Back action in quantum electro-optic sampling of electromagnetic vacuum fluctuations

Back action in quantum electro-optic sampling of electromagnetic vacuum fluctuations

T. L. M. Guedes, I. Vakulchyk, D. V. Seletskiy, A. Leitenstorfer, A. S. Moskalenko and Guido Burkard

Department of Physics and Center for Applied Photonics, University of Konstanz, Germany; Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon, Republic of Korea; Basic Science Program, Korea University of Science and Technology, Daejeon; Department of Engineering Physics, Polytechnique Montréal, Canada; Department of Physics, KAIST, Daejeon, Republic of Korea.

Physical Review Research 5, 013151 (2023). Received 7 February 2022; accepted 26 January 2023; published 27 February 2023.

Abstract

The influence of measurement back action on electro-optic sampling of electromagnetic quantum fluctuations is investigated. Based on a cascaded treatment of the nonlinear interaction between a near-infrared coherent probe and the mid-infrared vacuum, we account for the generated electric-field contributions that lead to detectable back action. Specifically, we theoretically address two realistic setups, exploiting one or two probe beams for the nonlinear interaction with the quantum vacuum, respectively. The setup parameters at which back action starts to considerably contaminate the measured noise profiles are determined. We find that back action starts to detrimentally affect the signal once the fluctuations due to the coupling to the mid-infrared vacuum become comparable to the base shot noise. Due to the vacuum fluctuations entering at the beam splitter, the shot noise of two incoming probe pulses in different channels is uncorrelated. Therefore, even when the base shot noise dominates the output of the experiment, it does not contribute to the correlation signal itself. However, we find that further contributions due to nonlinear shot-noise enhancement are still present. Ultimately, a regime in which electro-optic sampling of quantum fields can be considered as effectively back-action free is found.

I. Introduction

When performing quantum measurements, the interaction with the measurement device typically causes a perturbation of the quantum state. Even for experiments keeping the product of related uncertainties at their minimum, improving the accuracy with which one observable is measured inevitably increases the fluctuations in its canonically conjugate observable. This influence of the measurement device on a quantum system is called quantum back action. Often, the back action is undesired, but in some cases it underlies the functionality of quantum-information processing schemes.

Fluctuations in noncommuting observables persist even when the system reaches its ground state. In recent years, remarkable experiments have probed the zero-point fluctuations of a plethora of quantum systems, in particular, single-mode mesoscopic mechanical resonators and multimode electromagnetic radiation. Theoretical and experimental evidence points towards the inevitable presence of back action in quantum mechanical resonators probed by light in optical cavities. While the light affects the resonator through radiation pressure (Stokes and anti-Stokes scattering), the resonator imprints its phase-space signature on the photons in the cavity or, correspondingly, shifts the resonance frequency of the cavity. This back action, however, can be avoided by coupling the vibrational modes of two oscillators through the cavity photons, allowing the back-action contributions from the two modes to cancel each other.

Related arguments about mode coupling through back action were invoked to explain electro-optic measurements of correlations in the electromagnetic vacuum state. The potential effect of back action in such experiments, however, might considerably diverge from those seen in optomechanical cavities since the characteristic nonlinearity of the electro-optic interaction effectively couples optical modes between and within channels, each of them consisting of an infinite and continuous set of modes. The potential of this advanced scheme spans from the exploration of Hawking and Unruh effects to ultrafast quantum spectroscopy, thus calling for a formal and thorough description of the underlying physics.

In this paper, we theoretically study the back action in two experimental settings involving electro-optic sampling of the electromagnetic ground state. These measurements rely on a coupling that imprints information about a multimode quantum state on the ellipticity of a subcycle coherent pulse in a higher-frequency range. For a single-channel experiment, the interplay between shot noise and back action plays a crucial role in determining the optimal range of parameters. The situation is more complex when a second channel is included in the setup. Here, the contribution of the base shot noise to the signal correlation drops out when averaging over many readout events. We propose the working regimes most suitable to avoid undesired back-action contributions to the electro-optic signals (so that the measurement is considered effectively back-action free) and explain the subtle (yet fundamental) role that the population of the measured modes plays not only in the data presented in the two-channel experiment, but also in the general conceptual understanding of quantum electro-optic sampling.

Figure 1. (a) Nonlinear crystal and field components. The probe is polarized along one crystal axis and propagates with a given wave vector. The tensor components of the nonlinear susceptibility are such that only two field components can mix with the probe, generating a new propagating quantum-field component along a third direction. (b) Scheme of an electro-optic measurement. The probe changes its ellipticity in the nonlinear crystal due to the nonlinearly generated field component. A quarter-wave plate shifts the phase of one component by a quarter cycle, and a Wollaston prism spatially splits the two remaining components before independent photon counting. (c) Illustration of the state evolution under the nonlinear interaction. The states include both polarizations and two frequency bands, mid-infrared and near-infrared, with the near-infrared coherent probe represented as a pool of photons. Contours represent annihilation and creation of photons and the arrows show the directions of the energy transfers. The first diagram shows the lowest-order perturbation of the initial state, with mid-infrared and near-infrared photons being created through annihilation of a probe photon. The second and third diagrams show the second-order processes that lead to no back action in the mid-infrared, since photons created by the first-order process are annihilated. The last two diagrams show the remaining second-order processes, which cause additional back action in the mid-infrared via generation of extra photons.

