experimental study of microwave photon statistics under
TRANSCRIPT
PHYSICAL REVIEW A 88, 053806 (2013)
Experimental study of microwave photon statistics under parametric amplificationfrom a single-mode thermal state in a cavity
G. Galeazzi, A. Lombardi, and G. Ruoso*
INFN, Laboratori Nazionali di Legnaro, Viale dell’Universita 2, I-35020 Legnaro, Italy
C. BraggioDip. di Fisica e Astronomia, Via F. Marzolo 8, I-35131 Padova, Italy and INFN, Sez. di Padova, Via F. Marzolo 8, I-35131 Padova, Italy
G. CarugnoINFN, Sez. di Padova, Via F. Marzolo 8, I-35131 Padova, Italy
F. Della ValleDip. di Fisica and INFN, Sez. di Trieste, Via A. Valerio 2, I-34127 Trieste, Italy
D. ZanelloINFN, Sez. di Roma, Piazzale A. Moro 1, I-00185 Roma, Italy
V. V. DodonovInstituto de Fısica, Universidade de Brasılia, Caixa Postal 04455, 70910-900 Brasılia, Distrito Federal, Brazil
(Received 2 August 2013; published 6 November 2013)
In this paper we present theoretical and experimental studies of the modifications of the thermal spectruminside a microwave resonator due to a parametric amplification process. Both the degenerate and nondegenerateamplifiers are discussed. Theoretical calculations are compared with measurements performed with a microwavecavity parametric amplifier.
DOI: 10.1103/PhysRevA.88.053806 PACS number(s): 42.50.Ar, 42.65.Yj, 44.40.+a, 07.57.−c
I. INTRODUCTION
Experimental studies of quantum statistical properties ofmicrowave radiation generated by different sources alreadyhave a rather long history. The first observations of squeezingin a microwave signal from the Josephson parametric amplifierat 19.4 GHz were reported in [1]. Recently, schemes of con-trolled generation of squeezed states in the (0.1–10) GHz rangewere proposed in [2], and new experiments on squeezing atmicrowave frequencies (using again the Josephson parametricamplifiers) were described, e.g., in [3–5]. Different methodsof quantum state reconstruction (quantum tomography) inthe microwave domain were proposed, e.g., in [4,6–8], andthey were realized in experiments [9–15], where the Wignerfunctions or density matrix (or covariance matrix) weremeasured. Using these functions one can find the photondistribution function, which was extracted explicitly from themeasured data in [16] and in experiments described in [17].However, in all those experiments the mean numbers ofphotons were rather small (maximum 17 in Ref. [16]), andhere we perform a measurement of the microwave photondistribution functions in cases of large mean photon numbers.
We took notice of this problem in connection with at-tempts to observe the so-called dynamical Casimir effect inmicrowave cavities with time-dependent properties of bound-aries, where we expect to detect microwave quanta generateddue to the parametric amplification of the initial thermal (inperspective, vacuum) fluctuations in the fundamental field
mode [18]. In this case the mean number of created quanta canbe very big: hundreds or thousands, or even bigger. Thereforea different type of parametric amplifier has been used to studythe single-mode thermal field inside a resonator [19]. Thisamplifier is based on a microwave resonant cavity whoseproper frequency can be modulated by means of a variablecapacitance diode (varicap). By proper tuning of the varicap,a parametric amplification process can be initialized and willlead to different results depending on the amount of powerused to drive the varicap. The aim of this paper is to show ourrecent experimental results and their theoretical explanation.
Thanks to the presence of secondary resonances in thepumping system, our parametric amplifier can be used both inthe degenerate and nondegenerate modes. Moreover, by usingthe system in a pulse mode, it is possible to study the timeevolution of the energy present in the fundamental mode ofthe resonator. It is of particular interest to study this behaviorfor a cavity loaded with thermal photons, i.e., for the thermalradiation kept at some temperature �.
