Universal and Ultrafast Quantum Computation Based on Free-Electron-Polariton Blockade

Cavity QED, wherein a quantum emitter is coupled to electromagnetic cavity modes, is a powerful platform for implementing quantum sensors, memories, and networks. However, due to the fundamental trade-off between gate fidelity and execution time, as well as limited scalability, the use of cavity QED for quantum computation was overtaken by other architectures. Here, we introduce a new element into cavity QED—a free charged particle, acting as a flying qubit. Using free electrons as a specific example, we demonstrate that our approach enables ultrafast, deterministic, and universal discrete-variable quantum computation in a cavity-QED-based architecture, with potentially improved scalability. Our proposal hinges on a novel excitation blockade mechanism in a resonant interaction between a free-electron and a cavity polariton. This nonlinear interaction is faster by several orders of magnitude with respect to current photon-based cavity-QED gates, enjoys wide tunability and can demonstrate fidelities close to unity. Furthermore, our scheme is ubiquitous to any cavity nonlinearity, either due to light-matter coupling as in the Jaynes-Cummings model or due to photon-photon interactions as in a Kerr-type many-body system. In addition to promising advancements in cavity-QED quantum computation, our approach paves the way towards ultrafast and deterministic generation of highly entangled photonic graph states and is applicable to other quantum technologies involving cavity QED.

Popular Summary

Quantum computing is a promising paradigm that harnesses quantum physics to solve hard problems, which otherwise cannot be solved efficiently with classical computers. One of the earliest proposals of physical systems realizing a quantum computer relied on the interaction between a photon (a light particle) and a stationary matter particle (for example, an atom) placed inside a resonating optical cavity. This approach is known as cavity QED. However, it turned out that cavity QED suffers from fundamental limitations. It is intrinsically slow and becomes even slower the better the quality of the cavity used, and in many instances, it is very hard to implement large-scale computation with it, as it is very sensitive to variations in parameters of the system. Therefore, other approaches for quantum computation hardware have since prevailed.

In this work, we discover that the addition of another quantum particle—a flying charged particle like an electron or an ion—can break the fundamental limitations on speed and scalability of cavity QED and has the potential to bring it back to the mainstream of research in quantum computing hardware. Our proposal relies on a new type of interaction between this flying charged particle and the hybrid system of light and matter in the cavity, which can act several orders of magnitude faster than previously used interactions while being far more robust to changes in the system parameters. We present a universal, ultrafast, and deterministic quantum gate set based on this new approach, allowing in principle the realization of any quantum computation routine using our approach.

Our findings thus extend decades-old models of cavity QED and related hybrid systems of light and matter by introducing a fundamentally new quantum entity in the form of a free charged particle. In addition to the promising potential for boosting speed and scaling of cavity-QED quantum computing, our predictions can pave the way toward new avenues in other quantum technologies involving cavity QED, such as quantum sensing, quantum communications, and quantum simulation.

Figure 1

Integrating flying charged-particle qubits into cavity QED. (a) In standard cavity QED, a stationary matter qubit (shown in green) interacts nonlinearly with a flying photonic qubit (red wave packet), in an interaction mediated by a cavity. Light-matter entanglement between the flying and stationary qubits is generated over slow timescales. (b) A flying charged-particle qubit (such as an electron or ion, blue wave packet) interacts linearly with a photonic qubit on an ultrafast timescale, resulting in entanglement between them. (c) When a flying charged-particle qubit drives a cavity-QED system, the light-matter interaction can be both nonlinear and ultrafast, resulting in three-qubit entanglement between the constituents of the system. (d) Concept illustration of cavity QED implemented in integrated photonic circuits and driven by a flying charged-particle qubit: ultrafast quantum gates and entanglement distribution between separate cavity polaritons is mediated by the path- and energy-encoded information of the flying charged-particle qubit (manipulated by matter wave beam splitters). Each excited cavity polariton comprises an entangled light-matter state, and its photonic part can still be used as a flying qubit, for long-distance communication between processors or over a quantum network.

