Canonical versus noncanonical equilibration dynamics of open quantum systems

In statistical mechanics, any quantum system in equilibrium with its weakly coupled reservoir is described by a canonical state at the same temperature as the reservoir. Here, by studying the equilibration dynamics of a harmonic oscillator interacting with a reservoir, we evaluate microscopically the condition under which the equilibration to a canonical state is valid. It is revealed that the non-Markovian effect and the availability of a stationary state of the total system play a profound role in the equilibration. In the Markovian limit, the conventional canonical state can be recovered. In the non-Markovian regime, when the stationary state is absent, the system equilibrates to a generalized canonical state at an effective temperature; whenever the stationary state is present, the equilibrium state of the system cannot be described by any canonical state anymore. Our finding of the physical condition on such noncanonical equilibration might have significant impact on statistical physics. A physical scheme based on circuit QED is proposed to test our results.

Figure 1

$\Gamma \left(t\right)$ (a) and ${\Gamma }^{\beta }\left(t\right)$ (b) in different $\eta$. (c) $T_{\mathrm{eff}}$ evaluated when $t=100/{\omega }_0$. ${\omega }_c=8.0{\omega }_0,\beta =0.1/{\omega }_0$, and $s=1$ have been used.

Figure 2

$\Gamma \left(t\right)$ (a) and ${\Gamma }^{\beta }\left(t\right)$ (b) in different ${\omega }_c$. (c) $T_{\mathrm{eff}}$ evaluated when $t=100/{\omega }_0$. $\eta =0.1,\beta =0.1/{\omega }_0$, and $s=1$ have been used.

Figure 3

$N\left(t\right)$ (a) and $\tilde{n}\left(t\right)\equiv N\left(t\right)/N_{\mathrm{con}}$ (b) in different $\eta$. $|{\alpha }_0|=1.0$ and other parameters are the same as Fig. 1. (c) $lnN\left(\infty \right)$ and the corresponding Markovian values shown by dashed, dotted, and dot-dashed lines in different $T$.

Figure 4

$N\left(t\right)$ (a) and $\tilde{n}\left(t\right)\equiv N\left(t\right)/N_{\mathrm{con}}$ (b) in different ${\omega }_c$. $|{\alpha }_0|=1.0$ and other parameters are the same as Fig. 2. (c) $lnN\left(\infty \right)$ and the corresponding Markovian values shown by dashed, dotted, and dot-dashed lines in different $T$.

Figure 5

$N\left(t\right)$ in different ${\alpha }_0$ when $\eta =0.05$ (a) and $0.2$ (b). Other parameters are the same as Fig. 1. The dashed line in (a) is the value evaluated from Eq. (7).

Figure 6

(a) Scheme on a TLR (formed by $C_{\mathrm{s}}$ and $L_{\mathrm{s}}$) with ${\omega }_0=1/\left(2\sqrt{L_{\mathrm{s}}{C}_{\mathrm{s}}}\right)$ interacting with an array of TLRs (formed by $C_i$ and $L_i$) with identical ${\omega }_C=1/\left(2\sqrt{L_i{C}_i}\right)$ coupled via $C_c$ in strength $\xi =2Z_0{C}_c{\omega }_C^2$ ($Z_0$ being the impedance). $\Gamma \left(t\right)$ (b) and ${\Gamma }^{\beta }\left(t\right)$ (c) in different ${\omega }_0$. $\xi =0.05{\omega }_C,g=0.02{\omega }_C,\beta =0.05/{\omega }_C$, and $N=200$ have been used.