Superradiance and Subradiance due to Quantum Interference of Entangled Free Electrons

When multiple quantum emitters radiate, their emission rate may be enhanced or suppressed due to collective interference in a process known as super- or subradiance. Such processes are well known to occur also in light emission from free electrons, known as coherent cathodoluminescence. Unlike atomic systems, free electrons have an unbounded energy spectrum, and, thus, all their emission mechanisms rely on electron recoil, in addition to the classical properties of the dielectric medium. To date, all experimental and theoretical studies of super- and subradiance from free electrons assumed only classical correlations between particles. However, dependence on quantum correlations, such as entanglement between free electrons, has not been studied. Recent advances in coherent shaping of free-electron wave functions motivate the investigation of such quantum regimes of super- and subradiance. In this Letter, we show how a pair of coincident path-entangled electrons can demonstrate either super- or subradiant light emission, depending on the two-particle wave function. By choosing different free-electron Bell states, the spectrum and emission pattern of the light can be reshaped, in a manner that cannot be accounted for by a classical mixed state. We show these results for light emission in any optical medium and discuss their generalization to many-body quantum states. Our findings suggest that light emission can be sensitive to the explicit quantum state of the emitting matter wave and possibly serve as a nondestructive measurement scheme for measuring the quantum state of many-body systems.

Figure 1

Super- and subradiance from quantum-correlated free charged particles. (a) A quantum current operator $\widehat{\mathbf{j}}(\mathbf{r},t)$, associated with the emission of light quanta by multiple charged particles in a general optical environment, is used to find the collective (super- or subradiant) emission by calculating current-current correlations. (b) Exemplifying the general concept, when a pair of quantum-correlated particles emits radiation, the quantum interference between the transition amplitudes can lead to enhancement or suppression of the emitted light intensity. (c) This system can be realized by two free electrons of opposite spins with kinetic energy levels spaced by $\delta E$, prepared in an entangled state (left). Then, upon emission of a single photon, the two-electron state recoils by $\hslash \omega =\delta E$. Overlap between recoiled final states results in quantum interference leading to super- and subradiance.

Figure 2

Shaping light using quantum correlations. (a),(b) Illustration of coherent cathodoluminescence from two correlated particles. In (a), a pair of coincident electrons with different momenta $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$ and spins $\uparrow$ and $\downarrow$ are prepared in a classically correlated (mixed) state and interact with an optical environment—giving classical emission. In (b), the electrons are instead prepared in a path-entangled Bell state, and the emission pattern is modified, depending explicitly on the phase angle $\zeta$ of the electrons’ quantum state. (c),(d) Quantum shaping of Cherenkov radiation by two-electron Bell states. (c) The rate of Cherenkov photon emission per unit time per unit frequency $\mathrm{\Gamma }$, normalized by the single-particle rate ${\mathrm{\Gamma }}_0=\alpha \beta {\sin }^2{\theta }_c/2\pi$. The normalized rate $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$ is calculated along the Cherenkov cone and plotted as a function of the azimuthal angle $\varphi$ (see the inset). For different phases $\zeta$ of the electron Bell state, the radiation pattern is no longer azimuthally symmetric on the cone as in the classical case (red full line) and is either enhanced ($\zeta =\pi$, green dotted line) or suppressed ($\zeta =0$, blue dash-dotted line). The wave packets that constitute the two-electron Bell state differ by a transverse wave vector $\mathrm{\Delta }\mathbf{k}=\sin {\theta }_{c}n\omega /c\widehat{\mathbf{x}}=q_T\widehat{\mathbf{x}}$, chosen to match the transverse photon momentum $q_T$. (d) Normalized emission rate $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$ vs normalized frequency in units of ${\omega }_0=v_0\mathrm{\Delta }k$, where now $\mathrm{\Delta }\mathbf{k}$ is chosen parallel to $\widehat{\mathbf{z}}$. Near $\omega =v_0\mathrm{\Delta }k={\omega }_0$, a resonance appears. The magnitude of the spectral feature is governed by the phase angle $\zeta$ of the Bell state, giving quantum super- and subradiance. Electron velocity $v_0=0.7c$, refractive index $n=2$, emitted photon energy $\hslash \omega =2\mathrm{eV}$, wave function dimensions in (c) $\mathrm{\Delta }{r}_T=200\mathrm{nm}$ and $\mathrm{\Delta }z=1\mathrm{nm}$ and in (d) $\mathrm{\Delta }{r}_T=10\mathrm{nm}$ and $\mathrm{\Delta }z=500\mathrm{nm}$, for the transverse and longitudinal sizes, respectively.