Experimentally separating vacuum fluctuations from source radiation

Domain: physics
Published: 2026-02-17
Editorial outcome: minor_revision

Abstract
The distinction between vacuum-field and source-radiation effects in phenomena such as the Lamb shift, Casimir forces, and spontaneous emission relies on additional assumptions at the theoretical level, and an experimental approach was never considered feasible. Fermi’s two-atom problem, a Gedankenexperiment on how atoms interact with the electromagnetic field via vacuum and source radiation, provides key theoretical insight. Advances in ultrafast optics now enable experimental analogues using two laser pulses in a nonlinear crystal. Here we demonstrate the detection of vacuum- and source-radiation-induced correlations between two pulses, separated by their causal properties. Specifically, vacuum fluctuations and source radiation are shown to correlate distinct quadratures of near-infrared pulses, enabling individual probing via phase-sensitive detection. Our results experimentally verify the time-domain fluctuation-dissipation theorem at the quantum level and open avenues for studying quantum radiation effects in time-dependent media, including entanglement harvesting from the vacuum and quantum field detection in curved-space analogues.

Introduction
Fermi’s two-atom setup consists of two atoms in empty space whose interaction with the vacuum radiation field is abruptly switched on. Studying the subsequent build-up of correlations between the atoms reveals fundamental properties of quantum fields in space and time1–4.

In the Heisenberg picture, independent of the initial state of the two atoms, two different terms contribute to these correlations5–8, as illustrated in Fig. 1a. On the one hand, correlations between the two atoms arise from their individual interaction with the ground-state electromagnetic field—the vacuum fluctuations—whereby correlations intrinsically existing in the vacuum field are swapped to the atoms1–4. On the other hand, the two atoms can exchange a photon. The emission of this photon is triggered by the interaction of one atom with its own radiative field (radiation reaction). This radiation is known as source radiation1,6 and interacts with the other atom, which leads to correlations between the atoms. Mathematically, individual contributions of vacuum fluctuations and source radiation to fundamental processes in quantum electrodynamics, such as spontaneous emission or the Lamb shift, are in general not uniquely determined, since they depend on the initial ordering of the atomic and field operators in the light-matter interaction Hamiltonian6,9,10.

While for most of the phenomena associated with interactions with the ground state of light, vacuum fluctuations and source radiation can therefore be seen as inseparable “two sides of the same quantum mechanical coin”10, they have a different character regarding causality in Fermi’s two-atom problem: The vacuum-induced correlations arise instantaneously upon turning on the interaction, while the exchange of a photon leads to correlations only after the propagation time δt=r/c7,8. However, different choices of the aforementioned operator ordering can lead to a non-causal description of source radiation, making its interpretation ambiguous7. This issue remains purely theoretical unless the two contributions can be observed independently. Yet, an experimental investigation has long been considered unfeasible due to the challenge of initiating the two atoms, rapidly switching their interaction with the field on ultra-short timescales, and detecting the resulting correlations11. Here, we overcome these limitations using an all-optical version of Fermi’s two-atom setup and probe the contributions from source radiation and vacuum field fluctuations, separated by causality, individually for the first time.

In contrast to atoms, the interaction of laser pulses with a surrounding electromagnetic field can be initiated and terminated on ultrashort timescales as the pulses enter and leave a nonlinear material. This unique property has enabled an all-optical analog of Fermi’s two-atom setup, where vacuum-induced correlations between atoms are mapped onto two spatially separated laser pulses interacting with the vacuum field inside a nonlinear crystal12 (see Fig. 1b). The implementation relies on electro-optic sampling, in which near-infrared probe pulses couple to a terahertz or mid-infrared field (possibly in the vacuum state and restricted to the spectral width of the probe), generating a second-order nonlinear field polarized perpendicular to the probe13–15. This nonlinear interaction modifies the polarization state of the two probe pulses, directly paralleling the transition of the atoms in the Fermi setup. Here, however, the resonant single-mode transition of two-level atoms is replaced by an off-resonant, broadband coupling of multimode near-infrared pulses to terahertz field modes within the spectral width of the probes. The use of femtosecond pulses thus not only allows the interaction to be switched on and off with high temporal precision, but also enables the application of advanced quantum optics detection techniques to analyze the polarization state of the probes (see refs. 7, 12, 16 and Supplementary Information Note 1.A for further comparison between Fermi’s two-atom setup and two-beam electro-optic sampling).

Recently, an experiment similar to the one in ref. 12 has been theoretically suggested7, which would be capable of individually detecting correlations caused by source radiation as well (see Fig. 1c), and in the following, we demonstrate its experimental implementation.

Results
The idea relies on the fact that in the electro-optic sampling analog of Fermi’s two-atom setup, vacuum-field fluctuations (given by the correlation function of the THz vacuum field C(r,r',t,t')=1/2h[Eb(r,t),Eb(r',t')]i) and source radiation (given by the response function R(r,r',t,t')=(i/ℏ)θ(t-t')h[Eb(r,t),Eb(r',t')]i) correlate different quadratures of the two near-infrared pulses, which allows one to individually probe them by changing the phase plate in the employed homodyne detection scheme (see Fig. 1b, c). This distinction mirrors the situation in Fermi’s two-atom problem, where different Pauli matrices are correlated depending on whether the correlation originates from vacuum fluctuations or from source radiation7.

In the vacuum-field contribution, the near-infrared field of the two pulses individually interacts with the vacuum field at terahertz frequencies. Due to the retardance of π/2 occurring in the non-linear process, the generated field that contains the information about the THz field is out-of-phase with the probe field. As a consequence, correlations induced by terahertz vacuum fluctuations can be detected through quarter-wave plates in the homodyne detection schemes of both probe beams, as schematically illustrated in Fig. 1b. The instantaneous correlation of the two pulses induced by the terahertz vacuum fluctuations is obtained from the individually detected signals as a function of different time delays between the two probe pulses12. In the source radiation contribution schematically depicted in Fig. 1c, the electric field of one probe pulse interacts with the copropagating ground-state fluctuations in the near-infrared, resulting in the emission of source radiation. The second pulse then interacts with that terahertz source radiation in the same way as with the terahertz vacuum field again leading to a nonlinear field contribution that has a quadrature out-of-phase with the probe field. In contrast, to trigger the source radiation, the probe field of the first pulse interacted with the in-phase quadrature of the near-infrared vacuum fluctuations. To observe the correlations resulting from source radiation, one thus has to probe the first pulse with a half-wave plate—thereby accessing the quadrature that is in-phase with the pulse—and the second with a quarter-wave plate as in the vacuum field contribution (see ref. 7 for details and Supplementary Information Note 1.B).

For the experimental realization, a similar optical setup as in ref. 12 is utilized and schematically shown in Fig. 2a. To compensate for dispersion and obtain 110 fs-short pulses inside the detection crystal, the probe signal is pre-stretched in a prism compressor before being separated into the two individual probe signals (see the “Methods” section about experimental setup and Supplementary Information Note 2.D). The correlations between different quadratures of the field amplitudes of the two laser pulses emerging from the crystal are detected by probing the correlations between two sets of balanced detectors, each of which has a tunable waveplate determining the probed quadrature (see Fig. 1b, c and the “Methods” section). In agreement with ref. 12, the vacuum-induced correlation measured with two beams at a target distance of 50 μm and with a quarter-wave plate on both detection lines oscillates symmetrically around zero time delay δt with the maximum correlation for δt=0 (blue line in Fig. 2b). Compared to previous results, the reduced pulse duration increases the observed peak correlation to more than 5 V2/m212,15.

For the setup sensitive to correlations stemming from source radiation (combination of a half-wave plate on one and a quarter-wave plate on the other detection line), the temporal correlation (green line in Fig. 1b) is asymmetric. Confirming the causal nature of the correlations induced by source radiation, the signal occurs mainly for positive time delays δt > 0, while the residual correlations δt ≲ 0 arise from the finite spatio-temporal Gaussian extension of the laser pulses12.

The Fourier transform of the correlation traces for vacuum fluctuations and source radiation are plotted in Fig. 2c. Compared to our previous results, the signal bandwidth now extends to 4.5 THz due to the shorter pulse length12. The initial increase in the signal with frequency up to 2 THz is due to the increasing strength of the vacuum fluctuations with frequency. Above 2.5 THz, the two signals change sign as the distance between the two beams becomes comparable to the wavelength inside the crystal. The slight shift of the zero crossing of the experimentally observed vacuum-induced correlation compared to the theoretical model might be caused by a slight shift of the beam distance towards 30 μm for the measurement in a configuration sensitive to vacuum fluctuations (details in the Supplementary Information Note 2.C).

The vacuum field and source radiation signals are further not independent from each other but connected via the fluctuation-dissipation theorem in the quantum limit (zero temperature). It predicts a connection between the correlation that stems from vacuum fluctuations and source radiation in the time domain through the Hilbert transform, which is validated in the experimental data in Fig. 2b through the phase shift of π/2 between the two correlations7,17. In the frequency domain, this leads to a direct relation between the real part of the vacuum-induced correlation (blue solid line) and the imaginary part of the correlation caused by source radiation (green solid line), which are plotted in Fig. 2c. Theoretic calculations (plotted in dashed lines) predict that the imaginary part of the source radiation signal is equivalent to the vacuum-induced correlations multiplied by a factor of -1/27. When comparing the experimentally observed correlations, the imaginary part of the correlation caused by source radiation (green line) matches well the expectation based on the vacuum-induced correlation (gray line). The slight deviation between those two curves and between the experimental and theoretical data is mainly caused by the distance between the two beams inside the crystal, which deviated from the target value of 50 μm and between the two measurements (details in the Supplementary Information Note II.C).