II. Single-channel model

The paper derives the cascaded contributions to the quantum electric field that build up within the nonlinear medium and then converts them into the electro-optic signal noise upon detection. An incoming near-infrared ultrashort probe pulse copropagates with a mid-infrared vacuum-state electric-field component along a principal axis of a zinc-blende-type nonlinear crystal. The wave vector of the probe pulse is perpendicular to the crystal axis that is parallel to the probe electric field. Since the probe pulse is in a coherent state, its field operator separates into a classical amplitude plus the zero-point quantum fluctuations. The cascaded contribution to the second-order nonlinear polarization arising in the crystal comes from the mixing between the probe and any quantum-field contribution already present in the crystal, and is proportional to the vacuum permittivity, the nonlinear coefficient, the quantum field and the probe field.

(The remainder of Section II, and Appendices A to G, are omitted for length; they carry the operator algebra, the phase-matching and dispersion treatment, the third-order susceptibility contribution and the quantum-state evolution. The complete text is at the source.)

III. Single-channel results

To evaluate the back-action effect on the measurement results, we derive all contributions to the electro-optic signal variance up to fourth order. The cross term between the shot-noise signal and the fourth-order signal operator vanishes within our approximations. The remaining probe-dependent contribution, which also enhances the base shot noise, results from the square of the second-order signal operator and scales as the cube of the probe photon number. Concurrent to this contribution is the cross term between the first-order and third-order signal operators, which scales the same way but has an opposite sign, and consequently leads to variance reduction.

Figure 2 illustrates the total root-mean-square signal per probe photon as a function of the number of photons per probe pulse. For comparison, we provide plots corresponding to the experimental parameters of two earlier experiments: set 1 and set 2. For the probe pulses we assume central frequencies of 247 THz for set 1 and 375 THz for set 2, and spectral bandwidths of 150 THz for set 1 and 2.77 THz for set 2, with rectangular spectral amplitude distributions and flat phase. We consider beam waist radii of 3 micrometres for set 1 and 125 micrometres for set 2. For the nonlinear crystal we use a length of 7 micrometres for set 1 and 3 millimetres for set 2, an electro-optic coefficient of 3.9 picometres per volt, refractive indices of 2.76 and 2.85 respectively, group indices of 2.9 and 3.18, and a refractive index varying slightly within the relevant terahertz frequency range. We do not include contributions from four-wave mixing since they only affect the probe.

Figure 2 reveals a considerable deviation from the result determined solely by the shot noise and the main electro-optic contribution when photon numbers are larger than about a hundred billion. Minima in the root-mean-square signal per probe photon occur at 1.6 hundred billion photons per pulse for set 1 and 6.4 hundred billion for set 2, that is, roughly where the main electro-optic signal variance becomes comparable to the base shot-noise variance. Contrary to naive expectations that the main electro-optic term would generally dominate the total root-mean-square signal for large photon numbers, our results show that increasing the probe intensity beyond a certain value has a rather detrimental effect since both the base shot noise and main electro-optic signal variance are rapidly overtaken by the cascaded effects. Therefore, a reliable minimally disturbing quantum-state sampling may be achieved only for photon numbers considerably smaller than their value at those minima. Figures 2c and 2d show the normalized detected root-mean-square signal on top of the base shot-noise contribution. Departure from zero allows for a clear visualization of the electro-optic contributions, with the onset of the fourth-order terms resulting in a divergence between the solid and the dashed lines in each figure.

Figure 2. (a), (b) Ratio of the root-mean-square signal to the probe photon number, in dependence of the number of photons per probe pulse. The solid lines represent the total root-mean-square signal per photon, the dotted lines represent the base shot-noise contribution, the dashed lines show the main electro-optic root-mean-square signal, and two further lines account for the square of the second-order term and for the first-order-times-third-order cross term. The background gradient illustrates the transition between the effectively back-action-free, shot-noise dominated, and back-action dominated regimes. (c), (d) Increase of the excess signal over the shot-noise signal with photon number. The solid lines contain contributions up to fourth order, the dotted lines up to second order. Panels (a) and (c) correspond to parameter set 1; panels (b) and (d) to set 2.

