Structure of Quantum Mean Force Gibbs States for Coupled Harmonic Systems

In this subsection, we apply the method developed in the previous section to a chain of coupled oscillators one end of which is in contact with a heat bath. We consider a system described by Hamiltonian

The parameter $\lambda$ describes the coupling between the neighboring oscillators. For convenience, we assume that the system satisfies the periodic boundary condition, $\hat{x}_{N}=\hat{x}_{0}$. The oscillators thus form a ring (see Fig. 1 (a) for schematic description). The matrix $\bm{\Lambda}$ in Eq. (4) is given by

where

As shown in Appendix D, the eigenvalues of $\bm{\Lambda}$ are given by $M\Omega^{2}_{k}$, $k=0,1,\cdots,N-1$, where

Note that there are two-fold degeneracies as $\Omega_{N-k}=\Omega_{k}$. For stability we require $2|\lambda|<\Omega^{2}$. The eigenvectors of $\bm{\Lambda}$ can be used to construct an orthogonal matrix $\bm{R}$, which diagonalizes $\bm{\Lambda}$ as $\bm{R}^{\rm T}\bm{\Lambda}\bm{R}=\mathrm{diag}(M\Omega^{2}_{0},\cdots,M\Omega^{2}_{N-1})$. We therefore have

In Appendix D, we present explicit eigenvectors and consequently $\bm{R}$ which diagonalizes $\bm{V}$ and $\bm{\Lambda}$.

Now the full Green’s function $\bm{G}$ in the presence of the coupling to the bath can be obtained using Eq. (32). The effect of coupling to the bath is characterized by $\zeta^{(\alpha)}_{r}$ in Eq. (17) and the bath spectral density $J_{\alpha}(\omega)$ in Eq. (6). We will again drop the index $\alpha$ as there is only one bath. We consider an Ohmic bath with the cutoff $\omega_{D}$ and the coupling strength $\gamma$, which is given by

Putting this into Eq. (17), we have

In the following, we present explicit expressions for the MFG state for general values of $N\geq 2$. As we have seen above in Eqs. (27) and (28), the MFG state is completely characterized by its covariances. First, using the eigenvectors obtained in Appendix D and Eq. (43), we can write

We then obtain the covariances with respect to the system Gibbs state $\hat{\rho}_{\rm G}$, which we denote by $\langle\cdots\rangle_{\rm G}$. This can be done by using Eqs. (27) and (28) with $\bm{G}$ replaced by $\bm{G}^{(0)}$. Using the summation formula

we obtain

For the momentum covariances, we first note from Eq. (43) that

We have from Eq. (28)

Now the covariances with respect to the MFG state can be obtained from the full Green’s function in Eq. (32) with Eq. (33). We therefore have

where the differences compared to those obtained with respect to the system Gibbs state are given in terms of $\Delta\bm{G}$ in Eq. (33) by

We evaluate the infinite sums in Eqs. (53) and (54) numerically and calculate the covariances with respect to the MFG state. For the numerical calculations, we present $\langle\hat{x}_{i}\hat{x}_{j}\rangle$ and $\langle\hat{p}_{i}\hat{p}_{j}\rangle$ in units of $\hbar/(M\Omega)$ and $\hbar M\Omega$, respectively. In Figs. 2 and 3, we show the results for $\langle\hat{x}^{2}_{i}\rangle$ and $\langle\hat{p}^{2}_{i}\rangle$ calculated with respect to the MFG state and compare them with those obtained from the system Gibbs state for various values of the coupling strength $\gamma$ (in the unit of $\Omega$) to the bath and of the inter-oscillator coupling constant $\lambda$ (in the unit of $\Omega^{2}$). The calculations were done for a ring of $N=20$ coupled oscillators where $j=0$ oscillator is coupled with the heat bath at inverse temperature $\beta$. We take the Ohmic bath described by Eqs. (44) and (45) with $\omega_{D}=100$ (in the unit of $\Omega$).

At relatively high temperature, $\beta=1$ (The inverse temperature is in the unit of $1/(\hbar\Omega)$.), Fig. 2 (a) shows that $\langle\hat{x}^{2}_{j}\rangle_{\rm mfG}$ exhibits very little variation compared to those obtained from the system Gibbs state, which is independent of $j$ (see Eq. (48)). The small difference between the MFG and the system Gibbs averages occurs only near the contact point $j=0$ between the system and the bath. On the other hand, the momentum covariance $\langle\hat{p}^{2}_{j}\rangle_{\rm mfG}$ shows a different behavior as one can see from Fig. 2 (b). The value $\langle\hat{p}^{2}_{0}\rangle_{\rm mfG}$ right at the point of contact with the bath increases quite rapidly as the coupling strength $\gamma$ to the bath increases. As we will see below in the next subsection, this is a characteristic behavior of a quantum system where the coupling between the position operators of the system and the heat bath gets stronger. As we can see in the inset of the same figure, away from the point of contact, the effect of the coupling to the bath quickly disappears as in the position covariance.