Figure 2

Free-electron–polariton blockade. (a) A quantum free electron traverses a linear cavity while being phase matched to the cavity mode, conserving both momentum and energy. If the interaction is strong enough, the electron is able to emit multiple photons into the cavity, and its energy becomes highly entangled with the cavity photon number. (b) When the same electron traverses a cavity-QED-like system, which is comprised of a nonlinear cavity, the inherent polariton blockade effect allows for phase matching only in the process of a single polariton emission, while consecutive emission events will be phase mismatched. As a result, the electron energy becomes entangled in a minimal polariton-number Hilbert space. The dynamics then reduces to an effective two-level system driven by a quantum free electron. (c),(d) Cavity-QED models for a free-electron polariton blockade. In both settings, the electron grazes the cavity in vacuum and interacts with the polariton nearfield. The first model is a Kerr nonlinear system (c), characterized by the interaction Hamiltonian $H_{\mathrm{nl}}=\hslash \kappa a^{†}{a}^{†}aa$ and realized by interacting exciton polaritons with creation and annihilation operators $a^{†},a$ in microcavities or waveguides [98, 99, 100, 101]. Inset: the polaritonic energy levels as a function of the nonlinearity ratio $\kappa /\omega$, with spacing linearly increasing with polariton number n. The second model is a Jaynes-Cummings system (d), characterized by the interaction Hamiltonian $H_{\mathrm{nl}}=\hslash \kappa {\sigma }_{+}a+\hslash \kappa {\sigma }_{-}{a}^{†}$ and realized, for example, by situating a two-level system with ladder operators ${\sigma }_{+},{\sigma }_{-}$ inside a nanobeam single-mode microcavity [23, 24, 102] (typical mode lengths ranging between several microns to tens of microns [103, 104, 105, 106, 107]), with photonic creation and annihilation operators $a^{†},a$. Inset: polaritonic energy as a function of $\kappa /\omega$. While the Rabi splitting increases as $\sqrt{n}$, the spacing between adjacent energy levels decreases as $\sqrt{n}-\sqrt{n-1}$.

Figure 3

Polariton statistics and electron energy-loss spectra of the free-electron–polariton blockade: dependence on cavity nonlinearity and phase-matching selectivity. Simulation results of electron energy-loss spectra (EELS) and polariton-number statistics, via the numerical integration of the Lindblad master equation [Eq. (3)]. (a),(b) Continuous transformation of the EELS (a) and photon-number statistics (b) for free-electron spontaneous emission into a Kerr nonlinear cavity as a function of the coupling ratio $\kappa /\omega$. As the nonlinearity increases, complete population inversion from the Kerr-polariton ground state $|\overline{0}⟩=|0⟩$ and a single Kerr-polariton Fock state $|\overline{1}⟩=|1⟩$ occurs, whereas a Poissonian distribution is apparent in the linear case. Parameters used in the simulation are $E=200\mathrm{keV}$, $L=40\mu \mathrm{m}$, $\gamma /\omega ={10}^{-5}$, $v=\omega /q_0=0.6953c$, $q_0=2\pi /532\mathrm{n}{\mathrm{m}}^{-1}$, $g_{\mathrm{Q}}=\pi /2$. (c),(d) EELS (c) and polariton-number statistics (d) for free-electron spontaneous emission into a Jaynes-Cummings nonlinear cavity as a function of the velocity ratio $v/v_0$ (where $v_0=\omega /q_0=0.2719c$). $\kappa /\omega =0.02$, $g_Q=\pi /\sqrt{2}$, $E=20\mathrm{keV}$ and all other parameters are the same as in (a),(b). In the JC example, we consider the single excitation manifold of $|\overline{0}⟩=|0^{\ast }⟩=|0,g⟩$ and $|\overline{1}⟩$, which can be either the upper $(|1+⟩)$ or lower $(|1-⟩)$ polariton state, $\mathrm{where}|1\pm ⟩=(|g,1⟩\pm |e,0⟩)/\sqrt{2}$, with $|0⟩,|1⟩$ denoting the bare photon-number states and $|g⟩(|e⟩)$ the bare emitter ground (excited) state. Only when the free-electron velocity is tuned to phase match with a specific polariton $[v=v_{\pm }=(\omega \pm \kappa )/q_0=(1\pm 0.02)v_0]$, does a full population inversion take place. In this manner, the electron velocity can be used to selectively determine the polariton type chosen for excitation. The difference in $g_Q$ between (a),(b) and (c),(d) arises from our assumption that in the JC model, the electron couples only to the photonic degrees of freedom of the cavity.