In the pulse mode the amplification is kept only for a finiteamount of time tF , that can be chosen at will. By performingrepeated measurements it is possible to make a histogram ofthe measured energy values Ei(tF ) reached inside the cavityat time tF : these values are different each time, since the inputpower has the thermal Planck (Bose-Einstein) distribution withaverage energy kB� (which is much bigger than the energyof microwave quanta hω for temperatures higher than 1 K).In this way we are able to study the “coarse-grained” energydistribution in the amplified cavity mode (since we cannotresolve discrete energy levels).
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G. GALEAZZI et al. PHYSICAL REVIEW A 88, 053806 (2013)
The plan of the paper is as follows. In Sec. II weprovide theoretical backgrounds, giving explicit formulasfor the photon distribution functions (PDFs) arising in theprocess of degenerate and nondegenerate quantum parametricamplification in the case of strong damping (compared withthe parametric modulation depth). In many available papersand textbooks these processes were considered under theassumption of small (or zero) dissipation, but in our casethe dissipation was strong. Special attention is paid to thederivation of the coarse-grained distributions in the two cases,since these distributions can be verified in the experiment.The experimental scheme is discussed in Sec. III, and theexperimental results are shown in Sec. IV. The last section isdevoted to conclusions.
II. THEORETICAL BACKGROUND
Our aim was to measure the energy distribution in thefundamental cavity mode, excited from the initial thermalstate by means of degenerate or nondegenerate parametricamplification. The key point for the theoretical description isthe fact that the initial state is the special case of the family ofGaussian quantum states. It is well known that initial Gaussianstates remain Gaussian in the process of linear amplification.The photon distribution function for the most general Gaussianstates was derived in [20–22]. For zero mean values of thequadrature field components, the probability P(k) to discoverk quanta depends on two parameters only: the mean quantumnumber N ≡ 〈n〉 = ∑∞
k=0 kP(k) and the purity of quantumstate μ = Tr(ρ2), where ρ is the statistical operator describingthe mixed quantum state of the field mode. The exact formulacan be expressed in terms of the Legendre polynomialsPk(x) as
P(k) = 2Dk/2−
D(k+1)/2+
Pk
( M − 1√D+D−
), (1)
where M = μ−2 � 1 and
D± = 1 + M ± 2G, G ≡ 1 + 2N . (2)
The parameter M for the Gaussian states with zero first-ordermoments of the quadrature operators can be expressed interms of the second-order moments 〈a2〉 and 〈a†a〉 ≡ N ofthe annihilation and creation operators as
M = G2 − 4|〈a2〉|2. (3)
In the thermal states we have 〈a2〉 = 0, so that M = G2. ThenD± = (G ± 1)2, M − 1 = √
D+D−, and formula (1) turnsinto the well-known Planck distribution,
Ptherm(k) = N k
(N + 1)k+1, (4)
with N = [exp(hω/kB�) − 1]−1, where ω is the modefrequency, � is the temperature of the thermal reservoir, andkB is the Boltzmann constant. But the evolution of this initialdistribution is different for degenerate and nondegenerateamplifiers.
A. Degenerate amplifier
The degenerate parametric amplifier can be described by themodel of a single harmonic oscillator, whose eigenfrequencyis modulated in time at twice the fundamental frequency[23]: ω(t) = ω0[1 + 2κ cos(2ω0t)]. For a small modulationdepth |κ| � 1 one can make additional simplifications, ar-riving at the Hamiltonian in the interaction picture [24]Hint = (ihχ/2)(a†2 − a2) (where χ = κω0), which yields theHeisenberg equations
da/dt = χa†, da†/dt = χa. (5)
The simplest formal way to include dissipation is to followthe approach by Lax [25], adding to the right-hand sides ofEqs. (5) two new terms [25–27]:
da/dt = χa† − γ a + ξ (t), (6)
da†/dt = χa − γ a† + ξ †(t). (7)
The terms −γ a and −γ a† (with γ � 0) can be interpretedas friction forces, whereas the stochastic noise operator ξ (t)with zero average values is necessary to save the commutationrelations. The solution to the above set of equations has theform
a(t) = e−γ t [a(0) cosh(χt) + a†(0) sinh(χt)] + Y (t), (8)
where [hereafter we use the notation ξ (t) ≡ ξt ]
Y (t) =∫ t
0eγ (τ−t){ξτ cosh[χ (t − τ )] + ξ †
τ sinh[χ (t − τ )]}dτ.
