# Correlations in photon-numbers and integrated intensities in parametric processes involving three optical fields

## Abstract

Two strongly-pumped parametric interactions are simultaneously realized in a single nonlinear crystal in order to generate three strongly correlated optical fields. By combining together the outputs of two of the three detectors measuring intensities of the generated fields, we obtain the joint photocount statistics between the single field and the sum of the other two. Moreover, we develop a microscopic quantum theory to determine the joint photon-number distribution and the joint quasi-distributions of integrated intensities and prove nonclassical nature of the three-mode state. Finally, by performing a conditional measurement on the single field, we obtain a state endowed with a sub-Poissonian statistics, as testified by the analysis of the conditional Fano factor. The role of quantum detection efficiencies in this conditional state-preparation method is discussed in detail.

###### pacs:

42.50.-p, 42.50.Dv, 42.50.Ar, 03.65.Ud, 03.67.Bg## I Introduction

The quantum properties of photon pairs generated in the process of spontaneous parametric down-conversion (1); (2); (3) have been investigated in many experimental and theoretical works during the last 30 years. Pairwise character of the fields generated in this process has been used, e.g., for testing fundamental laws of quantum mechanics (3), as well as for quantum teleportation (4), quantum cryptography (5) and in metrology applications (6). The theory that describes this kind of interaction has been elaborated from several points of view for fields containing just a fraction of a photon pair (7); (8); (9) and for fields composed of many photon pairs (10). Also stimulated emission of photon pairs has been addressed (11); (12); (13).

More recently, the development of the field of quantum-information processing (14) has drawn attention to three-field quantum correlations. In fact, interesting correlations can be reached when strongly-pumped parametric down-conversion and parametric amplification are combined together through a common field. Such a system can be built in a single nonlinear crystal oriented in such a way that phase-matching conditions for both the interactions are fulfilled together (15). It has been demonstrated that the tripartite state generated in this way is endowed with entanglement in the number of photons. In particular as the constant of motion admitted by the hamiltonian that describes the process suggests, we obtain that the number of photons in one of the three generated fields is always equal to the sum of the photons in the other two fields (16). These properties make the system useful for the generation of nonclassical states by means of a suitable conditional state preparation. In more detail, if a given number of photons in field is detected then the remaining two fields and are ideally left in a state entangled in photon numbers, means the Fock state with photons in field and coefficients depend on the nonlinear interaction.

Here we present the experimental realization of this scheme and in particular we study the joint photocount statistics between one field (single field) and the sum of the other two (compound field) in order to find out the experimental conditions in which it is possible to realize a deterministic source of states entangled in the number of photons. We note that this source requires photon-number resolving detectors (17); (18); (19); (20); (21); (22).

Note that fields composed of photon pairs have been experimentally investigated (23); (24); (25); (26); (27) under conditions that allowed having up to several thousands of photon pairs per pump pulse. A detailed theory based on a multi-mode description of the generated fields has been developed for spontaneous (28) as well as for stimulated processes (29) with the aim to interpret the experimental data. In this work, as the photocounts from two of the three detectors are combined together, we also use this theory in order to correctly interpret and process the measurements. In particular, we can determine the joint single-compound-field (JSCF) photon-number distribution and the JSCF quasi-distributions of integrated intensities from the measured JSCF photocount distribution. Moreover, the conditional photocount and photon-number distributions, together with the corresponding Fano factors, are derived in order to point out the versatility of the scheme as a source of nonclassical states. To this aim, the quantum detection efficiency is an important parameter.

Theoretical description of the nonlinear process is presented in Sec. II. Photon-number distributions and quasi-distributions of integrated intensities as well as other properties of the measured fields are derived in Sec. III. Experimental results are discussed in Sec. IV. Sec. V gives conclusions.

