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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 due to Heisen-berg’s uncertainty principle. These fluctuations are ex-plained best in analogy with a harmonic oscillator of mass m, resonance frequency Ω and total energy

Quantization results in non-commuting operators for momentum p and displacement x. The Gaussian wave func-tion of the ground state exhibits a root-mean-square (rms) standard deviation of Δx = (ħ/2Ωm)1/2 (7, 8). The total energyof a radiation field of wave vector k in free space with elec-tric and magnetic amplitudes E and B, respectively, and vec-tor potential A in the Coulomb gauge is (9)

Considering one polarization direction and the trans-verse character of electromagnetic waves, Eq. 1 maps onto Eq. 2 by replacing x → A, m → ε0V and Ω → ck ≡ Ω. Instead ofx 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 ΔA = (ħ/2Ωε0V)1/2 results. In contrast tothe mechanical case where Δx is known, understanding ΔA

is less clear: outside any cavities, there are no obvious boundaries defining a normalization volume V. This situation raises the ques-tion whether direct measure-ment of the vacuum field amplitude in free space is physi-cally meaningful and feasible.

The quantum properties of light (10) are typically analyzed either by photon correlation (11–14), homodyning (15–18) or hy-brid measurements (19). In those approaches information is aver-aged over multiple cycles and accessing the vacuum state re-quires amplification. Femto-second studies still rely on pulse envelopes that are varying slowly as compared to the carrier fre-quency (20–23). Here, we direct-ly probe the vacuum noise of the electric field on a sub-cycle time-scale using few-femtosecond la-ser pulses. In ultrabroadband electro-optic sampling (24–27) a horizontally polarized electric-

field waveform (red) propagates through an electro-optic crystal (EOX) inducing a change Δn of the linear refractiveindex n0 proportional to its local amplitude ETHz (Fig. 1A and fig. S1). The geometry is adjusted such that a new index el-lipsoid emerges under 45° to the polarization of ETHz with ny’ and nx’ = n0 ± Δn. An ultrashort optical probe pulse at muchhigher carrier frequency νp (green, intensity Ip, electric field Ep) co-propagates with ETHz at a variable delay time td. The envelope of Ip has to be in the order of half a cycle of light at the highest frequencies Ω/2π of ETHz that are detected. Weemploy probe pulses as short as tp = 5.8 fs, corresponding to less than 1.5 optical cycles at νp = 255 THz (fig. S2). Upon passage through the EOX, the x’ and y’ components of Ep acquire a relative phase delay proportional to Δn andETHz(td). The final polarization state of the probe is analyzed with ellipsometry. The differential photocurrent ΔI/I is pro-portional to the electric field ETHz(td). A radio-frequency lock-in amplifier (RFLA) serves for readout.

We adjust for optimal conditions to measure the vacuum signal by studying classical multi-THz transients which are synchronized to the probe (8). In Fig. 1B, ΔI/I is depicted inred versus delay time td. Figure 1C shows the amplitude spectrum (red line) and phase deviations (blue) below ±π, corroborating calculations (8) of an effective sampling bandwidth of Δν = ΔΩ/2π = 66 THz (figs. S3 and S4) arounda center frequency of νc = Ωc/2π = 67.5 THz (free-spacewavelength λc = 4.4 μm). The electric-field amplitude

Direct sampling of electric-field vacuum fluctuations C. Riek, D. V. Seletskiy, A. S. Moskalenko, J. F. Schmidt, P. Krauspe, S. Eckart, S. Eggert, G. Burkard, A. Leitenstorfer*

Department of Physics and Center for Applied Photonics, University of Konstanz, D-78457 Konstanz, Germany.

*Corresponding author. E-mail: alfred.leitenstorfer@uni-konstanz.de

The ground state of quantum systems is characterized by zero-point motion. Those vacuum fluctuations are generally deemed 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 found to be inversely proportional to the four-dimensional space-time volume sampled electro-optically with tightly focused few-femtosecond laser pulses. Sub-cycle 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 multi-terahertz frequencies, such techniques might also allow time-resolved studies of intrinsic fluctuations of elementary excitations in condensed matter.

( )2

2 2 2 2 21 (1)2 2HO

p mH m x x xm

= + Ω = +Ω

( ) ( )22 2 2 2 20 0 (2)2 2RFV VH E c B A cε ε

= + = + ×k A

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ĒTHz(td) is calibrated using (28–30) 3

41 0

341 0

2 ( )sin

2 ( )(3)

p cTHz

p cTHz

r n l RI EI c

r n l RE

c

π ν

π ν

Ω∆=

Ω

≈

r41 denotes the electro-optic coefficient and l the thickness of the EOX. The amplitude response \R(Ωc)\ includes thepulse duration of the probe and velocity matching to the multi-THz phase (8). The classical field transient in Fig. 1B is sampled with signal-to-noise ratio better than 103 at a detection bandwidth of the RFLA set to 94 Hz. From the confocal amplitude trace and cross section we estimate a mean photon number below 900. This result proves our ca-pability of characterizing ultrabroadband coherent wave packets containing less than 10−3 photons on average within one second.