(9)
The simplest way to preserve the commutation relation[a(t),a†(t)] = 1 is to suppose that all the second-order mo-ments of noise operators ξτ and ξ †
η are proportional to the δ
function δ(τ − η). Then it is easy to verify that the canonicalcommutation relation (strictly speaking, its average value withrespect to stochastic sources) is preserved exactly duringthe evolution, provided [ξτ ,ξ
†η ] = 2γ δ(τ − η). This relation
is fulfilled if
〈ξτ ξ†η〉 = 2γ (1 + ρ)δ(τ − η), 〈ξ †
τ ξη〉 = 2γρδ(τ − η),
(10)
where ρ is a non-negative parameter. It is possible to introducean additional complex parameter r (responsible for the degreeof squeezing of the reservoir), writing 〈ξτ ξη〉 = γ rδ(τ − η).However, here we assume that r = 0, confining ourselves tothermal reservoirs only. Supposing that the initial operatorsa(0) and a†(0) are not correlated with the noise operators, weobtain [hereafter N (0) ≡ N0 and G(0) ≡ G0]
G(t) = G0 cosh(2χt)e−2γ t
+∑s=±1
γ (1 + 2ρ)
2(γ − sχ )[1 − e−2(γ−sχ)t ], (11)
〈a2(t)〉 = G0
2sinh(2χt)e−2γ t
+∑s=±1
sγ (1 + 2ρ)
4(γ − sχ )[1 − e−2(γ−sχ )t ], (12)
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so that
M(t) = G20
χ2 − γ 2[2χγ sinh(2χt)e−2γ t − γ 2 + χ2e−4γ t ].
(13)
If χ = 0, then 〈a2(t)〉 ≡ 0 and N (t) → ρ when t → ∞;therefore parameter ρ can be interpreted as the equilibriummean number of quanta in the thermal reservoir at the giventemperature. Hence we assume that N0 = ρ. We are interestedhere in the case when χ > γ and (χ − γ )t 1. Then
N |t→∞ ≈ G0χ
4(χ − γ )e2(χ−γ )t , (14)
D±|t→∞ ≈ e2(χ−γ )t G20χγ
χ2 − γ 2(1 ± R), (15)
M − 1√D+D−
∣∣∣∣t→∞
→ (1 − R2)−1/2, (16)
R = χ + γ
γG0= lim
t→∞(2G/M). (17)
The value of parameter R (17) is crucial for the asymptoticalphoton statistics (when exp[2(χ − γ )t] 1). If R 1, thenthe argument of the Legendre polynomial in formula (1) isclose to zero, and the photon distribution function P(k) showsstrong oscillations, being close to zero for odd values of k andnonzero for even values. The limiting case is the initial vacuumstate (G0 = 1) and very small damping, when the degenerateparametric amplifier generates the squeezed vacuum state.On the other hand, if R is small (in particular, for initialhigh-temperature thermal states with G0 1 and γ ∼ χ ,as was the case in our experiment), then no oscillations ofP(k) are observed. Actually, formula (1) is not convenient forcalculations if k 1. Simple asymptotical expressions werefound in [28]. Remarkably, it depends on the mean number ofquanta only, but the distribution is different from the Planckdistribution (4):
P(k) ≈ exp[−(2k + 1)/(4N )]√π (2k + 1)N
. (18)
It is valid for N (t) 1.
