Creation of Optical Cat and GKP States Using Shaped Free Electrons

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Creation of Optical Cat and GKP States Using Shaped Free Electrons

Raphael Dahan, Gefen Baranes, Alexey Gorlach, Ron Ruimy, Nicholas Rivera, and Ido Kaminer

Phys. Rev. X 13, 031001 – Published 6 July 2023

Abstract

Cat states and Gottesman-Kitaev-Preskill (GKP) states play a key role in quantum computation and communication with continuous variables. The creation of such states relies on strong nonlinear light-matter interactions, which are widely available in microwave frequencies as in circuit quantum electrodynamics platforms. However, strong nonlinearities are hard to come by in optical frequencies, severely limiting the development and applications of quantum-information science with continuous variable in the optical range. Here we propose using the strong interaction of free electrons with light to implement the desired nonlinear mechanism, showing its implication by creating optical cat and GKP states. The key to our finding is identifying conditions on the electron for which its interaction mimics the conditional displacement quantum gate. The strong interactions can be realized by phase matching of free electrons with photonic structures such as optical waveguides and photonic crystals in an ultrafast transmission electron microscope (UTEM). Our approach enables the generation of optical GKP states with above 10 dB squeezing and fidelities above 90% at postselection probability of 10%, even reaching $> 30 %$ using an initially squeezed-vacuum state. We analyze the different factors that affect the fidelity, such as electron dispersion, inhomogeneity, nonideal interaction, and limited detection efficiency. Furthermore, the free-electron interaction allows two qubit gates between a pair of GKP states, which can entangle them into a GKP Bell state. We present a roadmap for realizing such experiments in a UTEM. Since electrons can interact resonantly with light across the electromagnetic spectrum, our approach could apply for a generation of cat and GKP states also in other platforms of free-electron radiation, from klystrons to free-electron lasers.

Received 17 June 2022
Revised 18 April 2023
Accepted 24 April 2023

DOI:https://doi.org/10.1103/PhysRevX.13.031001

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Non-gaussian quantum states Quantum optics Quantum states of light

Atomic, Molecular & Optical

Authors & Affiliations

Raphael Dahan ^1,†, Gefen Baranes ^2,†, Alexey Gorlach¹, Ron Ruimy ¹, Nicholas Rivera^2,3, and Ido Kaminer ^1,*

¹Technion—Israel Institute of Technology, Haifa 32000, Israel
²Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
³Harvard University, Cambridge, Massachusetts 02138, USA

^*kaminer@technion.ac.il
^†R. D. and G. B. contributed equally to this work.

Popular Summary

Realizing full control of the quantum state of light is a long-standing problem in quantum optics. This state can store quantum information as a continuous variable, in contrast to more conventional quantum bits that serve as discrete variables. Substantial efforts have been invested toward an efficient generation of quantum light states that could enable fault-tolerant quantum computing, but no one has yet demonstrated the most desirable of these states at optical frequencies. We offer a promising new approach to this problem by relying on the strong effective nonlinearity arising from the interaction of photons with free electrons.

The fundamental new realization underlying this study is that the most basic interaction in physics—that between free electrons and photons—provides the necessary ingredient to create the most desirable quantum states of light. Specifically, we show that the interaction of a free electron whose wave function was prepared in a superposition state called a comb naturally implements a fundamental type of operator in quantum information processing known as the conditional displacement gate. Multiple operations of such a gate enable the generation of Schrödinger cat and Gottesman-Kitaev-Preskill optical states—states that have been specifically designed to exhibit robustness against displacement errors and photon loss, which are the primary sources of noise in optical systems.

Our work offers exciting prospects for continuous-variable quantum information processing, with applications in communication and computing. More generally, the tunability of free-electron interactions and their applicability over the entire electromagnetic spectrum suggests a route toward applications of quantum information science in new spectral ranges even up to the x-ray region.

