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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 12550–12558
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Analysis of the possibility of analog detectors calibration by exploiting Stimulated Parametric Down Conversion

Giorgio Brida, Maria Chekhova, Marco Genovese, and Ivano Ruo-Berchera  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 12550-12558 (2008)
http://dx.doi.org/10.1364/OE.16.012550


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Abstract

Spontaneous parametric down conversion (SPDC) has been largely exploited as a tool for absolute calibration of photon-counting detectors, i.e detectors registering very small photon fluxes. In [J. Opt. Soc. Am. B 23, 2185 (2006)] we derived a method for absolute calibration of analog detectors using SPDC emission at higher photon fluxes, where the beam is seen as a continuum by the detector. Nevertheless intrinsic limitations appear when high-gain regime of SPDC is required to reach even larger photon fluxes. Here we show that stimulated parametric down conversion allow one to avoid this limitation, since stimulated photon fluxes are increased by the presence of the seed beam.

© 2008 Optical Society of America

1. Introduction

Prompted by the necessity of a precise calibration of photo-detectors both in the analog and photon-counting regime [1

1. H. A. Bachor, A guide to experimental quantum optics (Wiley-VCH, New York, 1998).

, 2

2. M. Genovese, “Research on hidden variable theories: A review on recent progresses,” Phys. Rep. 413, 319–398 (2005) and references therein. [CrossRef]

, 3

3. See, for a general review, N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2000), and references therein. [CrossRef]

], we presented recently [4

4. G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, “The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study,” J. Opt. Soc. Am. B 23, 2185–2193 (2006). [CrossRef]

] a detailed theoretical analysis on the possibility to use the correlations of spontaneous parametric down conversion (SPDC) light [5

5. B. Y. Zel’dovich and D. N. Klyshko, “Statistics of field in parametric luminescence,” Sov. Phys. JETP Lett. 940–44 (1969).

, 6

6. D. C. Burnham and D. L. Weinberg, “Observation of Simultaneity in Parametric Production of Optical Photon Pairs,” Phys. Rev. Lett. 25, 84–87 (1970). [CrossRef]

] for calibrating analog detectors, as an extension of the technique developed in photon counting regime [7

7. D. N. Klyshko, “Use of two-photon light for absolute calibration of photoelectric detectors,” Sov. J. Quant. Elect. 10, 1112–1116 (1980). [CrossRef]

, 8

8. A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field,” Sov. Phys. JETP Lett. 33, 477–480 (1981).

, 9

9. D. N. Klyshko and A. N. Penin, “The prospects of quantum photometry,” Phys. Uspekhi 30, 716–723 (1987). [CrossRef]

, 10

10. A. Migdall, “Correlated-photon metrology without absolute standards,” Phys. Today January, 41–46 (1999). [CrossRef]

, 11

11. G. Brida, M. Genovese, and C. Novero, “An application of two photons entangled states to quantum metrology,” J. Mod. Opt. 472099–2104 (2000). [CrossRef]

, 12

12. V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, “Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect,” Opt. Eng. 32, 2911–2916 (1993). [CrossRef]

, 13

13. G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, “Conditioned Unitary Transformation on biphotons,” Phys. Rev. A 70, 032332 (2004). [CrossRef]

, 14

14. G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, “Single-photon detectors calibration by means of conditional polarization rotation,” J. Opt. Soc. Am. B 22, 488–492 (2005). [CrossRef]

, 15

15. G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, “Absolute quantum efficiency measurements by means of conditioned polarization rotation,” IEEE Trans Instrum. Meas. 54 NO. 2, 898–900 (2005). [CrossRef]

]. However, while the method is suitable for analog calibration at relatively low gain, which means a photon flux lower than 1010 photon/s, in higher-gain regime it is limited by the difficulty of collecting the same correlated modes in the two branches [4

4. G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, “The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study,” J. Opt. Soc. Am. B 23, 2185–2193 (2006). [CrossRef]

].

