Hamiltonian of mean force for damped quantum systems

We consider a quantum system linearly coupled to a reservoir of harmonic oscillators. For finite coupling strengths, the stationary distribution of the damped system deviates from the predictions of standard thermodynamics. With the help of the quantum Hamiltonian of mean force, we quantify this deviation exactly for a harmonic oscillator and provide approximations in the limit of high and low temperatures and weak and strong couplings. Moreover, in the semiclassical regime, we use the quantum Smoluchowski equation to obtain results valid for any potential. We finally give a physical interpretation of the deviation in terms of the initial system-reservoir coupling.

Figure 1

(a) Coefficient $MA$, Eq. (25), as a function of the dimensionless parameters $kT/\left(\hslash \omega \right)$ and $\gamma /\omega$. $A$ increases monotonically to zero with temperature and inverse coupling strength. The deviation from a Gibbs state in momentum is thus maximal in the strong coupling, low-temperature limit. (b) Cross sections of the $(T,\gamma )$ curve for $kT/\left(\hslash \omega \right)=0.5$ (red lower line),3 (blue middle line), and 15 (green upper line). In the limit of high damping, $A$ coincides with the semiclassical expression given by the quantum Smoluchowski equation [black dashed line for $kT/\left(\hslash \omega \right)=3$], Eq. (37). Parameters are ${\omega }_D=1000$, $\omega =1$, $M=1$.

Figure 2

(a) Coefficient $B/\left(M{\omega }^2\right)$, Eq. (26), as a function of the dimensionless parameters $kT/\left(\hslash \omega \right)$ and $\gamma /\omega$. $B$ increases monotonically to zero with temperature. However, it exhibits a minimum as a function of the coupling strength. (b) Cross sections of the $(T,\gamma )$ curve for $kT/\left(\hslash \omega \right)=0.1$ (red lower line), $0.5$ (blue middle line), and 1 (green upper line). For increasing temperatures, the dip gets less pronounced and eventually disappears. In the limit of high damping, $B$ coincides with the semiclassical expression given by the quantum Smoluchowski equation [black dashed line for $kT/\left(\hslash \omega \right)=0.1$], Eq. (45). The parameters are the same as in Fig. 1.

Figure 3

Function $\mathrm{arcoth}\left(2v\right)$ (red solid line) and $\sqrt{\langle p^2\rangle /\langle q^2\rangle }$ (blue dashed line) appearing in the definition of coefficient $B$, Eq. (26), as a function of the dimensionless parameter $\gamma /\omega$. Their different behaviors lead to a minimum for their product (green dotted line). The parameters are the same as in Fig. 1 and $kT/\hslash \omega =0.1$.

Figure 4

Relative mean deviation $\langle \Delta H_S\rangle /\langle H_S\rangle$, Eq. (24), showing the departure from a Gibbs state as a function of dimensionless temperature and coupling strength. The maximum deviation is observed in the low-temperature, strong coupling regime. The parameters are the same as in Fig. 1.