Direct sampling of electric-field vacuum fluctuations

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. Leitenstnrfer*

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.

V

acuum fluctuations give rise to a variety of phenomena, from spontaneous pbotnn emission (1, 2) and the Lamb shift (3) via the Casimir force (4) tn oosmological per turbations (5, 6). Representing the ground state, the quantum vacuum does not possess in tensity. However, finite noise amplitudes of elec tric and magnetic fields should exist because of Heisenberg's uncertainty principle. These fiuctu ations are best explained by analogy with a bar monic oscillatnr of mass m, resonance angular frequency

n,

and tntal energy

Q.uantization results in noncommuting oper atnrs fur momentum p and displacement

x.

The Gaussian wave function of the ground state exhib its a root mean square (RMS) standard deviation of M1 = (fl/2n:m)112 (7, B), where his the reduced Planck constant. The tntal energy of a radiation field of wavevector k in free space, with electric

Department of Physics and Center for Applied Photonics.

Uniwrsity of Konstanz, D 78457 Konstanz, Germany.

*Corresponding author. E mall: aHred.leltenstorfeo@

unllonstanz.de

420

and magnetic amplitudes E and B (respectively), and vector potential A in the Coulomb ~uge is (9)

{2)

Considering one polarization direction and the transverse character of electrcmagnetic waves, Eq. 1 maps ontn Eq. 2 by replacing x with A (amplitudeofvectorpotential A), m with £₀V(€o.

vacuum permittivity;

v;

spatial volume), and

n

with ck

= n

(c, speed of light; k = lkl). 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 varuum fluctuations M = (h/2ne0V)¹¹² results. In contrast tn the mecban ical case where M1 is known, understanding M is Jess 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 typi calJy analyzed either by phcton oorrelation (11 14), bomodyning (15 18), or hybrid measurements (19). In those approaches, information is averaged over multiple cycles, and aocessing the vacuum state requires amplification. Femtnsecond studies

still rely on pulse envelopes that vary slowly relative tn the carrier frequency (20 23). In our work, we directly probed the varuum noise of the electric field on a subcycle time scale using laser pulses lasting a few femtnseoonds. In ultrabroad band electro optic sampling (24 27), a horizon tally polarized electric field waveform (red in Fig.

lA) propagates through an electro optic crystal (EOX), inducing a change Lln of the linear re fractive index 11.o that is proportional to its local amplitude

Em:.

(Fig. lA and fig. SI). The geometty is adjusted so that a new index ellipsoid emerges under46°tothe polarization of

Ern.,

with

nv

and

nr

^{= 11.o :1::} !!:.n. An ultrashort optical probe pulse at a much higher carrier frequency vp (green in Fig.

1A; intensity, I p, electric field, E.J coprop~

with Em~ at a variable delay time td. The envelope

·of!Pbastn be on theorderofhalfacycle oflightat the highest frequencies il/2rt of En~ that are detected. We used probe pulses as short as tp = 5.8 fs, oorresponding tn Jess than L5 optical cycles at vp = 255 1Hz (fig. 82). Upon passage through the EOX, the a! andy' components of Ep acquire a relative phase delay proportional to Lln and Eml.,td). The final polarizatim state of the probe is analyzed with ellipsometry. The differential photn rurrent 111/I is proportional tn the electric field Eml.,t,V. We used a radio frequency lock in ampli tier (R.FLA) for readout.

