Glauber coherence of single-electron sources

Recently demonstrated solid-state single-electron sources generate different quantum states depending on their operation condition. For adiabatic and nonadiabatic sources, we determine the Glauber correlation function in terms of the Floquet scattering matrix of the source. The correlation function provides full information on the shape of the state and on its time-dependent amplitude and phase. The coherence properties of single-electron states are therefore essential for the production of quantum multiparticle states.

Figure 1

Schematic representation of an MZI, threaded by a magnetic flux $\Phi$. With a time-resolved measurement of the current in one of the output arms, one can access the first-order correlation function $G^{\left(1\right)}$ as a function of the time delay of the interferometer $\Delta \tau$ and time $t$. This allows us to reconstruct the incoming state emitted by the single-electron source. In the adiabatic regime, the single-electron current pulse has a Lorentzian shape, with a width $2{\Gamma }_{\mathrm{SES}}$.

Figure 2

Time-dependent potential $V$ driving the SES and induced current $I$ consisting of electron and hole pulses are shown for one period of $V$: (a) in the adiabatic regime and (b) in the nonadiabatic regime. The emission process takes place when $V$ makes the topmost occupied level cross the Fermi energy $\mu$. The emission times of an electron and a hole are, respectively, denoted $t^{-}=\pi /2\Omega$ and $t^{+}=3\pi /2\Omega$. The strong symmetric (asymmetric) shape of the current pulse is characteristic of the adiabatic (nonadiabatic) emission process.

Figure 3

Real (a) and imaginary (b) part of the first-order correlation function $G_{e,\text{ad}}^{\left(1\right)}(t^{\prime }-\Delta \tau /2,t^{\prime }+\Delta \tau /2)$ for electrons emitted adiabatically [Eq. (5) in units of $1/\left(\pi \Gamma v_D\right)$]. Here we set $t^{-}\equiv 0$. At $t^{\prime }=\Delta \tau =0$, the overlap of the wave packets is maximal ($G_{e,\text{ad}}^{\left(1\right)}=1$ in normalized units). The decrease of $G_{e,\text{ad}}^{\left(1\right)}$ as a function of $t^{\prime }$ and $\Delta \tau$ is set, respectively, by the lifetime $T_1=\Gamma$ and the coherence time $T_2=2\Gamma$. The real part of the $G_{e,\text{ad}}^{\left(1\right)}$ function at $\Delta \tau =0$ corresponds to the current pulse emitted by the source as a function of $t^{\prime }$, whereas its imaginary part is zero as expected.

Figure 4

First-order correlation function for electrons emitted nonadiabatically, $G_{e,\text{na}}^{\left(1\right)}(t^{\prime }-\Delta \tau /2,t^{\prime }+\Delta \tau /2)$, Eq. (7), in units of $1/\left({\tau }_D{v}_D\right)$. The exponential factor $e^{-i\pi \Delta \tau /\tau }$ is omitted as it sets the energy at which the single-particle state is emitted (see text). Here $t^{-}$ is set to 0. The correlation function clearly reflects the temporal shape of the single-electronic state emitted by the source, which is set by $T_1$ and $T_2$ as a function of $t^{\prime }$ and $\Delta \tau$, respectively.