Macroscopic quantum states, quantum phase transition for $N$ three-level atoms in an optical cavity -- Gauge principle and non-Hermitian Hamiltonian

Abstract

We study in this paper the quantum phase transition (QPT) from normal phase (NP) to superradiant phase (SP) for $N$ three-level atoms in a single-mode optical cavity for both Hermitian and non Hermitian Hamiltonians, where the $Ξ$-type three-level atom is described by spin-$1$ pseudo-spin operators. The long standing gauge-choice ambiguity of $\mathbf{A\cdot p}$ and $\mathbf{d\cdot E}$ called respectively the Coulomb and dipole gauges is resolved by the time-dependent gauge transformation on the Schrödinger equation. Both $\mathbf{A\cdot p}$ and $\mathbf{d\cdot E}$ interactions are included in the unified gauge, which is truly gauge equivalent to the minimum coupling principle. The Coulomb and dipole interactions are just the special cases of unified gauge. Remarkably three interactions lead to the same results under the resonant condition of field-atom frequencies, while significant difference appears in red and blue detunings. The QPT is analyzed in terms of spin coherent-state variational method, which indicates the abrupt changes of energy spectrum, average photon number as well as the atomic population at the critical point of interaction constant. Crucially, we reveal the sensitive dependence on the initial optical-phase, which is particularly useful to test the validity of three gauges experimentally. The non-Hermitian atom-field interaction results in the exceptional point (EP), beyond which the semiclassical energy function becomes complex. However the energy spectrum of variational ground state is real in the absence of EP, and does not become complex. The superradiant state is unstable due to the non-Hermitian interaction induced photon-number loss. Thus only the NP exists in the non-Hermitian Dicke Model Hamiltonian.

Figures (10)

• Figure 1: Diagram of the extended DM with N three-level atoms in an optical cavity. Atomic states$\left\vert 1\right\rangle$(bottom), $\left\vert 2\right\rangle$(middle), and $\left\vert 3\right\rangle$(top) are shown by black horizontal bars. Curved arrows denote photons of frequency $\omega$, while solid arrows indicate light-matter interactions between atomic and cavity modes, with dimensionless coupling constant G between the light field and atoms
• Figure 2: Energy spectrum $\varepsilon _{-}$ (1), photon number $n_{p\text{ }}$(2), and atomic population $n_{a}$ in the Coulomb gauge respectively for red detuning $\eta =0.5$ (a), resonance $\eta =1$ (b), blue detuning $\eta =1.5$ (c). The phase transition point $G_{c}$ shifts to the higher value with the increase of detuning.
• Figure 3: The energy spectrum $\varepsilon _{-}$ (1), photon number $n_{p}$ (2) and atomic population $\Delta n_{a}$ (3) in the dipole gauge respectively for red-detuning (a), resonance (b), and blue-detuning (c). The phase transition point $G_{c}$ shifts to the lower value with the increase of detuning.
• Figure 4: Phase diagrams in $G-\eta$ space for Coulomb gauge (a) and dipole gauge (b). The photon number $n_{p}$ in SP is indicated by color scale.
• Figure 5: The energy spectrum $\varepsilon _{-}$ (a), photon number $n_{p}$ (b) and atomic population $\Delta n_{a}$ (c) at fixed field phases $\phi$ =$\pi /6,\pi /4,\pi /3$ for red detuning $\eta =0.5$ (1), resonance $\eta =1$ (2), blue detuning $\eta =1.5$ (3).