Interference and multiparticle effects in a Mach-Zehnder interferometer with single-particle sources

We investigate a Mach-Zehnder interferometer fed by two time-dependently driven single-particle sources, one of them placed in front of the interferometer, the other in the center of one of the arms. As long as the two sources are operated independently, the signal at the output of the interferometer shows an interference pattern, which we analyze in the spectral current, in the charge and energy currents, as well as in the charge current noise. The synchronization of the two sources in this specifically designed setup allows for collisions and absorptions of particles at different points of the interferometer, which have a strong impact on the detected signals. It introduces further relevant time scales and can even lead to a full suppression of the interference in some of the discussed quantities. The complementary interpretations of this phenomenon in terms of spectral properties and tunable two-particle effects (absorptions and quantum exchange effects) are put forward in this paper.

Figure 1

(a) Schematic of an electronic MZI. Transport takes place along edge states (black lines; arrows indicate their chirality). Quantum point contacts ${\mathrm{QPC}}_{\text{L}}$ and ${\mathrm{QPC}}_{\text{R}}$ act as beam splitters. All the reservoirs are grounded and particles are injected into the system by two single-particle sources ${\mathrm{SPS}}_{\text{A}}$ and ${\mathrm{SPS}}_{\text{B}}$. Charge and energy currents are measured at reservoir 4. (b) Schematic of an SPS, which is realized by a mesoscopic capacitor. It is implemented as a circular edge state and periodically driven by a potential $U_k\left(t\right)$, emitting one electron and one hole per period.

Figure 2

Electronic part of the excess-energy distribution function $i_{\text{MZI,A}}^{\text{e}}(E,\Phi )$ as a function of the energy $E$ in units of $\hslash /{\sigma }_{\mathrm{A}}$ and the magnetic-flux-dependent phase $\Phi$. (a) Almost perfectly balanced interferometer, with $\Delta \tau =0.01{\sigma }_{\text{A}}$. (b) Unbalanced interferometer, with $\Delta \tau =20{\sigma }_{\text{A}}$. (c) Cuts through the 3D plot of (b) at different energies $E$ and phases $\Phi$. In all plots, the transmission probabilities are given by $T_{\text{L}}=T_{\text{R}}=0.5$.

Figure 3

Electronic part of the average charge current ${\overline{I}}_{\text{MZI,A}}^{\text{e}}$ (full lines) and of the average energy current ${\overline{J}}_{\text{MZI,A}}^{\text{e}}$ (dashed lines) as a function of the phase $\Phi$ for different values of the detuning $\Delta \tau$. The transmission probabilities are $T_{\text{L}}=T_{\text{R}}=0.5$.

Figure 4

Energy-distribution function $i^{\text{eh}}(E,\Phi )$ shown for positive values of the energy $E$ only, in the regime where absorptions of electrons emitted by A are possible through the emission of holes from B depending on the time difference $\Delta t_{\text{d}}^{\text{eh}}$. Here, we take $\Delta t_{\text{d}}^{\text{eh}}=0.1{\sigma }_{\text{A}}$ and show $i^{\text{eh}}(E,\Phi )$ as a function of the energy $E$ in units of $\hslash /{\sigma }_{\mathrm{A}}$ and the magnetic-flux-dependent phase $\Phi$. The interferometer is almost perfectly balanced $\Delta \tau =0.01{\sigma }_{\text{A}}$, the pulse widths are assumed to be equal ${\sigma }_{\text{A}}={\sigma }_{\text{B}}$, and the transmission probabilities are given by $T_{\text{L}}=T_{\text{R}}=0.5$.

Figure 5

Classical part of the excess-energy distribution function $i^{\text{ee,cl}}\left(E\right)$, in the regime where collisions between particles of the same kind are possible depending on the time difference $\Delta t_{\text{d}}^{\text{ee}}$. We show the electronic contribution as a function of the energy $E$ in units of $\hslash /{\sigma }_{\mathrm{A}}$. We take ${\sigma }_{\text{A}}={\sigma }_{\text{B}}$ and the transmission probabilities are given by $T_{\text{L}}=T_{\text{R}}=0.5$.

Figure 6

Interference part of the excess-energy distribution function $i^{\text{ee,int}}(E,\Phi )$ in the regime where collisions between particles of the same kind are possible depending on the time difference $\Delta t_{\text{d}}^{\text{ee}}$. We show the electron contribution (a) as a function of the energy $E$ in units of $\hslash /{\sigma }_{\mathrm{A}}$ and the magnetic-flux-dependent phase $\Phi$ close to collision $\Delta t_{\text{d}}^{\text{ee}}=0.1{\sigma }_{\text{A}}$ and (b) as a function of the energy $E$ for three different flux values and for $\Delta t_{\text{d}}^{\text{ee}}=0.1{\sigma }_{\text{A}}$ (full lines) and $\Delta t_{\text{d}}^{\text{ee}}=2{\sigma }_{\text{A}}$ (dashed lines). The interferometer is almost perfectly balanced $\Delta \tau =0.01{\sigma }_{\text{A}}$, the pulse widths are assumed to be equal ${\sigma }_{\text{A}}={\sigma }_{\text{B}}$, and the transmission probabilities are given by $T_{\text{L}}=T_{\text{R}}=0.5$.

Figure 7

Charge and energy current detected at reservoir 4 (output of the interferometer), with symmetric transmission of ${\mathrm{QPC}}_{\text{L}}$ and ${\mathrm{QPC}}_{\text{R}}$. (a) Charge current of an MZI fed by ${\mathrm{SPS}}_{\text{A}}$ only, as a function of the MZI detuning $\Delta \tau$ in units of the current-pulse width ${\sigma }_{\text{A}}$ and as a function of the magnetic flux $\Phi$. (b) Charge current of a slightly detuned MZI, $\Delta \tau =0.5{\sigma }_{\text{A}}$ fed by an electron from ${\mathrm{SPS}}_{\text{A}}$ and a hole from ${\mathrm{SPS}}_{\text{B}}$ as a function of the time difference $\Delta t_{\text{d}}^{\text{eh}}$ in units of the pulse width ${\sigma }_{\text{A}}={\sigma }_{\text{B}}$ and the magnetic-flux-dependent phase $\Phi$. (c) Charge current of a slightly detuned MZI, $\Delta \tau =0.5{\sigma }_{\text{A}}$ fed by an electron both from ${\mathrm{SPS}}_{\text{A}}$ and ${\mathrm{SPS}}_{\text{B}}$ as a function of the time difference $\Delta t_{\text{u}}^{\text{ee}}$ in units of the pulse width ${\sigma }_{\text{A}}={\sigma }_{\text{B}}$ and the magnetic-flux-dependent phase $\Phi$. (d)–(f) Energy currents for the same situations shown in (a)–(c).