## Ii Three-mode parametric process

We consider a three-mode parametric process in which two strong coherent classical fields pump two interlinked nonlinear interactions: a frequency down-conversion and a parametric amplification. The corresponding interaction Hamiltonian can be written as follows (30); (15):

(1) |

where (), , are the corresponding photon annihilation (creation) operators and and are coupling constants for frequency down-conversion and parametric amplification, respectively. Symbol h.c. stands for the hermitian conjugated term. Analytical solutions of the corresponding Heisenberg equations for the annihilation and creation operators are given in (15). They can be used for the determination of the normal three-mode characteristic function of the spontaneous process:

(2) | |||||

where () are parameters, c.c. means the complex conjugated term and

(3) |

The functions , , and are defined in (15). In order to study the nonclassical nature of the three-mode state, we introduce the determinants

(4) |

According to Eqs. (4) fields and are classically correlated whereas correlations between fields and ( and ) can lead to nonclassical behavior.

In the experiment, the outputs of the detectors placed on fields and have been summed with the aim to measure photocount correlations between the single field and the compound field formed by fields and . These correlations are important to test the performance of a source of states entangled in the number of photons and obtained after a conditional measurement of photons in field . They can be derived from the following normal characteristic function:

(5) | |||||

where

(6) |

To obtain Eq. (5) we have assumed that is real. We note that information about the coupling constants and can be obtained from the reconstruction of the photocount distributions provided that the interaction time is known.

We assume that each field is composed of independent temporal modes (27). Then the variances of integrated intensities can be expressed as follows:

(7) |

If photocounts belonging to fields and are combined together, we can write the variances of the integrated intensities in the following form:

(8) |

Real value of has been again assumed.

Note that the above-presented analysis can be generalized to stimulated processes (31). However, single as well as compound fields have to be stimulated in order to support nonclassical effects by interference terms. As for the compound field, stimulation of one of its components (i.e. field or ) is sufficient to observe increased nonclassical effects. Stimulation of only one field (i.e. field or or ) results in increased values of noise only.

## Iii Photon-number distributions and quasi-distributions of integrated intensities

The experimental scheme used to generate two nonlinear interactions is depicted in Fig. 1.

The harmonics of a continuous-wave mode-locked Nd:YLF laser regeneratively amplified at a repetition rate of 500 Hz (High Q Laser Production, Hohenems, Austria) provided two pump fields. In particular, the third harmonic pulse at 349 nm ( ps pulse-duration) was exploited as the pump field in frequency down-conversion, whereas the fundamental pulse at 1047 nm ( ps pulse-duration) was used as the pump field in parametric amplification. The two interactions simultaneously satisfied energy-matching ( and ) and type I phase-matching (, ) conditions, in which are the angular frequencies, denote the wave vectors and suffixes and indicate ordinary and extraordinary field polarizations. As depicted in the inset of Fig. 1, we set the pump-field direction so that the wave vector was normal to the crystal entrance face and propagated along the -axis of the medium. We also aligned the wave vector of the other pump field in the plane (, ) containing the optical axis (OA) of the crystal and the wave vector . As the nonlinear medium we used a -BaBO crystal (BBO, Fujian Castech Crystals, China, 10 mm 10 mm cross section, 4 mm thickness) cut for type-I interaction ( deg), into which both pumps were strongly focused. Typical intensity values of the pump fields were GW/cm for and GW/cm for . The required superposition in time of the two pump fields was obtained by a variable delay line.

