Nonequilibrium thermal transport and photon squeezing in a quadratic qubit-resonator system

We investigate steady-state thermal transport and photon statistics in a nonequilibrium hybrid quantum system, in which a qubit shows longitudinal and quadratic coupling with an optical resonator. Our calculations are conducted with the method of the quantum dressed master equation combined with full counting statistics. The effect of negative differential thermal conductance is unravelled at finite temperature bias, which stems from the suppression of cyclic heat transitions and large mismatch of two squeezed photon modes at weak and strong qubit-resonator hybridizations, respectively. The giant thermal rectification is also exhibited at large temperature bias. It is found that the asymmetric composition of the hybrid system by spin and boson roles and negative differential thermal conductance show the cooperative contribution. Noise power and skewness, as typical current fluctuations, exhibit global maximum with strong hybridization at small and large temperature bias limits, respectively. Moreover, the effect of photon quadrature squeezing is discovered in the strong hybridization and low-temperature regime, which shows asymmetric response to two bath temperatures. These results would provide some insight to thermal functional design and photon manipulation in qubit-resonator hybrid quantum systems.

Figure 1

(a) Schematic of the nonequilibrium coupled qubit-resonator system. The harmonics and two-level system denote the optical resonator and qubit, respectively. The pair of double black crossing curves shows the quadratic qubit-photon interaction; the red (left) and blue (right) panels represent the bosonic thermal baths, characterized as the temperatures $T_{\text{R}}$ and $T_{\text{Q}}$, respectively.

Figure 2

(a) Influence of the temperature bias $\mathrm{\Delta }T$ on rescaled steady-state heat current $J_{ss}/{\lambda }^2$ from weak to strong qubit-photon hybridizations, e.g., $\lambda =0.001{\omega }_a,0.01{\omega }_a$, and $0.1{\omega }_a$; (b) bird's view of the heat current $J_{ss}$ by both modulating $k_B\mathrm{\Delta }T/\left(\hslash {\omega }_a\right)$ and $\lambda /{\omega }_a$; (c) and (d) are one-way cyclic heat transitions to depict the first term of $I_{m,1}$ and $I_{m,0}$, respectively; (e) effect of the temperature bias on heat current component $I_{m,\sigma }/{\omega }_a$ given by Eqs. (20a) and (20b) in the weak hybridization regime to describe the origin of the NDTC; (f) two different types of heat transfer processes between eigenstates at strong qubit-photon hybridization, i.e., $|{\psi }_{m+2k}^1\rangle \to |{\psi }_m^0\rangle$ and $|{\psi }_{m+2l}^0\rangle \to |{\psi }_m^1\rangle$ ($m,k,l\in \mathrm{N}$), where the thickness of the blue arrowed lines denotes the transition rate strengths. Other parameters are given by $T_{\mathrm{R}}=T_0+\mathrm{\Delta }T/2, T_{\mathrm{Q}}=T_0-\mathrm{\Delta }T/2, T_0=\hslash {\omega }_a/k_B, \varepsilon ={\omega }_a, {\alpha }_{\mathrm{R}}={\alpha }_{\mathrm{Q}}={10}^{-3}$, and ${\omega }_c=10{\omega }_a$.

Figure 3

Thermal rectification factor $\mathcal{R}$ defined in Eq. (23) versus the temperature bias $k_B\mathrm{\Delta }T/\left(\hslash {\omega }_a\right)$ and the qubit-photon hybridization strength $\lambda /{\omega }_a$. Other parameters are the same as those in Fig. 2.

Figure 4

(a) Noise power $J_{ss}^{\left(2\right)}$ and (b) skewness $J_{ss}^{\left(3\right)}$ defined in Eqs. (16b) and (16c) by tuning the bath temperature bias $k_B\mathrm{\Delta }T/\left(\hslash {\omega }_a\right)$ and qubit-photon hybridization strength $\lambda /{\omega }_a$. Other parameters are the same as those given in Fig. 2.

Figure 5

Photon quadrature squeezing factor ${\xi }_{\text{B}}^2$ defined in Eq. (24) (a) at equilibrium $T_{\mathrm{R}}=T_{\mathrm{Q}}=T$, with the inset showing the angle distribution of ${\left(\mathrm{\Delta }{\widehat{X}}_{\theta }\right)}^2$ at $\lambda =0.2{\omega }_a$, and (b) far from equilibrium by tuning two bath temperatures $T_{\mathrm{R}}$ and $T_{\mathrm{Q}}$ individually, with $\lambda =0.2{\omega }_a$. Other parameters are the same as those given in Fig. 2.