Quantum vacuum experiments with nonlinear optical crystals

in microwave cavities: A feasibility study

C. Braggio∗ and G. CarugnoDip. di Fisica e Astronomia and INFN Sez. di Padova, Via F. Marzolo 8, I-35131 Padova, Italy

A. F. BorghesaniCNISM, Dipartimento di Fisica e Astronomia and INFN Sez. di Padova, via F. Marzolo 8, I-35131 Padova, Italy

V. V. Dodonov†

Instituto de Fısica, Universidade de Brasılia, Caixa Postal 04455, 70910-900 Brasılia, Distrito Federal, Brazil

G. RuosoINFN, Laboratori Nazionali di Legnaro, Viale dell’Universita 2, I-35020 Legnaro, Italy

We present an idea for parametric amplification of the quantum fluctuations of the electromagneticvacuum in a three-dimensional microwave resonator and report preliminary measurements to testits feasibility. In the presented experimental scheme, the fundamental mode of a microwave cavityis non-adiabatically perturbed by modulating the index of refraction of a nonlinear optical crystalenclosed therein. Intense, multi-GHz laser pulses as those delivered by a mode-locked laser sourceimpinge on the crystal to accomplish the n-index modulation. We theoretically analyse the processof parametric generation, which is related to the third order nonlinear coefficient χ(3) of the nonlinearcrystal, and assess the suitable experimental conditions for generating real photons from the vacuum.We substantiate the role of the second-order nonlinear coefficient χ(2) as a source of spurious photons,which can arise during the process of laser excitation. The interplay between a real nonlinear crystalendowed by a χ(2) coefficient and a real mode-locked laser source (i.e. featuring an offset-from-carrier noise) is also experimentally investigated. This work opens up the possibility of alternativeexperimental approaches to observe the DCE in three dimensional cavities.

October 30, 2014

I. INTRODUCTION

The term dynamical Casimir effect (DCE) is strictlyused nowadays for the effect of generation of real pho-tons when time-dependent boundary conditions are im-posed to the vacuum state of the electromagnetic field.It was predicted for the first time by Moore [1], DeWitt[2], and Fulling & Davies [3]. More general cases, in-cluding time variations in bulk media, were consideredby Yablonovitch [4] and Schwinger [5], who coined thename. For extensive reference lists one can consult re-views [6–8]. DCE analogs in superconducting circuit de-vices, which can be considered as simulations of movingmirrors in one dimension, were experimentally demon-strated in [9, 10].In the present work we analyze a three-dimensional

scheme, in which the fundamental mode of a microwavecavity is non-adiabatically perturbed by modulatingthe dielectric properties of a nonlinear crystal enclosedtherein. The idea of generating quanta of the electro-magnetic field in media with time-dependent dielectricproperties was discussed by many authors for more than

∗ caterina.braggio@unipd.it† vdodonov@fis.unb.br

two decades[11–18]. However, the first realistic experi-ment was proposed only by Braggio et al. [19] under thename MIR (remembering the terms Mirror Induced Radi-ation or Motion Induced Radiation introduced in [20, 21],although they were related to different configurations). Itwas expected that the DCE in a cavity could be observedby modulating the conductivity of a semiconductor slabwith laser pulse trains delivered by a master oscillatorpower amplifier (MOPA) laser system [22, 23]. However,this approach has been demonstrated to be severely lim-ited by microwave absorption phenomena in the laser-excited proton-irradiated GaAs [24].The main mechanism of the DCE in cavities is para-

metric amplification of vacuum fluctuations due to fastvariations of instantaneous eigenfrequencies of the fieldmode [6]. This can be achieved, for example, with theaid of a variable capacitance diode (varicap) located in-side a microwave resonant cavity [25, 26]. An alternativeway to obtain parametric amplification of the vacuumfield is to use the same laser system developed for theMIR experiment in order to optically induce polariza-tion changes in a nonlinear crystal mounted inside thecavity, thereby modulating its dielectric properties. Con-ceptually similar approaches have been described alreadyin some theoretical papers. Dezael and Lambrecht [27]devised a scheme, in which optically induced refractionindex variations in a nonlinear crystal with the second-order nonlinearity change the effective length of an opti-cal parametric oscillator in the infrared domain, whereasFaccio and Carusotto [28] and Schutzhold [29] proposed

2

to exploit the third-order Kerr effect.In contrast to [27, 28], we consider a possibility of

DCE in a microwave cavity, enclosing a nonlinear crys-tal, whose refractive index is modulated by near infrared(NIR) high-intensity laser pulses. The repetition rate ofthe pulses fl is set at twice the microwave resonance cav-ity frequency to satisfy the parametric resonance con-dition [30]. The possibility to use NIR laser light tomodulate the microwave cavity eigenfrequency is a keypoint, because the excitation source frequency (for ex-ample 1µm corresponds to 300THz) is several ordersof magnitude greater than the expected DCE signal fre-quency that would arise in the microwave range, at thecavity resonant frequency. Another advantage is relatedto the lack of microwave absorption phenomena in thelaser-excited nonlinear dielectric material.In the fist part of the present paper the process of para-

metric generation of photons is theoretically analyzed.As shown in Sec. II, the amplitude of eigenfrequencyvariations in the microwave cavity is proportional to thethird order nonlinear susceptibility coefficient χ(3). It ap-pears, however, that many crystals possessing high valuesof χ(3) have also nonzero values of the second order non-

linear coefficients χ(2)xyz. The latter coefficient can be a

source of spurious photons and we study the problem ofa time-dependent forced oscillator to estimate its level insection IID.The second part of the paper is devoted to a feasi-

bility study, in which we experimentally investigate thelimitations related to the utilization of the laser systemthat allows repetition rates as high as those requested inthe DCE experiment (multi-GHz repetition rates), themode-locked laser. Moreover, we systematically studythe phenomenon of direct microwave generation relatedto the utilization of a nonlinear crystal endowed also bya χ(2) coefficient.

II. THEORETICAL BACKGROUND

The scheme of experimental setup is shown in Fig. 1.The main part of the gap in a reentrant cylindrical cavity,where the electric field is concentrated, is filled in with anonlinear dielectric material (crystal). We assume thatin the gap the field is almost uniform. This crystal isperiodically illuminated with ps duration, intense laserpulses within a macro pulse of total duration about afew microsecond. The repetition rate of the pulses isclose to twice the cavity fundamental eigenfrequency, ofthe order of a few GHz. The change of refractive indexof the crystal due to the χ(3) nonlinear coefficient leadsto periodical changes of the instantaneous cavity eigen-frequency. These periodical variations, in turn, shouldresult in the generation of microwave quanta from theinitial vacuum state. In this section we give the theoret-ical description of this effect, showing that the numberof generated quanta depends on the total energy of themacropulse. We also discuss how to diminish spurious

U, I, ∆t

L

train of pulses

NL crystal

hg

x

y

FIG. 1. The theoretical problem we consider: a reentrantmicrowave cavity is perturbed by laser pulses impinging on anonlinear crystal that is placed in the gap region. The lengthof the crystal along the laser propagation direction x is L andhg is the height of the gap. The train of pulses lasts Tt andits energy is U .

effects due to the presence of χ(2) nonlinearities.

A. Basic field equations

We start from the Maxwell equations in the CGS units

rotE = −1

c

∂B

∂t, (1)

rotH =1

c

∂D

∂t=

ε(r)

c

∂E

∂t+

4π

c

∂PNL

∂t. (2)

Taking into account the second and third order nonlinear-ities, we can write components of the polarization vectoras (using the standard notation [31] and summation ruleover repeated indices)

PNLi = χ

(2)ijk

(

E(R)j + E

(las)j

)(

E(R)k + E

(las)k

)

+χ(3)ijkl

(

E(R)j + E

(las)j

)(

E(R)k + E

(las)k

)(

E(R)l + E

(las)l

)

.