As we have already shown in Ref. (16), we decided to generate three fields at non-degenerate frequencies by choosing a phase-matching condition in the plane (, ) (32). In order to investigate the nature of the state obtained by the interlinked interactions, we selected a triplet of coherence areas by means of pin-holes, whose sizes and distances from the crystal were chosen by searching for the condition of maximum intensity correlations between the generated fields (33). In fact, we expect strong correlations not only between the number of photons in the field and the sum of the other two fields (compound field), but also singularly among the numbers of photons in all pairs of fields. By applying this criterion, we put two pin-holes of 30 m diameter at distances cm and cm from the BBO along the path of the signal beam at 632.8 nm and of the idler beam at 778.2 nm, respectively. Moreover, as the beam at 446.4 nm has smaller divergence compared to the other two fields, we selected it by means of a 50 m diameter pin-hole placed at a distance cm from the crystal. The light was suitably filtered by means of bandpass filters and focused on each detector. In particular, as we performed measurements in the macroscopic intensity regime (more than 1000 photons per coherence area), we used three p-i-n photodiodes (two, D and D in Fig. 1, S5973-02 and one, D, S3883, Hamamatsu, Japan) as the detectors. We obtained the same overall detection efficiency () on the three arms by inserting two adjustable neutral-density filters in the pathways of fields and . The current output of the detectors was amplified by means of two low-noise charge-sensitive pre-amplifiers (CR-110, Cremat, Watertown, MA) followed by two amplifiers (CR-200-4 s, Cremat): to this aim, we connected the detectors and to the same amplifier device by means of a T-adapter. The two amplified outputs were then integrated by synchronous gated-integrators (SGI in Fig. 1, SR250, Stanford Research Systems, Palo Alto, CA) sampled, digitized by a 12-bit converter (AT-MIO-16E-1, DAQ National Instruments) and recorded by a computer.

From the collected experimental data we obtained the first (, ) and the second (, ) moments of the photocount (photoelectron) distribution of the single and compound fields, respectively. Moreover, the correlation between the number of photoelectrons in the single field and that in the compound field was measured. The unavoidable contributions of additive noise present in the detection chain can be quantified in an independent measurement and then subtracted from the experimental data. By correcting the moments of the photoelectron distribution for the quantum detection efficiency, we can obtain the moments for photons. Here we present the first (, ) and the second ( , , ) moments of the photon-number distribution:

(9) |

In addition, the moments for the integrated intensities can be determined as follows:

(10) |

In a theory based on the generalized superposition of signal and noise properties of the fields can be quantified using coefficients , , and number of temporal modes. These quantities can be determined from the moments of integrated intensities (28):

(11) |

The mean number of photons in mode is given by () whereas quantifies the correlations between the single field and the compound one. The number of modes can be determined either from the experimental data measured in the single or in the compound field (see Eqs. (11)). Ideally, should be equal to (15). This cannot be reached in a real experiment because of non-perfect alignment and detection noise. However, a correct alignment of the experimental setup allows to have so that we can define . A more detailed analysis concerning the determination of can be found in (34).

The JSCF photon-number distribution , which can be derived from the normal characteristic function in Eq. (5), is written as follows (28):

(12) | |||||

where is the gamma function. If determinant defined in Eqs. (6) is negative, the given field cannot be described classically. The quantities and occurring in Eq. (12) cannot be negative and can be interpreted as components of fictitious noise. In the ideal lossless case, and then the joint photon-number distribution takes the form of the diagonal Mandel-Rice distribution. As frequency down-conversion produces couples of photons, we expect that the highest values of the elements of are near the diagonal . In addition certain classical inequalities can be violated in this region (24).

If , at the boundary between the classical and nonclassical behaviors, the compound Mandel-Rice formula for JSCF photon-number distribution can be simplified as follows

(13) |

In the ideal case only the diagonal elements of the distribution are different from zero: in this case the post-selection scheme for the preparation of a state entangled in the number of photons perfectly works. However, losses present in any experimental implementation cause discrepancy from this ideal situation. In order to quantify this discrepancy, we determine the conditional compound-field photon-number distribution provided that photons are detected in the single field:

(14) |

Fano factor of photon-number distribution can be expressed as follows:

(15) | |||||

Note that the last approximation in Eq. (15) holds for and highlights that negative values of determinant cause sub-Poissonian conditional photon-number distribution. For the ideal lossless case, and Fano factor is equal to 0.

The correlations in the number of photons can also be quantified by the distribution of the difference photon number defined as:

(16) |

where is the Kronecker symbol. The variance of the difference can be lower than the sum of the mean photon numbers in all fields. This happens for nonclassical fields and we obtain sub-shot-noise correlations in this case (26); (16). Under this condition the so-called noise reduction factor (26),

(17) |

is lower than 1.