In this paper we present a detailed theoretical analysis of a scheme based on the stimulated PDC (on the other hand a discussion of the uncertainty budget is left to a specifically addressed paper [16

16. G. Brida et al., J. Mod. Opt. to appear.

]). We show that this scheme allow to overcome the problem mentioned above. Indeed, in this case the photon fluxes can be varied by varying the power of the coherent seed beam, without increasing the parametric gain G. On the other hand, the SPDC non-classical correlation at single-photon level which enables the absolute calibration of counting detectors survives, in some form, when a coherent seed is injected and the photon flux becomes macroscopic.

2. Analog detection and quantum efficiency

The schematic set-up for calibration of photodetectors by using PDC is shown in Fig. 1. We model the photodetection process in the analog regime as a random pulse train [17

17. K. Soda, I. Nishio, and A. Wada, “Analysis of noises in photocurrent time-correlation spectroscopy of scattered light,” J. Appl. Phys. 47, 729–735 (1976). [CrossRef]

]

i(t)=nqnf(ttn),

Fig. 1. Scheme for absolute calibration of analog detectors by using stimulated PDC. The quantum efficiency of detector 2, collecting the seed beam and the stimulated emission, is estimated by the ratio between the cross-correlation function of the photocurrents and the auto-correlation of the photocurrent i 1.
ij=nqjnf(ttn)=dtnqjf(ttn)F̂j(tn)
(1)

where the factor 〈qj〉 is the average charge produced in a detection event. We have assumed the response function for the two detectors to be the same, f 1(t)=f 2(t)=f(t).

At the same time, the auto-correlation and the cross-correlation functions for the currents can be expressed as

ij(t)ik(t+τ)=n,mqjnqkmf(ttn)f(ttm+τ)
=dtndtmqjqkf(ttn)f(ttm+τ)F̂j(tn)F̂k(tm),
(2)

respectively for j=k where 〈j(tn)j(tm)〉 is the auto-correlation function of the photon flux at detector j, and for jk where 〈j(tn)k(tm)〉 is the cross-correlation between the fluxes incident on the two different detectors. By the substitution j≡〈j〉+δF̂j, it is convenient to express them as

F̂j(tn)F̂k(tm)=F̂jF̂k+F̂jδ(tntm)δjk
+:δF̂j(tn)δF̂k(tm):.
(3)

The second contribution, proportional to the photon flux when j=k, represents the the intrinsic and unavoidable component of the fluctuation that does not depend on the specific property of the field since it generates from the commutation relation of the quantum fields in the free space. The third one is the normally ordered correlation function of the fluctuation. Actually it is a function just of the difference tn-tm and its typical variation scale provides the coherence time τcoh of the PDC radiation.

Now we introduce the quantum efficiency ηj of detector Dj, defined as the number of pulses generated per incident photon. In [4

4. G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, “The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study,” J. Opt. Soc. Am. B 23, 2185–2193 (2006). [CrossRef]

] a real detector is modelled, as usual, with an ideal one (η=1) preceded by a beam splitter with the intensity transmission coefficient equal to the quantum efficiency of the real detector [20

20. D. N. Klyshko and A. V. Masalov, “Photon noise: observation, squeezing, interpretation,” Phys. Uspekhi 38, 1203–1230 (1995). [CrossRef]

]. Within this picture it is possible to take into account the quantum efficiency by the following substitutions:

F̂j(t)ηjF̂j(t)
:F̂j(t)F̂k(t):ηjηk:F̂j(t)F̂k(t):.
(4)

Thus, 〈j(t)〉 being time independent, according to Eq. (1) we obtain:

ij=ηjqjFj.
(5)

Eq. (2) becomes

ij(t)ik(t+τ)=ijik+ηjqj2𝓕(τ)Fjδjk
+ηjηkqjqkdtndtm
f(ttn)f(ttm+τ):δF̂j(tn)δF̂k(tm):,
(6)

where we introduced the convolution of the response function of detectors 𝓕(τ)=∫dtf(t)f(t+τ).

