Many-fermion generalization of the Caldeira-Leggett model

We analyze a model system of fermions in a harmonic oscillator potential under the influence of a dissipative environment: The fermions are subject to a fluctuating force deriving from a bath of harmonic oscillators. This represents an extension of the well-known Caldeira-Leggett model to the case of many fermions. Using the method of bosonization, we calculate one- and two-particle Green’s functions of the fermions. We discuss the relaxation of a single extra particle added above the Fermi sea, considering also dephasing of a particle added in a coherent superposition of states. The consequences of the separation of center-of-mass and relative motion, the Pauli principle, and the bath-induced effective interaction are discussed. Finally, we extend our analysis to a more generic coupling between system and bath, which results in complete thermalization of the system.

Figure 1

(Color online) Left: coupling the fermions in the oscillator to the fluctuating quantum force $\widehat{F}$ leads both to relaxation of excited fermions (decay rates depending on the bath spectrum ${⟨\widehat{F}\widehat{F}⟩}_{\omega }$), as well as some smearing of the level occupation even at $T=0$. Right: the bosonized version is equivalent to chiral fermions moving on a ring, subject to a transverse fluctuating force.

Figure 2

(Color online) Equilibrium density matrix. Left: populations ${\rho }_{nn}$ as a function of coupling strength. Right: full density matrix ${\rho }_{n{n}^{\prime }}$ for fixed $N\gamma ∕{\omega }_{00}=10$, magnitude indicated by radius, negative values indicated by different color-gray level. Note that off-diagonal elements have been enlarged by a factor of 3 for clarity. We have chosen ${\omega }_c∕{\omega }_{00}={10}^4$ in both plots.

Figure 3

(Color online) Green’s function of fermions in a damped harmonic oscillator (with an Ohmic bath): Fourier transform $G_{nn}\left(\omega \right)$ of $⟨{\widehat{c}}_n^{†}\left(t\right){\widehat{c}}_n⟩$ plotted vs $\omega$, for different values of $n$ (curves displaced vertically for clarity). Note that $\omega$ is measured with respect to $-n_F{\omega }_0$ and we have set ${\omega }_0=1$. The strength of the coupling is given by $N\gamma ∕{\omega }_{00}=0.4$. Inset: weight function $W\left(\omega \right)$ vs $\omega$. The height of the smaller $\delta$ peaks is an indication of their weight.

Figure 4

(Color online) Dependence of Green’s function on coupling strength: Fourier transform $G_{nn}\left(\omega \right)$ of $⟨{\widehat{c}}_n^{†}\left(t\right){\widehat{c}}_n⟩$ vs $\omega$, for different values of the coupling strength, indicated in the plot (curves displaced vertically for clarity). Note that $\omega$ is measured with respect to $-n_F{\omega }_0$ and plotted in units of ${\omega }_0$, and that ${\omega }_0$ changes with $\gamma$, for fixed ${\omega }_{00}$; see Eq. (46). This plot is produced for a fixed excitation number, $n=n_F-4$.

Figure 5

Weight $P_0$ of the center-of-mass ground state [Eq. (55)] as a function of excitation $\delta n$, for different particle numbers $N$, including the bosonization result from Eq. (50) (“$N=\infty$”).

Figure 6

(Color online) Time evolution of ${\rho }_{\text{heat}}(n,n^{\prime },t)$, describing heating around the Fermi surface due to an extra particle placed in a highly excited state above the Fermi sea [Eq. (62)]. See text for the connection to the two-particle Green’s function, Eq. (60). In this plot ${\omega }_0={\omega }_0^{\prime }$. The radius of each dot gives $\mid {\rho }_{n{n}^{\prime }}\left(t\right)\mid$ (being 1 for the two occupied states in the lower left corner).

Figure 7

(Color online) Time evolution of the populations ${\rho }_{nn}\left(t\right)$ [Eq. (59)] for an electron placed in an initial state $\tilde{n}=n_F+6$ above the Fermi sea (weak-coupling result) with ${\omega }_0={\omega }_0^{\prime }$ (left) or ${\omega }_0={\omega }_0^{\prime }+2N\gamma$ (right); see text. The initial and final states are the same in both cases. Different curves have been displaced vertically for clarity. Note the incomplete decay and heating around the Fermi surface.

Figure 8

(Color online) Time evolution of the single-particle density matrix ${\rho }_{n{n}^{\prime }}\left(t\right)$ [Eq. (63)] starting from an extra particle in a superposition of states $n_1=5$, $n_2=6$ above the Fermi sea $({\omega }_0={\omega }_0^{\prime })$. Levels $n,n^{\prime }=n_F-1,n_F,\dots ,n_F+6$ are indicated by grid lines.

Figure 9

Distribution of decay rates ${\Gamma }_{\mathrm{GF}}$ entering the Green’s function in the weak-coupling approximation given by Eq. (72) for the Ohmic bath at $T=0$, where the bath connects arbitrary oscillator levels. Decay rates are plotted as a function of excitation level $\delta n$, and the grey level visualizes the weight in Eq. (72). The solid lines represent the lower and upper bounds $N\gamma \delta n∕2$ and $N\gamma \delta n^2∕2$, while the dashed line is $N\gamma \delta n(\delta n+1)∕4$.

Figure 10

(Color online) Decay of populations for a bath that induces transitions between levels that are arbitrarily far apart (indicated by arrows), calculated in the weak-coupling approximation explained in the text, for the Ohmic bath at $T=0$ (see Sec. 8 and Appendix ).