Relaxation and Dephasing in a 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 fluctuating force generated by 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 Green’s functions and discuss relaxation and dephasing of a single extra particle added above the Fermi sea. We also extend our analysis to a more generic coupling between system and bath that results in complete thermalization of the system.

Figure 1

(Left) Fermions in an oscillator, coupled to a dissipative bath via a fluctuating force $\widehat{F}$. (Right) Approximately equivalent model of chiral fermions on a ring, subject to a transverse force.

Figure 2

Time evolution of the single-particle density matrix ${\rho }_{n{n}^{\prime }}(t)$ [Eq. (11)], after placing an extra particle in a superposition of two states. The radius of each circle gives $|{\rho }_{n{n}^{\prime }}(t)|$. Levels $n,n^{\prime }=n_F-1,\dots ,n_F+6$ are indicated by grid lines. For this example ${\omega }_0-{\omega }_0^{\prime }=2\Gamma$.

Figure 3

Generic time evolution of density matrix in the limit of high initial excitation [Eqs. (12, 13, 14)]; plot for ${\omega }_0^{\prime }={\omega }_0$).

Figure 4

Decay of populations for a bath inducing transitions between levels that are arbitrarily far apart (see text).