At lower temperatures, the effect of the coupling to the bath in $\langle\hat{x}^{2}_{j}\rangle_{\rm mfG}$ is more prominent as we can see in Fig. 3 (a) at the inverse temperature $\beta=100$. The effect, however, disappears as we move away from the point of contact with the bath. In fact, at lower temperatures $\beta\gtrsim 10$, we find that the difference defined in Eq. (51) is well described by an exponential decay $|\Delta\langle\hat{x}^{2}_{j}\rangle|\sim\exp(-j/\xi)$ with the characteristic length $\xi$. This is quite similar to the skin effect found in Ref. Burke et al. (2024), where the effect of coupling to the environment decays exponentially with the distance from the system-environment boundary. In Fig. 4 (a), we plot the skin depth $\xi$ as a function of the inter-oscillator coupling $\lambda$ for various values of the temperature and the coupling strength $\gamma$ to the bath. We note that $\xi$ is quite small ($\lesssim 1$) even at low temperatures and strong system-bath couplings, and that it shows only a moderate increase as a function of $\lambda$ and $\beta$. A somewhat surprising result is that the effect of the bath measured in terms of $\xi$ is almost independent of the coupling strength $\gamma$ to the bath for given values of $\beta$ and $\lambda$. The momentum covariance $\langle\hat{p}^{2}_{j}\rangle_{\rm mfG}$ at lower temperatures, on the other hand, behaves quite similarly to that at higher temperatures as can be seen from Fig. 3 (b). The main feature still is a sharp increase of $\langle\hat{p}^{2}_{0}\rangle_{\rm mfG}$ as $\gamma$ increases. The effect of the coupling to the bath quickly disappears as can be seen in the inset of Fig. 3 (b).

In order to study the effect of the coupling to the bath on the MFG state further, we look at another type of covariances, $\langle\hat{x}_{0}\hat{x}_{j}\rangle_{\rm mfG}$ and $\langle\hat{p}_{0}\hat{p}_{j}\rangle_{\rm mfG}$, which connect operators at different sites, and compare them with those obtained with respect to the system Gibbs state. We again study the effect of the coupling to the bath using the differences, $\Delta\langle\hat{x}_{0}\hat{x}_{j}\rangle$ and $\Delta\langle\hat{p}_{0}\hat{p}_{j}\rangle$ between the MFG and the system Gibbs states, which are plotted in Fig. 5 at inverse temperature $\beta=100$ for $N=20$ oscillators. We note that in this case the system Gibbs averages as well as the MFG ones show a spatial dependence on $j$ as one can see from Eqs. (48) and (50). The differences between them, however, show a similar behavior to the previous case of $\Delta\langle\hat{x}_{j}^{2}\rangle$ as can be seen from Fig. 5 (a). In fact, we find that $|\Delta\langle\hat{x}_{0}\hat{x}_{j}\rangle|$ is also well described by an exponential decay $\sim\exp(-j/\xi)$. We plot the skin depth $\xi$ for this covariance in Fig. 4 (b) for various values of the parameters. We note that the skin depths in this case are slightly bigger than those for $\Delta\langle\hat{x}_{j}^{2}\rangle$, but the overall behavior of $\xi$ as a function of the coupling constants and temperatures is quite similar to the previous case. The main point we make here is that the effect of the heat bath on the equilibrium state measured with $\xi$ in both cases is quite short-ranged reaching only over one site. The difference in the momentum covariance $\Delta\langle\hat{p}_{0}\hat{p}_{j}\rangle$ connecting different sites also shows a similar behavior to $\Delta\langle\hat{p}^{2}_{j}\rangle$. Apart from the increasing $\Delta\langle\hat{p}^{2}_{0}\rangle_{\rm mfG}$ at $j=0$ as $\gamma$ gets bigger, the effect of the coupling to the bath quickly disappears as we move away from the contact point as we can see in Fig. 5 (b).

