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arXiv:2106.04402 · Physical Review Letters

Enhanced Electro-optic Sampling with Quantum Probes

Stéphane Virally, Patrick Cusson, Denis V. Seletskiy

Published Dec 30, 2021• 24 citations
arXivHTMLPDF

Abstract

Employing electro-optic sampling (EOS) with ultrashort probe pulses, recent experiments showed direct measurements of quantum vacuum fields and their correlations on subcycle timescales. Here, we propose a quantum-enhanced EOS where bright photon-number entangled twin beams are used to derive conditioned nonclassical probes. In the case of the quantum vacuum, this leads to a sixfold improvement in the signal-to-noise ratio over the classically probed EOS. In addition, engineering of the conditioning protocol yields a reliable way to extract higher-order moments of the quantum noise distribution and robust discrimination of the input quantum states, for instance, a vacuum and a few-photon cat state. These improvements open a viable route toward robust tomography of quantum fields in space-time, an equivalent of homodyne detection in energy-momentum space, and the possibility of precise experiments in real-space quantum electrodynamics.

DOI
10.1103/physrevlett.127.270504

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Employing electro-optic sampling (EOS) with ultrashort probe pulses, recent experiments showed direct measurements of quantum vacuum fields and their correlations on subcycle timescales. Here, we propose a quantum-enhanced EOS where bright photon-number entangled twin beams are used to derive conditioned nonclassical probes. In the case of the quantum vacuum, this leads to a sixfold improvement in the signal-to-noise ratio over the classically probed EOS. In addition, engineering of the conditioning protocol yields a reliable way to extract higher-order moments of the quantum noise distribution and robust discrimination of the input quantum states, for instance, a vacuum and a few-photon cat state. These improvements open a viable route toward robust tomography of quantum fields in space-time, an equivalent of homodyne detection in energy-momentum space, and the possibility of precise experiments in real-space quantum electrodynamics.
In many states of the quantum electromagnetic field, including the vacuum state, entanglement exists between different space-time regions---even spacelike separated ones. These correlations can be harvested and, thereby, detected by quantum systems which locally interact with the field. Here, we propose an experimental implementation of such an entanglement-harvesting scheme which is based on electro-optic sampling (EOS). We demonstrate that state-of-the-art EOS experiments enable one to harvest entanglement from the vacuum field and to study quantum correlations within general THz fields. We further show how Bell nonlocality present in the vacuum field can be probed. Finally, we introduce an approach to mitigate shot noise in single-beam EOS configurations. These findings pave the way for experimental inquiries into foundational properties of relativistic quantum field theory, and empower EOS as a diagnostic tool in THz quantum optics.
Following recent progress in the experimental application of electro-optic sampling to the detection of the quantum fluctuations of the electromagnetic-field ground state and ultrabroadband squeezed states on a subcycle scale, we propose an approach to elevate broadband electro-optic sampling from a spectroscopic method to a full quantum tomography scheme, able to reconstruct a free-space quantum state directly in the time domain. By combining two recently developed methods to theoretically describe quantum electro-optic sampling, we analytically relate the photon-count probability distribution of the electro-optic signal to a transformed phase-space quasiprobability distribution of the sampled quantum state as a function of the time delay between the sampled midinfrared pulsed state and an ultrabroadband near-infrared probe pulse. We catalog and analyze sources of noise and show that in quantum electro-optic sampling with an ultrabroadband probe pulse one can expect to observe thermalization due to entanglement breaking. Mitigation of the thermalization noise enables a tomographic reconstruction of broadband quantum states while granting access to its dynamics on a subcycle scale. Published by the American Physical Society 2024
Direct detection of vacuum fluctuations and analysis of subcycle quantum properties of the electric field are explored by a paraxial quantum theory of ultrafast electro-optic sampling. The feasibility of such experiments is demonstrated by realistic calculations adopting a thin ZnTe electro-optic crystal and stable few-femtosecond laser pulses. We show that nonlinear mixing of a short near-infrared probe pulse with the multiterahertz vacuum field leads to an increase of the signal variance with respect to the shot noise level. The vacuum contribution increases significantly for appropriate length of the nonlinear crystal, short pulse duration, tight focusing, and a sufficiently large number of photons per probe pulse. If the vacuum input is squeezed, the signal variance depends on the probe delay. Temporal positions with a noise level below the pure vacuum may be traced with subcycle resolution.
