## Probing the fluctuating vacuum

According to quantum mechanics, a vacuum is not empty space. A consequence of the uncertainly principle is that particles or energy can come into existence for a fleeting moment. Such vacuum or quantum fluctuations are known to exist, but evidence for them has been indirect. Riek *et al.* present an ultrafast optical based technique that probes the vacuum fluctuation of electromagnetic radiation directly.

*Science*, this issue p. 420

## Abstract

The ground state of quantum systems is characterized by zero-point motion. This motion, in the form of vacuum fluctuations, is generally considered to be an elusive phenomenon that manifests itself only indirectly. Here, we report direct detection of the vacuum fluctuations of electromagnetic radiation in free space. The ground-state electric-field variance is inversely proportional to the four-dimensional space-time volume, which we sampled electro-optically with tightly focused laser pulses lasting a few femtoseconds. Subcycle temporal readout and nonlinear coupling far from resonance provide signals from purely virtual photons without amplification. Our findings enable an extreme time-domain approach to quantum physics, with nondestructive access to the quantum state of light. Operating at multiterahertz frequencies, such techniques might also allow time-resolved studies of intrinsic fluctuations of elementary excitations in condensed matter.

Vacuum fluctuations give rise to a variety of phenomena, from spontaneous photon emission (*1*, *2*) and the Lamb shift (*3*) via the Casimir force (*4*) to cosmological perturbations (*5*, *6*). Representing the ground state, the quantum vacuum does not possess intensity. However, finite noise amplitudes of electric and magnetic fields should exist because of Heisenberg’s uncertainty principle. These fluctuations are best explained by analogy with a harmonic oscillator of mass *m*, resonance angular frequency Ω, and total energy

Quantization results in noncommuting operators for momentum *p* and displacement *x*. The Gaussian wave function of the ground state exhibits a root-mean-square (RMS) standard deviation of Δ*x* = (*ħ*/2Ω*m*)^{1/2} (*7*, *8*), where *ħ *is the reduced Planck constant. The total energy of a radiation field of wavevector **k** in free space, with electric and magnetic amplitudes *E* and *B* (respectively), and vector potential **A** in the Coulomb gauge is (*9*)

Considering one polarization direction and the transverse character of electromagnetic waves, Eq. 1 maps onto Eq. 2 by replacing *x* with *A* (amplitude of vector potential **A**), *m* with ε_{0}*V *(ε_{0}, vacuum permittivity; V, spatial volume), and Ω with *ck* ≡ Ω (*c*, speed of light; *k*= |**k**k|). Instead of *x* and *p*, an uncertainty product now links *E* and *B* or the amplitudes and phases of *E*, *B*, or *A*. An RMS amplitude of vacuum fluctuations Δ*A* = (*ħ*/2Ωε_{0}*V*)^{1/2} results. In contrast to the mechanical case where Δ*x* is known, understanding Δ*A* is less straightforward: Outside any cavities, there are no obvious boundaries that define a normalization volume *V*. This situation raises the question of whether direct measurement of the vacuum field amplitude in free space is physically meaningful and feasible.

The quantum properties of light (*10*) are typically analyzed either by photon correlation (*11*–*14*), homodyning (*15*–*18*), or hybrid measurements (*19*). In those approaches, information is averaged over multiple cycles, and accessing the vacuum state requires amplification. Femtosecond studies still rely on pulse envelopes that vary slowly relative to the carrier frequency (*20*–*23*). In our work, we directly probed the vacuum noise of the electric field on a subcycle time scale using laser pulses lasting a few femtoseconds. In ultrabroadband electro-optic sampling (*24*–*27*), a horizontally polarized electric-field waveform (red in Fig. 1A) propagates through an electro-optic crystal (EOX), inducing a change Δ*n* of the linear refractive index *n*_{0} that is proportional to its local amplitude *E*_{THz} (Fig. 1A and fig. S1). The geometry is adjusted so that a new index ellipsoid emerges under 45° to the polarization of *E*_{THz,} with *n _{y}*

_{'}and

*n*

_{x}_{'}=

*n*

_{0}± Δ

*n*. An ultrashort optical probe pulse at a much higher carrier frequency ν

_{p}(green in Fig. 1A; intensity,

*I*

_{p}; electric field,

*E*

_{p}) copropagates with

*E*

_{THz}at a variable delay time

*t*

_{d}. The envelope of

*I*

_{p}has to be on the order of half a cycle of light at the highest frequencies Ω/2π of

*E*

_{THz}that are detected. We used probe pulses as short as

*t*

_{p}= 5.8 fs, corresponding to less than 1.5 optical cycles at ν

_{p}= 255 THz (fig. S2). Upon passage through the EOX, the

*x*' and

*y*' components of

*E*

_{p}acquire a relative phase delay proportional to Δ

*n*and

*E*

_{THz}(

*t*

_{d}). The final polarization state of the probe is analyzed with ellipsometry. The differential photocurrent Δ

*I*/

*I*is proportional to the electric field

*E*

_{THz}(

*t*

_{d}). We used a radio-frequency lock-in amplifier (RFLA) for readout.

