Two-particle entanglement in capacitively coupled Mach-Zehnder interferometers

We propose and analyze a mesoscopic device producing on-demand entangled pairs of electrons. The system consists of two capacitively coupled Mach-Zehnder interferometers implemented in a quantum Hall structure. A pair of electron wave packets is injected into the chiral edge states of two (of the four) incoming arms; scattering on the incoming interferometers splits the wave packets into four components, of which two interact. The resulting interaction phase associated with this component leads to the entanglement of the state; the latter is scattered at the outgoing beam splitter and analyzed in a Bell violation test measuring the presence of particles in the four outgoing leads. We study the two-particle case and determine the conditions to reach and observe full entanglement. We extend our two-particle analysis to include the underlying Fermi seas in the quantum Hall device; the change in shape of the wave function, the generation of electron-hole pairs in the interaction regime, and a time delay between the pulses all reduce the degree of visible entanglement and the violation of the Bell inequality, effects which we analyze quantitatively. We determine the device settings optimizing the entanglement and the Bell test and find that violation is still possible in the presence of the Fermi seas, with a maximal Bell parameter reaching $\mathcal{B}=2.18>2$ in our setup.

Figure 1

Two Mach-Zehnder interferometers capacitively coupled via the arms 1 and 2. Lorentzian-shaped pulses are injected into the left arms, entangled through the Coulomb interaction, and analyzed in a Bell test on the outgoing channels. In spite of the decoherence generated by the presence of the Fermi sea, the Bell inequality can be violated, demonstrating the possibility to generate “useful” entanglement correlations in this setup.

Figure 2

Bell parameters ${\mathcal{B}}_{\max }$ and ${\mathcal{B}}_0$ as a function of of width parameter $\gamma =\xi /a$ at ${\phi }_0=\pi$ for an optimal setting of Bell parameters (${\mathcal{B}}_{\max }$, solid line) as given by Eq. (7) and for the nonoptimal settings in Eq. (29) (${\mathcal{B}}_0$, dashed line).

Figure 3

Contour plot of the Bell parameter ${\mathcal{B}}_{\max }$ for optimal settings of the Bell parameters as a function of the interaction parameter ${\phi }_0$ (horizontal axis) and the width parameter $\gamma$ (vertical axis). For small-width, high-energy wave packets ($\gamma \to 0$), the Bell inequality is maximally violated for odd multiples of $\pi$.

Figure 4

von Neumann entropy $S\left({\widehat{\rho }}_{B{B}^{\prime }}\right)$ of measured outcomes of the double Mach-Zehnder interferometer quantifying the entanglement with the unobserved degrees of freedom in the system. The entanglement increases with the interaction strength and the system is fully entangled with the environment at large ${\phi }_0$. The entropy $S\left({\widehat{\rho }}_B\right)$ expresses the entanglement between the two Mach-Zehnder interferometers and thus the potential for violation of the Bell test. Increasing the interaction first entangles the two MZ interferometers with each other, but as the entire system gets entangled with the environment upon further increase of ${\phi }_0$, the entanglement between the interferometers decreases.

Figure 5

Bell parameter (57) for the nonoptimal settings of the Bell parameters in Eq. (29) as a function of interaction parameter ${\phi }_0$ (horizontal axis) and the parameter $\eta$ (vertical axis) describing the ratio of self- to mutual capacitance.

Figure 6

Maximal value of the Bell parameter as a function of the interaction parameter ${\phi }_0$ (horizontal axis) and the parameter $\eta$ describing the ratio of self- to mutual capacitance (vertical axis).

Figure 7

Bell parameter (57) as a function of interaction parameter ${\phi }_0$ (horizontal axis) and the width parameter $\gamma =\xi /a$ (vertical axis) at $\eta =0.58$.

Figure 8

Interaction region between edge states 1 and 2 in a quantum Hall device. The edge states are separated from the closest screening Fermi sea by a distance $d_s\approx {\ell }^2/\xi$ (top left). We model the capacitive interaction Hamiltonian (2) by analyzing two charges $q_1$ and $q_2$ separated by $s$ and screened by metallic plates at a distance $d_{\text{F}}$.