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

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.

where the dipole operator is defined as In addition, the light field is assumed as a single-mode plane wave, i.e., where is the complex polarization vector of the light field and a is the corresponding annihilation operator. Therefore, the Hamiltonian (S1) becomes where g 1 and g 2 are real atom-photon coupling strengths and we have defined The Hamiltonian (S4) shows that due to the complex nature of the polarization vector and dipole matrix elements, some phases, called ϕ a , ϕ b , ϕ c , and ϕ d , emerge. Furthermore, if the dipole matrix elements are assumed to be real, i.e., d 31 = d * 31 and d 32 = d * 32 (or equivalently, ϕ a = ϕ b = ϕ 1 and ϕ c = ϕ d = ϕ 2 ), the Hamiltonian (S4) turns into which reduces to the same form as the Hamiltonian (2) of the main text.

B. V-TYPE THREE-LEVEL PARTICLES INTERACTING WITH TWO-MODE LIGHT FIELDS
When the states |1 and |2 of the V-type three-level particles are not degenerate, and moreover, interact with two light fields, as shown in Fig. S1(a), the corresponding Hamiltonian is given by where a 1 (a † 1 ) and a 2 (a † 2 ) are annihilation (creation) operators of two-mode bosonic fields, respectively. In contrast to the Hamiltonian (2) of the main text, the phases ϕ 1 and ϕ 2 in the Hamiltonian (S7) belong to two different parametric spaces. Therefore, all phases, including ϕ 1 , ϕ 2 , ϕ s1 , and ϕ s2 , can be removed by a unitary transformation This means that these phases do not affect the symmetries and the ground-state properties of the Hamiltonian (S7), and thus we may directly set ϕ 1 = ϕ 2 = ϕ s1 = ϕ s2 = 0.
It was revealed that the Hamiltonian (S7) has three quantum phases, including a normal phase, a red superradiant phase (collective excitation of the light field a 1 ), and a blue superradiant phase (collective excitation of the light field a 2 ) [1]. Correspondingly, there exists two other Z 2 symmetries Z R 2 and Z B 2 , which can be broken separately: The breaking of these symmetries are associated with different phase transitions (from the normal phase to the red superradiant phase or the blue superradiant phase). However, in the case of ω 1 = ω 2 , ω 01 = ω 02 , and λ 1 = λ 2 [see Fig. S1(b)], we find a conserved quantity which can induce a nontrivial U (1) symmetry. Furthermore, when λ 1 = λ 2 > λ c in this degenerate case (ω 1 = ω 2 and ω 01 = ω 02 ), this nontrivial U (1) symmetry is broken, and a red and blue superradiant phase, with collective excitations of both the light fields a 1 and a 2 , can thus be predicted.

C. RESULTS OF THE ROTATING-WAVE APPROXIMATION
Under the rotating-wave approximation, the Hamiltonian (2) of the main text becomes [2] In contrast to the Hamiltonian (2) of the main text, all phases in the Hamiltonian (S12) can be easily removed by a unitary transformation Therefore, these phases are no longer effective parameters to control the system's symmetries.
In addition, for the Hamiltonian (S12), we can still introduce two new operators to rewrite it as Clearly, the Hamiltonian (S15), which is a standard Tavis-Cummings model [3], shows that in the presence of the rotating-wave approximation, the symmetries and ground-state properties cannot be controlled by the phases of the