In this subsection, we investigate the MFG state obtained above, which is an exact result, in the limit when the coupling between the system and the bath is arbitrarily large. This is to make a connection with the ultra strong coupling (USC) limit obtained recently in Ref. Cresser and Anders (2021) of the MFG state for a general system coupled with a bath. If the interaction Hamiltonian between system and bath is given by $\kappa\hat{X}\hat{B}$ with a system operator $\hat{X}$ and a bath operator $\hat{B}$, and the coupling strength $\kappa$, it was shown in Ref. Cresser and Anders (2021) that the MFG state in the limit $\kappa\to\infty$ is diagonal in the eigenstates of $\hat{X}$. More explicitly, the ultra strong coupling limit is given by Cresser and Anders (2021)

where $\hat{P}_{n}$ is the projection operator on the eigenstates $|X_{n}\rangle$ of $\hat{X}$ with eigenvalue $X_{n}$. In other words, the USC limit is characterized by the effective Hamiltonian of mean force $H_{\rm USC}\equiv\sum_{n}\hat{P}_{n}\hat{H}_{\rm S}\hat{P}_{n}$, which is just the system Hamiltonian but projected onto the eigenstates of the system operator taking part in the coupling to the bath.

The reason we study our present system in this limit is twofold. First, for the case of harmonic oscillators, the system operator coupled to the bath is just the position operator $\hat{x}_{0}$ at $j=0$ thus has a continuous spectrum. Therefore, the USC limit of the exact MFG state for the quantum harmonic oscillators can provide a generalization of Eq. (55) which was originally obtained in Ref. Cresser and Anders (2021) for the coupling system operator with a discrete spectrum to a continuous one.

To illustrate this point, we consider the simplest case of a single ($N=1$) damped harmonic oscillator Grabert et al. (1984); Hilt et al. (2011) in the USC limit. Putting $N=1$ in Eqs. (21), (27) and (28), we have

Taking the coupling strength $\gamma$ in $\zeta_{r}$ to infinity (see Eq. (45)), we see that the infinite sum in $\langle\hat{p}^{2}\rangle_{\rm mfG}$ diverges. We also find that only the $r=0$ term in Eq. (56) survives, which gives $\langle\hat{x}^{2}\rangle_{\rm mfG}\to 1/(M\beta\Omega^{2})$ as $\gamma\to\infty$. As can be seen from Eqs. (23), the diverging momentum covariance indicates that the matrix element $\langle x|\rho_{\rm mfG}|x^{\prime}\rangle$ is proportional to $\delta(x-x^{\prime})$ i.e. diagonal in eigenstates of $\hat{x}$ in the USC limit. This point was recognized recently in Ref. Kumar and Ghosh (2024). The actual value of the position covariance in the USC limit indicates that $\rho_{\rm USC}$ is given by the Gibbs state of an effective Hamiltonian of the form $\hat{H}_{\rm USC}=(1/2)M\Omega^{2}\hat{x}^{2}$. This is exactly the system Hamiltonian projected onto the position eigenstate as done in Eq. (55). It is, however, different from another form of the USC limit suggested in Refs. Goyal and Kawai (2019); Orman and Kawai (2020), which is equal to $\sum_{n}\hat{P}_{n}\hat{\rho}_{\rm G}\hat{P}_{n}=\int dx|x\rangle\langle x|\hat{\rho}_{\rm G}|x\rangle\langle x|$ in our case. Therefore it would give the position covariance as $(\hbar/(2M\Omega))\coth(\beta\hbar\Omega/2)$ as one can see from the diagonal element in Eq. (23) with $G^{(0)}$ for $\hat{\rho}_{\rm G}$. But this is clearly in disagreement with the actual USC limit obtained from the exact expression except for very high temperatures.

The second point why we study the USC limit in our model is the following. We note that the USC limit suggested in Eq. (55) is characterized by the effective Hamiltonian which is a diagonal projection onto the eigenstates of the system operator that couples to the bath. In our model, this system operator is only local as only a part of the system couples to the bath. Therefore, using our exact result, we can check whether the USC limit in Eq. (55) still holds for the case where the coupling operator is local in space. To do this we focus on the simplest case $N=2$, where the positions of the oscillator are $\hat{x}_{0}$ and $\hat{x}_{1}$ and only $\hat{x}_{0}$ is coupled to the bath. The normal mode frequencies from Eq. (42) are $\Omega^{2}_{0}=\Omega^{2}-2\lambda$ and $\Omega^{2}_{1}=\Omega^{2}+2\lambda$, and from Eq. (46), we have

The USC limit of the two-oscillator system is obtained by taking $\gamma\to\infty$ in Eqs. (37) and (38). We have

and

For the momentum covariances, we can easily see from Eq. (38) that $\langle\hat{p}^{2}_{0}\rangle_{\rm mfG}\to\infty$ and $\langle\hat{p}_{0}\hat{p}_{1}\rangle_{\rm mfG}\to 0$ as $\gamma\to\infty$ and