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.
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.
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.
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.
A new theoretical framework to describe the experimental advances in electro-optic detection of broadband quantum states, specifically the quantum vacuum, is devised. Electro-optic sampling is a technique in ultrafast photonics which, when transferred into the quantum domain, can be utilized to resolve properties of a sampled quantum state via its interaction with a strong coherent probe pulse at ultrafast timescales. By making use of fundamental concepts from quantum field theory on spacetime metrics, the nonlinear interaction behind the electro-optic effect is shown to be equivalent to a stationary Unruh-DeWitt detector coupled to a conjugate field during a very short time interval. When the coupling lasts for a time interval comparable to the oscillation periods of the detected field mode (i.e., the subcycle regime), virtual particles inhabiting the field vacuum are transferred to the detector in the form of real excitation. We demonstrate that this behavior can be rigorously translated to the scenario of electro-optic sampling of the quantum vacuum, in which the (spectrally filtered) probe works as an Unruh-DeWitt detector, with its interaction-generated photons arising from virtual particles inhabiting the electromagnetic vacuum. Our analysis accurately encapsulates the quantum nature of the vacuum, and we propose the specific working regime in which we can experimentally verify the existence of virtual photons with quantum correlations in the electromagnetic ground state.
Entanglement harvesting provides an operational means of probing the nonlocal correlation structure of quantum fields through localized detector-field interactions. Although harvesting has been investigated in a wide variety of spacetime backgrounds and quantum field states, existing studies largely focus on individual detector configurations or specific physical settings. In this work, we develop a unified perturbative framework for entanglement harvesting with localized Unruh-DeWitt detectors that is applicable to arbitrary detector trajectories and quantum field states through the corresponding Wightman function. By expressing the detector response in terms of dimensionless variables and analytically reducing the perturbative integrals to numerically efficient representations, the framework enables a systematic exploration of multidimensional harvesting landscapes, phase boundaries, and harvesting robustness. As representative applications, we investigate inertial detectors in the Minkowski vacuum, thermal field states, and uniformly accelerated observers. We show that finite temperature enhances both the local detector response and the nonlocal exchange amplitude, leading, within the perturbative Gaussian-switching framework considered here, to an enlargement of the harvesting region in parameter space. The resulting harvesting landscapes provide a global characterization of detector-field interactions and demonstrate how temperature and observer motion reshape the competition between local detector noise and nonlocal field correlations. The framework developed here provides a unified and readily extensible methodology for investigating entanglement harvesting across a broad class of relativistic quantum systems.
We study how two fully featured hydrogenlike atoms harvest entanglement from the electromagnetic field vacuum, even when the atoms are spacelike separated. We compare the electromagnetic case---qualitatively and quantitatively---with previous results that used scalar fields and featureless, idealized atomic models. Our study reveals the new traits that emerge when we relax these idealizations, such as anisotropies in entanglement harvesting and the effect of exchange of angular momentum. We show that, under certain circumstances, relaxing previous idealizations makes vacuum entanglement harvesting more efficient.
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.