Methods
Experimental setup
The optical setup to detect electro-optic correlations is shown schematically in Fig. 2a of the main text. We are using the pulsed signal of a Ti:sapphire laser. To compensate for the dispersion of the transmission optics, we implemented a prism compressor. The beam position is actively stabilized to reduce long-term drifts. A beam splitter divides the intensity of the pulse into two individual probe signals. The delay stage controls the time delay δt between the two beams. The two beams are placed in close proximity to each other, and two piezo-controlled mirrors set the angle between the two beams. Afterwards, the two beams are directed inside a dilution fridge.

Inside the cryostat, a 50 mm lens focuses the beams into a 〈110〉-cut zinc-telluride crystal. The angle of the two beams determines the distance of the two beams in the focal plane and is adjusted to a distance of 50 μm in the current experiment. Since the beam waist measures 10 μm, the two beams are well-separated inside the nonlinear crystal. A second lens re-collimates the beams afterwards. Both lenses and the nonlinear crystal are thermally connected to the 4 K-plate of the cryostat to suppress thermal radiation in the terahertz regime. The two beams are propagating along the 〈110〉-axis of the zinc telluride and are polarized along the 〈001〉-direction.

After leaving the cryostat, the two probe beams are separated, and the change in polarization is analyzed individually using the combination of a wave plate, a polarizing beam splitter, and a balanced detector. The correlation of the two signals is calculated for each individual pulse pair using a fast acquisition card. Further details about the setup and data acquisition, which is suppressing slowly varying correlations, are reported in ref. 15.

Experimentally observed correlation
In general, the signal observed in electro-optic sampling can be expressed in terms of a difference in photon-number S=Nperp - Npara of the two outputs of the polarizing beam splitter (perpendicular Nperp and parallel Npara polarized) measured by the balanced detector13. For a wave plate with arbitrary retardance γ, the signal is given by:

Sγ=4πϵ0c/ℏ ∫ dω P(γ) Ebx(r?,ω)Ebz(r?,ω) + H.c., (1)

where Ebz(r?,ω) denotes the NIR field at the end face of the detection crystal polarized along the crystallographic 〈001〉-axis and is given by the field of the probe signal, and Ebx(r?,ω) is the field along the 〈110〉, which is determined by the NIR vacuum fluctuations and the nonlinear field induced by the THz vacuum field and the source radiation. The influence of the wave-plate retardance γ is described by the following function:

P(γ) = -cos(γ) - 2i cos(γ/2) / 2. (2)

Within the experiment, we measure two electro-optic signals, Stγ(0) and Sτγ(δt), where δt denotes the delay between the two probe pulses. For vacuum fluctuations, both detection schemes employ a quarter-wave plate (γt=γτ=π/2), and the experimentally observed correlation is given by:

G(1)VF(δt) = h0|Stπ/2(0), Sτπ/2(δt)|0i. (3)

For source radiation, the detection configuration consists of a half-wave plate for the first probe (γt=π) and a quarter-wave plate for the second (γτ=π/2), yielding

G(1)SR(δt) = h0|Stπ(0), Sτπ/2(δt)|0i. (4)

Here, Nt and Nτ denote the total number of detected photons in the two probe signals and the refractive index n=2.85 at frequency ωp=2π×375 THz, the electro-optic coefficient of zinc telluride r41=3.9 pm V−1, the speed of light c, and the length of the detection crystal l=1 mm, and converts the unitless correlation of photon numbers into a field correlation as presented in Fig. 2.

Theoretical model
To model the experiments, we use the theoretical framework of two-beam electro-optical sampling developed in ref. 7. It builds on previous works in refs. 13, 26.

Applying the paraxial wave-approximation to all involved near-infrared fields but not the THz field26, the electro-optic sampling signal with two quarter-wave plates reads7

G(δt) = ∫ d3r dt d3r0 dt0 L(r,t)L(r0,t0) C(r,r0,t,t0). (5)

With a quarter-wave plate for pulse one and a half-wave plate for pulse two it is in turn given by:

G(δt) = -ℏ ∫ d3r dt d3r0 dt0 L(r,t)L(r0,t0) R(r,r0,t,t0). (6)

Here, V is the volume of the nonlinear crystal and L(r,t)=L(r-δre x,t-δt)=(2/π)3/2 e-2(t-ngz/c)2/τ2 σ e-2r2/w2 [r∥=(rx,rz), w: beam waist, τσ: pulse duration, ng: group refractive index in the near-infrared] are the Gaussian-shaped pulse envelopes of the two laser pulses. The response and correlation functions of the x-polarized THz field Eb(r,t) inside the crystal accounting for dispersion and absorption are defined in the main text and can be evaluated using macroscopic quantum electrodynamics27. In frequency domain, they are given by R(r,r0,Ω)=μ0Ω2D(r,r0,Ω)/(2π) and C(r,r0,Ω)=ℏμ0sgn[Ω]Ω2Im[D(r,r0,Ω)]/(2π), respectively, with the xx component of the dyadic Green tensor of the vector Helmholtz equation D and the vacuum permeability μ0.

We further follow ref. 7 to numerically evaluate Eqs. (5) and (6). The resulting frequency domain correlation signals defined by G(Ω)=(1/2π) ∫ dδt G(δt) eiΩδt are shown in Fig. 2c. We used the following parameters in the simulation, which, except for ng, have all been independently determined experimentally: pulse duration τ=110 fs, Gaussian beam width w=10 μm, beam separation δr=50 μm, and the group refractive index in the near-infrared ng=3.18. The refractive index in the THz is given by:

n(Ω) = sqrt(ϵ∞ + (ℏωLO)2 - (ℏωTO)2 / (ℏωTO)2 - (ℏΩ)2 - iℏΩγ), (7)

with all parameters as measured in ref. 28 for a ZnTe crystal at temperature T=300 K, except that we use ϵ∞=7.38 to match the real part of the refractive index measured at T=4 K reported in ref. 12.

Note that the influence of the phonon resonance at 5.3 THz starts to increase above 4 THz, which in turn leads to an enhancement of the local (longitudinal) field fluctuations26, but also to an increase in absorption. As a result of the absorption and the longitudinal nature of these fluctuations, the spatial range of correlations in the quantum vacuum as well as the propagation of source radiation from one beam to the other are delimited, and the detected electro-optic correlation in Fig. 2c vanishes in that frequency range. This means that the bandwidth of our detection is actually not limited by the bandwidth of the laser pulse but by the onset of increasing losses due to the phonon resonance above ≈4 THz, and makes it crucial that our theoretical framework accounts for absorption via a complex permittivity.