The crucial point is that the total field in the cavity is thesum of two fields: the RF field E

(R) and the near infraredlaser field E

(las). Since we are interested in the creationof quanta of the RF field, then (taking into account fourto five orders of magnitude difference between the fre-quencies of RF and laser fields) we can replace all thefield vectors with their average values , calculated overtime intervals of the order of 10−13 − 10−12 s, which aremuch longer than the period of oscillations of the laserfield, but much shorter than the period of oscillations ofthe RF field. The RF field components practically donot change after such an averaging. On the other hand,if the laser field is quasi-monochromatic, whose ampli-tude varies slowly within the RF time scale, we can useimportant relations

E(las)j = E

(las)j E

(las)k E

(las)l = 0,

3

which result in eliminating the laser field componentsfrom Eq. (1). But these components give contributionsto the average nonlinear polarization vector PNL:

PNLi = χ

(2)ijkE

(las)j E

(las)k + aijE

(R)j , (3)

aij = E(las)k E

(las)l

(

χ(3)ijkl + χ

(3)ikjl + χ

(3)ilkj

)

. (4)

We have neglected here small terms proportional to

E(R)j E

(R)k and E

(R)j E

(R)k E

(R)l (since the RF field is sup-

posed to be much weaker than the laser one).We consider the laser beam propagating in the x direc-

tion (see Fig. 1), whose electric field vector is polarizedalong the vertical z axis. For this geometry, the part ofthe polarization vector arising due to the second ordernonlinearity can be written as

P(2) = (4π/c)χ(2)maxb(r)I(r; t),

where I = (c/4π)[

E(las)]2

is the time-average Poynting

vector of the laser beam, χ(2)max is the maximal value of

coefficients χ(2)ijk inside the nonlinear crystal, and vector

b(r) has the following components:

bi = χ(2)i33(r)/χ

(2)max. (5)

These components can be different from zero in crystalswithout central symmetry. Actually, the electric fieldinside the beam has also a small x component, due tothe equation divE = 0. However, the ratio |Ex/Ez | is ofthe order of [32] λz/d2, where λ is the laser wavelength,z is the vertical displacement from the beam axis, andd the effective width of the beam. Therefore |Ex/Ez| ∼λ/d ≪ 1 even for z ∼ d, provided the relation d ≫ λ issatisfied.The tensor aij defined by Eq. (4) has the following

components in the specific case under consideration:

aij = (4π/c)I(r; t)[

χ(3)ij33 + χ

(3)i3j3 + χ

(3)i33j

]

.

Let us neglect for simplicity the anisotropy of coefficients

χ(3)ijkl. Then aij = 0 if i 6= j [31] . Assuming that χ

(3)ii33 =

χ(3)i3i3 = χ

(3)i33i = χ

(3)3333 ≡ χ(3), we can rewrite Eq. (2) as

rotH =1

c

∂

∂t[ε(r; t)E] +

4π

cJ(r; t), (6)

with

ε(r; t) = ε(r) + 48π2χ(3)(r)I(r; t), (7)

J(r; t) = (4π/c)χ(2)maxb(r)∂I(r; t)/∂t. (8)

Hereafter E and H in all equations mean the RF electricand magnetic fields, whereas the laser field is hidden in

the intensity I(r; t). We see that the third order nonlin-earity results in a change of the effective dielectric con-stant inside the nonlinear crystal, which is proportionalto the laser intensity. The second order nonlinearity givesrise to an effective current, which is proportional to thetime derivative of laser intensity ∂I/∂t.It is convenient to rewrite Eqs. (1) and (6) in terms of

the electric induction vector D and magnetic field vectorH (assuming B = H for non-magnetic media):

rot [D/ε(r; t)] = −(1/c)∂H/∂t, (9)

rotH = (1/c)∂D/∂t+ (4π/c)J(r; I(r; t)). (10)

Excluding vector H, we arrive at the second order equa-tion

rotrot

(

D

ε(r; t)

)

= − 1

c2∂2

D

∂t2− 4π

c2∂J(r; I(r; t))

∂t. (11)

B. Reduction to forced nonstationary oscillators

The time variable t enters Eq. (11) through the func-tion I(r; t) only, therefore we can look for the solution inthe form of the expansion over “instantaneous” basis,

D(r, t) =∑

n

Qn(t)Dn(r; t), (12)

where functions Dn(r; t) satisfy the homogeneous part(with J = 0) of Eq. (11) with “frozen” functionε(r; I(r; t)):

rot rot

(

Dn(r; t)

ε(r; I(r; t))

)

=ω2n(t)

c2Dn(r; t). (13)

Note that in this way variable t enters as a parameteronly. It is important that functions Dn(r; t) form com-plete orthogonal sets for any fixed value of parameter t,with respect to the following scalar product:

∫

Dn(r; t)Dm(r; t)

ε(r; I(r; t))d3r = δmn. (14)

Therefore equations for coefficients Qn(t) can be ob-tained, provided we put expansion (12) in Eq. (11) withaccount of (13), multiply the equation thus obtained bythe function Dm(r; t)/ε(r; I(r; t)), and integrate over thecavity volume, using the orthogonality property (14).

Writing Dn(r; t) = D(0)n (r) + Kn(r; t), where D

(0)n (r)

is the solution to Eq. (13) with time independent func-tion ε(r) instead of ε(r; I(r; t)) (i.e., without perturba-tions caused by the laser illumination), we obtain thefollowing set of coupled ordinary differential equationsfor coefficients Qn(t):

Qn + ω2n(t)Qn +

∑

m

[

2QmGmn +Qmhmn

]

= Fn,

4

where time-dependent functions Gmn(t), hmn(t), andFn(t) are given by some integrals containing scalar prod-ucts of functions Dn(r; t) with time derivatives of func-tionsKm(r; t) and J(r; I(r; t)). However, it can be shownthat the terms containing small coefficients Gmn(t) andhmn(t) can be neglected. It is worth noticing that thesecoefficients are small, because functions Kn(r; t) are pro-

portional to χ(3)max. They are important for cavities with

equidistant frequency spectra, but for realistic 3D cavi-ties the intermode coupling is insignificant in the case ofparametric resonance, when laser intensity I(t) varies pe-riodically at twice the frequency of some selected mode,as was shown in [30, 33]. Therefore we finally obtainthe set of uncoupled equations for the forced harmonicoscillators with time-dependent frequencies

Qn + ω2n(t)Qn = Fn(t), (15)

where

Fn(t)

(4π)2= −

∫

d3rχ(2)max

[

∂2I(r; t)/∂t2]

[

b(r)D(0)n (r)/ε(r)

]

c∫

d3r[

D(0)n (r)

]2

[ε(r)]−1.

(16)It is convenient to normalize the basis functions

D(0)n (r) in such a way that the total energy of the field

equals ~ωc (formally, “one photon” inside the cavity).This means that the electric energy equals ~ωc/2, i.e.,

∫

d3r

[

D(0)n (r)

]2

8πε(r)=

1

2~ωc. (17)

In such a case, the variable Q(t) is dimensionless, as wellas the quantity

N =1

2

(

Q2 + Q2/ω2c

)

, (18)

which has the meaning of the number of quanta in theselected field mode (between and after the laser pulses).