Figure 4

Effective photonic two-level system dynamics driven by free electrons. Polariton-number statistics for both the Kerr (a),(b) and JC (c),(d) models, as a function of the dimensionless electron-light coupling $g_{\mathrm{Q}}$. A single Rabi oscillation is apparent in all figures within a manifold preselected by the electron velocity, producing phase matching with a specific polariton transition [(a) $v=\omega /q$; (b) $v=(\omega +2\kappa )/q$; (c) $v=(\omega +\kappa )/q$; (d) $v=(\omega -\kappa )/q$]. Each transition is illustrated in the corresponding inset nonlinear cavity energy-level diagram. Stimulated emission in the cavity produces a faster population inversion in (b), which occurs at $g_{\mathrm{Q}}=\pi /\left(2\sqrt{2}\right)$. The coupling ratio $\kappa /\omega$ was set to 0.02 in all simulations. Otherwise, all parameters in (a),(b) and (c),(d) are the same as in Figs. 3,(b) and (c),(d), respectively.

Figure 5

Quantum state fidelity as a function of coupling strength and cavity loss. Quantum state fidelity for different photon-loss rates and coupling strengths, calculated between the total density matrix ${\rho }_I$ of Eq. (3) and the target state generated by the analytic scattering matrix of Eq. (4). (a) Jaynes-Cummings cavity, with parameters $E=20\mathrm{keV}$, $v=(\omega -\kappa )/q_0=0.2719c$, $q_0=2\pi /532\mathrm{n}{\mathrm{m}}^{-1}$, $L=40\mu \mathrm{m}$, and $g_{\mathrm{Q}}=\pi /\sqrt{2}$. (b) Kerr cavity, with parameters $E=200\mathrm{keV}$, $v=\omega /q_0=0.6953c$, $q_0=2\pi /532\mathrm{n}{\mathrm{m}}^{-1}$, $L=40\mu \mathrm{m}$, and $g_{\mathrm{Q}}=\pi /2$. (c) Comparison of the polariton state fidelity in each model, for the maximal coupling $\kappa /\omega =0.02$. Maximal fidelity for both models exceeds $97\mathrm{\%}$. Apart from the cavity-loss rate, the limiting factor for the fidelities is the interplay between the cavity nonlinearity $\kappa$ and the phase-matching bandwidth, proportional to $v/L$. If the latter is not much smaller than the former, the electron couples to higher-order polariton states that lie outside of the target Hilbert space.

Figure 6

Universal free-electron-driven single-qubit gates for the polariton state. (a) Electron-controlled z rotation on the polariton Bloch sphere, implemented by letting an electron interact twice with the same cavity but at different transverse locations within the nearfield. The coupling has the same magnitude $|\mathrm{\Omega }|=\pi /2$ for each pass, yet the two paths can have a phase difference $\varphi$. (b) Electron-controlled transverse rotation gate about a general transverse axis $\hat{\rho }=(\cos \phi ,\sin \phi )$ on the polariton Bloch sphere, implemented by preparing an electron in a comb state $|\mathrm{comb}(\phi )⟩$ and letting it interact with the cavity. (c) Transformation of the physical qubit basis from the number basis $\{|0^{\ast }⟩,|1+⟩\}$ (top inset) to the polariton-type basis $\{|1-⟩,|1+⟩\}$ (bottom inset), by applying the gate $C_{\mathrm{ep}}{X}_{-}$ with a comb electron, having a velocity tuned with the transition $|0^{\ast }⟩\leftrightarrow |1-⟩$.

Figure 7

Two-polariton controlled-Z gate mediated by a free electron. (a) controlled-path $(C_{\mathrm{pe}}\mathrm{Path})$ gate between a polariton control qubit and a free-electron target qubit, prepared in an initial state $|E{⟩}_1$. The electron interacts with the cavity with coupling strength $\mathrm{\Omega }=\pi /2$, either losing or gaining an amount $\hslash \omega$ of energy. Then, the electron passes through an electron spectrometer, which translates the resulting different electron energies into two different output paths, both interacting again with the same cavity, restoring the original electron energy. The result is an entangled state wherein the polariton controls the output path of the electron, with an equivalent circuit as shown in the inset. (b) A free-electron Hadamard gate (realized, for example, by a ponderomotive beam splitter, Kapitza-Dirac elastic diffraction, a PINEM interaction, etc.). (c) The quantum circuit of the two-polariton controlled-Z gate and its equivalent form. The first polariton qubit controls the free-electron ancilla by $C_{\mathrm{pe}}\mathrm{Path}$. Then, the electron drives a $C_{\mathrm{ep}}Z$ gate acting on the second polariton qubit. The electron is disentangled from the polaritons by application of an electron Hadamard gate H followed by a second $C_{\mathrm{ep}}Z$ acting on the first qubit, and a final Hadamrd, leaving the ancilla qubit in state $|E{⟩}_0$. In the equivalent circuit (derived in Fig. 8 in Appendix pp4), it is clear that a controlled-Z gate is applied to the two bottom qubits, mediated by the top ancilla qubit.