We a<ljusted for optimal conditions tn measure the vacuum signal by studying classical multi

·terabertz transients, which were synchronized tn the probe (8). In Fig. lB, M/I is plotted in red against delay time f.!. Figure lC shows the am plitude spectrum (red) and phase deviations (blue) within ±It, corroborating calculations (8) of an effective sampling bandwidth of Ll v = till./2rt

= 66 THz (figs. 83 and 84) around a center frequency of vc : flc/2rt = 67.5 THz (free space wavelength Ac = 4.4 J.tlll). The electric field am plitude itmz(td) is calibrated using (28 30)

M I

r 41 denotes the electro optic coefficient, and lis thethidmess of the FDX. The amplitude response

IR(OJI

includes the pulse duration of the probe and velocity matching tn the multiterabertz phase (8). The classical field transient in Fig. lB was

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-301750

Erschienen in: Science ; 350 (2015), 6259. - S. 420-423 https://dx.doi.org/10.1126/science.aac9788

sampled with a signal to noise ratio better than 1<fl at a RFIA detection bandwidth set to 94 Hz.

From the confocal amplitude trace and cross sec tion, we estimated a mean phota:t nUIIJber below 900 per pW;e. This result proves the capabilizy of our approach to characterize uhrabroadband ro herent wave packets containing Jess than 10-³ photons, on average, within 1 s.

But can we directly access the ground state

<1>₀ of the radiation field? Wrth the pump branch

switched off, electro optic phase shifts might still be caused by vacuum fluctuations copropa gating with the probe. This effect should lead to a statistical distribution of the signal around the average of

<Ewe>

⁼ 0. The ground state expectation value ofthe squared operator for the

X 2

<i

(ij c:

A

l, n,=n0-An n,.=n0+An

~ OF-Pr+H+H-¥1

0

'E.

^-1

i

_Q) ^-2

a; ~~~~~~~~~~~~~~ 60

-40 -20 0 20 40

delay time f.! (Is)

1 0 0.5

e

~

g

00 ~

""

0

-0.5~

a;

-1.0

ele<tric field in free space (31) yields the RMS amplitude

do

and ~ are the operators for annihilation and creation of a photon with angular frequency Q, respectively ( v, frequency; h, Pland< ronstant).

c

1.0 3

Q) 2

~0.8 ₁

Ci.

:u

~ 0.6 0 ~

-g

^-1

i

~ 0.4 ^-2~

(I) a.

§o2 -3

c: -4

0.~0 _{50 60 70 80 90 100 110 120} ^-5 frequency (THz)

Fig. 1. Our experimental principle, with a demonstration of ultrahigh bandwidth and sensitivity.

(A) Scheme of electro-optic sampling of an electric-field wavefomn (red) by an ultrafast probe pulse (green).

consisting of an EOX. a quarter-wave plate (A/4). a Wollaston polarizer (WP). and a differential photocurrent detector (DO). (B) Classical electro-optic signal 61/1 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 multiterahertz amplitude (red) and phase (blue) obtained by Fourier transfomn.

A B

electnc r.eld (V/cm)

-200 ·100 0 100 200

E:Ol( 1.0

2.0

I

~0.6 short pulse _I4'

0(E)I²

lp_

.v

₉t t

0=5.8fs '0

:0 1.5 ;

~

^~

e

^0.6 lol1g pulse ^{'0 Ql}

g0.4 t 1.0 ~

0=100fs

a.

0 0.5 .. E

c: 0.2

t

^E

0.0 0.0

g

·1.5 1.5

Fig. 2. Studying vacuum fluctuations via statistic readout and lo~itudinal modification of the probed space-t me volume. (A) Diagram showing lcngitudinal expansion of the probe volume: Stretching the samplirg pulse from 5.8 fs (green) to 100 fs (black) causes temporal averaging over the vacuum field (red).

leading to a reductim of the detected noise amplitude (t,. pulse duration: v_{8 •} group velocity). (B) Nomnalized ca.mting probability as a function of electro-optic readout by 1he short pulse (green) and lmg pulse (stretched to 100 fs; black). The deconvolved wave function

l'1'ol

² of the electric-field grm.nd state is shown in red.

Because of the rommutation relation [tio,aiJI = 1, only ~provides a nonvanishing rontnbution.