B. Nondegenerate amplifier
If the resonant system under study (with frequency ν1)has another resonant mode at frequency ν2 (or it is coupledto another system having resonance frequency ν2), then it ispossible to achieve an amplification of the initial field, varyingthe system parameters (e.g., the varicap capacitance) at thefrequency νvar = ν1 + ν2. The theory of such a nondegenerateparametric amplifier was developed by many authors since thefirst paper [29], but mainly in the absence of losses or forsmall losses. The starting point in this case is the two-modeHamiltonian
H = hω1a†1a1 + hω2a
†2a2 + ihχ (a†
1a†2e
−2iωt − a1a2e2iωt ),
where a1 (a2) is the annihilation operator for the signal (idler)mode, and 2ω = ω1 + ω2 (we assume that a small couplingconstant χ is real). The resulting Heisenberg equations ofmotion in the interaction picture (removing fast oscillations at
the frequencies ω1,2) are [24]
da1/dt = χa†2, da
†2/dt = χa1, (19)
and their exact solutions read
a1(t) = a1(0) cosh(χt) + a†2(0) sinh(χt), (20)
a2(t) = a2(0) cosh(χt) + a†1(0) sinh(χt). (21)
Formulas (20) and (21) show that 〈a2j (t)〉 ≡ 0 for the initial
thermal uncorrelated states of the two modes [with 〈a2j (0)〉 =
〈a1(0)a†2(0)〉 = 0]. Consequently, the relations Mj = G2
j aremaintained during the amplification process (so that the purityof each mode decreases due to the intermode interaction),and the photon probability distribution in each mode has thePlanck form (4), with the time-dependent functions Nj (t).This remarkable result was obtained for the first time (in amore complicated way) by Mollow and Glauber [30].
What can happen in the presence of dissipation? Inprinciple, different situations can arise, depending on theconcrete mechanisms of dissipation. Using the Lax approachagain [25], we modify Eqs. (19) as [26]
da1/dt = χa†2 − γ a1 + ξ1(t), (22)
da†2/dt = χa1 − γ a
†2 + ξ
†2 (t). (23)
The solutions to Eqs. (22) and (23) are similar to (8):
a1(t) = e−γ t [a1(0) cosh(χt) + a†2(0) sinh(χt)] + Y1(t),
a†2(t) = e−γ t [a†
2(0) cosh(χt) + a1(0) sinh(χt)] + Y†2 (t),
where responses to the noise sources are given by a formulasimilar to (9):
Y1,2(t) =∫ t
0eγ (τ−t){ξ1,2(τ ) cosh[χ (t − τ )]
+ ξ2,1(τ )† sinh[χ (t − τ )]}dτ. (24)
Assuming that operators of the first mode are uncorrelated withthe operators of the second mode at t = 0 (and uncorrelatedwith the noise operators), we can see that the commutationrelations [aj (t),a†
j (t)] = 1 are maintained provided
[ξ1τ ,ξ†1η] = [ξ2τ ,ξ
†2η] = 2γ δ(τ − η), [ξ1τ ,ξ2η] ≡ 0.
These relations are satisfied if we suppose that the noiseoperators of the two modes are uncorrelated, 〈ξ1τ ξ2η〉 =〈ξ1τ ξ
†2η〉 = 0, and the only nonzero δ-correlated second-order
moments of the noise operators are
〈ξjτ ξ†jη〉 = 2γ (1 + ρj )δ(τ − η), 〈ξ †
jτ ξjη〉 = 2γρj δ(τ − η),
where j = 1,2. Note that non-negative values ρ1 and ρ2 canbe different. Imposing the additional condition 〈ξjτ ξjη〉 = 0(unsqueezed reservoir), we arrive at the identity 〈a2
j (t)〉 ≡ 0for the initial thermal uncorrelated states of the two modes.Thus we have generalized the Mollow-Glauber result, havingdemonstrated that for nondegenerate amplifiers with arbitrarydamping coefficients, the initial Planck distribution over thephoton numbers maintains its form, although with time-dependent mean photon numbers. It can be shown that thisresult remains true if one introduces two damping coefficients,
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γ1 and γ2, instead of the single coefficient γ (only the formulasbecome more complicated). However, the situation will bedifferent if one adds to the right-hand sides of Eqs. (22)and (23) the terms −γ1a
†1 and −γ2a2, respectively. Also, the
Planck distribution will be deformed in nonthermal reservoirswith nonzero values 〈ξjτ ξjη〉, or in the case of nonzerocorrelations between the noise operators in different modes.