C. Parametric amplification due to the χ(3)

nonlinearity

Let us consider first the case of a crystal with χ(2) = 0.Then F (t) ≡ 0, and we have the problem of parametri-cally excited oscillator. Considering the excitation of thefundamental cavity mode, we omit hereafter index n inEq. (15). The instantaneous frequency ω(t) can be writ-ten as ω(t) = ωc [1 + β(t)], where ωc is the unperturbedfundamental cavity angular eigenfrequency (ωc = 2πfc),and the small correction β(t) is proportional to χ(3)

and to the instantaneous pulse power P (t). Thereforeβ(t) = 0 during intervals between the pulses.The maximal number of quanta generated from the

initial vacuum quantum state after n pulses, is given bythe formula [34] (in the ideal case of lossless cavity)

Nn = sinh2 (nν) , (19)

where parameter ν is determined by the shape of func-tion β(t). This expression is valid in the ideal case of alossless cavity and for strictly periodic laser pulses withrepetition rate fl ≈ 2fc.Under the condition |β(t)| ≪ 1, the coefficient ν can

be represented as the Fourier transform of function β(t)at twice the cavity resonance frequency [35],

ν = ωc

∣

∣

∣

∣

∫ tf

0

β(t)e−2iωctdt

∣

∣

∣

∣

, (20)

provided the repetition rate is chosen as

fl = 2fc(1 − ϕ/π), ϕ = −ωc

∫ tf

0

β(t)dt (21)

(the sign of ϕ was corrected in [36]). Here we assumethat β(t) = 0 at t = 0 and t = tf .To evaluate possible values of parameter ν, we use the

known formula [37–39] for small variations of the eigen-frequency of an ideal cavity due to small variations ofdielectric permeability ε inside the cavity,

δω

ω≈ −

∫

δε(r)E2 dV

2∫

εE2 dV, (22)

where E(r) is the unperturbed electric field of the chosenresonance mode and ε(R) is the dielectric function in theunperturbed cavity. As we assumed a uniform electricfield in the gap region hosting the nonlinear crystal, wecan write β(t) = RY (t), where

R = − E2gap

2∫

εE2 dV, Y (t) =

∫

δε(r; t) dV. (23)

Here the integral defining coefficient R is taken over thewhole cavity, whereas the integral defining function Y (t)is taken over the part of nonlinear crystal occupied bythe laser pulse. For the Kerr type nonlinearity withδn = n2I we have (in view of the relation ε = n2)δε(r; t) = 2nGn2I(r; t), where nG is the refractive in-dex in the microwave domain. If the transverse size ofthe laser beam is smaller than that of the crystal, then

Y (t) = η

∫ L

0

dx

∫

I(r; t) dS = η

∫ L

0

dxP (x; t). (24)

Here η = 2nGn2, P (x; t) =∫

I(r; t) dS is the pulse powerat point x and instant t (the integral of the laser inten-sity over the crystal transverse section x is a constant).Consequently, Eq. (20) can be written as:

ν = ωcRηG, G =

∣

∣

∣

∣

∣

∫ tf

0

dt

∫ L

0

dxP (x; t)e−2iωct

∣

∣

∣

∣

∣

. (25)

The geometrical coefficient R can be estimated if thederivative ∂ω/∂ε for the given geometry of cavity andcrystal is known through numerical simulations. If theperturbation δε is constant throughout the crystal vol-ume, we can write the two equalities β = RδεLScr and

5

β = (∂ω/∂ε)δε/ωc, where Scr is the crystal cross sectionarea. Consequently, R = (∂ω/∂ε)/ (ωcLScr).Let us consider sharp rectangular laser pulses of dura-

tion τ and power P0. Then function P (x, t) equals P0 orzero, depending on the relation between τ and the prop-agation time through the crystal Tp = L/vg, where vgis the group velocity of the laser pulse. If τ < Tp, thenP (x, t) = P0 within the following space-time intervals:

0 < t < τ, 0 < x < tvg,τ < t < Tp, (t− τ)vg < x < tvg,Tp < t < Tp + τ, (t− τ)vg < x < L.

Outside these intervals P (x, t) = 0. If τ > Tp, then thenonzero intervals are

0 < t < Tp, 0 < x < tvg,Tp < t < τ, 0 < x < L,τ < t < Tp + τ, (t− τ)vg < x < L.

In both cases, the value of integral in (25) is given by:

G = upL |sinc (ωcτ) sinc (ωcTp)| , (26)

where up = Pτ is the energy of a single laser pulse andsinc(x) ≡ sin(x)/x.As stated by Eq. (19), the total number Nn of mi-

crowave quanta that can be produced from the initialvacuum state after n pulses is determined by the prod-uct nν. According to Eqs. (25) and (26), the coefficientν is proportional to the energy of an individual pulse up,therefore Nn depends on the total energy of macropulseU = nup:

Nn = sinh2 (|n2|UK) , (27)

where

K =2nG

Scr

∂ω

∂ε|sinc (ωcτ) sinc (ωcTp)| . (28)

We see that if the laser beam passes totally inside thecrystal, then Nn does not depend on the transverse shapeof the beam. Moreover, for short laser pulses and shortcrystals, satisfying the conditions ωcτ < 1/2 and ωcTp <1/2, the quantity Nn does not depend on the individualpulse duration or the peak pulse power because func-tions sinc (ωcτ) and sinc (ωcTp) become close to unity.Actually, this conclusion holds under the additional re-striction that the intensity I(t) of each individual pulseis not too small, otherwise the fundamental assumptionthat the laser electric field must be much stronger thanthe microwave field would not be fulfilled. This restric-tion and the assumption about the rectangular form ofthe pulse are not crucial for realistic laser pulses. Shortpulses satisfy the condition ωcτ ≪ 1, then one can writeexp(−2iωct) = 1 in the argument of integral (25), so thatthe integral equals upL for any temporal pulse shape,and the simplified formula K = (2nG/Scr) (∂ω/∂ε) canbe written.

Eq. (27) shows that to generate, say, 10 photons fromthe vacuum, one needs the product n2UK ≈ 2. Takingn2 = 3 × 10−14 cm2/W for ZnSe, and U = 200mJ, weneed K ≈ 3× 1014 s−1cm−2. To evaluate realistic valuesof the coefficient K, we notice that if the gap height hg ismuch smaller than the diameter of the post, the standardLC-formula for the resonance frequency ω = 1/

√LC can

be used, where L is the inductance of the cavity andC is the capacitance, approximated by the simple plaincapacitor formula. Assuming that the dielectric crystaloccupies all the gap, we can write (in the SI units) C ≈εε0Scap/hg, where Scap is the area of the crystal sideparallel to the post. The transverse area of crystal isScr ≈ L⊥ · hg, where L⊥ is the crystal width (it is closeto the post diameter D). Using the chain of equalities

∂ω

∂ε= −1

2

(

LC3)−1/2 ∂C

∂ε= − ω

2C

C

ε= − ω

2ε,

we get

K ≈ ω/(hgL⊥

√ε). (29)

The cavity we used for our preliminary measurements,characterized by ω = 1.44× 1010 rad/s, hg = L⊥ = 4mmand with a ZnSe crystal in the gap (ε ≈ 9), has a verysmall value of K ≈ 3 × 1010 s−1cm−2. We estimate thatthe parametric amplification sets in when a smaller mi-crowave resonator is considered, i.e. a much higher laserrepetition rates to satisfy the parametric resonance con-dition fl ≈ 2fc. Eq. (29) shows that, in order to obtainK ≈ 3 × 1014 s−1cm−2, the cavity should be rescaled bya factor 20. Possibly, a smaller scaling factor would beallowed by a proper design of the cavity shape and by in-creasing the total energy of laser pulses U , although theoptimal gap hardly should be smaller than 0, 4mm. Notethat the inductance is approximately preserved when thelength of the post and the external diameter of the cavity

are unchanged. In this case coefficient K scales as h−5/2g

and the gap should be diminished to 0.1mm, but this sizecan be increased if higher values of the derivative ∂ω/∂εare found for some design of the cavity.