The fluctuation dissipation theorem implies C(r,r0,Ω)=ℏsgn(Ω)Im[R(r,r0,Ω)], leading to the relation between G(Ω) and G(Ω) discussed in the main text7.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Quantum electrodynamics effects such as spontaneous emission, Lamb shift, and Casimir-Polder forces can be attributed to\nvacuum fluctuations, source reaction, or a combination of both. Theoretically, this ambiguity was settled by Dalibard,\nDupont-Roc, and Cohen-Tannoudji, who proposed that the symmetric operator ordering between the variables of the atoms\nand the field should be adopted in order to ensure that the effective Hamiltonians originating from vacuum fluctuations and\nradiation reaction are separately Hermitian. However, it was generally believed that only the total effect is experimentally\nobservable, and not the individual contributions. In this work, the authors propose a method to experimentally distinguish\nthese two contributions. Based on recent advances in ultrafast optics, they present an optical analog of Fermi’s two-atom\nproblem using two laser pulses propagating through a nonlinear crystal. Through phase-sensitive detection, they\ndemonstrate that vacuum fluctuations and source radiation correlate different quadratures of the laser pulses, allowing for\ntheir individual detection. This approach also provides an experimental verification of the time-domain fluctuation-dissipation\ntheorem at the quantum level. Before recommending the paper for publication in Nature Communications, I have the\nfollowing questions and comments.\n1. In the abstract, the authors wrote: “The unique distinction between vacuum-field and source-radiation induced effects in\nprocesses such as the Lamb shift, Casimir forces or spontaneous emission, remains unresolved even at the theoretical\nlevel, and an experimental approach was never considered feasible” and “Beyond resolving a longstanding theoretical\nambiguity”. While I agree that experimentally separating these two contributions was long considered unfeasible, it should\nbe noted that, theoretically, this ambiguity has been solved by Dalibard, Dupont-Roc, and Cohen-Tannoudji in the 1980s.\n2. In Line 41, the authors wrote: ”On the other hand, the two atoms can exchange a real photon.” It would be clearer to\nspecify the initial state of the atoms before this statement.\n3. Naively, the vacuum field contains fluctuating quantum fields across all frequencies. It is not clear to me why in this\nexperiment the relevant vacuum fluctuations are specifically in the THz regime.", "reviewer_2": "The paper of Herter et al. “Experimentally separating vacuum fluctuations from source radiation” presents an important\nmilestone in developing time-domain characterization techniques based on electro-optic sampling (EOS). The authors\nsuccessfully implement a scheme where they can measure not only correlations between the electric field operators at\nvarious times and/or positions but also between the electric field and its conjugate quadrature in the time domain, being pi/2-\nCEP-shifted electric field. By fixing the spatial separation between the probe beams, i.e. by fixing the difference in the\nposition in the measured correlations, and analyzing the temporal correlation behavior the authors experimentally verify\ntheory predictions they made in Ref. [14]. The idea of the theory proposal was that the corresponding two types of\ncorrelations can be associated with the so-called vacuum and source contributions in the dynamics of two spatially\nseparated atoms coupled via the surrounded quantum electro-magnetic environment. Given that analogy, the authors\nmanage to show that the suggested measurements can be indeed performed. Their results quite well match the results of the\nmodelling of the corresponding correlations as they are captured in the two-beam EOS setup. The discussion about the role\nof those vacuum and source contributions, appearing in the perturbative treatment of the dynamics of the interacting atoms\nand field up to the second order, was quite important in the past. It might be somewhat outdated in the context of modern\nquantum field theoretical treatments of the interaction of matter with electro-magnetic field. Nevertheless, the demonstration\nthat this discussion can be illuminated by an analogue measurement seems to be interesting and also highly valuable.\nThus, I would be generally in favor of publication of this paper.\nSaid that, I think that for giving a definitely positive recommendation the paper still lacks certain clarity and scientific rigor in\nterms of the presentation and explanations. I am listing the corresponding particular issues that I noticed below:\n- lines 28-29: The authors write “Beyond resolving a longstanding theoretical ambiguity, …”\nThis sentence looks good for the motivation but I think it contradicts to the fact that according to the current understanding\nthere is actually no “ambiguity” remained but rather indeed “vacuum fluctuations and source radiation can therefore be seen\nas inseparable ”two sides of the same quantum mechanical coin””, as the authors also write in lines 47-48. The initial\nconfusions leading to the “ambiguity” seem to be rather of technical matter (ordering of the operator terms in the second\norder of the perturbation theory). When the physical process is considered as a whole, there is no such issue of sorting out\nthe terms.\n- In lines 41-43, the authors discuss the situation of two atoms. They write “On the other hand, the two atoms can exchange a\nreal photon.” but continue “This process, triggered by the interaction of the emitting atom with its own radiative field” jumping\nin the description to a situation of just one atom. Where is then the exchange (!) of a real photon? The authors think about\none situation but seem to describe a relevant but still different another situation. That is confusing.\n- In lines 43-43, the authors write “Mathematically, the individual contributions of vacuum fluctuations and source radiation\nare in general not uniquely determined, since they depend on the initial ordering of two commuting operators.” Here the\ndescription becomes sloppier. Contributions to what are meant by the authors? I understand that various atomic observables\ncan be meant, as in the paper of Dalibard, but here the description becomes so abstract that, in my opinion, it just does not\nreally make sense to state if some contributions are “uniquely determined” or not. Further, the explanation becomes even\nmore unclear with mentioning of “the initial ordering of two commuting operators”. This is a very technical description of the\nthings, being neither some physical phenomena, nor effects or variables, which are not even introduced in any appropriate\nway. Which operators? Which type ordering? I doubt that this is clear enough and thus suitable for the\nmotivation/introductory part of the paper. Otherwise, it has to be explained clearer.\n- The “operator ordering” appears again without an appropriate context in line 52.\n- In line 58, “probe source radiation and vacuum field fluctuations individually for the first time” – with the use of the word\n“individually” the sentence might leave an impression as if it were possible to separate the corresponding effects from each\nother, leading to confusion.\n- Starting from line 74 the authors introduce an analogy between the version of the two-beam EOS setup\nused by them in the present paper and the two-atom problem. I understand that this analogy was introduced and discussed at length in a very\nthorough preceding theory paper, Ref. [14]. However, should the reader consider that paper as a kind of “Supplementary” to\nthe present paper? Even so, I think that the key points of the analogy should be further simplified/purified and explained in a\nclear way, mentioning also the main differences in the systems and arguing why they would not be significant in the present\ncontext. For example, a few differences: the fields considered as counterparts of atoms in the two-atom problem have a\ncontinuous set of possible transition (photon) energies and not a discrete one. Moreover, for each transition energy many\nexcitations are possible and not just one, since we deal in that case with bosons. The explanations about all those things\ncan be organized, e.g., as a comparison table.\n- The only quantities defined before showing the correlation plots in Fig. 2, are two quantities C and R in lines 76 and 76.\nHowever, which “correlations” are then exactly plotted in Fig. 2b? Obviously, they cannot be both represented by C and R,\nbecause R is an imaginary quantity. Moreover, R has a step in the time dependence, which is not observed in Fig. 2b. Of\ncourse, some explanations by words are given in lines 78-94, from which one can guess what was measured. Also, one can\ncheck again Ref. [14]. However, I think that the measured quantities have to be precisely defined in the main text of the\nmanuscript, before discussing the figures and what they show. From the description stating that actually the correlations\nbetween the electric field and the pi/2-CEP-shifted quadrature are measured, I wonder why the authors do not cite the paper\nof Leitenstorfer’s group, Physical Review A 101, 033821 (2020), where such pi/2-CEP-shifted quadrature was measured (of\ncourse, yet classically and without evaluation of the correlations). I know there is a limitation on references and there are\nmany citations of that group anyway but still. Moreover, I guess that the definition of the measured correlations would be\nprobably expressible via E and that E_(pi/2). This would, in turn, make obvious that since E_(pi/2) is the Hilbert transform of\nE the same should be probably valid for the corresponding two different types of measured correlations in Fig. 2. I think this\nshould be made clear before discussing the consequences when interpreting the measurement in terms of the two-atom\nanalogy.\n- In line 82, in which sense the term “orthogonal” is used here?\n- In line 99 it is written that the beams are separated by δr=50 um. However, when the source radiation that supposed to be\ncausal is discussed, it is written “The signal occurs mainly for positive time delays δt > 0”. Why not for δt > δr/(c/n)~ 0.5 ps,\ntaking into account the spatial separation between the beams (“atoms”) and the time required to travel between them?\n- It is difficult to distinguish 3 lines in Fig. 2b neither on my screen nor in the printed version.\n- In Fig. 2c, comparison between the experimental data on both types of correlations and the theoretically simulated\ncounterparts is provided. What about comparing the experimental “vacuum” and “source” contributions between each other,\nsince they have to be related as given by the equation in line 415? Does this equation well hold when we reconstruct one\nfrom another (this can be also illustrated in the plot)?\n- What should “Moreover” indicate in line 121?\n- In lines 129-138, again the discussion about “separation” of contributions might be confusing, since the corresponding\nprocesses always happen simultaneously.