Ultrafast non-destructive measurement of the quantum state of light with free electrons

Abstract

Since the birth of quantum optics, the measurement of quantum states of nonclassical light has been of tremendous importance for advancement in the field. To date, conventional detectors such as photomultipliers, avalanche photodiodes, and superconducting nanowires, all rely at their core on linear excitation of bound electrons with light, posing fundamental restrictions on the detection. In contrast, the interaction of free electrons with light in the context of quantum optics is highly nonlinear and offers exciting possibilities. The first experiments that promoted this direction appeared over the past decade as part of photon-induced nearfield electron microscopy (PINEM), wherein free electrons are capable of high-order multi-photon absorption and emission.Here we propose using free electrons for quantum-optical detection of the complete quantum state of light. We show how the precise control of the electron before and after its interaction with quantum light enables to extract the photon statistics and implement full quantum state tomography using PINEM. This technique can reach sub-attosecond time resolutions, measure temporal coherence of any degree (e.g., g (1) , g (2) ), and simultaneously detect large numbers of photons with each electron. Importantly, the interaction of the electron with light is non-destructive, thereby leaving the photonic state (modified by the interaction) intact, which is conceptually different from conventional detectors. By using a pulse of multiple electrons, we envision how PINEM quantum detectors could achieve a single-shot measurement of the complete state of quantum light, even for non-reproducible emission events. Altogether, our work paves the way to novel kinds of photodetectors that utilize the ultrafast duration, high nonlinearity, and nondestructive nature of electron-light interactions. Section I -IntroductionQuantum optics 1,2 has seen significant advancements in the past decades, allowing for fundamental tests of quantum mechanics 3,4 , observations of nonclassical states of light [5][6][7] , and realization of quantum technologies [8][9][10][11][12][13] . Much of these achievements were made possible thanks to the development of new concepts for measuring photons: with examples ranging from the Hanbury Brown-Twiss (HBT) experiment 14 and coincidence measurements 4 , to photonnumber resolving detectors 15,16 and homodyne detection 17,18 , used for quantum state tomography 2,19-23 , which enables the reconstruction of the entire photonic state. All current quantum optical detectors are based on the interaction of light with bound-electron systems, such as avalanche photodiodes 24,25 , superconducting nanowires 26 , and photomultiplier tubes 27 . While many of these platforms utilize electrons bound in solids, we also see electrons bound in atoms (placed in optical cavities) proposed for precise quantum optical detectors [28][29][30][31][32] .The state-of-the-art quantum-optical detectors are based on the linear interaction between bound electrons and light through d...

Imprinting the quantum statistics of photons on free electrons

“…2 for the state evolution, we find the coefficients ckn=gq−1−1neiφgkWn+12k+1,12kgq2/n+k!n!, where W is the Whittaker function and ϕ g is the phase of g q (supplementary text S2). In the limit of weak interaction and a large number of photons, the coefficients become ckn≈eiφgkJk2gqn ( 23 , 26 ), which is reminiscent of the PINEM theory ( 17 , 18 ).…”

“…( 1)) but requires a quantum-optics theory. Below, we present the formulation of the quantum interaction of light with a highly-paraxial free-electron beam, i.e., the Q-PINEM theory (22,(24)(25), which has the scattering matrix:…”

“…where 𝑊𝑊 is the Whittaker function (SM S2). In the limit of weak interaction and a large number of photons, the coefficients become 𝑐𝑐 𝑘𝑘 𝑛𝑛 ≈ 𝑒𝑒 𝑖𝑖𝜑𝜑 𝑔𝑔 𝑘𝑘 𝐽𝐽 𝑘𝑘 �2�𝑔𝑔 q �√𝑛𝑛� (22,25), reminiscent of the PINEM theory (16)(17).…”

“…1) but requires a quantum-optics theory. Below, we present the formulation of the quantum interaction of light with a highly paraxial free-electron beam—the Q-PINEM theory ( 23 , 25 , 26 ), which has the scattering matrixS=expgqba†−gq*b†awhere g q is the quantum coupling constant; a and a † are the photonic annihilation and creation operators, respectively; and b and b † are the free-electron energy ladder operators describing the electron losing or gaining a single-photon energy quantum, respectively. The resulting joint electron-photon state is generally nonseparable, and its density matrix can be written asρel−ph=∑n,m=0∞ρphn,mψnel−phψmel−phwhere ρphn,m=nρphm is the density matrix element of the light in Fock space.…”

“…The free-electron interaction enables us to characterize the amplifier output, showing its transition from a coherent state with Poissonian statistics in the regime of deep saturation to super-Poissonian statistics owing to amplified spontaneous emission and eventually to a thermal state with Bose-Einstein statistics. Our experiments offer a pathway for using free electrons for quantum-state tomography of light, with high resolution in both time and frequency ( 23 ).…”