The fascinating nature of light has driven many discoveries in physics, from the theory of special relativity to quantum mechanics. As a carrier of information, light is at the core of high-speed communication over the internet and facilitates the study of numerous physical properties of materials by the interaction of light and matter. Advances in generating and detecting ultrashort light pulses enabled the spectroscopic investigation of matter with unprecedented detail. Meanwhile, the development of more precise and sensitive measurements, leveraging quantum properties of light, has expanded the experimental frontiers of the physical world. Naturally, the question arises: Could the combination of these two breakthroughs push the frontiers even further? A promising candidate to achieve this combination is electro-optic sampling, an electromagnetic-field measurement comprising an up-conversion and a phase-space measurement based on homodyne detection. While it is predominantly used in classical terahertz time-domain spectroscopy, the high sensitivity and spatio-temporal resolution of electro-optic sampling recently enabled the detection of vacuum fluctuations in the ground state of the electromagnetic field by probing an adjustable space-time volume. Yet, the generation and characterization of more intricate quantum pulses on ultrafast timescales is still challenging, especially in the terahertz frequencies. Exploiting the quantum properties of a light pulse over a broad frequency range requires sophisticated knowledge about the prepared quantum pulse as well as the measurement setup at use. However, Heisenberg's uncertainty relation fundamentally prevents the complete determination of an arbitrary quantum state from a single measurement. Instead, the quantum state has to be reconstructed from the statistics of many measurements — a process known as quantum state tomography. In this thesis, we establish the theoretical foundations for a comprehensive tomographic reconstruction of pulsed quantum states of light, encompassing both the statistics and dynamics of the state. Additionally, we introduce tools to simplify the description of the nonlinear interaction underlying electro-optic sampling and in turn the tomography setups presented here. As a first step, we investigate the application of electro-optic sampling to the time-local reconstruction of various pulsed quantum states of light on ultrafast timescales. We identify two types of noise, which impact the quality of the reconstruction: shot noise, arising from the nonlinear interaction in electro-optic sampling, and thermal noise, due to the localization of the (correlated) quantum state within a small space-time volume during the measurement. To improve the reconstruction, we propose to use a strong probe pulse, suppressing the shot noise, and to select an appropriate detection frequency band, reducing thermal noise. Thus, the proposed tomography enables the characterization of quantum states in terahertz frequency range on ultrafast timescales. However, the time-local nature of the measurement imposes a trade-off between the reconstruction of the quantum state in phase space and the time domain. In a second step, we propose a reconstruction method based on two-time field-correlation measurement, which we call correlation tomography, able to overcome the restriction to time-local states. While the tomography of quantum states in an optimal pulse basis has been demonstrated, it requires prior knowledge about the temporal structure of the quantum state. We introduce an algorithm for reconstructing high-dimensional multimode Gaussian quantum states, which are completely characterized by the covariance matrix of the quadratures of fully distinguishable pulses. The reconstruction is achieved by orthogonalizing the overlapping time windows of the correlation measurement in post-processing. We compare a homodyne-based implementation to electro-optic sampling and develop an optimization scheme for the latter, to mitigate the shot noise in the non-perturbative regime of strong nonlinear interactions. Furthermore, we derive the full-counting statistics of the field-correlation measurement and theoretically demonstrate the reconstruction of pulsed Fock states. This opens an avenue to the full tomography of highly multimode non-Gaussian states. In the third and last step, we derive a formalism to simplify the description of the up-conversion required in electro-optic sampling. If a pulsed quantum states is up-converted and localized in a small space-time region, the transformation of the localized modes during the nonlinear interaction corresponds to an open quantum system, rendering the standard tools of quantum optics inapplicable. By introducing the generalized Bloch-Messiah decomposition, we simplify the input-output relation of the nonlinear interaction to a set of effective modes relevant for subsequent processing, such as detection. Using this framework, we derive an analytic phase-space input-output relation for the nonlinear interaction of pulsed quantum states, applicable to open quantum systems. We study the resulting dynamics using two example input states: a Fock state and a two-mode squeezed state. Based on the von Neumann entropy of the output state, we develop an intuitive understanding of the generalized Bloch-Messiah decomposition. The rich structure of pulsed quantum states of light can be a powerful resource, but also challenging to operate with, in theory as well as in experiment. The characterization methods and theoretical tools developed in this thesis, tailored to pulsed quantum light and the nonlinear interaction thereof, can pave the way to the application of this resource to fundamental research as well as quantum technologies.
Electro-optical sampling of Terahertz fields with ultrashort pulsed probes is a well-established approach for directly measuring the electric field of THz radiation. This technique usually relies on balanced detection to record the optical phase shift brought by THz-induced birefringence. The sensitivity of electro-optical sampling is, therefore, limited by the shot noise of the probe pulse, and improvements could be achieved using quantum metrology approaches using, e.g., NOON states for Heisenberg-limited phase estimation. We report on our experiments on THz electro-optical sampling using single-photon detectors and a weak squeezed vacuum field as the optical probe. Our approach achieves field sensitivity limited by the probe state statistical properties using phase-locked single-photon detectors and paves the way for further studies targeting quantum-enhanced THz sensing.