\nMinor issues:\n- Grammar issues in lines 82, 84-85, lines 8-9 (as well as lines 12 and 13) of the caption to Fig. 2, lines 335-336\nIf the issues mentioned above are resolved successfully the paper is definitely worth publishing in Nature Communications,\nin my opinion.", "reviewer_3": "In their work titled “Experimentally separating vacuum fluctuations from source radiation”, Alexa Herter et al report their\nlaboratory findings on probing vacuum-source correlations through the two electro-optically sampled space-time volumes, in\nturn interrogated by two tightly focused near-infrared femtosecond laser beams with a controlled separation in space and\ntime. As the authors suggest, by using different probing schemes, namely by employing a quarter- and a half-wave plate\nanalysis of the EOS signal in two probing volumes can be used to separate vacuum-vacuum and vacuum-source\ncontributions, which has been a long-standing problem in modern theoretical physics, and has not yielded a successful\nexperimental confirmation to date. Following the works of Fermi and Dalibard, the team of authors has offered a theoretical\nprediction of an experimental method of testing such correlations by means of electro-optic sampling and this present work\nrepresent, at large, a validation of those theoretical predictions.\nI find this work creative, daring, exciting and a stellar example of how experimental science should be used to lead\nexploration of fundamental physics. I congratulate the authors on these heroic efforts, and I generally support the publication\nof these findings, if they are proven convincing at the highest standards of rigor. Having said that, I am not yet convinced that\nthe reported data supports the interpretation offered by the team, at least in the form currently reported. I offer my constructive\ncriticism for authors’ consideration in the hope that it is taken as my attempt to help them to identify the weak points of the\npresentation and to improve them, if possible, in order to strengthen the arguments.\nGeneral comments:\n1. The authors provide a compelling proposal of using EOS with QWP/HWP configurations to separate vacuum-source\ncontributions. I find the analogy between Fermi’s two-atom setup and the two-beam EOS sampling to be innovative and\nintriguing. However, I struggle with one concept which I believe should be thoroughly clarified.\nLine 68-72: Here, the analogy between a Fermi’s two-atom setup and a two-pulse EOS experiment is built. I believe this is a\ncrucial place where precision in the analog should be increased as much as possible and warrants and additional\ndiscussion/references. If I understood correctly, in the EOS analog of the Fermi’s setup, the role of two atoms is played by\nthe coherent states of the two probing beams. If such, the analogy is not 1-to-1 in the sense that the two states of an atom\ncan only lead to sequential absorption-emission vs emission-absorption events, while the harmonic oscillator nature of the\nbeams can admit sequential absorption and sequential emission processes. It is possible that the two are equivalent in the\nlowest order of perturbation analysis, but this should be explicated.\n2. In the Fermi’s setup, the source contribution is isolated to the processes where one atom emits a “real” photon, which is\nthen absorbed by the second atom. In the EOS setup, can we think of the source-based exchange mediated via real photon\nemission, or rather taking place via field correlations? I believe this is an important distinction and should be clarified. In my\nunderstanding, the EOS setup is used to detect (correlations in) THz fields, and as such, one should be careful when\napplying “particle-vs-wave” interpretations. It is true that when we think of the EOS detection, we sometimes speak of\n“orthogonally-polarized photons” which are emitted, but I believe the issue is much more subtle than that and in fact points to\nthe main difference between the Fermi’s proposal and the current experiment. In fact, I view the current experiment as an\nUPGRADE to the Fermi’s setup, where instead of probing the source term through the causal photon emission, in the EOS\nsetup, the authors offer a novel way of probing those terms through field correlation measurements. Whether my\ninterpretation is correct or not is less relevant to the fact that a careful attention to the field vs particle interpretation should be\ngiven. If I’m correct, this is a foundational contribution going beyond Fermi and Dalibard, instead of an experimental\nverification, as stated by the authors up to now. Another crucial distinction between the experiment and the those\nproposals is that the experimental involves TRAVELLING detectors, namely space-time volumes where correlations are sampled are\nof propagating nature, vs. the Fermi's experiment involving stationary atoms.\n3. Line 88-94: Discussion of the source radiation contribution. The pi/2 shift is not explained in great detail, however it is of\ncrucial importance. In the context of the Fermi’s setup, the pi/2 shift is naturally expected arising from on/off resonant\ncontributions by the different influences (vacuum field vs emission), while in the EOS context, which presumably lacks\n“resonant” excitations, this is less clear. Perhaps this is the place where the analog of the two setups should be explained\nmore.\n4. Line 96: Transition to the experimental section is too abrupt. No technical comment here, I was just stunned by it and\nwould like to communicate to the team that an improvement is in order.\n5. Line 112: The extension of EOS bandwidth to 4.5 THz in a ZnTe crystal is impressive but raises the question of whether\nunaccounted-for phase shifts in the EOS response function are introduced due to the proximity of the phase-matching band\nto the Reststrahlen band, starting with the LO phonon at a frequency of 5.3 THz. The fact that the crystal is held at 4K may\nperhaps alleviate this issue if the phonon (phonon-polariton) lineshape function sharply decreases between the 4.5 THz and\n5.3 THz frequencies; however, this warrants careful justification and discussion, given that the main finding relies on\nphase-shifted (delay-dependent) results. This point is especially exacerbated by the extent of the abscissa range to 5 THz in Fig.\n2c.\n6. Line 274-278: Similar to the previous comment, the delay between the two signals (QWP vs HWP) is of crucial importance\nto the conclusions derived from the data. In this sense, there should be more discussion allocated to the fact that the time-\nzero of the vacuum correlation signal is determined only from a narrow frequency window, approximately between 2-2.5\nTHz, while the conclusions on the delay of the vacuum and source contributions are drawn across a wider frequency span.\nThis is particularly important, as the change of sign between the two signals occurs at a frequency which is nearly centered\non the frequency window analyzed. The authors should give due diligence in ensuring that the choice of the analysis\nwindow does not in fact introduce the aforementioned correlations.\n7. Line 362-371: Similar to the above remark, the delay of the two signals needs to be established with great precision. To\nthe great credit of the authors, they provide an honest report of the temporal drifts of the signal during the data acquisition.\nMore work should be done to dispel any doubts about whether systematic drifts in the experiment cause the observed delay.\nTo this extent, acquisition time should be reported, perhaps along with statistical measurements of temporal drifts between\nthe two laser beams during such times.\n8. Line 380-389: Similar to the above comment, the interpretation of the correlation structure in the data relies on the\nseparation of the two beams. It seems that the discussion of acquisition times and beam angle sensitivities suggests that the\nbeam separation might have drifted during the experiments. If so, this should be clarified more explicitly and perhaps instead\nof insisting on a single separation value of 50um, the simulations should be averaged over the statistical sampling of the\nseparations, as predicted by the drifts of the system during the measurement run. In addition, it would help to report the error\nbars both on the 50um separation and/or on the spot size measurement of the two beams.\nIf the authors manage to provide convincing discussions, details, explanations, arguments, calculations to the points raised\nabove, in my opinion the demonstration of the claimed effect would be accomplished beyond the reasonable doubt (of at\nleast this reviewer)."}, "author_response": "Response to referees\nWe would like to thank all reviewers for their positive assessment of our work. Given the suggestions and\ncomments we received in the first round of reviews, we have decided to undertake the following major revisions\nto the initially submitted version of the manuscript and supplementary information:\n1. We reduced the length of the abstract to less than 150 words to fulfill the guidelines of Nature Commu-\nnications.\n2. We revised the data analysis and added further investigations on the stability of the beam distance and\nthe zero time delay position reported in the Supplementary Information.\nFurthermore, we made the following small revisions to the manuscript:\n1. During the review process we refined the data analysis, resulting in a slight shift of the time-axis of\nthe correlation induced by source radiation by 16.8fs. Instead of the initially reported shift of 50.2fs\noccurring during acquisition of the source radiation, we corrected this value now to 67fs. The discussion\nand statements made based on the data are not influenced of this change. In contrast, the agreement\nbetween expectations and experimental observations improved slightly.\n2. We made some changes to the manuscript to fulfill better the guidelines of Nature Communications,\ne.g. formulate one heading per figure instead of a heading for each panel of the figures, move additional\ninformation from the Methods section of the main text into a separate Supplementary Information file,\netc.\nIn the following, the questions of the referees are shown in blue, the answers are shown in black and changes to\nthe manuscript are shown in green.\n1. In the abstract, the authors wrote: “The unique distinction between vacuum-field and source-radiation\ninduced effects in processes such as the Lamb shift, Casimir forces or spontaneous emission, remains un-\nresolved even at the theoretical level, and an experimental approach was never considered feasible” and\n“Beyond resolving a longstanding theoretical ambiguity”. While I agree that experimentally separating\nthese two contributions was long considered unfeasible, it should be noted that, theoretically, this ambi-\nguity has been solved by Dalibard, Dupont-Roc, and Cohen-Tannoudji in the 1980s.\nWe agree with the referee that one can say that the ambiguity in distinguishing source-radiation and\nvacuum-field induced effects was solved theoretically by Dalibard, Dupont-Roc, and Cohen-Tannoudji in\nthe1980s. Still,thisapproachreliesonamathematicallymotivatedchoiceofoperatororderingofthelight\nand matter operators in the light-matter coupling Hamiltonian. As we outline in the manuscript, this is\ndifferent in Fermi’s two-atom setup, where the physically motivated causality requirement for the source\nradiation contribution removes the ambiguity. To avoid confusion and clarify this issue, we rephrased the\ncorresponding sentences in the abstract (see also comment 1 by Reviewer 2).\nSpecifically, we rephrased ”The unique distinction between vacuum-field and source-radiation induced ef-\nfectsinprocessessuchastheLambshift,Casimirforcesorspontaneousemission,remainsunresolvedeven\nat the theoretical level, and an experimental approach was never considered feasible” to “The distinction\nbetween vacuum-field and source-radiation effects in phenomena such as the Lamb shift, Casimir forces,\nand spontaneous emission relies on additional assumptions at the theoretical level, and an experimen-\ntal approach was never considered feasible”. The phrase “Beyond resolving a longstanding theoretical\nambiguity [...]” has been removed from the manuscript.