Summing frequencies over our finite sensitivity interval ensures convergence of Fq. 4. The lateral extension of the volume Vis now identified with

'the effe<tive cross section A..«, defined by the

Gaussian intensity profile of the near infrared probe beam inside the EOX. Theoretical model ing based on Laguerre Gaussian modes (30) yields Aar = WQ2

1t, where w0 is the probe spot radius (8).

Because V = A,lf L, only the length L remains to be determined. Periodic boundary ronditioos are ap plicable when the EOX is short relative to the Rayleigh range of the muhiterahertz transverse mode, resulting in a density of free space modes Lfc. Summing over all longitudinal modes within a bandwidth of tJ.v eliminates L, and we obtain

20.2-

v

cm {5)

A tactor of n₀ -1/2 accounts for dielectric screen ing inside the EOX (8). Thus, the vacuum ampli tude is maximized when averaging over a minimal space time volume, determined in the transverse directions by w₀ = 4.25 Jl.ID (fig. S5). The Jongi tudinal cross section cndCvct!.v) is defined by the Fourier transform of R(Q), containing the inten sity envelope of the 5.8 fs probe pulse and phase matming ronditioos within the EOX (8).

Are such fluctuations discernible on top of the shot noise due to the Poissonian photon sta tistics of the coherent probe? An average num ber of NP = 5 x 10⁸ photons detected per pulse causes a relative RMS shot noise current of t!JSNII =Np-¹¹²• With Eq. 3, we obtain the noise equivalent field

c

21tVpT4t~lyiNp !R(ilc)l

65.0 ~

em {6)

Because the shot noise of the near infrared probe, which is centered around vp, and thevacu urn fi uctuations at multiterahertz frequencies Q .are unrorrelated with each other and Jack spectral

overlap, the two contributions add up in quadra ture. Therefore, the RMS width of the total de tected noise distnbution is expected to rise by a tactorof

L047 (7)

rorresponding to a 4.7% increase, due to the multiterahertz vacuum noise.

To experimentally access the statistics of the . quantum vacuum, we extended the RFIA band

width to L6 MHz and sampled the probability distnbution of the electric field P(EtotaJ) every 5 (.IS.

The contribution of the multiterahertz vacuum can be modified to discriminate ~the shot noise baseline by longitudinal or transverse expan sion of the probed space time volume (Eq. 5). In the first approadl, we decreased vc and tJ.v by chirping the probe pulse to 100 fs (fig. 83), via

421

A B

^0.05 _~

Wo (J.Jm) 0.04 :0

- 4.25 ^(I)

0.03 ^J:l

- 17

e

0.02

co

_-a ^0.

0.01 ^c _~ 0.00 ~ '6 -0.01

g

-0.02

g

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 electro-optic signal Alii x 10⁴

Fig. 3. Detection by transverse expansion of the space-t me segment. (A) Sketch showing the lateral increase of the sampling cross section. which leads to averaging over noise patterns within the circled areas. (B) Differential histograms obtained by subtracting the result for the confocal detector position with wo = 4.25 JlfT1 from the results for positions with the beam diameter at 4.25 JlfT1 (black).l7 11m (purple).

25~tm (cyan). 50 JlfT1 (orange). and 85 JlfT1 (red).

§ >

-10 u.J~

<I

Q)

"0

:E a.

E ro E

::I ::I u

ro

>

1

10

probe spot radius w

₀

(I.Jm)

Fig. 4. Dependence of the varuum amplitude on transverse extension of the probed space-time volume. Relative excess noise of electro-optic signal lll/1 (right vertical axis) and RMS vacuum amplitude

f!.Evac

(left vertical axis) versus probe radius w0 (blue squares). Red lines represent a theoretical assessment based on Eq. 5.