Figure 8

Equivalent quantum circuit to the two-polariton controlled-Z gate mediated by an electron ancilla qubit.

Figure 9

Approximate single-mode interaction with periodic-structure-assisted phase-matched electron-photon coupling. (a) Illustration of phase matching to the first Fourier order of a periodic structure. The free-electron dispersion line $\mathrm{\Delta }E/\hslash$ (in green) wraps around the Brillouin zone (BZ). Each time the line passes the right BZ edge, a new line segment continues from the left edge. The mth line segment is described by $\mathrm{\Delta }E/\hslash =(q+2\pi m/\Lambda )v$, for $q\in \mathrm{BZ}$, where $v$ is the electron velocity and $\Lambda$ the structure’s period. If the mth segment intersects the photon dispersion such that $\omega (q)=(q+2\pi m/\Lambda )v$, this corresponds to phase matching with the mth Fourier order of that periodic mode. The zeroth-line segment $(m=0)$, shown by the dashed line, does not intersect the photon dispersion anywhere in the BZ. The first-line segment $(m=1)$ intersects a guided (or cavity) mode at negative q, having a negative group velocity, as required by Eq. (F5). This line segment may also intersect radiating modes outside the material’s transparency window (with a much smaller emission probability). (b) Illustration of the phase-matching bandwidth for a typical system as in (a), satisfying Eq. (F6), where only a single cavity mode is present within the phase-matching bandwidth. The parameters used for the illustration in (b) are $n_g=2$, $E=80\mathrm{keV}$, $\lambda =532\mathrm{nm}$, $Q={10}^4$, $L=40\mu \mathrm{m}$.

Figure 10

Proposal for a Bragg hollow nanofiber cavity for phase-matched electron-photon coupling. (a) Illustration of the proposed structure, comprising a periodic nanofiber ($a=40\mathrm{nm},b=200\mathrm{nm}$, $\Lambda =177.7\mathrm{nm}$, $n_1=2.4$, e.g., as in $\mathrm{Ga}\mathrm{N}$ in the visible range, and $n_2=1.44$, as in a glass material, or using air corrugations; inside the transparency range, we assume a lossless and nondispersive material for simplicity); band structure and eigenmodes are found using a Fourier-expansion method with $N=8$ spatial harmonics; dielectric constant profile is smoothed using a super-Gaussian function $ϵ(z)=n_1^2+(n_2^2-n_1^2)\exp (-{[(z-\Lambda /2)/\sigma ]}^p/2)$, with $\sigma =57.7\mathrm{nm}$ and $p=10$). A cavity could be formed from this structure using, e.g., hollow Bragg mirrors (not shown; a different configuration, using a photonic crystal defect, is also possible). The electron (energy $100\mathrm{keV}$, beam waist $6\mathrm{nm}$) is passing through the hollow core. (b) Band diagram of the TM mode inside the first Brillouin zone shown in blue, light line shown in black, electron dispersion lines corresponding to the zeroth- and first-order Fourier phase matching, shown in green (higher-order phase-matching dispersion lines lie above the material transparency cutoff, which for $\mathrm{Ga}\mathrm{N}$, is about 3.44 eV, or 833 THZ). Solutions to the phase-matching equation $\omega =(q+2\pi m/\Lambda )v$ correspond to intersections between the electron dispersion and the band structure curves. Only the first-order electron dispersion line intersects the band (at $\lambda =521.4\mathrm{nm}$), in the negative-q part as illustrated in Fig. 9, with $v_g=-v$ allowing for a single-mode interaction. The fact that there are no intersections between the electron dispersion for $m=0$ (dashed line) implies that there are no TM Bloch modes whose zeroth-order Fourier component is phase matched with the electron. (c),(d) Profiles of the z component of the electric field of the TM mode: transverse profile at $z=\Lambda /2$ (c) and the longitudinal profile in the unit cell (d), with a modulation ratio $M_1=27\mathrm{\%}$.