\n2. In Line 41, the authors wrote: ”On the other hand, the two atoms can exchange a real photon.” It would\nbe clearer to specify the initial state of the atoms before this statement.\nWe thank the referee for raising this point. In the manuscript, the coupling of the atoms to the field is\nswitchedonandoffatshorttime-scales(sub-wavelength),suchthatthetwoatomscanexchangeaphoton\neven if they are initially in the ground state. So both processes, individually interacting with the vacuum\nfield and exchanging a photon, can take place even if the two atoms are initially in the ground state; see\nRefs. [14,15]. So we believe that this discussion is independent of the initial state of the atoms.\nTo avoid confusion of what is meant by a real photon in this context, we decided to remove ”real” from\nthe sentence and write “On the other hand, the two atoms can exchange a photon”. We also changed ”In\n1\ntheHeisenbergpicture,twodifferenttermscontributetothesecorrelations”to“IntheHeisenbergpicture,\nindependent of the initial state of the two atoms, two different terms contribute to these correlations”.\n3. Naively, the vacuum field contains fluctuating quantum fields across all frequencies. It is not clear to me\nwhy in this experiment the relevant vacuum fluctuations are specifically in the THz regime.\nIndeed, vacuum fluctuations are present at all frequencies, such that the laser pulse can interact not just\nwith the vacuum fluctuations in the THz regime but also at higher frequencies. However, the nonlinear\ninteraction of the laser pulse with vacuum fluctuations above ∼ 5THz, will result in photons that have\na frequency outside the bandwidth of the laser and they are therefore not detected in our setup. This\nway, we only probe the mixing of the laser pulse with vacuum fluctuations that have frequencies that are\nsmaller than the bandwidth of the laser pulse.\nTo clarify this important point raised by the referee, we write now “The implementation relies on electro-\nopticsampling, inwhichnear-infraredprobepulsescoupletoaterahertzormid-infraredfield(possiblyin\nthe vacuum state and restricted to the spectral width of the probe) generating a second-order nonlinear\nfield polarized perpendicular to the probe [1, 2, 3].”\n1. lines28-29: Theauthorswrite“Beyondresolvingalongstandingtheoreticalambiguity,...”Thissentence\nlooks good for the motivation but I think it contradicts to the fact that according to the current under-\nstanding there is actually no “ambiguity” remained but rather indeed “vacuum fluctuations and source\nradiation can therefore be seen as inseparable ”two sides of the same quantum mechanical coin””, as the\nauthors also write in lines 47-48. The initial confusions leading to the “ambiguity” seem to be rather of\ntechnical matter (ordering of the operator terms in the second order of the perturbation theory). When\nthe physical process is considered as a whole, there is no such issue of sorting out the terms.\nWeagreewiththeReviewerthattheambiguityindistinguishingsource-radiationandvacuum-fieldcanbe\nconsideredbeingmainlyoftechnicalnature,whenconsideringtheprocessasawhole,whichisindependent\nof the initial operator ordering. If as we do in this work, one attempts to individually observe the source\nradiation and vacuum-induced contributions this mathematical freedom becomes a physical ambiguity(as\nonlyoneuniqueanswercanbeobserved)whichcanonlyberesolvedbyamathematicallymotivatedchoice\nof operator ordering of the light and matter operators in the light-matter coupling Hamiltonian. As we\noutline in the manuscript, this is different in Fermi’s two-atom setup, where the causality requirement for\nthe source radiation contribution removes the ambiguity. To avoid confusion and clarify this issue, we\nrephrased the corresponding sentences in the abstract (see also comment 1 by Reviewer 1).\nWe rephrased ”The unique distinction between vacuum-field and source-radiation induced effects in pro-\ncesses such as the Lamb shift, Casimir forces or spontaneous emission, remains unresolved even at the\ntheoreticallevel,andanexperimentalapproachwasneverconsideredfeasible”to“Thedistinctionbetween\nvacuum-fieldandsource-radiationeffectsinphenomenasuchastheLambshift, Casimirforces, andspon-\ntaneous emission relies on additional assumptions at the theoretical level, and an experimental approach\nwas never considered feasible”. The phrase “Beyond resolving a longstanding theoretical ambiguity [...]”\nhas been removed from the manuscript.\n2. In lines 41-43, the authors discuss the situation of two atoms. They write “On the other hand, the two\natoms can exchange a real photon.” but continue “This process, triggered by the interaction of the emit-\nting atom with its own radiative field” jumping in the description to a situation of just one atom. Where\nis then the exchange (!) of a real photon? The authors think about one situation but seem to describe a\nrelevant but still different another situation. That is confusing.\nFor the exchange of a real photon, one of the two atoms has to emit a photon. Therefore, we describe the\npossible effects leading to the emission of a photon by one single atom. To clarify the connection to the\nphoton exchange, we reformulated the mentioned part in the text to:\n“On the other hand, the two atoms can exchange a photon. The emission of this photon is triggered by\nthe interaction of one atom with its own radiative field (radiation reaction). This radiation, known as\nsource radiation [4, 5], can interact with the other atom, which leads to correlations between the atoms.”\n2\n3. Inlines43-43,theauthorswrite“Mathematically,theindividualcontributionsofvacuumfluctuationsand\nsource radiation are in general not uniquely determined, since they depend on the initial ordering of two\ncommuting operators.” Here the description becomes sloppier. Contributions to what are meant by the\nauthors? I understand that various atomic observables can be meant, as in the paper of Dalibard, but\nhere the description becomes so abstract that, in my opinion, it just does not really make sense to state\nif some contributions are “uniquely determined” or not. Further, the explanation becomes even more\nunclear with mentioning of “the initial ordering of two commuting operators”. This is a very technical\ndescription of the things, being neither some physical phenomena, nor effects or variables, which are not\neven introduced in any appropriate way. Which operators? Which type ordering? I doubt that this is\nclear enough and thus suitable for the motivation/introductory part of the paper. Otherwise, it has to be\nexplained clearer.\nWe thank the Reviewer for raising these points. For more clarity, we now rephrased and changed the\nreferences: “Mathematically, the individual contributions of vacuum fluctuations and source radiation to\nfundamental processes in quantum electrodynamics, such as spontaneous emission or the Lamb shift, are\nin general not uniquely determined, since they depend on the initial ordering of the atomic and the field\noperators in the light-matter interaction Hamiltonian [2-4].”\n4. The “operator ordering” appears again without an appropriate context in line 52.\nWe thank the referee for pointing this out. As we clarified this issue now in the previous paragraph, we\nsimply changed ”operator ordering” to “the aforementioned operator ordering”.\n5. In line 58, “probe source radiation and vacuum field fluctuations individually for the first time” – with\ntheuseoftheword“individually”thesentencemightleaveanimpressionasifitwerepossibletoseparate\nthe corresponding effects from each other, leading to confusion.\nIndeed, for our setup it was found that contributions from vacuum fluctuations and source radiation,\nseparated by the causality requirement, correlate different field quadratures. So by probing correlation\nbetween different field quadratures we can individually probe the contributions from vacuum fluctuations\nand source radiation, separated by the causality requirement.\nTo address these subtleties, we rephrased the sentence quoted by the referee to “...and probe the contri-\nbutions from source radiation and vacuum field fluctuations, separated by causality, individually for the\nfirst time.”\n6. Starting from line 74 the authors introduce an analogy between the version of the two-beam EOS setup\nused by them in the present paper and the two-atom problem. I understand that this analogy was intro-\nduced and discussed at length in a very thorough preceding theory paper, Ref. [14]. However, should the\nreader consider that paper as a kind of “Supplementary” to the present paper? Even so, I think that the\nkey points of the analogy should be further simplified/purified and explained in a clear way, mentioning\nalso the main differences in the systems and arguing why they would not be significant in the present\ncontext. For example, a few differences: the fields considered as counterparts of atoms in the two-atom\nproblem have a continuous set of possible transition (photon) energies and not a discrete one. Moreover,\nfor each transition energy many excitations are possible and not just one, since we deal in that case with\nbosons. The explanations about all those things can be organized, e.g., as a comparison table.\nWe agree with the Reviewer that the comparison between Fermi’s two-atom setup and our EOS setup\nremainsvagueinthesubmittedversionofthemanuscript. Asthishasbeendiscussedindifferentprevious\npublications, we decided to add a chapter to the Supplemental Material, in which we outline the compar-\nison. Moreover, we reformulated the transition from the two atom picture to the two-beam experiment\nto: “In contrast to atoms, the interaction of laser pulses with a surrounding electromagnetic field can be\ninitiatedandterminatedonultrashorttimescalesasthepulsesenterandleaveanonlinearmaterial. This\nunique property has enabled an all-optical analogue of Fermi’s two-atom setup, where vacuum-induced\ncorrelationsbetweenatomsaremappedontotwospatiallyseparatedlaserpulsesinteractingwiththevac-\nuumfieldinsideanonlinearcrystal[6](seeFig.1b). Theimplementationreliesonelectro-opticsampling,\nin which near-infrared probe pulses couple to a terahertz or mid-infrared field (possibly in the vacuum\nstate) and generate a second-order nonlinear field polarized perpendicular to the probe and lying within\nits bandwidth [1, 2, 3]. This nonlinear interaction modifies the polarization state of the two probe pulses,\n3\ndirectlyparallelingthetransitionofatomsinFermi’ssetup. Here,however,theresonantsingle-modetran-\nsition of two-level atoms is replaced by an off-resonant, broadband coupling of multimode near-infrared\npulses to terahertz field modes within the spectral width of the probes. The use of femtosecond thus\nnot only allows the interaction to be switched on and off with high temporal precision, but also enables\nthe application of advanced quantum optics detection techniques to analyze the polarization state of the\nprobes(seeRefs.[7,6,8]andSupplementalInformationNote1.AforfurthercomparisonbetweenFermi’s\ntwo-atom setup and two-beam electro-optic sampling). ”\nFurthermore, we added an additional section to the Supplemental Material with the title ”Comparison\nbetween Fermi’s two atom and our electro-optic sampling setup”.