But can we directly access the ground state Φ0 of the ra-diation field? With the pump branch switched off, electro-optic phase shifts might still be caused by vacuum fluctua-tions co-propagating with the probe. This effect should lead to a statistical distribution of the signal about the average of <Ēvac> = 0. The ground-state expectation value of the squared operator for the electric field in free space (31) yields the rms amplitude

âΩ and âΩ† are the operators for annihilation and creation of

a photon with frequency Ω, respectively. Owing to the com-mutation relation [âΩ,âΩ

†] = 1, solely âΩâΩ† provides a non-

vanishing contribution. Summing frequencies over our fi-nite sensitivity interval ensures convergence of Eq. 4. The lateral extension of the volume V is now identified with the effective cross section Aeff defined by the Gaussian intensity profile of the NIR probe beam inside the EOX. Theoretical modeling based on Laguerre-Gaussian modes (30) yields Aeff = w0

2π where w0 is the probe spot radius (8). Since V = Aeff L, only the length L remains to be determined. Periodic boundary conditions are applicable when the EOX is short compared to the Rayleigh range of the multi-THz transverse mode, resulting in a density of free-space modes L/c. Sum-ming over all longitudinal modes within a bandwidth of Δνeliminates L and we obtain

A factor of n0-1/2 accounts for dielectric screening inside

the EOX (8). Thus, the vacuum amplitude is maximized when averaging over a minimal space-time volume deter-mined in the transverse directions by w0 = 4.25 μm (fig. S5).The longitudinal cross section cn0/(νc Δν) is defined by theFourier transform of R(Ω) containing the intensity envelopeof 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 Np = 5×108 photons detected per pulse causes a relative rms shot-noise current of ΔISN/I =Np

-1/2. With Eq. 3, we obtain the noise-equivalent field

Because the shot noise of the NIR probe centered around νp and the vacuum fluctuations at multi-THz frequencies Ωare 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 ex-pected to rise by a factor of

corresponding to a 4.7% increase due to the multi-THz vac-uum noise.

To experimentally access statistics of the quantum vacu-um, we extend the RFLA bandwidth to 1.6 MHz and sample the probability distribution of the electric field p(Etotal) every 5 μs. The contribution of the multi-THz vacuum can bemodified to discriminate against the shot-noise baseline by longitudinal or transverse expansion of the probed space-time volume, see Eq. 5. In the first approach, we decrease νc and Δν by chirping the probe pulse to 100 fs (fig. S3) viatranslating an SF10 prism in the compressor stage (see Fig. 2A). A distinct reduction in peak counts around p(Etotal = 0) is observed when comparing the probability distribution obtained with the 5.8-fs probe (green line in Fig. 2B) to the measurement with a stretched pulse (black). Also, the prob-abilities in the wings of the distribution including the multi-THz vacuum (green) are consistently higher than the corre-sponding values in the black histogram. 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 de-convolution algorithm searching for the best link between distributions p(Etotal) obtained with and without vacuum noise. This result directly mirrors the ground-state wave function |Ψ0(E)|2 of the electromagnetic field in the polariza-tion plane and space-time volume which we probe. From |Ψ0(E)|2, an rms standard deviation of ΔĒvac = 18 V/cm is ob-tained in good coincidence with the theoretical prediction of 20.2 V/cm in Eq. 5.

341 0

V65.0 (6)cm2 ( )SN

p p c

cEr n l N Rπ ν

∆ = =Ω

20 0 0

V20.2 (5)2 cm

cvac

hEc n wν ν

π ε∆

∆ = =

( )

( )

/2 2†0 0

/2 0

/22 † † †2

0 0/2 0

/2

/2 0

ˆ ˆ2

ˆ ˆ ˆ ˆ ˆ ˆ2

(4)2

c

c

c

c

c

c

vacE a aV

a a a a a aV

hV

ν ν

ν ν ν

ε

ωε

νε

Ω +∆Ω

Ω ΩΩ=Ω −∆Ω

Ω +∆Ω

Ω Ω Ω Ω Ω ΩΩ=Ω −∆Ω

+∆

= −∆

− Ω∆ = Φ − Φ

= Φ − + + − Φ

=

∑

∑

∑

)7(047.122

=∆

∆+∆=

∆∆

SN

vacSN

SN

total

EEE

EE

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In the transverse option, we keep the short pulse dura-tion and expand the probe radius w0 by translating the EOX out of the confocal plane (Fig. 3A). Averaging over a larger cross section causes a decrease of fluctuation amplitude pro-jected onto the transverse mode of the gate. The effect of progressive narrowing is emphasized with differential prob-abilities obtained by subtracting a histogram at w0 = 4.25 μm from p(Etotal) sampled at increasing spot radii (Fig. 3B).Due to normalization of all original histograms, the maxi-mum change of probability Δp(Etotal = 0) of 0.04 ≡ 4% direct-ly corresponds to the difference of the relative noise amplitude measured with and without multi-THz vacuum fluctuations, in quantitative agreement with Eq. 7. The de-pendence of the vacuum rms amplitude on the transverse extension of the probed space-time volume is studied in Fig. 4. Here, the normalized increase of total noise measuredwith respect to bare shot noise (right vertical axis) is depict-ed versus probe spot radius w0 (blue squares). Conversion to the vacuum electric amplitude ΔEvac (left vertical axis) hasbeen carried out in analogy to |Ψ0(E)|2 in Fig. 2B. The func-tional dependence expected from Eq. 5 is depicted as a red line. The inset of Fig. 4 shows the data recorded at low beam cross sections on a linear scale to highlight the hyper-bolic increase of vacuum fluctuations for the smaller space-time volumes we probe.