The explicit formula for the mean number of photons in thefirst mode N1 ≡ 〈a†
1a1〉 is as follows (for initial thermal anduncorrelated states of both modes):
N1(t) = e−2γ t {N1(0) cosh2(χt) + [1 + N2(0)] sinh2(χt)}+
∑s=±1
γ (1 + ρ1 + ρ2)
4(γ − sχ )[1 − e−2(γ−sχ)t ]
+ 1
2(ρ1 − ρ2 − 1)(1 − e−2γ t ). (25)
The mean energy of the mode is E1 = hω1(N1 + 1/2).
C. Binning
In order to have a measurable energy value, the ampli-fication factor is extremely large and the average number ofphotons that are present in the resonant system can be as high as1012. If we want to reproduce exactly the histograms describedby formulas (4) and (18), a huge number of measurementsmust be performed. Since this is not feasible, we measured a“coarse-grained” distribution, obtained by binning together b
adjacent values of k from the original distribution:
F (k) =(k+1)b−1∑
m=kb
P(m), 1 � b � N . (26)
In this way, even if the total number of measurements is smallerthan the amplified average number of photons, all bins aredifferent from zero. From an experimental point of view, it isconvenient to use the distribution normalized by the probabilityof the first bin F (0), i.e., F (k) ≡ F (k)/F (0).
In the case of the Planck distribution (4), the sum in (26)can be easily calculated exactly:
F (k) = (1 + 1/N )−bk = exp[−kb ln(1 + 1/N )].
Since N 1, this formula can represented as
F (k) = exp(−kb/N ) = exp(−k�E/E), (27)
where �E = bhω is the bin size and E = hωN is the meanenergy (shifted by the zero point energy hω/2) accumulatedin the mode. The remarkable feature of formula (27) is itsscaling property: it maintains the same analytical form for anybin size.
In the case of a degenerate amplifier, we inserted Eq. (18) inEq. (26) and replaced the summation by the integral over dm,assuming that 1 � b � N . Moreover, in this approximation,the quantity m + 1/2 can be replaced simply by m, as soonas the resulting integral does not diverge for m → 0. Thus wefind
F (k) ≈ �(√
b(k + 1)/(2N )) − �(√
bk/(2N )), (28)
where
�(x) = 2√π
∫ x
0exp(−t2)dt (29)
is the error function. Note that the coarse-grained distribu-tion (28) maintains the normalization exactly:
∞∑k=0
F (k) = �(∞) − �(0) = 1.
Moreover, since b/(2N ) � 1, we can replace the differenceof two error functions with close arguments by the first term ofthe Taylor expansion. Writing
√k + 1 ≈ √
k + 1/(2√
k) (thisapproximation is not bad even for k = 1, since
√2 = 1.41,
whereas the approximation yields√
2 ≈ 1.5), we obtain
F (k) ≈√
b
2πN kexp
(− bk
2N
)for k � 1, (30)
F (0) = �
(√b
2N
)≈
√2b
πN . (31)
The normalization is preserved again, in the sense that byneglecting F (0) and integrating function (30) from 0 to ∞we obtain exactly the unit value. For the function F (k) ≡F (k)/F (0) we obtain the expression (for k � 1, since F (0) ≡1 by definition)
F (k) = 1
2√
kexp
(− bk
2N
)= 1
2√
kexp
(−k�E
2E
),
(32)
which also does not change its analytical form under the energybin scaling.
