Linear Coupling

In many applications, one assumes (often for simplicity) a linear coupling to the bath,

$$H_B[x]+H_{SB}[xq] = \sum_{\alpha=1}^N \left[ \frac{p_\alpha^2}{2M_\alpha}+ \frac{1}{2... ...ha^2 \left(x_\alpha - \frac{c_\alpha}{M_\alpha\Omega_\alpha^2}q\right)^2\right]$$

$$= \sum_{\alpha=1}^N \left[ \frac{p_\alpha^2}{2M_\alpha}+ \frac{1}{2... ...pha q x_\alpha +\frac{1}{2}\frac{c_\alpha^2}{M_\alpha\Omega_\alpha^2}q^2\right]$$

$${\cal F}[q_{t'},q'_{t'}] = \exp \left\{-\Phi[q_{t'},q'_{t'}] \right\}$$

$$\Phi[q_{t'},q'_{t'}] = \int_{0}^{t}dt'\int_{0}^{t'}ds \left\{ q_{t'} - q'_{t'} \right\} \left\{ L(t'-s) q_{s} - L^*(t'-s) q'_{s}\right\}$$

$$+ i \frac{\mu}{2}\int_0^t dt'\left\{ q_{t'}^2-(q'_{t'})^2 \right\}$$ (208)

Here, the kernel $L(t)$ and the spectral density $J(\omega)$ are

$$\fbox{$ \begin{array}{rcl} \displaystyle L(\tau)&\equiv& \frac{1}... ...rac{\beta\omega}{2} \cos \omega \tau - i \sin \omega \tau\right)\end{array}$\ }$$

$$\fbox{$ \begin{array}{rcl} \displaystyle J(\omega)&\equiv& \frac{... ..._\alpha^2}{M_\alpha\Omega_\alpha} \delta(\omega-\Omega_\alpha). \end{array}$\ }$$ (209)

Note that in this form, an additional term appears in $H_{SB}$ as a potential

$$V_{\rm counter}(q) \equiv \frac{1}{2}\mu q^2,\quad \mu \equiv \fr... ...\alpha\Omega_\alpha^2}=\frac{2}{\pi} \int_{0}^{\infty}\frac{J(\omega)}{\omega}.$$ (210)

Since the action $S$ appears as $\exp(iS[q])$ in the path integral for $q$ and $\exp(-iS[q])$ in the path integral for $q'$, we could absorb the counter term into the influence phase as

$$\exp (i \frac{\mu}{2}\int_0^t dt'\left\{ q_{t'}^2-(q'_{t'})^2 \right\}).$$

Note that the entire information on the coupling to the bath is now contained in the spectral density $J(\omega)$, which we have defined following the notation of Weiss, `Quantum Dissipative Systems'.