3. Correlation functions of stimulated PDC

Here we consider a type I PDC process. In the limit of monochromatic and plane-wave pump approximation, only pairs of modes with opposite transverse wave vectors, q and -q, and with conjugate frequencies, ω 1=ωpump/2-Ω and ω 2=ωpump/2+Ω, are coupled such that the energy and momentum conservation hold. The equations describing the down conversion process are the input-output transformations

â1out(q,Ω)=U1(q,Ω)â1in(q,Ω)+V1(q,Ω)â2in(q,Ω)
â2out(q,Ω)=U2(q,Ω)â2in(q,Ω)+V2(q,Ω)â1in(q,Ω)
(7)

linking the fields âout 1 and âout 2 after the non-linear interaction to the fields âin 1 and âin 2 before the interaction started. The coefficients Uk(q,Ω) and Vk(q,Ω) must satisfy the properties

Uk(q,Ω)2Vk(q,Ω)2=1,(k=1,2)
U1(q,Ω)V2(q,Ω)=U2(q,Ω)V1(q,Ω),
(8)

which guarantee the conservation of the free-field commutation relations

[a^jout(q,Ω),a^kout(q,Ω)]=δj,kδq,qδΩ,Ω[a^jout(q,Ω),a^kout(q,Ω)]=0

Here, Uk and Vk define the strength of the process and at the same time the range of transverse momentum and frequencies in which it takes place and thus are named gain functions. In fact, the finite length of the crystal introduces a partial relaxation of phase matching condition concerning the third component of the momenta. By selecting a certain frequency, the transverse momentum uncertainty, i.e. the angular dispersion of the emission direction, is not null (Δq~(ltanϑ¯)-1 for the non-collinear PDC, where l is the crystal length and ϑ¯ is the central angle of propagation with respect to the pump direction). On the contrary, by fixing the transverse momentum q, or equivalently a direction of propagation ϑ, the spectral bandwidth ΔΩ is proportional to l -1/2, and typically 1/ΔΩ~10-13s for type I.

Hereinafter we will focus on the properties of the far field, observed at the focal plane of a thin lens of focal length f, placed at distance f from the crystal. The spatial distribution of the far field is, in this case, the Fourier transform of the field just at the output face of the crystal. This special imaging configuration is convenient to show the basic of calculation, nonetheless the validity of the results is more general. Thus, any transverse mode q is associated with a single point x in the detection (focal) plane according to the geometric transformation q→2π x/(λf). The far field operator in the space-temporal domain is therefore

B̂k(x,t)=ΩeiΩtâkout(2πλfx,Ω).
(9)

The mean value of the operator Îk(x,t)≡ k(x,t)k(x,t) is

Îk(x,t)=ΩΩei(ΩΩ)t
âkout(2πλfx,Ω)âkout(2πλfx,Ω),
(10)

representing the intensity profile of the emission in the detection plane. This mean value has to be calculated over the following initial state, in which the field 1 is in the vacuum state whereas the field 2 is in a multi-mode coherent state:

ψin=01q,Ωα(q,Ω)2,
(11)

where

α(q,Ω)=α(q)δΩ0Ω

is the complex parameter associated to this state. The seed beam is therefore represented as a coherent state of field 2 with fixed frequency Ω0 and a certain distribution of the transverse momentum q. Here we assume that, for the modes that are stimulated by the seed, the spontaneous component of the emission is negligible with respect to the stimulated one. This corresponds to the assumption |Vk(q0)|2≪1 and, at the same time, |α(q)|2≫1, that is a typical experimental situation in which a few millimeters length crystal is pumped by a continuous pump and the seed has a power just around the microwatt or more. The intensities (Eq. (10)) of the two stimulated beams after the crystal can be evaluated by using Eq. (7), with the substitution q→2π x/(λf). This leads to

Î1(x,t)V~1(x,Ω0)2α~(x)2
(12)
Î2(x,t)U~2(x,Ω0)2α~(x)2
=(V¯2(x,Ω0)2+1)α~(x)2
(13)

where we defined

U~k(x,Ω)=Uk(2πλfx,Ω),
V~k(x,Ω)=Vk(2πλfx,Ω),
α~(x)=α(2πλfx).
(14)

We used the first property of the gain functions in Eq. (8) into the last line of Eq. (13), in order to show explicitly the two contributions to the beam 2, one given by the original coherent seed and the other from the stimulated emission, proportional to the parametric gain Gmax|Vk(q0)|2. We notice that, aside from the spontaneous emission that is neglected, the generated beams conserve the original momentum distribution of the seed but weighted according to the gain function.