III. EXPERIMENTAL APPARATUS
Principle experimental scheme of the apparatus is shownin Fig. 1. It is a much improved version of the waveguideparametric amplifiers built mainly in the 1960s [31]. A detaileddescription will be given elsewhere [32]. It is based on areentrant geometry microwave resonant cavity with circularsection, with a diameter of 42 mm and a height of 50 mm.Its resonance frequency νr is about 1.5 GHz, with a quality
FIG. 1. (Color online) A principle scheme of the microwaveparametric amplifier. The copper resonant cavity is kept inside acryogenic system that can be used both with liquid nitrogen andliquid helium. The varicap (Macom Ma 46470-91) is a cylinder witha height about 2.4 mm and 1 mm diameter.
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factor Q = πνrτc ≈ 2000 (so that the field decay time isτc ≈ 400 ns). The reentrant cylinder, having a diameter of4 mm, leaves a 5.2-mm gap in front of one of the cavity ends.This gap almost entirely determines the capacitance Cr of theequivalent circuit of the resonator. In front of the reentrantcylinder, a varicap diode is mounted, with a zero bias capacityof 1.5 pF. The varicap can be seen as a capacitance added inseries to Cr to give a total capacitance C ′
r and hence a differentvalue of the resonance frequency. By driving the varicap withan ac voltage, it is thus possible to periodically modulate theresonance frequency of the system, and the modulation depthstrongly depends on the position of the varicap itself inside theresonator.
By using the three ports P1, P2, and P3, it is possible tocharge the cavity, to measure the stored energy, and to drivethe capacitance of the varicap, respectively. Port P1 holdsa weakly coupled antenna with a coupling c1 ∼ 2 × 10−4
whose position is kept fixed. The antenna is connected tothe oscillator RF-I used to charge up the system in thecalibration measurements. Port P2 holds an antenna whoseposition can be varied to obtain critical coupling. This antennais connected to a 6-GHz bandwidth oscilloscope with inputimpedance Rg = 50 �. If the energy in the cavity is constant,the antenna delivers a stationary signal on the oscilloscope. Anac signal Vvar = V0 cos(2πνvart + θ ) applied by the oscillatorRF-P through port P3 is used to drive the varicap.
The two oscillators RF-I and RF-P are gated by thepulse generator PG; the relative phase of their outputs iscontrolled by a phase-locking system, permitting a completecharacterization of the parametric amplification process alsoat very large gain.
By driving the varicap at a frequency νvar ≈ 2νr , adegenerate parametric amplification of the energy stored inthe resonator can be realized. We have identified in our systema few more resonances ν(i) < νr . It is then possible to obtaina nondegenerate parametric amplification process by drivingthe varicap at a frequency νvar ≈ νr + ν(i). In the presence ofa parametric amplification process, the signal on the scopehas an exponential-like growth. The growth rate depends onthe efficiency of the parametric process and deviates towardssaturation in a short time.
As we do not work in a saturation regime, the oscillatorRF-P is activated only for a limited amount of time. We studythe time behavior of the amplitude Va at frequency νr of theantenna signal on port P2 when the pump RF-P is present. Theoutput power is then Pout = V 2
a /(2Rg), for a correspondingenergy in the cavity:
Ecav = c2Poutτc = c2V 2
a τc
2Rg
. (33)
For a critically coupled antenna the coefficient c2 equals unity.Analogously, for the input port, the energy loaded in the cavityis Ecav = c1Pinτc, with Pin the power delivered by the oscillatorRF-I.
The general trigger for a measurement is the leadingedge of a square pulse of the pulse generator PG: this is atransistor-transistor logic (TTL) signal with a duration of about40 μs that enables the output of the pump oscillator RF-P andstarts the acquisition by the oscilloscope. When performing
calibrations with a known input on port P1, the oscillator RF-Iis switched off at the start of the pump signal in order not tohave a constant input in the cavity during amplification. Theacquisition system of the oscilloscope limits the repetitionrate of the measurements to about 10 Hz, also ensuring thatthe cavity is again in a stationary state after the pump isswitched off.