Generation of 1064nm wavelength train of pulses oftotal energy U = 400mJ has been achieved at 5GHzrepetition rates [22]. In principle, it should be possibleto generate macro pulses with, say, U ∼ 80mJ at a repe-tition rate of 20GHz. If the macro pulse lasts Tt = 2µs,approximately 40 000 pulses are delivered with energyup = 2µJ. Considering a pulse duration of τ ≈ 5 ps(the relation ωcτ < 0.5 is satisfied), the intensity isImax ∼ 250MW/cm2 for hg = 0.4mm. By increasingthe macro pulse duration, the intensity could be furtherreduced to limit the second order spurious effects dis-cussed in the next section, while Nn would remain thesame.

6

D. “Spurious” quanta due to the χ(2) nonlinearity

We now consider χ(3) = 0 and χ(2) 6= 0 in order toestimate the spurious photons generated during the laserexcitation. In fact, in this case ω = ωc = const, and wea have the standard problem of excitation by an externaltime dependent force f(t), which can be considered asclassical, because it does not depend on the existenceof quantum fluctuations. As the modes are not coupledwhen χ(3) = 0, we consider again the excitation of thesingle fundamental mode, whose evolution is describedby Eq. (15) with ω(t) = ωc = const.It is convenient to introduce the complex amplitude

ξ(t) = Q + iωcQ, whose evolution is given by the wellknown formula

ξ(t) = e−iωct [ξ0 +A(t)] . (30)

The constant amplitude ξ0 is determined by the ini-tial conditions (e.g., some pre-loaded field or a thermalstochastic field), and

A(t) =

∫ t

0

F (t′)eiωct′

dt′. (31)

Then the quantity N (t), defined by Eq. (18), reads asN (t) = |ξ(t)|2/(2ω2

c ). In particular, in the case of aninitial thermal field, the phase of the complex number ξ0is random, thus after averaging over many measurementswe have 〈N (t)〉 =

[

|ξ0|2 + |A(t)|2]

/(2ω2c ).

Assuming again that the electric field is homogeneousin the crystal volume, we can write Eq. (16) as

F (t) = χ(2)maxR

∫ L

0

dx∂2P (x; t)

∂t2, (32)

where

R = − (4π)2 (b · Egap)

c∫

εE2 dV

and the function P (x; t) is the same calculated in theprevious section: P (x; t) =

∫

I(r; t) dS. Finally, makingtwo integrations by parts in Eq. (31), we get

A(t) = χ(2)maxR

∫ L

0

dx

{

eiωct

[

∂P (x; t)

∂t− iωcP (x; t)

]

−ω2c

∫ t

0

P (x; t′) exp (iωct′) dt′

}

. (33)

Note that P (x; t) ≡ 0 during the intervals between pulsesand after the complete series of pulses. Consequently,after n identical and strictly periodic pulses of durationτ and periodicity T , the standard formula for the “timediffraction” on n “time slits” is obtained

|An|2 = |A1|2[

sin (nωcT/2)

sin (ωcT/2)

]2

, (34)

where

A1 = −χ(2)maxRω2

c

∫ L

0

dx

∫ tf

0

P (x; t) exp (iωct) dt. (35)

For short pulses, such that ωcτ ≪ 1, we obtain A1 =

−χ(2)maxRω2

cLup. The actual pulse shape P (x; t) is notimportant for short pulses.The pulse repetition rate in the DCE experiments must

be close to the twice cavity frequency, but with with somesmall deviation [34]. Writing T = (1+φ)Tc/2 with |φ| ≪1, we can neglect the small phase φ in the denominator offraction in Eq. (34). Therefore the number of “spurious”quanta, that can be added to the initial level |ξ0|2/(2ω2

c)due to the χ(2) nonlinearity after n laser pulses, can beevaluated as

N (2)n =

1

2

{

χ(2)maxRωcLup sin [nπ(1 + φ)/2]

}2

. (36)

The microwave signal due to the χ(2) nonlinearity, canthus exhibit some kind of “beats” with period ∆T ≈Tc/φ. Actually, the real temporal behavior of the mi-crowave signal can be even more complicated than pre-dicted by this simple model, as shown in Sec. IVB. In or-der to estimate the spurious quanta, we need to know thevalue of the coefficient R, which strongly depends on themutual orientation of vectors b (5) and Egap, i.e., on theorientation of the crystal optical axes. We can estimatethe maximal possible value |R|max by assuming that thevectors are parallel and that the electric field is concen-trated in an “effective cavity volume” Vc. The cavity fieldin the gap region can be evaluated from the normalizationintegral (17): |Egap| ≈

√

4π~ωc/εVc, where Vc = LhgL⊥

is the crystal volume (assuming that the crystal occu-

pies all the gap). Then |R|max ≈[

~ωcεVcc2/(4π)3

]−1/2.

Therefore we obtain for the parallel vectors b and Egap

N (spur)max ≈ 32π3ωcL

~εc2L⊥hg

(

χ(2)maxup

)2

. (37)

This quantity depends on the energy of each short pulseup, but it does not depend on the number of pulses n andtheir total energy U . Let us compare the value (37) withthe results obtained in the case χ(3) 6= 0 and χ(2) = 0.taking hg = 0.4mm, L = L⊥, ε ≈ 10, up = 2µJ, and

ωc = 2π × 10GHz. The value χ(2)max can be estimated

as [31] χ(2)max ∼ E−1

at ∼ 10−7CGS units. Then we ob-

tain N (spur)max ∼ 1010. Although realistic numbers can

be several orders of magnitude smaller (for example, the“effective cavity volume” used in the normalization inte-gral can be significantly bigger than Vc), they still canbe very big, this result shows that the presence of χ(2)-nonlinearity can be dangerous for attempts to observepure quantum effects in nonlinear crystals.The influence of the χ(2)-nonlinearity can be signifi-

cantly suppressed by diminishing the value of the scalarproduct bEgap. This can be realized by properly choos-ing the crystal orientation in the cavity, as demonstrated

7

A

scope

ZnSe

Hh

D

d

λ/2

AOM

M-OSC

SAM

CM

CW train

macropulse

PD

spectrum

analyzer

FIG. 2. Scheme of the experimental apparatus. The lasermaster oscillator (M-OSC) delivers a CW train of infraredpulses. An AOM selects a finite number of pulses to be am-plified in A, a double stage optical amplifier. The ultrafastphotodiode PD detects the loss of the curved mirror CM tomonitor the laser pulses repetition rate. A reentrant cylindercavity hosts the nonlinear crystal (ZnSe). Its dimensions are:H = 26.2, D = 38 and h = 22.8 , d = 6 respectively externaland internal cylinder height and diameter in mm.

in Sec. IVA. Another possibility is to properly tailor theduration of the macro pulse for the maximum U experi-mentally achievable, reminding the fact that the numberof DCE quanta grows exponentially whereas the numberof spurious quanta linearly diminishes when the total en-ergy U is shared between an increased number of pulses.