The normal-ordered correlation function of the intensities fluctuation is defined as

:δÎk(x,t)δÎj(x,t):=:Îk(x,t)Îj(x,t):
Îk(x,t)Îj(x,t)
(15)

The calculation leads to the following normal-ordered mean values:

:δÎ1(x,t)δÎ1(x,t):=0,
(16)
:δÎ2(x,t)δÎ2(x,t):Ωei(Ω0Ω)(tt)α~(x)2
V~2(x,Ω)2U~2(x,Ω0)2δxx
+c.c.
(17)

Fluctuations of the stimulated field 1 are negligible, because they have the same order of magnitude of the spontaneous emission. Concerning the cross-correlation we have:

:δÎ1(x,t)δÎ2(x,t):Ωei(Ω0+Ω)(tt)α~(x)2
U~2(x,Ω0)U~2(x,Ω)
V~1(x,Ω0)V~1(x,Ω)δxx
+c.c.
(18)

Let us consider two detectors, D 1 and D 2, to register the photons crossing the regions R 1 and R 2 centered on the two correlated bright beams in the detection plane. Let the two areas be much larger than the dimension of the seed beam on the detection plane. This simply means to collect all the bright components of the emission, corresponding to the modes stimulated by the seed.

We move now from the discrete modes representation of the electromagnetic field, in which it is artificially enclosed in a fictitious cube of side L, to a continuous set of modes for L→∞. Thus, in the following, the Ω-sum and the Kronecker function δ xx in Eqs. (17) and (18) are substituted respectively with the integral over Ω and the Dirac function δ (x-x′). The fact that the second-order correlation function has an unphysical spatial behaviour, dominated by a delta function, comes from the starting assumption of a plane-wave pump, having infinite transverse dimension. In a realistic case the delta function is replaced by the Fourier transform of the pump transverse profile.

The quantum mean values of the photon fluxes FjRjIĵ(x,t)dx reaching detectors 1 and 2 can be obtained integrating Eqs. (12) and (13) respectively on R 1 and R 2. By applying relations (Eq. (8)) we have

F1R1dxV~1(x,Ω0)2α~(x)2,
(19)
F2R2dx(V~2(x,Ω0)2+1)α~(x)2.
(20)

At the same time integrating Eqs. (16) and (17) over R 1×R 1 and R 2×R 2 respectively, we have the following auto-correlation functions for the photon fluxes

:δF1(t)δF1(t):0
(21)
:δF2(t)δF2(t):Ωei(Ω0Ω)(tt)
R2dxV~2(x,Ω)2α~(x)2+c.c.,
(22)

Here and in the following, we have assumed that in the limit of small G the gain functions |Ũj(x,Ω)|~1, according to the relations (Eq. (8)). Then, integrating Eq. (18) over R 1×R 2 gives the following cross-correlation:

:δF1(t)δF2(t):Ωei(Ω0+Ω)(tt)R1dx
α~(x)2V~1(x,Ω0)V~1(x,Ω)+c.c..
(23)

Although the seed beam is monochromatic with frequency Ω0, the temporal width of both autocorrelation and cross-correlation in a fixed point x still depends on the spectral bandwidth ΔΩ of the gain functions. This explains why they originate from the correlation between the monochromatic seed and the spontaneous emission that is broadband. Therefore, the coherence time of the stimulated emission is still of the order of τcoh=1/ΔΩ, analogously to the spontaneous case.

4. Quantum efficiency estimation

The auto-correlation of the current fluctuations δi 1(t)=i 1(t)-〈i 1〉 can be calculated by introducing the result in Eq. (21) in Eq. (6), obtaining

δi1(t)δi1(t+τ)=η1q12𝓕(τ)F1.
(24)

Under the condition of small gain (G≪1) and large intensity of the seed (|α|2≫1) the fluctuation of the current at detector D 1 is dominated by the shot noise component.