\n7. The only quantities defined before showing the correlation plots in Fig. 2, are two quantities C and R\nin lines 76 and 76. However, which “correlations” are then exactly plotted in Fig. 2b? Obviously, they\ncannot be both represented by C and R, because R is an imaginary quantity. Moreover, R has a step in\nthetimedependence,whichisnotobservedinFig. 2b. Ofcourse,someexplanationsbywordsaregivenin\nlines78-94,fromwhichonecanguesswhatwasmeasured. Also,onecancheckagainRef. [14]. However,I\nthinkthatthemeasuredquantitieshavetobepreciselydefinedinthemaintextofthemanuscript, before\ndiscussing the figures and what they show. From the description stating that actually the correlations\nbetweentheelectricfieldandthepi/2-CEP-shiftedquadraturearemeasured,Iwonderwhytheauthorsdo\nnot cite the paper of Leitenstorfer’s group, Physical Review A 101, 033821 (2020), where such pi/2-CEP-\nshifted quadrature was measured (of course, yet classically and without evaluation of the correlations).\nI know there is a limitation on references and there are many citations of that group anyway but still.\nMoreover, I guess that the definition of the measured correlations would be probably expressible via E\nand that E . This would, in turn, make obvious that since E is the Hilbert transform of E the same\nπ/2 π/2\nshould be probably valid for the corresponding two different types of measured correlations in Fig. 2. I\nthinkthisshouldbemadeclearbeforediscussingtheconsequenceswheninterpretingthemeasurementin\nterms of the two-atom analogy.\nWe thank the Reviewer for point this out. What is measured in our two-beam EOS setup, are the cor-\nrelations between two EOS signals resulting from ellipsometry with different wave-plates in the detection\nsetup. This has been outlined in detail in our previous works [3, 6]. The measured signals are related\nto C(r,r′,t,t′) and R(r,r′,t,t′) evaluated over the space-time volumes (r,t) and (r′,t′) defined by the\ntemporalandspatialextendofthetwoprobepulses. Duetothetemporalextendoftheprobesignals,the\nmeasured correlationrelated to R(r,r′,t,t′) does notshowa sharpstep buta continuous riseofthe signal\nas soon as a causal overlap between the two space-time volumes occurs. Further theoretical discussion\nontheinfluenceofthepulseshapeontothetemporalshapeoftheobservedcorrelationcanbefoundin[8].\nTo state more clearly what is actually measured and shown in Fig. 2, we added the following sentence\nto the main text (second paragraph on page 5). “...(see Methods and Extended Data Fig. 1). The\ncorrelations between different quadratures of the field amplitudes of the two laser pulses emerging from\nthe crystal are detected by probing the correlations between two sets of balanced detectors, each of which\nhas a tunable waveplate determining the probed quadrature (see Methods and Fig. 1).”\nIn the caption of Fig.2, we also added “Correlations between the signal from the two balanced detectors,\nwhich is sensitive to the correlations caused by terahertz vacuum fluctuations (blue, two quarter wave\nplates in the detection setup) and source radiation (green, one quarter and one half-wave plate in the\ndetection setup), is plotted depending on the time-delay between the two pulses δt.”\nMoreover,weaddedasmallparagraphtotheMethodsSection,alsoincludingadefinitionofthemeasured\nsignal.\nInthepaperofLeitenstorfer’sgroup,PhysicalReviewA101,033821(2020)[9],alsodifferentfieldquadra-\ntures are seperated. We thank the Reviewer for bringing this up. The way the different quadratures are\nseparatedin[9]reliesonfrequencyfiltering, whichisdifferentfromourapproachbasedondifferentwave-\nplates in the balanced detection scheme. To clarify this connection, we added a remark on the use of a\nhalf-wave plate in electro-optic sampling to the Supplementary Information.\n8. In line 82, in which sense the term “orthogonal” is used here?\n”Orthogonal” in this case means the field quadratures that are in-phase and out-of-phase with the initial\nprobefield. Wereferheretotheorthogonalityofthep-andq-quadrature,butwanttokeepthestatement\n4\nindependent of the initial phase properties of the probe signal by avoiding assigning a specific quadrature\nto the probe and nonlinear field. To avoid confusion, we now use the wording in- and out-of phase\nquadratures.\n9. In line 99 it is written that the beams are separated by δr =50 um. However, when the source radiation\nthat supposed to be causal is discussed, it is written “The signal occurs mainly for positive time delays\nδt > 0”. Why not for δt > δr/(c/n) ≈ 0.5 ps, taking into account the spatial separation between the\nbeams (“atoms”) and the time required to travel between them?\nAssumingthepulsesaspoint-likeobjectsthatonlyinteractwiththevacuumfieldbyexchangingaphoton\northogonaltotheirpropagationdirection,thecorrelationwouldoccurindeedonlyforδt>δr/(c/n)≈0.5\nps. First, however, the Gaussian beam waist and the finite duration of the laser beams lead to a finite\nextent of the laser pulses. The same applies to Fermi’s two-atom setup, in which a finite size of the\natoms is typically considered via a smearing function [10]. The Gaussian shape of the atoms or laser\npulses leads to a very small, but non-vanishing spatial overlap of the two laser pulses. Second, the two\nlaser pulses mainly exchange photons that propagate diagonally from one laser pulse to the other, as\nthe crystal length L = 1mm is much longer than the spatial extent of the laser pulses, and modes that\nare propagating diagonally are favored by phase-matching [6, 8]. Still, as discussed in reference [6], the\nvacuum induced correlation observed for a beam distance of 50µm contain contributions from time-like\nand space-like separated space-time regions, but the signal is dominated by contributions from space-like\nseparated regions. Similarly, the source radiation leads to a small, but non-vanishing correlation also\nfor δt < δr/(c/n). In the theoretical study [8], it has been shown, that for rectangular pulses that are\ncompletely space-like separated, the source radiation contribution vanishes exactly, as expected.\nTo clarify these issues, we write now “Confirming the causal nature of correlations induced by source\nradiation, the signal occurs mainly for positive time delays δt > 0, while the residual correlations δt ≲ 0\narise from the finite spatio-temporal Gaussian extension of the laser pulses [6].”\n10. It is difficult to distinguish 3 lines in Fig. 2b neither on my screen nor in the printed version.\nWe agree with the reviewer the lines being hard to distinguish. As a consequence, we decided to only\nshow the pure data including the high frequency noise, since the noise is low enough to identify well the\ncorrelation signal.\n11. In Fig. 2c, comparison between the experimental data on both types of correlations and the theoretically\nsimulated counterparts is provided. What about comparing the experimental “vacuum” and “source”\ncontributionsbetweeneachother,sincetheyhavetoberelatedasgivenbytheequationinline415? Does\nthis equation well hold when we reconstruct one from another (this can be also illustrated in the plot)?\nWethanktheReviewerforthissuggestionandaddedthespectralcorrelationofsource-radiationexpected\nfrom the experimentally observed vacuum-induced correlation and added to the discussion: “Comparing\ntheexperimentallyobservedcorrelations,theimaginarypartofthecorrelationcausedbysourceradiation\n(green line) matches well the expectation based on the vacuum-induced correlation (gray line).”\n12. What should “Moreover” indicate in line 121?\nWe agree that ”Moreover” is a bit misleading here. To improve the clarity in general, we reformulated\n“Thevacuum-fieldandsourceradiationsignalsarefurthernotindependentfromeachotherbutconnected\nvia the fluctuation-dissipation theorem in the quantum limit (zero temperature). It predicts a connection\nbetween correlation stemming from vacuum fluctuations and source radiation in the time domain via the\nHilbert transform, which is validated in the experimental data in Fig. 2 b through the phase shift of π\n2\nbetween the two correlations [11, 8]. In frequency domain, this leads to a direct relation between the real\npart of the vacuum-induced correlation (blue solid line) to the imaginary part of the correlation caused\nby source radiation (green solid line), which are plotted in Fig. 2 c. ”\n13. In lines 129-138, again the discussion about “separation” of contributions might be confusing, since the\ncorresponding processes always happen simultaneously.\nWe refer back to our response to comment 5 by the Reviewer. We think that we have clarified this issue\nby the corresponding changes outlined there.\n5\n1. The authors provide a compelling proposal of using EOS with QWP/HWP configurations to separate\nvacuum-source contributions. I find the analogy between Fermi’s two-atom setup and the two-beam EOS\nsampling to be innovative and intriguing. However, I struggle with one concept which I believe should be\nthoroughly clarified. Line 68-72: Here, the analogy between a Fermi’s two-atom setup and a two-pulse\nEOSexperimentisbuilt. Ibelievethisisacrucialplacewhereprecisionintheanalogshouldbeincreased\nas much as possible and warrants and additional discussion/references. If I understood correctly, in the\nEOSanalogoftheFermi’ssetup,theroleoftwoatomsisplayedbythecoherentstatesofthetwoprobing\nbeams. If such, the analogy is not 1-to-1 in the sense that the two states of an atom can only lead to\nsequential absorption-emission vs emission-absorption events, while the harmonic oscillator nature of the\nbeams can admit sequential absorption and sequential emission processes. It is possible that the two are\nequivalent in the lowest order of perturbation analysis, but this should be explicated.\nWe agree with the reviewer that it is important to highlight the similarities and differences between our\ntwo-beam electro-optic sampling setup and Fermi’s two atom setup. However, there are several previous\nworks that have already addressed this issue, see e.g. [7, 12]. To clarify the connection in the current\nmanuscript, we reformulated the corresponding paragraph in the main text “In contrast to atoms, the\ninteraction of laser pulses with a surrounding electromagnetic field can be initiated and terminated on\nultrashorttimescalesasthepulsesenterandleaveanonlinearmaterial. Thisuniquepropertyhasenabled\nan all-optical analogue of Fermi’s two-atom setup, where vacuum-induced correlations between atoms are\nmapped onto two spatially separated laser pulses interacting with the vacuum field inside a nonlinear\ncrystal [6] (see Fig. 1 b). The implementation relies on electro-optic sampling, in which near-infrared\nprobe pulses couple to a terahertz or mid-infrared field (possibly in the vacuum state) and generate a\nsecond-order nonlinear field polarized perpendicular to the probe and lying within its bandwidth [1, 2, 3].\nThis nonlinear interaction modifies the polarization state of the two probe pulses, directly paralleling\nthe transition of atoms in Fermi’s setup. Here, however, the resonant single-mode transition of two-level\natoms is replaced by an off-resonant, broadband coupling of multimode near-infrared pulses to terahertz\nfield modes within the spectral width of the probes. The use of femtosecond thus not only allows the\ninteraction to be switched on and off with high temporal precision, but also enables the application\nof advanced quantum optics detection techniques to analyze the polarization state of the probes (see\nRefs. [7, 6, 8] and Supplemental Information Note 1.A for further comparison between Fermi’s two-atom\nsetup and two-beam electro-optic sampling).”\nWealsoaddedanewChaptertotheSupplementalMaterialdedicatedtothecomparisonbetweenelectro-\noptic sampling and Fermi’s two-atom setup.