However, things become quite different when an extra energy level is introduced. So, it is a trivial variable in the standard two-level Dicke model. This means that the phase factor does not affect the system symmetries as well as the superradiance phase transitions 36. However, this phase factor e iφ can be removed by a simple unitary transformation. Where J i ( i = ±) are the collective spin ladder operators. In this sense, the interaction Hamiltonian of the standard two-level Dicke model becomes a phase-factor-dependent form, i.e., 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 phase 33, 34, 35 φ. Recently, based on this two-level Dicke model, novel transitions between different symmetries 30, 31, 32, especially from the discrete to the continuous 31, 32, have been revealed. The above important symmetry and symmetry-broken physics of the Dicke and Tavis-Cummings models have been explored experimentally 24, 25, 26, 27, 28, 29. In its corresponding U(1)-broken superradiant state, an infinitely-degenerate ground state can be anticipated. In contrast, under the rotating-wave approximation, the Dicke model reduces to the Tavis-Cummings model 23, with a continuous U(1) symmetry. In the Z 2-broken superradiant state, the ground state is doubly degenerate. When increasing the collective coupling strength, this model exhibits a second-order quantum phase transition from a normal state to a superradiant state 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, with the breaking of the discrete Z 2 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 general, this model possesses a discrete Z 2 symmetry. 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 field 5. More importantly, different symmetry-broken phases usually exhibit different ground-state properties. It is the emergence of a new phase that breaks an intrinsic symmetry of the system. Symmetry and spontaneous symmetry breaking are central concepts in modern many-body physics 1, 2, 3, due to their natural and clear relations with quantum phase transitions 4. Our work provides a way to explore phase-factor-induced nontrivial physics by introducing additional particle levels. Finally, we propose a possible scheme to experimentally probe the predicted physics of our model. When these symmetries are breaking separately, rich quantum phases emerge.
Specifically, we find that the phase factors affect dramatically the system symmetry. We mainly establish an important relation between the phase factors and the symmetry or symmetry-broken physics. 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. To gain a better understanding of light-matter interaction, it is thus necessary to explore the phase-factor-dependent physics in such a 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.
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