Quantum Susceptibilities in Time‐Domain Sampling of Electric Field Fluctuations

Abstract

Electro‐optic sampling has emerged as a new quantum technique enabling measurements of electric field fluctuations on subcycle time scales. In a second‐order nonlinear material, the fluctuations of a terahertz field are imprinted onto the polarization properties of an ultrashort probe pulse in the near infrared. The statistics of this time‐domain signal are calculated, incorporating the quantum nature of the involved electric fields right from the beginning. A microscopic quantum theory of the electro‐optic process is developed adopting an ensemble of noninteracting three‐level systems as a model for the nonlinear material. It is found that the response of the nonlinear medium can be separated into a conventional part, which is exploited also in sampling of coherent amplitudes, and quantum contributions, which are independent of the state of the terahertz input. Interactions between the three‐level systems which are mediated by terahertz vacuum fluctuations are causing this quantum response. Conditions under which the classical response serves as a good approximation of the electro‐optic process are also determined and how the statistics of the sampled terahertz field can be reconstructed from the electro‐optic signal is demonstrated. In a complementary regime, electro‐optic sampling can serve as a spectroscopic tool to study the pure quantum susceptibilities of matter.

A complete POVM description of multi-channel quantum electro-optic sampling with monochromatic field modes

“…As is the case for any quantum-mechanical indirect measurement, electro-optic sampling makes use of an ancillary system (in the present case, the high-frequency NIR modes of the electric field), which becomes correlated with the low-frequency mode of the field (here the MIR) through interactions in a nonlinear crystal [33][34][35][36][37][38][39][40][41][42][43][44]. We consider the specific case of an optical parametric oscillator consisting of a zincblende-type nonlinear crystal in a cavity, labelled as (i) in Fig.…”

“…One such possibility is to use eightport homodyne detection, in which the sampled mode is split into two modes using a beam splitter, then the two quadratures can be measured simultaneously using a four-port homodyne detection scheme [17]. Another possibility to measure the field quadratures is provided by quantum electro-optic sampling (EOS) [33][34][35][36][37][38][39][40][41][42][43]. EOS is an indirect measurement of low-frequency modes, usually in the mid-infrared (MIR), mediated by higher frequency modes, usually in the near-infrared (NIR).…”

Self‐Referenced Subcycle Metrology of Quantum Fields

“…One of the easiest ways to realize this, suitable for our discussion here, is based on the cuts in the spectra of the detected photons that can be implemented via the corresponding frequency bandpass filters. [ 25,30 ] Looking at the second line of Equation () one can anticipate that, for example, for a bandpass filter cutting the frequencies above the central frequency of the LO, $$\mathcal {G}_-(\Omega )$$ dominates over $$\mathcal {G}_+(\Omega )$$ in terms of the absolute magnitude and $$\mathcal {G}_\phi (\Omega )$$ becomes complex. In the easiest case, it can be written as $$\mathcal {G}_\phi (\Omega )=|\mathcal {G}_\phi (\Omega )| \text{e}^{i\theta }$$, where the phase $$\theta =\theta (\phi )$$ is uniquely determined by the phase ϕ (and vice versa) whereas being independent of frequency.…”

“…[23] Such developments promise direct routes toward MIR and THz quantum sensing technologies, while the time-domain character motivates new metrology protocols [24][25][26] as well as a path toward experimental quantum electrodynamics in space-time. [23,[27][28][29][30] Despite this progress, time domain quantum photonics faces a few outstanding challenges. On the one hand, strong phononpolariton dispersion in the 𝜒 (2) crystal used in EOS makes the detection of signals around the Reststrahlen band frequencies challenging.…”