factor of []. Under these conditions, the collective coupling strength λ₁ and λ₂ can reach the order of several MHz, which is much large than the cavity and atomic decays, placing the system in a hamiltonian dynamics dominated regime. Furthermore, by properly tuning the frequencies of the cavity field and Raman lasers, it is not hard to achieve the superradiant condition .

Another issue to be specified is the experimental observation of different quantum phases, which lies in the measurement of the introduced order parameters, such as 〈A₁₁〉, 〈A₂₂〉, 〈X¹〉, and 〈X²〉. In a practical experiment, for example, the atomic population 〈A_ii〉 (i = 1, 2) can be straightforwardly obtained by detecting the transmission of certain probe field⁵⁴,⁵⁵, whereas the quadrature 〈Xⁱ〉 (i = 1, 2) could also in principle be measured by probing the cavity output using the technique of homodyne detection⁵⁶.

The above proposal with ultracold atoms provides just one example of the potential experimental implementations. Recent advances in circuit QED system make it another alternative candidate¹⁰,¹⁶,⁵⁷,⁵⁸,⁵⁹,⁶⁰. We hope our report could stimulate related works in that area.

Discussion

We briefly discuss the results of a deviation of the two levels and . A deviation of the two levels just adds an extra term δA_nn (n = 1 or 2) to the Hamiltonian (2), where δ is the deviation. In such case, Eq. (4) still remains invariant, and the discrete Z₂, , and symmetries can thus emerge. That is, the Z₂-broken phases can still be predicted. In contrast, due to existence of δ, no similar conserved quantity and decoupled state, as defined in the Result section, can be found. It implies that neither the trivial nor nontrivial U(1) continuous symmetries of the system can be found. Correspondingly, the U(1) electromagnetic superradiant state disappears.

Another point we should notice is that a thorough understanding of the Hamiltonian (2) demands paying more attention on the non-equilibrium properties. However, a complete description of the non-equilibrium features, which require more detailed and sophisticated analysis¹⁴,¹⁵, goes beyond the purpose of the present report. We leave this interesting problem for future investigation.

In summary, we have studied the V-type three-level particles, whose two degenerate levels are degenerate, interacting with a single-mode quantized light field, with a tunable ϕ. Upon using this model, we have made a bridge between the phase difference, i.e., ϕ = φ₂ − φ₁, of the light field and the system symmetry and symmetry broken physics. Specifically, when ϕ = π/2 or 3π/2, we have found and symmetries as well as a nontrivial U(1) symmetry. If these symmetries are broken separately, three quantum phases, including an electric superradiant state, a magnetic superradiant state, and a U(1) electromagnetic superradiant state, emerge. When ϕ = 0 or π, we have revealed a Z₂ symmetry and a trivial U^tr(1) symmetry. When ϕ ≠ 0, π/2, π, and 3π/2, only the Z₂ symmetry can be found. If this Z₂ symmetry is broken, we have predicted a Z₂ electromagnetic superradiant state, in which both the electric and magnetic components of the quantized light field are collective excited and the ground state is doubly degenerate. Finally, we have proposed a possible scheme, in which the relative parameters can be tuned independently over a wide range, to probe the predicted physics of our introduced model. Our work demonstrates that the additional particle level can highlight significant physics of the phase factor of the quadrature of the quantized light field, which can’t be captured in the two-level cavity QED systems.

Methods

To obtain stable quantum phases, we should introduce a 6 × 6 Hessian matrix , whose matrix elements can be calculated as (Y_i = α_1,2, β_1,2, γ_1,2). If the Hessian matrix is positive definite (i.e., all eigenvalues of are positive), quantum phases are stable, and vice versa.

Additional Information

How to cite this article: Fan, J. et al. Phase-factor-dependent symmetries and quantum phases in a three-level cavity QED system. Sci. Rep. 6, 25192; doi: 10.1038/srep25192 (2016).