translation of an SFIO prism in the compressor stage (Fig. 2A). A distinct reduction in peak counts around P(Et<131 = 0) is omerved when romparing the probability distribution obtained with the 5.8 fs probe (green in Fig. 2B) to the m~t with a stretched pulse (black). Also, the proba bilities in the wings of the distribution including the multiterahertz vacuum (5.8 fs probe) are con sistently higher than the corresponding values in the stretdJ.ed pulse distribution. The total change of the normalized nciseamplitude amountsto4%, in good agreement with the theoretical consid erations underlying Eqs. 5 to 7. The red histogram in Fig. 2B emerges from adecmvolutim algorithm that seardles fur the best link between distnbu tions of P(E₁₀w) obtained with and without vacuum noise. This result directly mirrors the ground state wave function l'f' o(E)I² of the electro magnetic field in the polarization plane and space time volume that we probed. From l'f'o(E)I², a RMS standard deviation of

t1E.oc

= 18 V/cm is obtained, in good agreement with the theoretical prediction of 20.2 Vfcm in Eq. 5.

In the transverse option, we kept the short pulse duration and expanded the probe radius w₀ by

422

translating the EOX out of the confocal plane (Fig. 3A). Averaging over a larger cross section causes a decrease in fluctuation amplitude, whidl is projected onto the transverse mode of the gate.

The effect of progressive narrowing is emphasized with diifurent:W probabilities obtained by subtracting a cfuttibution at Wo = 4.25 !1ID frcm ~)sampled at increasing spot radii (Fig. 3B). When all original histograms are normalized, the maximum change in probability LV'(Eu,ru = 0) of 0.04

=

4% directly rorresponds to the difference between the relative noise amplitudes measured with and without multiterabertz vacuum fluctuations, in quantitative agreement with Eq. 7. The dependence of the varuum RMS amplitude on the transverse exten sion of the probed space time volume is shown in Fig. 4. The normalized increase r:i total noise, measured with respect to bare shot noise (right vertical axis~ is plotted against the probe spot radius w0 (blue squares). Ccnversion to the vacuum eledric amplitude

!!.Evst:

(left vertical axis) ha<> been carried out analogously to I 'f' o(E)I² in Fig. 2B. The functional dependence expected from Eq. 5 is shown a<> a red line. The inset in Fig. 4 inustrates 1he data rerorded at low beam cross sections on a

linear scale to highlight the hyperbolic increase of vacuum fluctuatims for the smaller space time volumes that we probed.

In our study, we directly monitored vacuum fluctuations without amplifYing them The only effective part

L:o,

0 ~ of the operator that extracts the variance of the field in Eq. 4 indicates that vacuun1 fluctuations correspond to photons, which spontaneously arise and vanish in the ground state <Po-Time energy uncertainty de mands that virtual excitations have a limited life time on the order of their oscillation cycle (32).

The subcycletemporal resolution provided by the ultrashort probe ensures that we can directly detect effects originating from purely virtual photons. Phase matched copropagation of the varuum field and probe inside the EOX maxi mi1es those signals. But does this measurement influence the quantum vacuum at all? Ba<>ed on the electro optic dlange of the refractive index

~np-

r.aEm...

the local multiterabertz field im prints a phase shift onto the ultrashort probe, whidl we detected Because sum and difference frequency mixing ocrur 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 fre quencies Qj2rr « v.,. Our second order nonlinear element operates far from resonance. Virtual driving of the transitions avoids problenlS with decoherence, distinguishing our experiment from detection approaches in quantum optics or cirwit quantum electrodynamics in which resonant two level systems are involved (33). In ronsequence, our approach may be used to study the multi terahertz ground state while imposing negligible influence on it. Back action might arise only in third order: The nonlinear refractive index~ gen erates a local anomaly of phase velocity ropropa gating with the intensity envelope of the probe, because ~no

-

~11'1>1²• When N

rfwo

²tp 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

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ACKNOWLEDGMENTS

Support by the European Resea-ch Council {Advanced Grant 290076 "UitraPhase'), by Deutsche Faschungsg€111E!inschan (SFB767), and by NSF via a postdoctoral fellowship for D.VS. (award no.1160764) is grateftJiy ackrowledged.

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