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Vol. 13, Iss. 3 — July - September 2023

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Figure 1
Generating optical cat and GKP states using free electrons. A scheme of free-electron-based cat- and GKP-state generation divided into three main stages. (a) Preparation stage: A monoenergetic electron is shaped into a comb with an energy spacing of $N ℏ ω$ . The case illustrated here shows $N = 2$ . The shaping of free electrons can be implemented using a few harmonics [75] or a few points of interaction [76]. Further preparation of the electron, such as collimation for an extended interaction, can be done with electron lenses and deflectors that are common in transmission electron microscopes. (b) Light-emission stage: The electron then interacts with a photonic structure, emitting photons into an optical mode. The coupling efficiency $g_{Q} = g_{Q} (x, y)$ can be tuned by changing the electron-beam transverse location (in $x y$ ) relative to the structure. The preferred structures are ones designed to guide light in a waveguide or cavity such that the guided mode is phase matched with the electron, achieving a stronger coupling constant (higher $g_{Q}$ ). (c) Postselection stage: The electron is measured, heralding the generated photonic state. In the case illustrated here, if the measured energy is even ( $k = 0$ ), an even cat state is created $| α ⟩_{ph} + | - α ⟩_{ph}$ . If the measured energy is odd ( $k = 1$ ), an odd cat state is created $| α ⟩_{ph} - | - α ⟩_{ph}$ . The measurement can be done via EELS equipped with a fast detector [41, 77]. (d) Examples of Wigner functions for the photonic states that can be generated via our scheme: A cat state can be generated using an electron comb and a postselection of its energy. A GKP state can be generated using multiple comb electrons with a specific sequence of postselections as described in Table 1. ( $g_{Q} = 4$ and $\sqrt{π / 2}$ for the top and bottom panels, respectively).
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Figure 2
Characterization of the $N$ -component cat states emitted from a free electron. (a) When a comb electron with an energy spacing of $ℏ ω$ emits photons of energy $ℏ ω$ , (b) the Wigner function of the photon takes the form of a coherent state. (c) When a comb electron with an energy spacing of $2 ℏ ω$ is postselected for even or odd energies after emitting photons of $ℏ ω$ , (d) the photonic Wigner function takes the form of even or odd cat states. (e) When a comb electron with an energy spacing of $4 ℏ ω$ is postselected after emitting photons of $ℏ ω$ , (f) the photonic Wigner function takes the form of the different four-component cat states. (g) Energy spectra of Gaussian electron combs (energy spacing of $2 ℏ ω$ ) with a standard deviation of $σ = 4$ (red) and $σ = 8$ (black) in units of photon energy ( $g_{Q} = \sqrt{π / 2}$ ). Such states can be created with high fidelity by three PINEM-type interactions with classical laser light, as described in Ref. [76]. (h) The fidelity of the postselected even cat states after interaction with the Gaussian comb electron. The average fidelity for $σ = 4$ is 0.97 and for $σ = 8$ is 0.99 (Supplemental Material [85] Sec. V. 2). ( $g_{Q} = 4$ for all panels in this figure. Additional examples with lower $g_{Q}$ values are provided in the second panels of Figs. 3, 4, and 4 showing the creation of various kitten states.).
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Figure 3
A scheme for creation of the GKP state: squeezing and postselection probability. (a) The evolution of the Wigner function of the photonic state after each electron interaction and postselection. The first interactions all have the same coupling constant $g_{Q 1} = i \sqrt{π / 8}$ , together squeezing the vacuum state. We then shift the phase of the interaction by $π / 2$ , so the later interactions all have a coupling constant $g_{Q 2} = \sqrt{π / 2}$ transforming the squeezed-vacuum state into a GKP state. (b) Creation of GKP state directly from an initial squeezed-vacuum excitation, i.e., seeding the electron-photon interaction with a squeezed vacuum in the optical mode. The GKP state is alternating between the approximated ${| 0 ⟩}_{ph}^{GKP}$ and ${| 1 ⟩}_{ph}^{GKP}$ , showing that each interaction resembles the $X$ gate for the GKP states. (c) The coefficients of the photonic state at every step of the process are described analytically using a Pascal triangle. This description simplifies the calculation of the postselection probabilities in Eq. (6) (Supplemental Material [85] Sec. 4). (d),(e) The squeezing parameter and postselection probability of the final GKP state as a function of the number of electron interactions: comparing photonic initial conditions of vacuum (a) and squeezed vacuum (b).
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Figure 4
Schemes for the creation of different GKP states and their characterization. (a) Creation of the magic GKP state ${| H ⟩}_{ph}^{GKP}$ (third row in Table 1). (b) Creation of the hexagonal-GKP state ${| 0 ⟩}_{L}$ (sixth row in Table 1). (c) Creation of the hexagonal magic state ${| T ⟩}_{ph}^{GKP}$ (seventh row in Table 1). (d),(e) The squeezing parameter and postselection probability of the GKP states in (a)–(c) as a function of the number of electrons.
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