III. EXPERIMENTAL RESULTS

In Fig. 2 we show the experimental scheme that hasbeen used to investigate possible spurious effects in aDCE experiment based on the modulation of the indexof refraction of a nonlinear crystal. The laser systemthat has been used, described in section III B, has a rep-etition rate in the range 4.6 ≤ fl ≤ 4.7GHz, thus thecondition of parametric resonance is met through a mi-crowave resonator whose gap is of a few mm. Therefore,in these preliminary studies we do not expect any de-tectable parametric amplification of photons through theχ(3) coefficient.

A. The microwave cavity

The heart of our experimental scheme, shown in Fig. 2,is a copper reentrant cylinder cavity. Its dimensions areH = 26.2, D = 38 and h = 22.8 , d = 6 respectively ex-ternal and internal cylinder height and diameter in mm.In order to maximize the cavity perturbation for a givenamount of the macro pulse energy U , we designed thecavity in such a way that the nonlinear crystal almost

fills up the gap, in which the electric field is concen-trated. Actually, the reentrant part of the cavity is com-posed of a cylinder and of a cone to further concentratethe field lines in the gap, as shown in Fig. 3. The dis-played magnetic and electric field profiles of the lowestfrequency TM010 mode that we considered have been ob-tained through numerical simulations based on the finiteelement method (FEM).

E (V/m)

H (A/m)

9 e-01

1.1 e-01

3 e-05

6.2 e-01

3 e-04

4 e-01

5.6 e-02

2 e-04

1.1 e-04

5.7 e-05

4.4 e-07

FIG. 3. Electric (left) and magnetic (right) field profile inthe reentrant cylinder cavity used. The reentrant part of thecavity is composed of a cylinder and a cone, whose smallerdiameter (on the ZnSe crystal side) is of 4mm.

We measured a quality factor QL ≈ 860 correspondingto a decay time of the stored field [40] τL = QL/πν0 =120 ns, with ν0 ∼ 2.31756GHz. The dominant contri-bution to the microwave losses is related to the ZnSetangent loss (tanδ = 0.0017 at room temperature) ratherthan to the copper cavity walls. The empty cavity is infact characterized by a quality factor of ≈ 2500.Light can be made to directly impinge on the nonlinearcrystal through an aperture whose diameter is compara-ble with the laser beam waist. Opposite to the entrancehole, an equivalent aperture allows light to exit the cavity.The field in the cavity during and after the laser actionis studied by means of a coaxial transmission line endedby an inductive loop. A 33 dB gain Miteq microwaveamplifier is used to amplify the microwave signals to alevel greater than the noise floor of a high-rate samplingdigital oscilloscope (6GHz bandwidth).

B. The laser oscillator

Stable emission of laser pulses at high repetition rateis a necessary requirement to satisfy the parametric res-onance condition in a 3D microwave cavity and can beobtained from passively mode-locked lasers [41].In the present work we use a multi-GHz passively

mode-locked Nd:YVO4 laser (M-OSC, master oscillator)

8

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

rectangular pulsesinc pulse

(b)

f / f0

10-4

10-3

10-2

10-1

100

0.0 0.5 1.0 1.5 2.0 2.5 3.0

f / f0

fl = 2f

c

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0.99 0.995 1 1.005 1.01

inte

nsi

ty (

a.u.)

f/fl

FIG. 4. Optical pulses in the frequency domain. A train ofpulses has a spectrum composed of several harmonics whoserelative amplitude is determined by the single pulse shape.Inset (a) shows for example the difference between trains ofrectangular and sinc pulses of 10 ps duration, separated byapproximately 200 ps (fl = 5GHz). When a finite numberof pulses (N = 2000) is considered, secondary maxima areobserved between each harmonic. For the aims of the presentwork, we show in inset (b) their calculated envelope amplitudearound the condition of parametric resonance, whereby thelaser pulses are repeated at twice the frequency of a microwaveresonator.

that has been described elsewhere [22]. Its repetitionrate can be changed in the interval 4.6 ≤ fl ≤ 4.7GHzby properly adjusting the position of the curved mir-ror (CM) and of the saturable absorbing mirror (SAM)shown in Fig. 2. As the M-OSC output pulse energy(≈ 5 pJ) is not sufficient to induce significant changesof the dielectric properties in a few cubic square millime-ters nonlinear crystal, several amplification stages (diode-pumped Nd:YVO4 and Nd:YAG flash-lamp pumpedmodules) are employed in order to enhance the pulse en-ergy up to 10µJ at 1064nm wavelength.

The amplified, 10 ps-long pulses are delivered in groupsof N ≈ 2000 pulses, separated by tl = 1/fl ≈ 200 ps, soas to form 500 ns-long pulse trains that can repeated onceper second. The finite pulse train duration Tt = 500 nssets the minimum width ∆f = 1/Tt ≈ 2MHz of theharmonics in the pulse train spectrum.

The ideal spectrum of a macro pulse consisting ofN = 2000, 10 ps-long, rectangular pulses is shown inFig. 4. In the frequency domain it is analogous to theinterference figure obtained by a diffraction grating withN = 2000 rectangular slits. It consists of several har-monics (the principal maxima) at frequencies fm = mfl,which are integer multiples of the fundamental frequencyfl. The width of each harmonic is set by the duration ofthe macro pulse Tt.

The amplitude of the principal maxima, i.e., of the

harmonics of the fundamental frequency, decreases withincreasing order of the harmonic in much the same wayas the amplitude of the principal maxima in the interfer-ence figure obtained with a diffraction grating made ofrectangular slits. The envelope Am of the amplitude ofthe principal maxima is given by

Am =

∣

∣

∣

∣

sin 2πmflτ

2πmflτ

∣

∣

∣

∣

(38)

where τ = 10 ps is the duration of each individual pulsein the train.N − 1 secondary maxima appear between the harmon-

ics and can be observed as side lobes surrounding eachprincipal maximum, as can be noted in the central partof the figure.In the inset (a) of Fig. 4 we show the envelope of

all maxima, i.e., including the secondary ones, of thediffraction-like spectrum for the first two harmonics ofthe train. It has been calculated for the macro pulse du-ration we used during the present measurements (N =2000, ∆t ∼ 450 ns). It is worth noting that, at the cav-ity frequency fc = fl/2, the secondary maxima envelopeestablishes a floor at the level of 10−3. A cw pulsed laserexcitation would instead produce a sequence of perfectlyisolated harmonics. Unfortunately, there are no mode-locked oscillators that can deliver in CW the requestedlaser/energy per pulse, and we need to optically selectand amplify only a finite number of pulses.For the duration of the optical pulses (10 ps) at hand,

the principal maxima envelope is very wide but only thefirst few harmonics fit within the typical detection band-width of a fast photodiode [42]. Actually, the individualpulses are not rectangular. Their shape is rather givenby a squared hyperbolic secant, thus yielding a differentenvelope. In the inset (b) of Fig. 4 we compare the tworesulting envelopes that show that the relative amplitudeof the harmonics of the train is determined by the indi-vidual pulse shape.In any case, the difference between the two envelopes

for the first few harmonics is hardly detectable.The actual spectrum of our mode-locked oscillator is

characterized by a rather complicate spectrum that cansignificantly differ from the ideal situation, as shown inthe example in Fig. 5. We divide the logarithmic am-plitudes observed in the optical signal spectrum by thenoise floor of the spectrum analyzer to get the displayedamplitudes AN . A resolution bandwidth of 30 kHz wasset and the internal preamplifier (whose bandwidth ex-tends up to 3GHz) is switched on in order to obtain anincreased sensitivity to secondary laser oscillations. Weobserve that, in addition to fl ≈ 4.6GHz and the secondharmonic 2fl of the train of pulses, there exist secondaryoscillations at f1,2 < fl. The carrier frequency at fl ismodulated by f1,2 that in turn appear as sidebands toeach harmonic of the train.In order to obtain a laser spectrum characterized by