The JSCF photon-number distribution is given by Mandel’s photo-detection formula (3); (35); (28), which is defined in terms of the -ordered quasi-distribution of integrated intensities. This formula can be inverted in such a way that the quasi-distributions of integrated intensities can be written in terms of the distribution in Eq. (12) with the following results. If we have the -ordered determinant (, , ), the -ordered JSCF quasi-distribution of integrated intensities is a non-negative ordinary function (28):

(18) | |||||

where is the modified Bessel function.

On the other hand, for , the JSCF quasi-distribution of integrated intensities takes the form of a generalized function. It can be approximated using the following expression (28):

(19) | |||||

in which and . The quasi-distribution derived in Eq. (19) typically oscillates and has negative values in some regions. The threshold value giving the boundary between the expressions in Eqs. (18) and (19) can be determined from the condition :

(20) |

in which .

The variances of the difference of single- and compound-field photon numbers () and the difference of the single- and compound-field integrated intensities () are linked by the following formula:

(21) |

Equation (21) shows that negative values of the quasi-distribution of integrated intensities are needed to observe sub-shot-noise correlations in single- and compound-field photon numbers, i.e. in Eq. (17).

## Iv Experimental results and their interpretation

From the intensity measurements of the single and compound fields we calculated the following moments for photoelectrons: , , , , . Moreover, the knowledge of the quantum detection efficiencies allowed us to derive the corresponding moments for photons: , , , , .

In the case of photoelectrons the calculated values of the coefficients in Eqs. (11) are:

(22) |

We also note that the number of modes and relative to the single and compound fields, respectively, are almost the same. In the case of photons the values of coefficients , and are the following:

(23) |

The number of modes is the same for photocount and photon-number distributions.

Determinant is negative both for photoelectron- () and photon-number () distributions. At the same time, the fluctuations in the difference between the single and the compound fields both for photoelectrons () and for photons () are below the shot-noise level. In accordance with Eq. (21), this means that the wave variance of the difference between the single and compound field integrated intensities is negative: in fact we have the wave variance equal to for photoelectrons and for photons. These negative values are due to the fact that frequency down-conversion emits the same number of photons into the single field and in the compound field formed by fields and . For the same reason we obtain that the covariance for photoelectrons and the analogous expression for photons assume values very close to 1: for photoelectrons and for photons. Moreover, we note that even the principal squeezing parameter (36) indicates a nonclassical behavior: we obtained for photoelectrons and for photons; holds for coherent states. On the basis of the above mentioned quantities, we want to emphasize that the quantum nature of the state produced by the nonlinear process is more evident in terms of photons than in terms of photoelectrons.

The JSCF photocount, , and photon-number, , distributions calculated along the formula in Eq. (12) with values of parameters appearing in Eqs. (22) and (23) are shown in Fig. 2 and Fig. 3, respectively. There are strong photon-number correlations between the single field and the compound one. In particular, we note that the elements of and assume the highest values in the vicinity of the diagonal, for and (see contour plots in Figs. 2 and 3).

Values of Fano factors of the conditional photocount and photon-number distributions are important from the point of view of the conditional state-preparation scheme. We remind that a nonclassical state requires , i.e. sub-Poissonian statistics of the conditional distributions. In our case this requirement is not fulfilled by the photocount distribution, as shown in Fig. 4. This behavior is due to the relatively low quantum detection efficiency (). On the other hand, if the experimental data are corrected for the quantum efficiency, we can obtain a sub-Poissonian photon-number distribution for (see Fig. 5). The greater the number of detected single-field photons the smaller the value of Fano factor . Actually, the values of Fano factor do not change for numbers of and greater than 100.