By substituting Eq. (23) in Eq. (6), under the same condition, for j=1 and k=2, one has

δi1(t)δi2(t+τ)=2η1η2q1q2𝓕(τ)F1
(25)

where we assumed τpτcoh. We stress that it is the usual situation, the coherence time of SPDC being on the order of picoseconds or less and the typical resolving time of detectors on the order of nanosecond. In this case any fluctuations in the light power are averaged over τp.

The cross-correlation function of the fluctuations has the same form as the one obtained in the case of spontaneous down-conversion in our earlier work [4

4. G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, “The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study,” J. Opt. Soc. Am. B 23, 2185–2193 (2006). [CrossRef]

]. This is the evidence that quantum correlations do not disappear when the emission is stimulated. In fact, the down-converted photons are still produced in pairs, although in the beam 2 the photons of the pairs are added to the bright original coherent beam propagating in the same direction. The factor 2, appearing in Eq. (25) can be interpreted as due to the fact that, for any down-converted photon of a pair propagating along direction 2, there is also the original photon of the seed that stimulated the generation of that pair. Formally it emerges from the time integration of two identical contribution in Eq. (23), the explicit one and its complex conjugated.

Since we are interested in a relatively large power of incident light, we can at first consider detectors without internal gain, and assume that the charge produced in any detection event is equal to the single electron charge q, i.e. 〈qk〉=q and 〈q 2 k〉=q 2. Therefore, according to Eqs. (24) and (25), the quantum efficiency can be evaluated as

η2=12δi1(t)δi2(t+τ)δi1(t)δi1(t+τ).
(26)

For detectors in which electrons multiplication occurs the statistic fluctuations of this process do not allow to use Eq. (26) for absolute calibration. However, it can be performed by integrating Eq. (25) over time τ [4

4. G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, “The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study,” J. Opt. Soc. Am. B 23, 2185–2193 (2006). [CrossRef]

]. It corresponds to evaluating the power spectrum of the fluctuations at frequencies around zero, namely much smaller than 1/τp. We would like to stress that in this case, the assumption f 1(t)=f 2(t)=f(t) is not necessary. Since dτ𝓕(τ)=1 we obtain

η2q2=12dτδi1(t)δi2(t+τ)i1,
(27)

In Eq. (27) the statistic of the electron gain does not play any role.

5. Conclusion

Acknowledgments

The Turin group acknowledges the support of Regione Piemonte (ricerca scientifica applicata E14) and San Paolo Foundation. The Russian group acknowledges the support of Russian Foundation for Basic Research (grant # 06-02-16393). Both acknowledge the joint project of Associazione Sviluppo del Piemonte by Grant RFBR-PIEDMONT # 07-02-91581-APS.

References and links

1.

H. A. Bachor, A guide to experimental quantum optics (Wiley-VCH, New York, 1998).

2.

M. Genovese, “Research on hidden variable theories: A review on recent progresses,” Phys. Rep. 413, 319–398 (2005) and references therein. [CrossRef]

3.

See, for a general review, N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2000), and references therein. [CrossRef]

4.

G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, “The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study,” J. Opt. Soc. Am. B 23, 2185–2193 (2006). [CrossRef]

5.

B. Y. Zel’dovich and D. N. Klyshko, “Statistics of field in parametric luminescence,” Sov. Phys. JETP Lett. 940–44 (1969).

6.

D. C. Burnham and D. L. Weinberg, “Observation of Simultaneity in Parametric Production of Optical Photon Pairs,” Phys. Rev. Lett. 25, 84–87 (1970). [CrossRef]

7.

D. N. Klyshko, “Use of two-photon light for absolute calibration of photoelectric detectors,” Sov. J. Quant. Elect. 10, 1112–1116 (1980). [CrossRef]

8.

A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field,” Sov. Phys. JETP Lett. 33, 477–480 (1981).

9.

D. N. Klyshko and A. N. Penin, “The prospects of quantum photometry,” Phys. Uspekhi 30, 716–723 (1987). [CrossRef]

10.

