Equations of motion approach to decoherence and current noise in ballistic interferometers coupled to a quantum bath

We present a technique for treating many particles moving inside a ballistic interferometer, under the influence of a quantum-mechanical environment (phonons, photons, Nyquist noise, etc.). Our approach is based on solving the coupled Heisenberg equations of motion of the many-particle system and the bath, and it is inspired by the quantum Langevin method known for the Caldeira-Leggett model. As a first application, we treat a fermionic Mach-Zehnder interferometer. In particular, we discuss the dephasing rate and present full analytical expressions for the leading corrections to the current noise, brought about by the coupling to the quantum bath. In contrast to a single-particle model, both the Pauli principle as well as the contribution of hole-scattering processes become important, and are automatically taken into account in this method.

Figure 1

(Color online) (a) The Caldeira-Leggett model (single particle and oscillator bath) and (b) a ballistic many-particle system subject to a quantum noise potential $\widehat{V}(x,t)$.

Figure 2

(Color online) Schematic of the Mach-Zehnder setup, with beam splitters $A,B$, input ports 1, 2, and output ports 3, 4.

Figure 3

(Color online) Energy-resolved dephasing rate for an optical phonon mode (at ${\omega }_0$), as a function of transport voltage applied to the Mach-Zehnder, for two different temperatures: (a) $T=0.1{\omega }_0$; (b) $T=0.5{\omega }_0$. Here ${\Gamma }_0={\Gamma }_{\phi }$ ($ϵ\to \infty$, $T=0$).

Figure 4

(Color online) Contributions to the decoherence rate in a Keldysh-diagrammatic treatment. Left: Diagram involving both thermal and quantum fluctuations of the bath, but no Fermi distributions. This diagram is the same in a single-particle situation. Right: Diagram corresponding to the “backaction” term discussed in the equations of motion approach. It involves the fermionic Keldysh Green’s function [that contains the Fermi distribution, $G^K(ϵ-\omega )\propto 1-2f(ϵ-\omega )$] and the bath’s retarded propagator (describing the response).

Figure 5

(Color online) Contribution of particle- and hole-scattering processes to the dephasing rate in a many-fermion interferometer.

Figure 6

(Color online) (a) Correction to the flux-averaged current noise power $S_0$ for a damped optical phonon mode (strength $\alpha$, time-of-flight $\tau$), normalized with respect to unperturbed value. (b) and (c): The corrections to the first and second harmonics $S_1$ and $S_2$ are negative, revealing the loss of interference contrast in $S\left(\varphi \right)$. Energies are plotted in units of the phonon mode $({\omega }_0=1)$, and the MZ setup has been chosen asymmetric ($T_A=0.3$, $T_B=0.4$).

Figure 7

(Color online) Corrections to the phase shifts $\delta {\varphi }_1$ (a) and $\delta {\varphi }_2$ (b) of the first and second harmonics in the shot noise pattern $S\left(\varphi \right)$, in an asymmetric MZ setup (parameters as in Fig. 6). These phase shifts will, in general, be different from the effective phase shift $\delta \overline{\phi }$ in the current pattern, leading to the situation schematically depicted in (c).