IV. EXPERIMENTAL RESULTS
Choosing the varicap modulation frequency as νvar = 2νr ,we observed the degenerate amplification of the initial thermalfield in the cavity. As follows from formulas (11) and (14),the time behavior of the output power Pout(t) is described byan exponential law after the transient time of the order ofτp = [2(χ − γ )]−1:
Pout(t) = Ke(t−tref )/τp , (34)
where K is the value of the power at some reference timetref τp. Note that the constant K for each realizationhas different values, since it is proportional to the factor[19] cos2(θ − φ), where φ is the uncontrolled phase of thefluctuating thermal cavity field at t = 0 and θ is the initialphase of the varicap ac voltage. By repeating the measurementsat least several thousand times we obtained average values thatcan be compared with the quantum-mechanical average valuesfor the initial thermal state.
Note that the exponential growth given by Eq. (34) cannotlast forever, since the amplitude of the field saturates finally toa value related to the physical characteristics of the electricalcomponents. But for our purposes, the saturation part of Pout(t)was not used. The time constant τp of the parametric processis measured by feeding some known amount of power in thecavity through port P1 [19].
Data
In order to increase the dynamic range of the measureddata, measurements were performed as follows: a conventionalvalue PT was chosen so that the saturation part of Pout(t)begins well after this value is reached. The time tT at whichPout(tT ) = PT determines the overall amplifier gain etT /τP .Subsequent measurements with different input levels, as inthe case of a fluctuating thermal input, would then havedifferent values of tT and consequently of gain. To obtain thepower distribution at a fixed reference time tR , formula (34)is reversed to calculate for each measurement the valueP (tR) = PT e−(tT −tR )/τp . To ensure the validity of formula (34)we choose tR τp. This procedure is necessary since theoscilloscope reading the antenna output does not have largedynamic range. If we would turn on the pump exactly for thetime tR , in the case of a low input the amplified signal wouldstill be buried inside the detector noise.
All the collected power values can now be binned intohistograms. As discussed in the previous section, thesehistograms are normalized to the value of the first bin. This isdone in order to avoid the scale error on the determination ofthe total gain of the system, as was explained in Ref. [19]. Bydoing this, it is perfectly the same using power levels insteadof energy since they are related just by a constant factor.
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FIG. 2. (Color online) Coarse-grained distribution of the initialthermal field amplified with a degenerate parametric process. Com-parison of the experimental data (black dots with error bars) with theresults of formula (32). The values for k = 1,2 have been calculatedmore precisely. The bin size for the measured data was 1.1 × 1010
photons, which corresponds to an energy of 1.1 × 10−14 J.
The results are given in Figs. 2 and 3. Figure 2 showsthe experimental data obtained in the case of a degenerateamplification process: the cavity was kept at 77 K, andits resonance frequency was νr = 1541.25 MHz, so thatN (0) = kB�/(hνr ) ≈ 103. The pump frequency was νvar =3082.5 MHz with an amplitude of 19 dBm at the input ofthe cryostat port. We measured τp = 800 ns. Together withthe value τc = 400 ns, this gives γ = τ−1
c = 2.5 × 106 s−1
and χ = (2τp)−1 + γ = 3.125 × 106 s−1. Therefore param-eter R (17) equals R ≈ N−1
0 ≈ 10−3, so that we are inthe nonoscillating regime of the PDF, given by Eqs. (18)and (32). The reference time is tR = 17.9 μs. The total numberof measured values is 153 000. The measured probability,normalized to the value of the first bin, is then compared withthe formula (32). This formula is approximate for k 1, where
FIG. 3. (Color online) Coarse-grained distribution of the initialthermal field amplified with a nondegenerate parametric process.Comparison of the experimental data (black crosses with errorbars) with the results of formula (27). The bin size for themeasured data was 3.5 × 1011 photons, corresponding to an energy of3.5 × 10−13 J.
values are calculated more precisely. We see a good agreementwith (32). The distribution of an unperturbed thermal field ina cavity is given by the Planck formula (4), which would havethe form of a straight line in the logarithmic scale used in theplot of Fig. 2. Even taking into account the fact that data hasbeen rebinned, and we are now showing a coarse-grained dis-tribution, this would definitely not be able to fit the measureddata.