A. Migdall, “Correlated-photon metrology without absolute standards,” Phys. Today January, 41–46 (1999). [CrossRef]

11.

G. Brida, M. Genovese, and C. Novero, “An application of two photons entangled states to quantum metrology,” J. Mod. Opt. 472099–2104 (2000). [CrossRef]

12.

V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, “Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect,” Opt. Eng. 32, 2911–2916 (1993). [CrossRef]

13.

G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, “Conditioned Unitary Transformation on biphotons,” Phys. Rev. A 70, 032332 (2004). [CrossRef]

14.

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, “Single-photon detectors calibration by means of conditional polarization rotation,” J. Opt. Soc. Am. B 22, 488–492 (2005). [CrossRef]

15.

G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, “Absolute quantum efficiency measurements by means of conditioned polarization rotation,” IEEE Trans Instrum. Meas. 54 NO. 2, 898–900 (2005). [CrossRef]

16.

G. Brida et al., J. Mod. Opt. to appear.

17.

K. Soda, I. Nishio, and A. Wada, “Analysis of noises in photocurrent time-correlation spectroscopy of scattered light,” J. Appl. Phys. 47, 729–735 (1976). [CrossRef]

18.

R. G. McIntyre, “Multiplication Noise in Uniform Avalanche Diodes,” IEEE Trans. Electron. Devices , ED-13, 164–168 (1966). [CrossRef]

19.

The condition is fulfilled if the electronics is the same for the two detector and has a pass band much smaller than the band of the detectors.

20.

D. N. Klyshko and A. V. Masalov, “Photon noise: observation, squeezing, interpretation,” Phys. Uspekhi 38, 1203–1230 (1995). [CrossRef]

OCIS Codes
(120.1880) Instrumentation, measurement, and metrology : Detection
(120.3940) Instrumentation, measurement, and metrology : Metrology
(270.4180) Quantum optics : Multiphoton processes

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: May 30, 2008
Revised Manuscript: July 9, 2008
Manuscript Accepted: July 10, 2008
Published: August 5, 2008

Citation
Giorgio Brida, Maria Chekhova, Marco Genovese, and Ivano Ruo-Berchera, "Analysis of the possibility of analog detectors calibration by exploiting stimulated parametric down conversion," Opt. Express 16, 12550-12558 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12550


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References

  1. H. A. Bachor, A guide to experimental quantum optics (Wiley-VCH, New York, 1998).
  2. M. Genovese, "Research on hidden variable theories: A review on recent progresses," Phys. Rep.  413, 319-398 (2005) and references therein. [CrossRef]
  3. See, for a general review, N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2000), and references therein. [CrossRef]
  4. G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, "The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study," J. Opt. Soc. Am. B 23, 2185-2193 (2006). [CrossRef]
  5. B. Y. Zel�??dovich and D. N. Klyshko, "Statistics of field in parametric luminescence," Sov. Phys. JETP Lett. 9, 40-44 (1969).
  6. D. C. Burnham and D. L. Weinberg, "Observation of Simultaneity in Parametric Production of Optical Photon Pairs," Phys. Rev. Lett. 25, 84-87 (1970). [CrossRef]
  7. D. N. Klyshko, "Use of two-photon light for absolute calibration of photoelectric detectors," Sov. J. Quant. Elect. 10, 1112-1116 (1980). [CrossRef]
  8. A. A. Malygin, A. N. Penin, and A. V. Sergienko, "Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field," Sov. Phys. JETP Lett. 33, 477-480 (1981).
  9. D. N. Klyshko and A. N. Penin, "The prospects of quantum photometry," Phys. Uspekhi 30, 716-723 (1987). [CrossRef]
  10. A. Migdall, "Correlated-photon metrology without absolute standards," Phys. Today 52, 41-46 (1999). [CrossRef]
  11. G. Brida, M. Genovese, and C. Novero, "An application of two photons entangled states to quantum metrology," J. Mod. Opt. 47, 2099-2104 (2000). [CrossRef]
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