Emission of Polarization-Entangled Microwave Photons from a Pair of Quantum Dots

We describe a mechanism for the production of polarization-entangled microwaves using intraband transitions in a pair of quantum dots. This proposal relies neither on spin-orbit coupling nor on control over electron-electron interactions. The quantum correlation of microwave polarizations is obtained from orbital degrees of freedom in an external magnetic field. We calculate the concurrence of emitted microwave photon pairs and show that a maximally entangled Bell pair is obtained in the limit of weak interdot coupling.

Figure 1

Schematic of the microwave entangler. Left panel: The four-dot arrangement between two electron reservoirs. Right panel: Positions of dot levels, with arrows indicating the transitions discussed in the text.

Figure 2

Fock-Darwin spectrum of dots $A$ and $B$: the six lowest levels with their quantum numbers $n_{\pm }$ are shown for each dot. The dot sizes are tuned such that ${\omega }_{-}^A={\omega }_{+}^B=\Omega$. Selection rule allowed transitions are shown, with solid (dashed) arrows denoting on (off) resonant transitions. The circular polarization, ${\mathrm{CP}}^{+}$ or ${\mathrm{CP}}^{-}$, of the emitted microwaves is also shown. In the entangler, electrons are injected into the two levels at $E_1$, and decay via the transitions shown in the dotted rectangle. In dot $A$, this produces a cascade of two ${\mathrm{CP}}^{+}$ photons, whereas in dot $B$ two ${\mathrm{CP}}^{-}$ photons are produced.

Figure 3

Dynamics of the microwave entangler. Plotted is the time dependence of the mean number of photon pairs in the cavity $N_2$ and their degree of entanglement $\mathcal{C}$ (the concurrence). For long times, the field approaches a steady state with, for the weak coupling $T_A=T_B=0.05\Omega$ shown here, a high proportion of photon pairs $N_2^{\infty }\approx 0.7$ and almost maximal concurrence ${\mathcal{C}}^{\infty }\approx 1$. Model parameters were ${\omega }_c=g={\Gamma }_R=0.1\Omega$.

Figure 4

Performance of the microwave entangler. Upper panel: The asymptotic entanglement of the photon pairs as a function of the asymmetry in the couplings of dots $A$ and $B$ to the other two dots. The solid curve is the analytic result (7) for $T_A,T_B\ll {\omega }_c$. We fixed $g={\omega }_c$. Lower panel: Probability $N_2^{\infty }$ of successful operation of the entangler as a function of the dot-microwave coupling $g$. The symbols correspond to the same values of $T_A$ as in the upper panel, and we fixed $T_A=T_B$. In both plots, we took ${\omega }_c={\Gamma }_R=0.1\Omega$.

Figure 5

Effect of decoherence, with rate ${\Gamma }_{\varphi }$, on the asymptotic entanglement of the photon pairs. We fixed $T_A=T_B=0.05\Omega$, ${\omega }_c={\Gamma }_R=0.1\Omega$, and varied $g$.