the smallest secondary oscillations, tiny adjustments of

9

-100

-80

-60

-40

4.63 4.632 4.634 4.636 4.638 4.64

A (

dB

m)

frequency (GHz)

frx

frx

- +

1

1.5

2

2.5

0 2 4 6 8 10 12

AN

frequency (GHz)

fl

2 fl

f1

f2 f

l - f

2

fl - f

1 fl + f

1

fl + f

2

2 fl - f

22 f

l + f

2

2 fl + f

1

FIG. 5. Representative spectrum of the master oscillator acquired by a fast photodetector with bandwidth greater than 15GHz(EOT InGaAs pin detector ET-3500). The spectrum extends up to 13.2 GHz, the upper frequency of the spectrum analyzer.The displayed amplitudes AN are obtained by means of dividing the logarithmic amplitudes observed in the optical signalspectrum by the noise floor of the spectrum analyzer. Secondary oscillations, in addition to the first and second harmonic ofthe train of pulses (respectively fl and 2fl), are indicated by f1 and f2. Since these frequencies represent a modulation of theintensity of the train of pulses delivered by the master oscillator, they also appear as sidebands (fl ± f1,2) to each harmonic. Itis worth noticing that the amplitude of the peaks at f1,2 is more than 30 dB smaller than the carrier frequency at fl. On theright side our typical level of the relaxation oscillations f±

rx is shown (50 dB smaller than the first harmonic peak amplitude),acquired with 30 kHz resolution bandwidth.

the master oscillator curved mirror and the SAM are per-formed while observing its photodiode signal at a spec-trum analyzer (Agilent ESA-E E4405B). We note thatto thoroughly suppress the secondary oscillations, someQ-switching instabilities necessarily arise around the car-rier, as proved for several M-OSC different alignmentsand fl frequencies. In Fig. 5 (b) we also show the relax-ation oscillations (f±

rx ∼ fl ± 3MHz) in the spectrum ofour passively mode-locked laser oscillator. These oscil-lations are in general considered acceptable when theiramplitude is at least 30 dB smaller than that of the car-rier [43].

In the present work we maintain the cleanest modelocking condition, whereby both the secondary peaks andthe relaxation oscillations are kept at least 40 dB be-low the amplitude of the carrier frequency fl. Once thiscondition is met, the master oscillator repetition rate islocked to a microwave oscillator by means of a feedbackcircuit described elsewhere [23] and is thus stabilized dur-ing the measurements.

Until now, there have been many reports of noise mea-surements on mode-locked lasers [44–46], but almost al-ways the power density of phase noise up to a few MHzis reported whereas the secondary oscillations we observeoccur at a much higher frequency, from a few hundredsMHz to approximately 2GHz. The actual physical ef-fects that are responsible for the oscillations in additionto the expected harmonics are not discussed here. Suchan investigation is beyond the scope of this paper.

TABLE I. Second- and third-order nonlinear optical coeffi-cients of selected materials [47].

Material n2 (cm2/W) χ(3) (esu) d (pm/V)

diamond 1.3 · 10−15 1.8 · 10−13

GaAs 3.3 · 10−13 1.0 · 10−10 d14 = 90.4

ZnSe 3 · 10−14 4.4 · 10−12 50− 60

C. The nonlinear crystal

As described in section II, high speed switching bound-ary conditions to the EM field can be realized by exploit-ing the intensity dependent refractive index n = n0+n2I,where I is the laser intensity and n2 is related to the non-linear susceptibility by means of [31]

n2 = 12π2χ(3)/n20. (39)

Therefore a primary requirement is to find a material,in which χ(3) is sufficiently large. Semiconductors oftenpossess a large third order susceptibility, typically in therange 10−13− 10−10 esu, while insulating solids are char-acterized by a nonlinear coefficient χ(3) of the order of10−13 − 10−14 [47]. Eq. (39) applies both to a semicon-ductor and an insulating solid as diamond for an incidentphoton energy smaller than the band gap energy. In Ta-ble I we report values of χ(3), n2 and the scalar secondorder nonlinear coefficient d for some materials that weconsidered. The most attractive material for such typeof experiment appears to be a semiconductor (GaAs),whose χ(3) is nearly two orders of magnitude greater indiamond, the latter being used in many nonlinear op-tics applications [48, 49]. However, we point out that

10

-0.04

-0.02

0

0.02

0.04

-200 -100 0 100 200 300 400 500 600

amp

litu

de

(V)

time (ns)

FIG. 6. Microwave signal detected in the cavity. The train ofpulses impinges on the ZnSe crystal at t = 0 and lasts approx-imately 500 ns. The optical pulse intensity is ∼ 100MW/cm2.

infrared photon (λ = 1064nm) absorption was observedin semi-insulating GaAs, and this limits the possibility touse it in our experiment, in which laser-excited carriers,responsible of ohmic losses for the cavity microwave field,are not allowed [24]. Absorption of photons with energysmaller than the band gap is in fact possible throughthe EL2−like defect [50]. In Table I we also report theproperties of ZnSe (zinc selenide), another semiconduc-tor belonging to the zinc-blende II-IV group whose bandgap is 2.7 eV at room temperature.

In the following sections we report measurements car-ried out with ZnSe and KTP crystals. The latter hasbeen chosen because it is endowed with big χ(2) and thusallows to experimentally study the interplay between thelaser non-ideal spectrum and the second-order coefficient,which can give rise to radiation at the cavity frequency.ZnSe is instead interesting due to its high χ(3) that makesit a proper crystal for DCE experiments. We have how-ever to keep in mind that, as any nonlinear material, italso displays a second order coefficient, that has to behandled with as described in the following sections.

IV. EXPERIMENTAL RESULTS

A. Measurements with a ZnSe crystal

To investigate the possible spurious photons generationrelated to the ZnSe χ(2) coefficient, we have used theexperimental scheme in Fig. 2. The ZnSe crystal used isa parallelepiped 3× 3× 2mm3 in size whose short axis isaligned along the direction of laser propagation (x axis).The signal in Fig. 6 is observed when at t = 0 a train oflaser pulses with total energy U = 70.2mJ impinges onthe ZnSe crystal. As the laser beam gaussian diameter atthe crystal position is of approximately 2mm, the pulseintensity is ∼ 100MW/cm2. In the lower part of Fig. 7we plot the spectrum of the microwave signal of Fig. 6,obtained by averaging over 100 signal FFTs (Fast Fourier

Transform) calculated for each signal. The noise floor ofthe instrument is removed as described earlier in thistext, in this case we divide the calculated amplitudes ofthe FFT of the signal registered when the laser impingeson the ZnSe crystal by the calculated FFT amplitudeswhen the laser is not allowed to enter the cavity.

0.8

1.2

1.6

2

2.4

2.8

3.2

0 1 2 3 4 5 6

AN

frequency (GHz)

CAVITY - microwave signal

PD - optical signal

fl

f3

f4

fl - f

3

fl - f

4

fc

1

1.2

1.4

1.6

1.8

2

2 2.1 2.2 2.3 2.4 2.5 2.6

FIG. 7. Laser oscillator spectrum (indicated by the label “PD- optical signal”) and frequency analysis of the signal detectedin the cavity up to 6GHz (CAVITY - microwave signal). Boththe spectra are normalized to the respective noise floors ofthe instruments used (the spectrum analyzer and the oscillo-scope), as described in detail in the text. The laser oscillatorrepetition rate was set to fl = 4.6347 GHz. In the inset, bothspectra are shown around fc, to evidence that there are nolaser oscillations at the cavity frequency of resonance. Thespectrum analyzer was set to 30 kHz resolution bandwidth.The M-OSC spectrum has been upwards shifted for visibilitypurposes.