The behavior of the JSCF quasi-distribution of integrated intensities depends on the value of the ordering parameter . In our case the threshold value of the ordering parameter, , is equal to 0.811 for photoelectrons. This means that the nonclassical behavior of the quasi-distribution is expected only for values of greater than . Indeed, in Fig. 6 we show that oscillations and negative values of occur for (); on the other hand, the nonclassical features do not appear for (). This means that the quantum noise present in the detection chain covers the nonclassical behavior. In the case of photons, the JSCF quasi-distribution has a lower threshold value of the ordering parameter, namely . For this reason, the JSCF quasi-distribution shown in Fig. 7 already shows nonclassical features like oscillations and negative values for .

It is clear from all these considerations that to appreciate the quantum features of the three-mode state and its usefulness for the preparation of a conditional state, the quantum detection efficiencies must be high enough. A typical dependence of the conditional Fano factor on the quantum detection efficiency () is plotted in Fig. 8 a) for . By decreasing the value of we obtain an increase in the value of . Moreover, it can be demonstrated that there is a threshold value of the efficiency below which the conditional compound-field photon-number distribution is no more sub-Poissonian. As shown in Fig. 8 b), the critical value decreases by increasing the number of the photons in the single field. We note that graphs in Figs. 8 and 9 have been obtained using the formula for Fano factor in Eq. (15) assuming substitution , , and and coefficients , , and for photons [Eqs. (23)]

In more detail, only the value of the quantum detection efficiency is crucial for the sub-Poissonian behavior of the conditional compound-field photon-number distribution, as shown in Fig. 9, where the contour plot of Fano factor is depicted as a function of detection efficiencies and for . In fact, for we have a sub-Poissonian behavior, which does not depend on the value of (we suppose that ). This means that when exceeds its critical value, the quantum detection efficiency corresponding to the conditionally prepared state will not affect the nonclassical behavior of the state itself.

Note that, from the experimental point of view, the nonclassical behavior of an optical state is usually exhibited by the condition . On the other hand, in accordance with the theoretical study presented in (28), nonclassical fields satisfy the condition . Actually, the two conditions can be linked together. In fact, if , negative values of the determinant , which testify a nonclassical character, lead to . By using Eq. (17) we thus obtain that , i.e. the sub-shot-noise reduction of fluctuations in the difference between the photons in the single and compound fields. However, it might happen that for and : in this case the noise reduction factor cannot be used to check the nonclassical nature of the measured state. However, when this condition occurs other quantities, such as the conditional Fano factor , can provide evidence of the nonclassical character of the fields.

## V Conclusions

The quantum properties of two interlinked nonlinear interactions generating a tripartite entangled state have been analyzed with special attention to the mutual correlations in the number of photons. In particular, we have taken into account the correlation between the photons in the first field and the sum of photons in the other two. The joint photon-number distribution, its conditional photon-number distribution as well as the joint quasi-distribution of integrated intensities have been determined to study the nonclassical properties of the measured fields. It has been shown that states entangled in photon numbers can occur in the second and third fields provided that a given number of photons is detected in the first field. In this entangled state, sum of photon numbers in the second and third fields equals the given number of photons in the first field and photon number in the second field (as well as in the third field) is not determined. The crucial role played by the quantum detection efficiencies in the production of the conditional state is widely discussed.

## Acknowledgments

This work was supported by projects KAN301370701 of Grant agency of AS CR, 1M06002 and MSM6198959213 of the Czech Ministry of Education.