We adjusted for optimal conditions to measure the vacuum signal by studying classical multiterahertz transients, which were synchronized to the probe (*8*). In Fig. 1B, Δ *I*/*I* is plotted in red against delay time *t*_{d}. Figure 1C shows the amplitude spectrum (red) and phase deviations (blue) within ±π, corroborating calculations (*8*) of an effective sampling bandwidth of Δν = ΔΩ/2π = 66 THz (figs. S3 and S4) around a center frequency of ν_{c} = Ω_{c}/2π = 67.5 THz (free-space wavelength λ_{c} = 4.4 μm). The electric-field amplitude *Ē*_{THz}(*t*_{d}) is calibrated using (*28*–*30*)

*r*_{41} denotes the electro-optic coefficient, and *l* is the thickness of the EOX. The amplitude response |*R*(Ω_{c})| includes the pulse duration of the probe and velocity matching to the multiterahertz phase (*8*). The classical field transient in Fig. 1B was sampled with a signal-to-noise ratio better than 10^{3} at a RFLA detection bandwidth set to 94 Hz. From the confocal amplitude trace and cross section, we estimated a mean photon number below 900 per pulse. This result proves the capability of our approach to characterize ultrabroadband coherent wave packets containing less than 10^{−3} photons, on average, within 1 s.

But can we directly access the ground state Φ_{0} of the radiation field? With the pump branch switched off, electro-optic phase shifts might still be caused by vacuum fluctuations copropagating with the probe. This effect should lead to a statistical distribution of the signal around the average of *<Ē*_{vac}*>* = 0. The ground-state expectation value of the squared operator for the electric field in free space (*31*) yields the RMS amplitude

*â*_{Ω} and *â*_{Ω}^{†} are the operators for annihilation and creation of a photon with angular frequency Ω, respectively (ν, frequency; *h*, Planck constant). Because of the commutation relation [*â*_{Ω},*â*_{Ω}^{†}] = 1, only *â*_{Ω}*â*_{Ω}^{†} provides a nonvanishing contribution. Summing frequencies over our finite sensitivity interval ensures convergence of Eq. 4. The lateral extension of the volume *V* is now identified with the effective cross section *A*_{eff}, defined by the Gaussian intensity profile of the near-infrared probe beam inside the EOX. Theoretical modeling based on Laguerre-Gaussian modes (*30*) yields *A*_{eff} = *w*_{0}^{2}π, where *w*_{0} is the probe spot radius (*8*). Because *V* = *A*_{eff} *L,* only the length *L* remains to be determined. Periodic boundary conditions are applicable when the EOX is short relative to the Rayleigh range of the multiterahertz transverse mode, resulting in a density of free-space modes *L*/*c*. Summing over all longitudinal modes within a bandwidth of Δν eliminates *L*, and we obtain

A factor of *n*_{0}^{–1/2} accounts for dielectric screening inside the EOX (*8*). Thus, the vacuum amplitude is maximized when averaging over a minimal space-time volume, determined in the transverse directions by *w*_{0} = 4.25 μm (fig. S5). The longitudinal cross section *cn*_{0}/(ν_{c}Δν) is defined by the Fourier transform of *R*(Ω), containing the intensity envelope of the 5.8-fs probe pulse and phase-matching conditions within the EOX (*8*).

Are such fluctuations discernible on top of the shot noise due to the Poissonian photon statistics of the coherent probe? An average number of *N*_{p} = 5 × 10^{8} photons detected per pulse causes a relative RMS shot-noise current of Δ*I*_{SN}/*I* = *N*_{p}^{–1/2}. With Eq. 3, we obtain the noise-equivalent field

Because the shot noise of the near-infrared probe, which is centered around ν_{p}, and the vacuum fluctuations at multiterahertz frequencies Ω are uncorrelated with each other and lack spectral overlap, the two contributions add up in quadrature. Therefore, the RMS width of the total detected noise distribution is expected to rise by a factor of (7)corresponding to a 4.7% increase, due to the multiterahertz vacuum noise.