The cavity data are supplemented with the spectrumof the laser master oscillator, shown in the upper partof the Fig. 7. The laser oscillator repetition rate was setto fl = 4.6347GHz and the displayed optical signal FFTis obtained as described in section III B. In the opticalspectrum of the M-OSC several secondary oscillations(f3,4 and the coupled f3,4 − fl) are detected, whereasthe microwave signal in the cavity displays only f4 andits coupled frequency fl − f4. Note that they lie morethan 200MHz far from fc and is therefore well outsidethe cavity line width (2.7MHz). Radiation at fl is alsodetected in spite of the narrow bandwidth of the amplifier(0.4GHz centered about 2.5GHz). Similar results havebeen obtained even with a different configuration for themaster oscillator as shown in Fig. 8.We then studied the influence of the M-OSC align-

ment on the photons detected at the cavity frequency fc.In Fig. 9 we show the spectra around fc, obtained withdifferent master oscillator alignment conditions. Notethat for the first two data series the laser oscillatorlength was adjusted in such a way that fl ∼ 2fc (withinthe cavity line width) whereas in the third data set

11

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6

AN

frequency (GHz)

fl

f5

f6

fl - f

6

fl - f

5

fl + f

6

fl + f

5

PD - optical signal

CAVITY - microwave signal

fc

f7 f

l - f

7

1

1.1

1.2

1.3

1.4

1.5

2 2.1 2.2 2.3 2.4 2.5 2.6

FIG. 8. Data are displayed as in Fig. 7 for a different mas-ter oscillator configuration, whose repetition rate was in thiscase set to fl = 4.6552GHz. As compared to the previousoscillator alignment, in the laser spectrum much smaller sec-ondary oscillations around fc (f7 and its mirrored f7−fl) aredisplayed. Secondary oscillations f5,6 at a smaller frequencythan those observed in the previous example are in this align-ment detected.

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

AN

frequency (GHz)

4.6552

4.6366

4.6347

FIG. 9. Frequency analysis of the signal detected in the cav-ity in a span of 800MHz approximately centered around thenominal cavity frequency of resonance. The arrow indicatesthe value of fc independently measured at a vector analyzer.The spectra obtained with the laser repetition rate set at thefrequencies 4.6652 and 4.6366GHz have been upwards shiftedof, respectively, 0.5 and 0.25 for visibility purposes.

(fl = 4.6552GHz) we set the laser excitation far from thecondition of parametric resonance (fl/2 − fc >10MHz).Independent of the different conditions, the area underthe peak at fc is verified to be constant.

To sum up the results shown on Figures 6, 7, 8, 9,we observe that microwave power is detected at f ∼ fc

0

1

2

3

4

5

6

7

8

200 240 280 320 360

Vrms (

V)

θL (deg)

FIG. 10. Dependence of the measured root-mean-square mi-crowave signal amplitude Vrms on the λ/2 plate rotation. Theoscilloscope root-mean-square noise has been subtracted tothe original data rms values (empty dots) to get the filleddots data.

in the cavity during the laser excitation, whose level isin first approximation independent of the frequency andthe amplitude of the M-OSC secondary oscillations. Insection IVB we instead describe a case in which the be-haviour of the M-OSC strongly influences the spectrumof the photons detected inside the microwave resonator.Finally, we report in Fig. 10 the root mean square am-

plitude of the microwave signal detected in the cavity forseveral positions of the half-wavelength plate set beforethe cavity entrance. The data exhibit an eight-fold peri-odicity modulated by a four-fold one and are fitted to afunction of the form y = a+ b1 cos(8x− c1)+ b2 cos(4x−c2). We observe that there are some orientations of theincident laser polarization whereby it is possible to di-minish the level of the microwave power detected in thecavity during the laser action to a level below the currentexperimental sensitivity.

B. Measurements with a KTP crystal

To better understand the role of the second ordernonlinear coefficient in the DCE experiment, we havemounted in the cavity gap a potassium titanyl phosphate(KTiOPO4, KTP) crystal, a material that is being widelyused for several second-order nonlinear optical applica-tions due to its high nonlinear d coefficient [51].The KTP χ(2)-related photons emitted at frequency flhave been previously studied by means of a cavity withproper frequency fc ∼ fl [52]. The amount of generatedradiation has been demonstrated to be so high that am-plification was not needed. In fact, direct observation ofa few hundred mV signals was achieved at the oscillo-scope input with a laser pulse intensity of the order of

12

70MW/cm2. Here we study the KTP χ(2)-related pho-tons detected in a cavity aimed at a DCE experimentwhose fundamental resonance is at fc ∼ fl/2, in the ex-perimental same scheme used in the previous section tostudy the ZnSe χ(2)-related photons. In this case, insteadof simply monitoring the master oscillator spectrum ata spectrum analyzer, we mount the device described inRef. 42 on the beam line before the cavity entrance. Thisdevice allows acquisition of the spectrum of the radia-tion emitted by a KTP crystal undergoing the train oflaser pulses without the bandwidth limitations of the cav-ity. The nonlinear crystal is in fact enclosed in a coaxialstructure, in which the χ(2)-related radiation is trans-ferred to the TEM (transverse-electromagnetic) mode oftransmission [42]. The inner and outer diameters of thecoaxial structure are such that the TEM mode cutoff fre-quency is approximately 8.5GHz, thus the device allowsindependent observation of the generated radiation in amuch wider bandwidth than that of the cavity. In thisway we can establish a direct experimental link betweenthe radiation detected in the cavity and the infrared lasertrain instabilities.

In Fig. 11 we show several modulated microwave pulsesobserved in the cavity when the laser pulse impinges inthe KTP crystal. The train of pulses starts at t = 0 andlasts for an interval of 400ns, during which a modulatedmicrowave field is observed. Afterwards the stored energyexponentially decays with the cavity time constant τL ∼200ns.

In Figure 12 we show the microwave spectra of thelaser macro-pulse recorded with the wide-bandwidthcoaxial device. Acquisition of these signals has been si-multaneous to the signals displayed in Fig. 11, thus thenumbers 1-6 that characterize each spectrum correspondto the frequency values described in the relative caption.As we plot the macro pulse spectrum, the bandwidthresolution cannot be better than 2MHz due to the finiteduration Tt ∼ 500 ns of the train. Therefore the detectedsecondary oscillations fi and the coupled fj cannot beresolved when they are very close, as shown in panel 1of Fig. 12. The output of the device described in Ref. 42(see Fig. 13 (a)) is recorded in coincidence with the mod-ulated signal detected in the cavity (see Fig. 13 (b)) whenthe laser repetition rate was set to fl = 4.6766GHz.

Finally we verify that the inverse of the period of themodulation T observed in the cavity (Fig. 11) exactlycorresponds to the difference between two secondary fre-quencies fi and fj detected in the master oscillator spec-trum obtained with the coaxial wide-bandwidth detector(Fig. 12). The sum between fi and fj is shown to beequal to fl. In table II the results of the comparison be-tween the macro pulse spectra and the cavity detectionare resumed.