### References

- D.F. Walls, G.J. Milburn, Quantum Optics (Springer, Berlin, 1994) chap. 5.
- L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995) chap. 22.4.
- J. Peřina, Z. Hradil, B. Jurčo, Quantum Optics and Fundamentals of Physics (Kluwer, Dordrecht, 1994) chap. 8.
- D. Bouwmeester, J.W. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Nature 390, 575 (1997).
- D. Bruß, N. Lütkenhaus, in Applicable Algebra in Engineering, Communication and Computing Vol. 10 (Springer, Berlin, 2000), p. 383.
- A. Migdall, Physics Today 1, 41 (1999)
- T.E. Keller, M.H. Rubin, Phys. Rev. A 56, 1534 (1997).
- J. Peřina Jr, A.V. Sergienko, B.M. Jost, B.E.A. Saleh, M.C. Teich, Phys. Rev. A 59, 2359 (1999).
- J. Peřina Jr., M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, Phys. Rev. A 73, 033823 (2006).
- E.M. Nagasako, S.J. Bentley, R.W. Boyd, G.S. Agarwal, J. Mod. Opt. 49, 529 (2002).
- A. Lamas-Linares, J.C. Howell, D. Bouwmeester, Nature 412, 887 (2001).
- F. De Martini, V. Bužek, F. Sciarrino, C. Sias, Nature 419, 815 (2002).
- D. Pelliccia, V. Schettini, F. Sciarrino, C. Sias, F. De Martini, Phys. Rev. A 68, 042306 (2003).
- D. Bouwmeester, A. Ekert, A. Zeilinger (Eds.) The Physics of Quantum Information (Springer, Berlin, 2000).
- A. Allevi, M. Bondani, A. Ferraro, M.G.A. Paris, Laser Physics 16, 1451 (2006).
- A. Allevi, M. Bondani, M.G.A. Paris, A. Andreoni, Phys. Rev. A 78, 063801 (2008).
- J. Kim, S. Takeuchi, Y. Yamamoto, H. H. Hogue, Appl. Phys. Lett. 74, 902 (1999).
- A.J. Miller, S.W. Nam, J.M. Martinis, A.V. Sergienko, Appl. Phys. Lett. 83, 791 (2003).
- O. Haderka, M. Hamar, J. Peřina Jr., Eur. Phys. J. D 28, 149 (2004).
- J. Řeháček, Z. Hradil, O. Haderka, J. Peřina Jr., M. Hamar, Phys Rev. A 67, 061801(R) (2003).
- D. Achilles, Ch. Silberhorn, C. Sliwa, K. Banaszek, I.A. Walmsley, J. Mod. Opt. 51, 1499 (2004).
- M.J. Fitch, B.C. Jacobs, T.B. Pittman, J.D. Franson, Phys. Rev. A 68, 043814 (2003).
- A. Agliati, M. Bondani, A. Andreoni, G. De Cillis, M.G.A. Paris, J. Opt. B: Quant. Semiclass. Opt. 7, S652 (2005).
- O. Haderka, J. Peřina Jr., M. Hamar, J. Peřina, Phys. Rev. A 71, 033815 (2005).
- O. Haderka, J. Peřina Jr., M. Hamar, J. Opt. B: Quantum Semiclass. Opt. 7, S572 (2005).
- M. Bondani, A. Allevi, G. Zambra, M.G.A. Paris, A. Andreoni, Phys. Rev. A 76, 013833 (2007).
- F. Paleari, A. Andreoni, G. Zambra, M. Bondani, Optics Express 12, 2816 (2004).
- J. Peřina, J. Křepelka, J. Opt. B: Quant. Semiclass. Opt. 7, 246 (2005).
- J. Peřina, J. Křepelka, Opt. Commun. 265, 632 (2006).
- E.A. Mishkin, D.F. Walls, Phys. Rev. 185, 1618 (1969).
- J. Peřina, J. Křepelka, Opt. Commun. 281, 4705 (2008).
- M. Bondani, A. Allevi, E. Gevinti, A. Agliati, A. Andreoni, Opt. Express 14, 9838 (2006).
- A. Allevi, M. Bondani, M. G. A. Paris, A. Andreoni, Eur. Phys. J. ST 160, 1 (2008).
- J. Peřina, J. Křepelka, J. Peřina Jr., M. Bondani, A. Allevi, A. Andreoni, Phys. Rev. A 76 043806, (2007).
- B.E.A. Saleh, Photoelectron Statistics (Springer-Verlag, New York, 1978).
- J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena (Kluwer, Dordrecht, 1991).