\nIn the new chapter of the supplemental material, we also address the specific points raised by the Referee\ninhiscomment1andcomment2,seebelow. ThecorrespondencebetweenFermi’stwoatomsetupandour\ntwo-beam electro-optic sampling setup is not 1-to-1 but there are certain differences. Concerning the fact\nthatinFermi’stwoatomsetuptwo-leveldetectorsareconsideredwhereasinourcasethedetectorsconsist\nofharmonicoscillatorsisaminordifferenceatthelevelofperturbationtheoryvalidforthecurrentsetup.\nAs anticipated by the Reviewer, at second order in perturbation theory that describes the experiments\nwell, there is no difference between having two-level systems or harmonic oscillators as detectors. This\nhas been discussed for instance in Ref. [13] for the case of entanglement harvesting, and we have added\nthis reference to the reference list of the Supplemental Material.\n2. In the Fermi’s setup, the source contribution is isolated to the processes where one atom emits a “real”\nphoton, which is then absorbed by the second atom. In the EOS setup, can we think of the source-based\nexchange mediated via real photon emission, or rather taking place via field correlations? I believe this\nis an important distinction and should be clarified. In my understanding, the EOS setup is used to de-\ntect (correlations in) THz fields, and as such, one should be careful when applying “particle-vs-wave”\ninterpretations. ItistruethatwhenwethinkoftheEOSdetection,wesometimesspeakof“orthogonally-\npolarized photons” which are emitted, but I believe the issue is much more subtle than that and in fact\npoints to the main difference between the Fermi’s proposal and the current experiment. In fact, I view\nthe current experiment as an UPGRADE to the Fermi’s setup, where instead of probing the source term\nthroughthecausalphotonemission,intheEOSsetup,theauthorsofferanovelwayofprobingthoseterms\nthrough field correlation measurements. Whether my interpretation is correct or not is less relevant to\nthe fact that a careful attention to the field vs particle interpretation should be given. If I’m correct, this\nis a foundational contribution going beyond Fermi and Dalibard, instead of an experimental verification,\nas stated by the authors up to now. Another crucial distinction between the experiment and the those\nproposals is that the experimental involves TRAVELLING detectors, namely space-time volumes where\ncorrelationsaresampledareofpropagatingnature,vs. theFermi’sexperimentinvolvingstationaryatoms.\nthecorrelationsstemmingfromvacuumfluctuationsandsourceradiationtofieldcorrelationsbetweenthe\ntwo near infrared fields emerging from the crystal. This makes it experimental feasible to detect these\ncorrelations with advanced detection schemes from quantum optics, as it is the case in two-beam electro-\noptic sampling. In this sense we agree that our setup is going beyond the two atom setup considered\npreviously and provides an experimental feasible version in which the source radiation and vacuum field\ncontributions are transferred to experimentally accessible field correlations.\nHowever, we still believe that the underlying process that are probed are the same: The exchange of\nsource radiation between the two pulses in electro-optic sampling as well as in Fermi’s two-atom setup\nis governed by the THz field response function (not its two-point correlation function). It is therefore a\nprocess in which source radiation is exchanged. That this is seen as a single photon that is exchanged\nis simply due to the fact that the interaction is weak and lowest-order perturbation theory applies. We\nincluded a discussion of this issue in the new section of the Supplemental Material.\n3. Line88-94: Discussionofthesourceradiationcontribution. Thepi/2shiftisnotexplainedingreatdetail,\nhoweveritisofcrucialimportance. InthecontextoftheFermi’ssetup,thepi/2shiftisnaturallyexpected\narising from on/off resonant contributions by the different influences (vacuum field vs emission), while in\nthe EOS context, which presumably lacks “resonant” excitations, this is less clear. Perhaps this is the\nplace where the analog of the two setups should be explained more.\nWe thank the reviewer for this comment, since it reveals some lack of clarity in our manuscript. There\nare two different pi/2 shifts discussed in the manuscript. In Line 88-94 we refer to a phase shift in the\nnear-infrared. As a result the information about the probed THz field is mapped onto the out-of phase\nquadrature of the NIR field (in case to the in-phase quadrature triggering the source radiation). Those\nquadratures are the analogue to different Pauli-matrices, that become correlated in the two-atom picture.\nOntheotherhand,thetwoexperimentallyobservedcorrelationsarealsophase-shiftedbypi/2(thistimea\nphaseintheTHzrange),whichrelatestoon/offresonantcontributionsmentionedbythereviewer. Similar\nto the two-atom problem, the emission of source radiation experiences a phase-shift of pi/2 (arising from\ntheretardationinthenonlineargenerationprocess), whilewhenusingtheQWPthepulsesprobethereal\npart of the electric THz field.\nTo clarify the role of the NIR quadratures and the related phase-shift in the NIR, we added after ”[...]\ncorrelate different quadratures of the two near-infrared pulses, which allows to individually probe them\nby changing the phase plate in the employed homodyne detection scheme (see Fig. 1 b,c). ”: “This\ndistinctionmirrorsthesituationinFermi’stwo-atomproblem,wheredifferentPaulimatricesarecorrelated\ndepending on whether the correlation originates from vacuum fluctuations or from source radiation [8].”\nIn the discussion about the different phase-relations we slightly reworded and use the terms in-phase and\nout-of phase quadrature consistently to keep the connection.\nFurthermore, when discussing the experimentally observed correlations, we changed ”It predicts a phase\nshift of π between the temporal correlation caused by source radiation and vacuum fluctuations [11, 8],\n2\nwhichisclearlyvisibleintheexperimentaldatainFig.2b”to“Itpredictsaconnectionbetweencorrelation\nstemmingfromvacuumfluctuationsandsourceradiationintime-domainviatheHilbert-transform,which\nisvalidatedintheexperimentaldatainFig.2bthroughthephase-shiftof π betweenthetwocorrelations\n2\n[11, 8].”.\n4. Line 96: Transition to the experimental section is too abrupt. No technical comment here, I was just\nstunned by it and would like to communicate to the team that an improvement is in order.\nWe thank the Reviewer for point this out. The first two sentences have been changed to:\n“For the experimental realization, a similar optical setup as in [6] is utilized and schematically shown\nin Fig.2a. To compensate for dispersion and obtain 110fs-short pulses inside the detection crystal, the\nprobe signal is pre-stretched in a prism compressor before being separated into the two individual probe\nsignals (see Methods). ”\n5. Line 112: The extension of EOS bandwidth to 4.5 THz in a ZnTe crystal is impressive but raises the\nquestion of whether unaccounted-for phase shifts in the EOS response function are introduced due to\n7\nthe proximity of the phase-matching band to the Reststrahlen band, starting with the LO phonon at a\nfrequency of 5.3 THz. The fact that the crystal is held at 4K may perhaps alleviate this issue if the\nphonon (phonon-polariton) lineshape function sharply decreases between the 4.5 THz and 5.3 THz fre-\nquencies; however, this warrants careful justification and discussion, given that the main finding relies on\nphase-shifted (delay-dependent) results. This point is especially exacerbated by the extent of the abscissa\nrange to 5 THz in Fig. 2c.\nIndeed, the new setup accesses THz fields up 5THz and is therefore within the range of the LO phonon.\nOur theoretical analysis takes this into account by modeling the material response via a complex and\ndispersive permittivity in the THz frequency range, compare Eq. (3) of the Supplemental Material and\nsee Ref. [14]. As a result, we find that due to absorption losses while propagating from one pulse to the\nother, the two-beam electro-optic sampling signal vanishes for frequencies bigger than 4THz. This is also\ntrue for the correlations stemming from vacuum-fluctuations. So absorption also limits the spatial range\nof correlations in the quantum vacuum. This means that the bandwidth of our detection is actually not\nlimited by the bandwidth of the laser pulse but by the onset of increasing losses due to the LO phonon\nresonance above ≈4THz.\nToaddressthispointinthemanuscript,wehaveaddedthefollowingsentencetotheChapter”Theoretical\nmodel”oftheSupplementalMaterial: “NotethatduetotheLOphononresonanceat5.3THz,absorption\nlosses start to increase above 4THz, which in turn leads to a vanishing two-beam correlation signal.\nThis is because absorption limits the spatial range of correlations in the quantum vacuum as well as the\npropagation of source radiation from one beam to the other. This means that the bandwidth of our\ndetection is actually not limited by the bandwidth of the laser pulse but by the onset of increasing losses\ndue to the LO phonon resonance above ≈ 4THz, and makes it crucial that our theoretical framework\naccounts for absorption via a complex permittivity. ”\n6. Line 274-278: Similar to the previous comment, the delay between the two signals (QWP vs HWP) is of\ncrucialimportancetotheconclusionsderivedfromthedata. Inthissense,thereshouldbemorediscussion\nallocatedtothefactthatthetime-zeroofthevacuumcorrelationsignalisdeterminedonlyfromanarrow\nfrequency window, approximately between 2-2.5 THz, while the conclusions on the delay of the vacuum\nand source contributions are drawn across a wider frequency span. This is particularly important, as the\nchange of sign between the two signals occurs at a frequency which is nearly centered on the frequency\nwindowanalyzed. Theauthorsshouldgiveduediligenceinensuringthatthechoiceoftheanalysiswindow\ndoes not in fact introduce the aforementioned correlations.\nAs reaction of this comment, we refined spectral window of the phase-fitting to span from 1.55THz to\n2.87THz. This frequency range contains 61% of the signals energy. The uncertainty in the zero-time\ndelay affects the observed spectral correlation mainly within this spectral window. The conclusions from\nthe observed spectral correlations are valid within the given uncertainty range.\n7. Line362-371: Similartotheaboveremark, thedelayofthetwosignalsneedstobeestablishedwithgreat\nprecision. To the great credit of the authors, they provide an honest report of the temporal drifts of\nthe signal during the data acquisition. More work should be done to dispel any doubts about whether\nsystematic drifts in the experiment cause the observed delay. To this extent, acquisition time should be\nreported, perhaps along with statistical measurements of temporal drifts between the two laser beams\nduring such times.\nInspired by the comment of the reviewer, we further investigated potential drifts in time-axis during the\ndataacquisitionandaddedtheanalysistotheSupplementarymaterial. Thecalculatedshiftsinduringthe\nacquisition of the source radiation remains within the expected uncertainty of ±16.7fs (corresponding to\nhalfatime-step),expectofonesignificantjumpof67fsduringthealreadyreportedincident. Weseparate\nthe data into ”before” and ”after” the incident and correct the time-axis accordingly. The conclusions\nfromtheobservedspectralcorrelationtakeintoaccounttheuncertaintyof±16.7fsinthezerotime-delay.\nFor details on the analysis, we refer to the Supplementary Information.