While in Fig. 12 we limited the spectrum to a rangearound the cavity frequency fc, in Fig. 14 we show anexample of full spectrum, where it is evident how fi andfj are 30dB smaller than the first harmonic of the trainof pulses at fl. It has been verified [42] that an equiva-

FIG. 11. Modulated microwave pulses detected in the cav-ity for a few different master oscillator alignments, whosecorresponding repetition rate fl is indicated in each frame.The laser excitation starts at instant 0 and lasts approxi-mately 500 ns. Exponential decay of the stored field is af-terwards observed. The repetition rate of the pulses fl wasset to 1 = 4.6766 GHz; 2 = 4.6767GHz; 3 = 4.67685 GHz;4 = 4.677GHz; 5 = 4.6761GHz; 6 = 4.6776GHz.

TABLE II. Identification of the pulsed microwave radiationobserved with the KTP crystal. In the first column we reportthe value of the laser repetition rate that characterizes eachalignment as described in the caption of Fig 11; fi,j are thefrequency of the secondary oscillations detected in the masteroscillator spectra displayed in Fig. 12. T is the period of themodulation measured from the microwave pulses in Fig. 11.

fl T−1 fi fj fi − fj

(GHz) (MHz) (GHz) (GHz) (MHz)

1 4.6766 ∼ 1/Tt – – 6 2

2 4.6767 4.4±0.4 2.3365 2.3402 3.7

3 4.67685 6.6±0.6 2.3352 2.3413 6.1

4 4.677 8.7±0.7 2.3338 2.3427 8.9

5 4.6761 15.6±0.3 2.3305 2.3462 15.7

6 4.6776 20.5±0.6 2.3285 2.3492 20.7

lent spectrum is obtained by calculating the FFT of anultrafast photodiode output when it monitors the macropulse.

It is then possible to find a condition in which no sec-ondary oscillations in the vicinity of the cavity frequencyare observed. In this case the signal reduces again to thesmall one observed with the ZnSe crystal (Fig. 6), with

13

-95

-90

-85

-80

-75

-70

-65

-601

A (

dB

)2 3

4.67685

-95

-90

-85

-80

-75

-70

-65

2.3 2.32 2.34 2.36 2.38

4

frequency (GHz)

A (

dB

)

2.3 2.32 2.34 2.36 2.38

5

frequency (GHz)

2.3 2.32 2.34 2.36 2.38

6

frequency (GHz)

FIG. 12. Coaxial structure wide-bandwidth detection. Themaster oscillator secondary oscillations fi and fj = fl − fi,detected in the master oscillator microwave spectrum in thevicinity of fc = 2.33525GHz (central frequency of each plot).Their corresponding frequency is resolved to the best of2MHz, due to the macro pulse duration (Tt = 500 ns). Forthis reason fi and fj are not resolved in the first plot. In plotnumber 5 additional frequencies are observed, but they do notinfluence the corresponding plot in Fig. 11 because they aretoo far from the cavity resonance.

-0.8

-0.4

0

0.4

0.8

0 200 400 600 800

time (ns)

A (

mV

)

(b)

-0.02

0

0.02

0.04

A (

mV

)

(a)

FIG. 13. Simultaneous detection of microwave radiation gen-erated by the laser pulses interaction in KTP (a) inside awide-bandwidth coaxial structure and (b) in the cavity res-onator. The signal is amplified by a factor of approximately33 dB in (b) while (a) is directly observed at the oscilloscope.

an instantaneous rise and no exponential decay of thestored field at the end of the laser excitation. We be-lieve that this signal is of electronic origin, a crosstalkbetween the cavity and the amplifier input that couldpossibly be reduced using a much higher quality factorcavity resonator.

-100

-90

-80

-70

-60

-50

-40

-30

-20

0 1 2 3 4 5

A (

dB

)

frequency (GHz)

fl

1

FIG. 14. Coaxial structure detection: full spectrum of thelaser macro pulse. Note that the amplitude of fi and fj is30 dB smaller than that of the carrier at fl. The same obser-vation is valid for the other signal examples 2 − 6 shown inFig. 12.

V. CONCLUSIONS

In this work we have studied both theoretically andexperimentally the parametric excitation of a microwavecavity resonator with of a train of multi-GHz laser pulsesacting on a nonlinear crystal enclosed within the cavityitself. Our final aim is to establish the suitable experi-mental conditions to study the DCE in the three dimen-sional case.In section II we have estimated the number of photons

that can be generated inside the cavity, starting fromthe vacuum state of the electromagnetic field, throughexcitation of the third order nonlinear response of thecrystal, proportional to χ(3). An optimized cavity (andnonlinear crystal) geometry has been proposed, which re-quires a much higher repetition rate laser system (and,as a consequence, a smaller cavity) than the one imple-mented in the preliminary measurements reported in thesecond part of the present work. Nonetheless, even thenon-optimized scheme enables us to investigate possiblespurious effects in dynamical Casimir experiments. Wesingled out three different processes of spurious photongeneration. The first one, studied theoretically in sectionIID, is related to the evaluation of the maximal possiblenumber of the spurious RF quanta, generated by period-ical laser pulses, whose influence can be described withina simple model of an oscillator, excited by a classical forceproportional to the χ(2) nonlinear coefficients.The second one, described in section III B, is related

to the finite duration of the excitation and it can be re-duced by increasing the laser excitation duration. Weought to consider here that in principle the laser pulsescannot have exactly the same energy, thereby creating aquantum noise floor between the main peaks at fm = mflalso in the case of a cw laser excitation [53].

14

Both in the first and the second problems the laser os-cillator is considered to be ideal (i.e. the laser spectrumcontains only the harmonics fm). The third problem isinstead related to the actual oscillator, that can exhibitsecondary oscillations in the vicinity of the cavity reso-nance, and its influence has been experimentally studiedin section IVB. We have seen that a very small modu-lation of the infrared train of pulses intensity (the levelof the detected secondary oscillation fi,j is found to bemore than 30 dB less than the carrier frequency fl) is adirect source of microwave photons. The effect is relatedto the second order coefficient d of the nonlinear crys-tal, in which a time-dependent polarization is producedas a consequence of optical rectification (OR) [52]. Dueto this phenomenon, the crystal behaves as if it were anantenna that radiates an electromagnetic field whose fre-quency content is directly related to the laser spectrum.Therefore any optical frequency component is to be reg-istered as a generation of photons inside the cavity.Weremark the fact that any optical crystal is characterizedby a second order nonlinearity. Therefore, regardless howsmall it is, its role has to be considered in the QED ex-periment.We have demonstrated that a possible way to reduce

the observed χ(2)-related photons under the current de-tection threshold is to set the laser polarization along acrystal axis characterized by a null d coefficient. Alter-natively, this axis might be orthogonal to the direction

of the electric field in the cavity [54]. This process is alsoworth further investigations at cryogenic temperaturesand improved electronic detection sensitivity.

Finally, we have shown that, even if the secondary os-cillations near the cavity resonance are reduced below oursensitivity, a signal is detected, whose shape suggests itspossible origin, i.e. cross-talk between the cavity andthe amplifier input. A higher quality factor cavity couldbe of help to reduce this noise. Moreover, we expect animproved detection scheme from the point of view of elec-tronic noise when the cavity and a low noise amplificationstage are both cooled to liquid helium temperatures, asdemonstrated in Ref. 55. In addition, at this tempera-ture the number of thermal photons kT/hν in the cavitymode is also very small, necessary condition to start aDCE experiment.

VI. ACKNOWLEDGEMENTS

C. B. gratefully acknowledges the financial support ofthe University of Padova under Progetto di Ateneo 2013(cod. CPDA135499/13). V. V. D. acknowledges a partialsupport from the Brazilian agency CNPq and the Visit-ing Professor grant from the University of Padova. Theauthors acknowledge the technical assistance of E. Bertoand F. Zatti from the University of Padova and INFN.

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