Quantum phase transitions triggered by a four-level atomic system in dissipative environments

We investigate the quantum phase transitions (QPTs) of a four-level system in a noisy environment. The system is composed of a four-level atom which is driven by classical fields and simultaneously coupled with a quantum field mode. A second-order dissipative phase transition (DPT) is observed by combining unitary processes with dissipation dynamics. The analytical expression for the DPT in the anisotropic Rabi model is given. We demonstrate that the QPT in an open four-level system is characterized as several boundaries, which is completely different from that in a closed system. In addition, our model is robust against strong atomic spontaneous emission, providing a promising platform for observing DPT in an open system and studying the related dynamics associated with DPT.

Figure 1

Schematic of a four-level atom trapped into an optical cavity. The inset shows the energy-level structure of the atom.

Figure 2

(a) The photon number and (b) ADR as a function of $g$. The parameters are set as ${\omega }_0^{\prime }=1, q=1,\eta =1000,\kappa /{\omega }_0^{\prime }=0.1$. Other parameters are solved using the conditions in the text.

Figure 3

The fidelity $F(g,g+\delta g)$ as a function of $g$ for different $\eta$ with $\delta g=0.01g$. The common parameters are the same as in Fig. 2.

Figure 4

The phase diagrams. (a)[(b)] $1-F(g,g+\delta g)$ and (c)[(d)] ${\langle {\widehat{a}}^{†}\widehat{a}\rangle }_s$ as functions of $g$ and $q$ for the full Hamiltonian $\widehat{H}$ [the Rabi Hamiltonian ${\widehat{H}}_{\text{Rabi}}$]. The parameters are $\eta =100,\delta g=0.01g,p=40$. Other parameters are the same as in Fig. 2. The black dotted lines are given by Eq. (11). (e), (f) ${\langle {\widehat{a}}^{†}\widehat{a}\rangle }_s$ as a function of $g$ with $q$ being fixed as (e) 0.2 and (f) 4, respectively.

Figure 5

The photon number as a function of $g$ for different ${\mathrm{\Delta }}_2$, with $q=1,p=21,\eta =100,\kappa /{\omega }_0^{\prime }=0.1$, and the remaining parameters are the same as in Fig. 2. The solid lines are obtained from the numerical simulation of the full Hamiltonian $\widehat{H}$ in Eq. (1), and the dashed lines are from the effective Hamiltonian ${\widehat{H}}_{\text{eff}}$ in Eq. (4). The phase transition peaks are marked with circles and the letter P.

Figure 6

(a) The photon number ${\langle {\widehat{a}}^{†}\widehat{a}\rangle }_s$ and (b) fidelity $1-F(g,g+\delta g)$ as a function of $g$ for different rates of atomic spontaneous emission $\gamma$. The parameters are $q=1,\eta =100,\kappa /{\omega }_0^{\prime }=0.1$, and the other parameters are the same as in Fig. 4. Here, we assume that all ${\gamma }_i$ are the same and denoted as $\gamma$.

Figure 7

(a) Schematic diagram of the possible evolution paths of the system. The energy differences are ${\mathrm{\Delta }}_n^{\left(1\right)}={\omega }_0^{\prime }+\mathrm{\Omega }, {\mathrm{\Delta }}_n^{{\left(1\right)}^{\prime }}=\frac{{\lambda }_1^2}{{\mathrm{\Delta }}_2}+{\omega }_0^{\prime }+\mathrm{\Omega }, {\mathrm{\Delta }}_n^{\left(2\right)}={\omega }_0^{\prime }-\mathrm{\Omega }, {\mathrm{\Delta }}_n^{{\left(2\right)}^{\prime }}=\frac{{\lambda }_2^2}{{\mathrm{\Delta }}_1}+{\omega }_0^{\prime }-\mathrm{\Omega }$. (b) Equivalent schematic diagram of the matrix form of the effective Hamiltonian ${\widehat{H}}_{\text{eff}}$ in the Fock-state basis with basis vectors of $\{...|g_2,n\rangle ,|g_1,n\rangle ,|g_2,n+1\rangle ,|g_1,n+1\rangle ...\}$. The purple squares denote the diagonal elements in the matrix of the Hamiltonian, while the blue and yellow squares denote the elements corresponding to evolution path 1 and path 2, respectively. (c) Schematic diagram of the matrix form of the effective Hamiltonian ${\widehat{H}}_{\text{eff}}$ in another basis (defined as $\left\{\right|{\psi }_n\rangle \}$) where all subspaces can be diagonalized.

Figure 8

(a), (b) The photon number $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ and (c), (d) eigenvalues as a function of $g$ for different values of ${\mathrm{\Delta }}_2$: (a),(c) ${\mathrm{\Delta }}_2/{\omega }_0^{\prime }=2\times {10}^{3.0}$ and (b),(d) ${\mathrm{\Delta }}_2/{\omega }_0^{\prime }=2\times {10}^{3.5}$. All results are obtained from the effective Hamiltonian ${\widehat{H}}_{\text{eff}}$. Other parameters are the same as in Fig. 5.

Figure 9

Time evolution of (a), (b) the photon number $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ and (c), (d) the probability of the atom being in the state $|g_1\rangle$ [$P_1\left(t\right)$] and $|g_2\rangle$ [$P_2\left(t\right)$] for different values of $\gamma$: (a), (c) $\gamma /{\omega }_0^{\prime }=10$ and (b), (d) $\gamma /{\omega }_0^{\prime }=100$. The coupling strength is fixed as $g=1.6$. Other parameters are the same as in Fig. 6.