IV. Two-channel model

For a two-channel setup, the probe beam undergoes a beam-splitting operation before any of the processes discussed before. The two probe pulses released from a fifty-fifty beam splitter not only carry different phases — reflected and transmitted beams differ in phase by a quarter cycle — but also commuting annihilation and creation operators due to the admixture of uncorrelated vacuum noise. Once the first probe pulse meets the nonlinear crystal, its interactions with the mid-infrared vacuum will generate the back-action contributions as discussed above. The second probe pulse reaches the crystal with a time delay and interacts not only with that mid-infrared vacuum, but also with the back-action contributions generated by the passage of the first probe. It will also generate its own back-action contributions that can interact with the first probe. Each channel output undergoes its own ellipsometry detection. The signals from the two channels are then multiplied before readout, rendering a delay-dependent signal variance with the properties of a correlation function — the symmetrized vacuum expectation value of the two channel signal operators, normalized by a constant built from the refractive index, the crystal length, the probe frequency, the electro-optic coefficient and the photon number.

We consider a setup in which the directions of the central wave vectors of the beams in the two channels deviate only slightly from each other. This allows for consideration of effectively coplanar beam waists in the crystal, as well as nearly collinear phase matching for the wave-mixing processes. Therefore, the treatment of fields in terms of the paraxial decomposition remains well justified. To avoid considerable deviations of output wave vectors from either channel direction, only the second set of parameters will be considered. Due to the limited beam waist, mixing between modes from different channels during the ellipsometry step is avoided. For the quantum fluctuations of the probe pulses, the cross-channel commutator vanishes, and consequently the anticommutator of the two channels’ shot-noise signal operators vanishes as well. Consequently, while shot noise still affects the measurements, the two-channel equivalent of the base shot noise does not contribute to the correlation function. In general, the time-dependent signal operators are given by equations similar to those of the single-channel case. The cascaded contributions are now composed of convolutions with either channel’s probe field, splitting each single-channel contribution into a growing number of terms. Ellipsometry conducted with either of these fields then leads to the corresponding channel signal operator. Since oscillations in the delay at near-infrared frequencies can neither be resolved nor are of major interest in such an experiment, only contributions to the correlation oscillating at mid-infrared frequencies will be considered.

Figure 3. (a) Electro-optic measurement with two channels, with the quarter-wave plate, the Wollaston prism and the correlation register. (b) Total second-order and fourth-order contributions to the normalized correlation function for a hundred billion photons per pulse, representing correlations in the two-channel measurements.

V. Two-channel results

Figure 3b shows the normalized correlation function with all terms up to fourth order. The main contribution, which depends on the product of the second-order fields in the two channels, is proportional to the square of the photon number times an integral over mid-infrared frequency of the refractive-index ratio and the squared response function, weighted by a cosine of frequency times delay. This behavior is also seen in the measured data of the two-channel experiment up to differences in spectral shape chosen for the probe. Contributions from higher-order terms are minor up to about a hundred billion photons per pulse, where they become comparable to the second-order contributions. For a hundred million photons per pulse, as in the experiments, higher-order contributions to the correlation function are negligible.

The characteristic timescale of the correlation function is similar to that of the two probe pulses since the multiplied signal is nonzero only when they share interactions with the same propagating mid-infrared modes. The oscillations of the correlation function in Figure 3b happen with a timescale approximately inverse to the average probed mid-infrared angular frequency and reflect the interference between modes from different channels. From a state-evolution perspective, the correlation function includes interchannel probe-probe correlations mediated by mid-infrared states, populated or not.

VI. Conclusions

Our results show that in single-channel quantum electro-optic measurements of the electromagnetic vacuum there is a suitable setup-dependent range of probe intensities to minimize the contributions from measurement-generated mid-infrared photons to the signal. For small probe intensities the results are inevitably contaminated by excessive shot noise; for large probe intensities, by back action. The photon numbers where the main electro-optic signal variance becomes comparable to the base shot-noise variance represent upper limits to the experimentally admissible probe intensities to avoid back-action predominance. The back action to the mid-infrared states is inherent to the electro-optic measurement but the detected mid-infrared signal variance can be considered effectively back-action free — a weak measurement — for small perturbations of the mid-infrared states. Creation of mid-infrared photons considerably changes the picture of electro-optic sampling as a potential quantum nondemolition measurement since, notwithstanding its indirect character with the mid-infrared as sample and the near-infrared as ancilla, a postselection scheme is needed to filter components of the sample without generated photons. Furthermore, the base shot-noise contribution does not show up in the averaged correlations measured with two channels, while enhanced shot-noise contributions appear for probe intensities orders of magnitude above the experimentally used. Future works might explore postselection to achieve further back-action evasion.

Acknowledgments

T.L.M.G., A.L., A.S.M. and G.B. acknowledge funding by the Deutsche Forschungsgemeinschaft, Project No. 425217212, SFB 1432. T.L.M.G. and A.S.M. gratefully acknowledge the funding by the Baden-Württemberg Stiftung via the Elite Programme for Postdocs. A.S.M. was also supported by the National Research Foundation of Korea grant funded by the Korean government, Grant No. 2020R1A2C1008500. I.V. acknowledges support by the Institute for Basic Science, Projects No. IBS-R024-D1 and No. IBS-R024-Y3. We thank P. Sulzer for the productive discussions at the early stage of this research.