To experimentally access the statistics of the quantum vacuum, we extended the RFLA bandwidth to 1.6 MHz and sampled the probability distribution of the electric field *P*(*E*_{total}) every 5 μs. The contribution of the multiterahertz vacuum can be modified to discriminate against the shot-noise baseline by longitudinal or transverse expansion of the probed space-time volume (Eq. 5). In the first approach, we decreased ν_{c} and Δν by chirping the probe pulse to 100 fs (fig. S3), via translation of an SF10 prism in the compressor stage (Fig. 2A). A distinct reduction in peak counts around *P*(*E*_{total} = 0) is observed when comparing the probability distribution obtained with the 5.8-fs probe (green in Fig. 2B) to the measurement with a stretched pulse (black). Also, the probabilities in the wings of the distribution including the multiterahertz vacuum (5.8-fs probe) are consistently higher than the corresponding values in the stretched-pulse distribution. The total change of the normalized noise amplitude amounts to 4%, in good agreement with the theoretical considerations underlying Eqs. 5 to 7. The red histogram in Fig. 2B emerges from a deconvolution algorithm that searches for the best link between distributions of *P*(*E*_{total}) obtained with and without vacuum noise. This result directly mirrors the ground-state wave function |Ψ_{0}(*E*)|^{2} of the electromagnetic field in the polarization plane and space-time volume that we probed. From |Ψ_{0}(*E*)|^{2}, a RMS standard deviation of Δ*Ē*_{vac} = 18 V/cm is obtained, in good agreement with the theoretical prediction of 20.2 V/cm in Eq. 5.

In the transverse option, we kept the short pulse duration and expanded the probe radius *w*_{0} by translating the EOX out of the confocal plane (Fig. 3A). Averaging over a larger cross section causes a decrease in fluctuation amplitude, which is projected onto the transverse mode of the gate. The effect of progressive narrowing is emphasized with differential probabilities obtained by subtracting a distribution at *w*_{0} = 4.25 μm from *P*(*E*_{total}) sampled at increasing spot radii (Fig. 3B). When all original histograms are normalized, the maximum change in probability Δ*P*(*E*_{total} = 0) of 0.04 ≡ 4% directly corresponds to the difference between the relative noise amplitudes measured with and without multiterahertz vacuum fluctuations, in quantitative agreement with Eq. 7. The dependence of the vacuum RMS amplitude on the transverse extension of the probed space-time volume is shown in Fig. 4. The normalized increase of total noise, measured with respect to bare shot noise (right vertical axis), is plotted against the probe spot radius *w*_{0} (blue squares). Conversion to the vacuum electric amplitude Δ*E*_{vac} (left vertical axis) has been carried out analogously to |Ψ_{0}(*E*)|^{2} in Fig. 2B. The functional dependence expected from Eq. 5 is shown as a red line. The inset in Fig. 4 illustrates the data recorded at low beam cross sections on a linear scale to highlight the hyperbolic increase of vacuum fluctuations for the smaller space-time volumes that we probed.

In our study, we directly monitored vacuum fluctuations without amplifying them. The only effective part ∑_{Ω>0} *â*_{Ω}*â*_{Ω}^{†} of the operator that extracts the variance of the field in Eq. 4 indicates that vacuum fluctuations correspond to photons, which spontaneously arise and vanish in the ground state Φ_{0}. Time-energy uncertainty demands that virtual excitations have a limited life-time on the order of their oscillation cycle (*32*). The subcycle temporal resolution provided by the ultrashort probe ensures that we can directly detect effects originating from purely virtual photons. Phase-matched copropagation of the vacuum field and probe inside the EOX maximizes those signals. But does this measurement influence the quantum vacuum at all? Based on the electro-optic change of the refractive index Δ*n*_{p}~*r*_{41}*E*_{THz}, the local multiterahertz field imprints a phase shift onto the ultrashort probe, which we detected. Because sum- and difference-frequency mixing occur simultaneously in this process (*29*), it requires no net transfer of energy, momentum, or angular momentum, and it even avoids modulation of the refractive index at frequencies Ω/2π << ν_{p}. Our second-order nonlinear element operates far from resonance. Virtual driving of the transitions avoids problems with decoherence, distinguishing our experiment from detection approaches in quantum optics or circuit quantum electrodynamics in which resonant two-level systems are involved (*33*). In consequence, our approach may be used to study the multiterahertz ground state while imposing negligible influence on it. Back-action might arise only in third order: The nonlinear refractive index *n*_{2} generates a local anomaly of phase velocity copropagating with the intensity envelope of the probe, because Δ*n*_{Ω} ~ *n*_{2}|*E*_{p}|^{2}. When *N*_{p}/*w*_{0}^{2}*t*_{p} suffices to induce phase shifts of the multiterahertz field during passage through the EOX, depletion of the vacuum amplitude in the sampled space-time volume and enhanced fluctuations in an adjacent interval are expected.

## Supplementary Materials

## REFERENCES AND NOTES

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**Acknowledgments:**Support by the European Research Council (Advanced Grant 290876 “UltraPhase”), by Deutsche Forschungsgemeinschaft (SFB767), and by NSF via a postdoctoral fellowship for D.V.S. (award no. 1160764) is gratefully acknowledged.