Effective Hamiltonian approach to the quantum phase transitions in the extended Jaynes-Cummings model

The study of phase transitions in dissipative quantum systems based on the Liouvillian is often hindered by the difficulty of constructing a time-local master equation when the system-environment coupling is strong. To address this issue, the complex discretization approximation for the environment is proposed to study the quantum phase transition in the extended Jaynes-Cumming model with an infinite number of boson modes. This approach yields a non-Hermitian effective Hamiltonian that can be used to simulate the dynamics of the spin. It is found that the ground state of this effective Hamiltonian determines the spin dynamics in the single-excitation subspace. Depending on the opening of the energy gap and the maximum population of excitations on the spin degree of freedom, three distinct phases can be identified: fast decaying, localized, and stretched dynamics of the spin. This approach can be extended to multiple excitations, and similar dynamics were found in the double-excitation subspace, indicating the robustness of the single-excitation phase.

Figure 1

Illustration of the path in complex plane used to approximate ${\int }_0^{\infty }\text{d}x$ as ${\int }_{\mathrm{\Gamma }}\text{d}z$, where $\mathrm{\Gamma }$ denotes the semi-circle with radius $R$, centered at $(R,0)$.

Figure 2

The plots for the computational errors defined in Eq. (21) vs the evolution time $t$ (in units of $1/\mathrm{\Delta }$). For all plots, $N=2000,R=6,\mathrm{\Delta }=1$, and ${\omega }_c=10$ are chosen.

Figure 3

(a) The plot for the computational errors vs the evolution time $t$ (in units of $1/\mathrm{\Delta }$) when ${\tilde{H}}_{\text{dis}}$ is used to evaluate the survival probability for $s=1$. It is stressed that the data for $\eta =0.05$ and 0.1 are enlarged by ten times for convenience of illustration. The other parameters are the same as those in Fig. 2. (b) The detailed plot for the oscillation of survival probability when $\eta =0.5$. The dashed line denotes the value 0.54744. The evaluation is obtained by ${\tilde{H}}_{\text{dis}}$ with $N=2000$. The other parameters are the same as those in Fig. 2.

Figure 4

The phase diagram for $H_{\text{dis}}$ in the single-excitation subspace. All points are determined by exact diagonalization of $H_{\text{dis}}$. Region I: delocalized phase. Region II: localized phase. Region III: intermediate phase. ${\eta }_I$ denotes the critical value, where the energy gap is zero. ${\eta }_{II}$ denotes the critical value, where the imaginary part of ground-state function on basis state ${|e\rangle |0\rangle }^{\otimes k}$ is zero. For this plot, $R=6,N=1000$ and $\mathrm{\Delta }=1,{\omega }_c=10$ are chosen.

Figure 5

The evolution of survival probability with time $t$ (in units of $1/\mathrm{\Delta }$) is plotted for selected $s$ and $\eta$. For panels (a) and (b), the evolution is obtained by $H_{\text{dis}}$. While for panel (c), ${\tilde{H}}_{\text{dis}}$ is used to obtain convergent results for $\eta =0.05,0.1$. As for $\eta =0.5$, the evolution is determined by the combination of $H_{\text{dis}}$ and ${\tilde{H}}_{\text{dis}}$, as stated at the end of Sec. 2b. The Roman numbers denote the different regions in Fig. 4, to which the chosen values of $s$ and $\eta$ belong. For all plots, $N=2000,R=6,\mathrm{\Delta }=1$, and ${\omega }_c=10$ are chosen.

Figure 6

Plots of the spectrum (in units of $\mathrm{\Delta }$) of $H_{\text{dis}}$ for different $s$ and $\eta$. For all plots, $N=2000,R=6,\mathrm{\Delta }=1$, and ${\omega }_c=10$ are chosen.

Figure 7

The PMP (orange line) and average value of ${\sigma }_z$ (blue line) for the ground state of $H_{\text{dis}}$ vs $\eta$. For P.M.P., the arabic numbers denote the basis state $|k\rangle$. The parameters are chosen the same in Fig. 6

Figure 8

The comparative plots of spin dynamics in the double-excitation subspace (empty circle) and single-excitation subspace (solid line). The spin dynamics in single-excitation subspace is evaluate by $H_{\text{dis}}$ for $s=0,0.2$ and ${\tilde{H}}_{\text{dis}}$ for $s=1$. For the calculation in double-excitation subspace, ${\tilde{H}}_{\text{dis}}$ is solved by choosing $N=200$ to obtain the convergent result. For all plots, $R=6,\mathrm{\Delta }=1,{\omega }_c=10$ are chosen. The evolution time $t$ is in units of $1/\mathrm{\Delta }$.

Figure 9

The comparative plots for the survival probability ${\left|\langle \psi \left(t\right)|\psi \left(0\right)\rangle \right|}^2$ with $|\psi \left(0\right)\rangle ={|e\rangle |0\rangle }^{\otimes k}$ for different $s$ and $\eta$. The solid line depict the exact result, whereas the empty circles represent results obtained by CDA. For all plots, $N=2000,R=6,\mathrm{\Delta }=1$, and ${\omega }_c=10$ are chosen. The evolution time $t$ is in units of $1/\mathrm{\Delta }$.

Figure 10

The comparative plot between the exact results (solid line) and the numerics (empty circle) obtained by ${\tilde{H}}_{\text{dis}}$. The other parameters are same to those in Fig. 9. The evolution time $t$ is in units of $1/\mathrm{\Delta }$.

Figure 11

Plots for CDA approach to the long-term feature of ${\left|\langle \psi \left(t\right)|\psi \left(0\right)\rangle \right|}^2$ when ${\tilde{H}}_{\text{dis}}$ and $H_{\text{dis}}$ are used, respectively. It is evident that the numerics by $H_{\text{dis}}$ would become convergent to the results of ${\tilde{H}}_{\text{dis}}$ when $N$ increases. The evolution time $t$ is in units of $1/\mathrm{\Delta }$.

Figure 12

The numerical fitting for the stretched-like evolution, shown in Fig. 5. The evolution time $t$ is in units of $1/\mathrm{\Delta }$.

Figure 13

${\eta }_I$ and ${\eta }_{II}$ vs the parameter $R$ and $N$. For panels (a1)–(a3), $R=6$ is chosen. While for panels (b1)–(b3), $N=1000$ is chosen. For all plots, $\mathrm{\Delta }=1,{\omega }_c=10$ are chosen.

Figure 14

The reevaluation of spin dynamics in the double-excitation subspace (empty circle) by Gauss quadratures method. The solid line corresponds to the result of ${\left|\langle \psi \left(t\right)|\psi \left(0\right)\rangle \right|}^2$ in single-excitation subspace. For all plots, $\mathrm{\Delta }=1$ and ${\omega }_c=10$ are chosen.