In discussing some fundamental aspects of our study first note that we directly monitor vacuum fluctuations without amplifying them. The only effective part ∑Ω>0 âΩâΩ

†

of the operator extracting the variance of the field in Eq. 4 indicates that vacuum fluctuations correspond to photons which spontaneously arise and vanish in the ground state Φ0. Time-energy uncertainty demands that virtual excita-tions have a limited lifetime in the order of their oscillation cycle (32). The sub-cycle temporal resolution provided by the ultrashort probe ensures that we can directly detect ef-fects originating from purely virtual photons. Phase-matched co-propagation of vacuum field and probe inside the EOX maximizes those signals. But does this measure-ment influence the quantum vacuum at all? Based on the electro-optic change of refractive index Δnp ~ r41ETHz, thelocal multi-THz field imprints a phase shift onto the ultra-short probe we detect. Since sum- and difference-frequency mixing occur simultaneously in this process (29), it requires no net transfer of energy, momentum or angular momen-tum and even avoids modulation of the refractive index at frequencies Ω/2π << νp. Our second-order nonlinear ele-ment operates far from resonance. Virtual driving of the transitions avoids problems with decoherence, distinguish-ing our experiment from detection approaches in quantum optics or cavity quantum electrodynamics where resonant two-level systems are involved (33). In consequence, we may study the multi-THz ground state while imposing negligible influence on it. Back action might arise only in third order: the nonlinear refractive index n2 generates a local anomaly of phase velocity co-propagating with the intensity envelope

of the probe since ΔnΩ ~ n2|Ep|2. When Np/w0

2tp suffices toinduce phase shifts of the multi-THz 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.

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ACKNOWLEDGMENTS

Support by ERC Advanced Grant 290876 “UltraPhase”, by DFG via SFB767 and by NSF via a Postdoc Fellowship for D.V.S. (Award No. 1160764) is gratefully acknowledged.

SUPPLEMENTARY MATERIALS www.sciencemag.org/cgi/content/full/science.aac9788/DC1 Materials and Methods Figs. S1 to S5 References (34–40)

8 July 2015; accepted 16 September 2015 Published online 1 October 2015 10.1126/science.aac9788

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Fig. 1. Experimental principle with demonstration of ultrahigh bandwidth and sensitivity. (A) Scheme of electro-optic sampling of an electric-field waveform (red) by an ultrafast probe pulse (green), consisting of an electro-optic crystal (EOX), a quarter-wave plate (λ/4), Wollaston polarizer (WP) and differentialphotocurrent detector (DD). (B) Classical electro-optic signal ΔI/I and corresponding electric-field amplitude versus delay time td (red line). The intensity envelope of the 5.8-fs probe pulse is shown in arbitrary units for comparison (green line). (C) Spectral multi-THz amplitude (red) and phase (blue) obtained by Fourier transform.

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Fig. 2. Studying vacuum fluctuations via statistic readout and longitudinal modification of the probed space-time volume. (A) A diagram shows longitudinal expansion of the probe volume: stretching the sampling pulse from 5.8 fs (green) to 100 fs (black) causes temporal averaging over the vacuum field (red), leading to reduction of the detected noise amplitude. (B) Green histogram: normalized counting probability as a function of electro-optic readout by the short pulse. Black histogram: probe pulse stretched to 100 fs. Red graph: de-convoluted wave function |Ψ0|2 of the electric field ground state.

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Fig. 3. Detection by transverse expansion of the space-time segment. (A) Sketch featuring lateral increase of sampling cross section which leads to averaging over noise patterns within circled areas. (B) Differential histograms obtained by subtracting the result for confocal detector position with w0 = 4.25 μm from a set of histograms at positions with beam diameter expanding from 4.25 μm (black) to 17 μm (purple), 25 μm (cyan), 50 μm (orange) and 85 μm (red).

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Fig. 4. Dependence of vacuum amplitude on transverse extension of probed space-time volume. Relative excess noise of electro-optic signal ΔI/I (right vertical axis) and rms vacuum amplitude ΔEvac (left vertical axis) versus probe radius w0 (blue squares). Red lines: theoretical assessment based on Eq. 5.

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Direct sampling of electric-field vacuum fluctuationsC. Riek, D. V. Seletskiy, A. S. Moskalenko, J. F. Schmidt, P. Krauspe, S. Eckart, S. Eggert, G. Burkard and A. Leitenstorfer

published online October 1, 2015

ARTICLE TOOLS http://science.sciencemag.org/content/early/2015/09/30/science.aac9788

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2015/09/30/science.aac9788.DC1

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