Figure 3 shows the experimental data obtained in thecase of a nondegenerate amplification process: the cavity waskept at 7 K, its resonance frequency was νr = 1541.6 MHz,and the pump frequency was νvar = 2908.8 MHz, with anamplitude of 12.4 dBm at the input of the cryostat port. Wemeasured τp = 351 ns. The reference time was tR = 8.4 μs.The total number of measured values is 10 000. The measuredprobability, normalized to the value of the first bin, is thencompared with formula (27).
As can be seen from both figures, the experimental data arein very good agreement with the theory. The measured distri-butions have been obtained at a single point in the temporalevolution of the amplification process. Due to the behavior ofthe parametric process, described by formula (34), the shapeof the distributions does not change after completion of thetransient, having duration of the order τp. The differentiationbetween a degenerate and nondegenerate process happensduring the transient time, where it is not possible to performmeasurements since the signal is too weak and still buried inthe noise.
The validity of formula (34) has been demonstrated in[19]. By using different conventional values PT during dataacquisition, we measured the thermal field distribution atseveral time values in the case of a nondegenerate amplificationprocess. The results are shown in Fig. 4. For these data thecavity was kept at room temperature, its resonance frequencywas νr = 1534.3 MHz, and the pump frequency was νvar =2909 MHz, with an amplitude of 17.3 dBm at the input ofthe cryostat port. We measured τp = 170 ns, and at room
FIG. 4. (Color online) Coarse-grained distribution of the initialthermal field amplified with a nondegenerate parametric process.Experimental data obtained at different times during amplificationsuperimposed with a fit to an exponential law. The bin size is thesame for all curves. The time for each measurement is indicated bythe proper label on each curve.
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EXPERIMENTAL STUDY OF MICROWAVE PHOTON . . . PHYSICAL REVIEW A 88, 053806 (2013)
temperature the cavity decay time was only τc = 250 ns.The shape of the distribution does not change with time; infact, all the data can be fitted with an exponential law of thetype given by formula (27). The slope of each distribution isproportional to the average energy E , and we have verified thatits time behavior follows the exponential growth described byformula (34).
V. CONCLUSION
We have measured the modification of coarse-grainedstatistics of microwave photons in the fundamental mode of aresonant cavity, generated from the initial thermal state in theprocesses of degenerate and nondegenerate amplification. Theexperimental results are in good agreement with theoreticalmodels, taking into account strong dissipation (comparedwith the modulation depth of the field mode frequency).The photon statistics are quite different in the two cases. Ithas the form of the Planck distribution with time-dependentmean number of quanta in the nondegenerate case (even inthe strong damping regime, which was not studied earlier),
whereas it is roughly the square root of the Planck distribution(multiplied by a slowly decaying factor) in the degenerate case.Our measurements were performed in the high-temperatureregime, where the parameter R (17), determining the formof the PDF, was very small. Our next goal is to use low-temperature (about a few Kelvin) superconducting cavitieswith high Q values (of the order of 104−105) with biggerresonance frequencies (about 5 GHz). Having the valuesN0 ∼ 10 and χ/γ ∼ 102, we will be able to reach the regime ofR � 1, where quantum oscillations of the photon distributionfunction could be seen.
ACKNOWLEDGMENTS
We thank S. Petrarca for stimulating the measurementand critical reading of the manuscript, as well as E. Bertoand F. Zatti for technical help. V.V.D. acknowledges thepartial support of the Brazilian funding agency CNPq andis thankful for the hospitality extended by Dipartimento diFisica e Astronomia of the University of Padova and LaboratoriNazionali di Legnaro of the INFN.
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