\n8. Line 380-389: Similar to the above comment, the interpretation of the correlation structure in the data\nrelies on the separation of the two beams. It seems that the discussion of acquisition times and beam\nangle sensitivities suggests that the beam separation might have drifted during the experiments. If so,\nthis should be clarified more explicitly and perhaps instead of insisting on a single separation value of\n50um, thesimulationsshouldbeaveragedoverthestatisticalsamplingoftheseparations, aspredictedby\nthe drifts of the system during the measurement run. In addition, it would help to report the error\nbars both on the 50um separation and/or on the spot size measurement of the two beams.\nWe thank the Reviewer for this remark. As a reaction, we further analyzed the stability of the beam\n8\ndistance during the acquisition of the vacuum-induced correlation. The zero-crossing of the spectral\ncorrelation serves as an indirect measure of the beam distance and fluctuates in a range corresponding to\n±5µm. Thesefluctuationsmightbecausedbytheweaksignal-to-noiseratioandarenotnecessarilycause\nby ”true” fluctuations of the beam distance. Within the given precision we do not see evidence for a slow\ndrift during the acquisition of several days. More likely the beam distance changed during the exchange\nof the wave plate. For further details, we refer to the corresponding analysis in the Supplementary\nInformation.\nWe changed ”..., the vacuum-induced correlation for the two beams separated by 50µm and measured\nwith a quarter-wave plate...” by “..., the vacuum-induced correlation measured with two beams at a\ntargetdistanceof50µmandwithaquarter-waveplate...”. Furthermoreweaddedtothediscussionofthe\nexperimentally observed spectral correlations: “The slight shift of the zero-crossing of the experimentally\nobserved vacuum-induced correlation compared to the theoretical model might be caused by a slight\nshift of the beam distance towards 30µm for the measurement in a configuration sensitive to vacuum\nfluctuations (details in Supplementary Information Note 2.C).”\nReferences\n[1] A. S. Moskalenko, C. Riek, D. V. Seletskiy, G. Burkard, and A. Leitenstorfer. “Paraxial Theory of Direct\nElectro-optic Sampling of the Quantum Vacuum.” Physical Review Letters, 115(26) (2015).\n[2] C. Riek, D. V. Seletskiy, A. S. Moskalenko, J. F. Schmidt, P. Krauspe, S. Eckart, S. Eggert, G. Burkard,\nand A. Leitenstorfer. “Direct sampling of electric-field vacuum fluctuations.” Science, 350(6259) (2015).\n[3] I. C. Benea-Chelmus, F. F. Settembrini, G. Scalari, and J. Faist. “Electric field correlation measurements\non the electromagnetic vacuum state.” Nature, 568(7751) (2019).\n[4] A. K. Biswas, G. Compagno, G. M. Palma, R. Passante, and F. Persico. “Virtual photons and causality\nin the dynamics of a pair of two-level atoms.” Physical Review A, 42(7) (1990).\n[5] J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji. “Vacuum fluctuations and radiation reaction :\nidentification of their respective contributions.” Journal de Physique, 43(11):1617–1638 (1982).\n[6] F. Settembrini, F. Lindel, A. Herter, S. Buhmann, and J. Faist. “Detection of quantum-vacuum field\ncorrelations outside the light cone.” Nature Communications, 13(1) (2022).\n[7] S. Onoe, T. L. Guedes, A. S. Moskalenko, A. Leitenstorfer, G. Burkard, and T. C. Ralph. “Realizing a\nrapidly switched Unruh-DeWitt detector through electro-optic sampling of the electromagnetic vacuum.”\nPhysical Review D, 105(5):056023 (2022).\n[8] F. Lindel, A. M. Herter, J. Faist, and S. Y. Buhmann. “Probing vacuum field fluctuations and source\nradiation separately in space and time.” Physical Review Research, 5(4):043207 (2023).\n[9] P. Sulzer, K. Oguchi, J. Huster, M. Kizmann, T. L. Guedes, A. Liehl, C. Beckh, A. S. Moskalenko,\nG. Burkard, D. V. Seletskiy, and A. Leitenstorfer. “Determination of the electric field and its Hilbert\ntransform in femtosecond electro-optic sampling.” Physical Review A, 101(3) (2020).\n[10] E. Tjoa and E. Mart´ın-Mart´ınez. “When entanglement harvesting is not really harvesting.” Physical\nReview D, 104(12) (2021).\n[11] N.PottierandA.Mauger. “Quantumfluctuation-dissipationtheorem: atime-domainformulation.” Phys-\nica A: Statistical Mechanics and its Applications, 291(1-4):327–344 (2001).\n[12] F. Lindel, A. Herter, V. Gebhart, J. Faist, and S. Y. Buhmann. “Entanglement harvesting from electro-\nmagnetic quantum fields.” Physical Review A, 110(2):022414 (2024).\n[13] E. Tjoa and E. Mart´ın-Mart´ınez. “Vacuum entanglement harvesting with a zero mode.” Physical Review\nD, 101(12):125020 (2020).\n[14] F. Lindel, R. Bennett, and S. Y. Buhmann. “Theory of polaritonic quantum-vacuum detection.” Physical\nReview A, 102(4):041701 (2020).\nWe would like to thank all reviewers for their positive assessment of our work and their support for its publi-\ncation. Motivated by the question of reviewer 3, we extended and refined the discussion about THz absorption\nin the Methods section (subsection Theoretical model):\n“Note that the influence of the phonon resonance at 5.3THz starts to increase above 4THz, which in turn\nleads to an enhancement of the local (longitudinal) field fluctuations [1], but also to an increase in absorption.\nAsaresultoftheabsorptionandthelongitudinalnatureofthesefluctuations,thespatialrangeofcorrelationsin\nthe quantum vacuum as well as the propagation of source radiation from one beam to the other are delimited,\nand the detected electro-optic correlation in Fig. 2 (c) vanishes in that frequency range. This means that\nthe bandwidth of our detection is actually not limited by the bandwidth of the laser pulse but by the onset of\nincreasing losses due to the phonon resonance above ≈ 4THz, and makes it crucial that our theoretical\nframework accounts for absorption via a complex permittivity.”\nQuestion reviewer 3\nI thank the authors for carefully addressing all of my comments and questions raised during the first round.\nAll of those have been addressed. The exception is the discussion on the phase shift. Something that I missed\nin the original manuscript but was revealed in this discussion is that the phase-matching function extends to 5\nTHz. If so, the onset of the restrahlen-band absorption above 4 THz should be treated with great care. While\nit is correct, as argued by the authors in the rebuttal, that the coherent amplitude of the field is suppressed\ndue to the absorption, the care should be nonethless excertised since of course what matters is not a coherent\nsignal, but fluctuations. Those are not suppressed, but carry associated phase-shifts wrt to the main driving\npulse, causally consistent with the time-domain picture of the process.\nWe thank the reviewer for the positive feedback and this question. Indeed, locally the amplitudes of the\nfluctuations close to the phonon resonance are not suppressed, but even enhanced, as discussed in Ref. [1]. In\nour experiment, the contribution of fluctuations above 4THz to the overall correlation signal is, however, very\nsmall. While the local amplitude of the fluctuations is not suppressed, these fluctuations are short ranged as\ntheyonlycorrelatenearbyspacetimeregions. Physically,thisoriginatesintheirlongitudinalandevanescentna-\nture[1]. Forthesourceradiationcontribution,informationhastopropagatefromoneprobemodetotheother,\nwhich is also suppressed by absorption close to the phonon-resonance. This is also confirmed by our theoretical\nmodel that fully incorporates longitudinal fluctuations due to the phonon resonance as well as absorption. To\naccount for these points, we have rewritten the paragraph below Eq. (7) in the methods section.\nMoreover, the associated phase-shift occurs on the THz signal. The phase relation between the probe and\nthe detected nonlinear field in the NIR is not affected by the THz phase, since the contributions of SFG and\nDFG are balanced in our configuration (in contrast to the experiments in reference [2]).\nThe complex phase of the spectral correlation (THz) is determined by the relative phase difference between\nthe two spatial regions accessed by the probe modes. The key difference between source radiation and vacuum\nfluctuation lies in the symmetric (non-causal) vs. asymmetric (causal) connection between the two spatial re-\ngions. This relation is represented by the connection between the imaginary part of the source radiation and\nthe real part of the vacuum fluctuation signal, which is not affected by the phase of the local field fluctuation\ncaused by the Reststrahlenband.\nReferences\n[1] F. Lindel, R. Bennett, and S. Y. Buhmann. “Theory of polaritonic quantum-vacuum detection.” Physical\nReview A, 102(4):041701 (2020).\n[2] P. Sulzer, K. Oguchi, J. Huster, M. Kizmann, T. L. Guedes, A. Liehl, C. Beckh, A. S. Moskalenko,\nG. Burkard, D. V. Seletskiy, and A. Leitenstorfer. “Determination of the electric field and its Hilbert\ntransform in femtosecond electro-optic sampling.” Physical Review A, 101(3) (2020)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The revisions have significantly improved the manuscript, and all my previous concerns have been properly addressed. I\nnow recommend the revised manuscript for publication.", "reviewer_2": "I have carefully examined the answers of the authors to my comments as well as to the comments of other reviewers. I think\nthat the authors did a very good job, providing quite convincing answers and additional explanations for the most of the\nraised issues. In my particular case, their answer to my question 7 in the rebuttal letter helped me to understand better the\nnature of their measured signal: I thought that technically the source term comes from correlations of the Hilbert transforms of\nthe THz vacuum electrical field, whereas it actually originates from a corresponding cascaded term containing the THz\nvacuum only as a mediator. This looks indeed right. In particular, that signal vanishes at zero delay as it follows from\nprevious existing works.\nThe authors have also produced calculations I asked for as well as measurements requested by other reviewers, confirming\ntheir previous results and conclusions. I think that the overall revision of the manuscript can be seen as satisfactory and the\npaper can be now accepted for publication.", "reviewer_3": "I thank the authors for carefully addressing all of my comments and questions raised during the first round. All of those have\nbeen addressed. The exception is the discussion on the phase shift. Something that I missed in the original manuscript but\nwas revealed in this discussion is that the phase-matching function extends to 5 THz. If so, the onset of the restrahlen-band\nabsorption above 4 THz should be treated with great care. While it is correct, as argued by the authors in the rebuttal, that\nthe coherent amplitude of the field is suppressed due to the absorption, the care should be nonethless excertised since of\ncourse what matters is not a coherent signal, but fluctuations. Those are not suppressed, but carry associated phase-shifts\nwrt to the main driving pulse, causally consistent with the time-domain picture of the process.\nRegardless, however, I believe that the interpretation, measurements and overall experimental rigor are of sufficiently high\nquality in this work to be considered for publication. I once more congratulate the authors on this achievement. If they would\nlike to address this last point in a modified version of the manuscript/supplementary, I would not need to see it, as my\ndecision \"to publish\" will not be affected by their response, either way."}, "author_response": ""}]

Decision letter