Voltage and dephasing probes in mesoscopic conductors: A study of full-counting statistics

Voltage and dephasing probes introduce incoherent inelastic and incoherent quasielastic scattering into a coherent mesoscopic conductor. We discuss in detail the concepts of voltage and dephasing probes and develop a full-counting statistics approach to investigate their effect on the transport statistics. The formalism is applied to several experimentally relevant examples. A comparison of different probe models and with procedures like phase averaging over an appropriate phase distribution shows that there is a perfect equivalence between the models for the case of one single-channel probe. Interestingly, the appropriate phase-distribution function is found to be uniform. A uniform distribution is provided by a chaotic cavity with a long dwell time. The dwell time of a chaotic cavity plays a role similar to the charge response time of a voltage or dephasing probe. For multichannel or multiple probes the transport statistics of voltage and dephasing probes differs and the equivalence with phase averaging is similarly lost.

Figure 1

(Color online) A voltage (dephasing) probe described by the potential $V_p$ (the occupation function $n_p$), connected to a mesoscopic conductor with $M$ terminals.

Figure 2

(Color online) (a) A voltage probe is a metallic contact with a floating potential $V_p$. It can be represented in an electrical circuit as a metallic region with capacitance $C$ to the ground and charge relaxation resistance $R$. The low-frequency current into the probe vanishes. (b) Model of a dephasing probe: the probe is described by a nonequilibrium occupation function $n_p$. The low-frequency current vanishes separately in each energy interval.

Figure 3

(Color online) Time-dependent potential $V_p\left(t\right)$ at a voltage probe. The current injected from the conductor into the probe gives rise to fast fluctuations on the time scale ${\tau }_s=h∕eV$. The response of the probe gives rise to modulations on a much longer time scale ${\tau }_d=RC$.

Figure 4

(Color online) A two-particle scattering event from channels $1\alpha$ and $1\beta$ into $2\beta$ and $3\beta$: the occupation probability is multiplied by the scattering probability and by the exponential factor containing the counting fields. The scattering probability ${\mid \det \left({\mathcal{S}}_{\{2\beta ,3\beta \}}^{\{1\alpha ,1\beta \}}\right)\mid }^2={\mid {\mathcal{S}}_{2\beta ,1\alpha }{\mathcal{S}}_{3\beta ,1\beta }-{\mathcal{S}}_{2\beta ,1\beta }{\mathcal{S}}_{3\beta ,1\alpha }\mid }^2$ represents the two indistinguishable processes shown above in the center, with the minus sign originating from the fermionic statistics of the particles.

Figure 5

(Color online) Mach-Zehnder interferometer: a four-terminal conductor with unidirectional transport channels (indicated by arrows), threaded by a magnetic flux ${\Phi }_{AB}$.

Figure 6

(Color online) The upper arm of the Mach-Zehnder interferometer is coupled to a dephasing or voltage probe with coupling strength $\epsilon$.

Figure 7

(Color online) (i) No particles are moving in or out of the probe. (ii) One particle enters, one leaves the probe, thus there is no net charge transfer into the probe. (iii) One particle is emitted from the probe. (iv) One particle enters the probe. (i) and (iv) are one-particle processes, and (ii) and (iii) are two-particle processes. Note the full red dots represent injected particles and filled (empty) green circles represent the one (two) final states.

Figure 8

(Color online) Phase averaging: a particle enters the elastic scatterer $\varphi$ with probability $\epsilon$ and obtains a uniformly distributed phase factor $e^{i\phi }$. After multiple internal reflections it leaves the scatterer with an overall phase $e^{i\tilde{\phi }}$. The inset shows a general conductor to which an elastic scatterer $\varphi$ is connected.

Figure 9

(Color online) Beam splitter with a probe: a three-terminal quantum Hall bar at filling factor 2 and one voltage or dephasing probe. The two quantum-point contacts QPC 1 and QPC 3 are open for the outer transport channel. A voltage is applied on terminal 1, thus the channels shaded in (light) gray are effectively empty and do not matter. The inset shows the setup for the case when the connection to the probe is closed, i.e., when there is no probe connected to the beam splitter.

Figure 10

(Color online) A four-terminal geometry with two probes attached. The direction of transport channels is indicated by arrows. The intersections marked with yellow bars represent beam splitters. Interference between paths passing through the probes in different order as shown on the right is destroyed by dephasing probes, but not by phase averaging.

Figure 11

(Color online) Three-barrier setup with two dephasing probes (a) or two elastic scatterers (b). While the probes destroy the phase memory in each loop individually, the elastic scatterer (phase averaging) allow coherent multiple loops.