The quantum-classical transition in thermally seeded parametric downconversion
We address the pair of conjugated field modes obtained from parametric-downconversion as a convenient system to analyze the quantum-classical transition in the continuous variable regime. We explicitly evaluate intensity correlations, negativity and entanglement for the system in a thermal state and show that a hierarchy of nonclassicality thresholds naturally emerges in terms of thermal and downconversion photon number. We show that the transition from quantum to classical regime may be tuned by controlling the intensities of the seeds and detected by intensity measurements. Besides, we show that the thresholds are not affected by losses, which only modify the amount of nonclassicality. The multimode case is also analyzed in some detail.
The boundary between quantum and classical physics has been controversial [1, 2, 3, 4, 5, 6, 7, 8, 9] ever since the early days of quantum mechanics. Nevertheless, the solution to this problem is very important for several fundamental issues in quantum and atomic optics and, more generally, in quantum measurement theory [10, 11]. More recently, with the development of quantum technology, the issue gained new interest since nonclassical features, in particular entanglement, represent practical resources to improve quantum information processing.
As a matter of fact, quantum decoherence, i.e. the dynamical suppression of quantum interference effects, cannot be the unique criterion to define a classical limit , which should emerge from an operational approach suitably linked to measurement schemes [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. To this aim here we address the bipartite system made by the pair of field modes obtained from parametric downconversion (PDC) as a convenient physical system to analyze the quantum-classical transition in the continuous variable regime. We consider a PDC process seeded by thermal radiation, a scheme that we already investigated in ghost-imaging/diffraction experiments [30, 31, 32] and where it has been shown that both entanglement and intensity correlations may be tuned upon changing the intensities of the seeds . This, in turn, puts forward the PDC output as a natural candidate to investigate the quantum-classical transitions in an experimentally feasible configuration. We focus on some relevant parameters employed to point out the appearance of quantum features, namely sub-shot-noise correlations, negativity and entanglement. We analyze at varying the mean photon numbers of the interacting fields the different nonclassicality thresholds that appear. Remarkably, the corresponding transitions from classical to quantum domain may be observed experimentally by means of intensity measurements, thus avoiding full state reconstruction by homodyne or other phase-sensitive techniques [33, 34].
The paper is structured as follows. In the next section we review the PDC process, establish notation and introduce the nonclassicality parameters we are going to analyze. In Section 3 we analyze the effect of losses, whereas in Section 4 we discuss the generalization of our analysis to the multimode case. Finally, Section 5 closes the paper with some concluding remarks.
2 Parametric downconversion with thermal seeds
The evolution of a pair of field modes under PDC is described by the unitary operator , where is the coupling constant and are the mode operators . In the following we consider the two modes initially in a thermal state, i.e. excited in a factorized thermal state , being a single-mode thermal state with mean number of photons, i.e. . The density matrix at the output is given by , whereas the output modes are given by (with and ) where , and is the coupling (i.e. pump) phase. The statistics of the two output modes, taken separately, are those of a thermal state , i.e. , , where and is the mean number of photons due to spontaneous PDC; the symbols and denote and , respectively. Notice that the case of vacuum inputs, , corresponds to spontaneous downconversion, i.e. to the generation of the so-called pure twin-beam state (TWB) .
2.1 Intensity correlations
The shot-noise limit (SNL) in a photodetection process is defined as the lowest level of noise that can be achieved by using semiclassical states of light [36, 37, 38], that is Glauber coherent states. On the other hand, when a noise level below the SNL is observed, we have a genuine nonclassical effect. For a two-mode system if one measures the photon number of the two beams and evaluates the difference photocurrent the SNL is the lower bound to the fluctuations that is achievable with classically coherent beams, i.e. .
Let us consider a simple measurement scheme where the modes at the output of the PDC crystal are individually measured by direct detection and the resulting photocurrents are subtracted from each other to build the difference photocurrent. We have quantum noise reduction, i.e. violation of the SNL, when , that is 
In order to quantify intensity correlations and to evaluate the amount of violation of the SNL we introduce the parameter
The value corresponds to noise at the SNL, whereas the presence of nonclassical intensity correlations leads to . For the pair of modes at the output of the PDC crystal we obtain
The maximal violation of SNL is achieved by the TWB state (), while upon increasing the intensity of at least one of the seeding fields the SNL is eventually reached.
The nonclassical behaviour of a set of light modes has been often related to the negativity of the Glauber-Sudarshan P-function, which, in turn, prevents the description of the systems as a classical statistical ensemble. Here, in order to quantify negativity in terms of the photon number distribution, we employ the criterion introduced by Lee [39, 40], which represents the two-mode generalization of the Mandel’s criterion of nonclassicality  for single-mode beams and, in turn, implies the negativity of the P-function. According to Lee [39, 40], a bipartite system shows nonclassical behaviour if the inequality
is satisfied. For the PDC output state, the condition in Eq. (4) corresponds to . As we did in the case of intensity correlations, we define a parameter quantifying the amount of negativity
We have , with corresponding to maximum nonclassicality. For the PDC output state we obtain
Again the most nonclassical state is the TWB state (), whereas by increasing the intensity of at least one of the seeding field, the positive P-function region is eventually reached.
The PDC process provides pairwise entanglement in the two modes. In the spontaneous process the output state is entangled for any value of , whereas for thermally seeded PDC the degree of entanglement crucially depends on the intensity of the seeding fields . For a bipartite Gaussian state, entanglement is equivalent to the positivity under partial transposition (PPT) condition , which may be written in terms of the smallest partially transposed symplectic eigenvalue. Thus, seeded PDC produces an entangled output state if and only if 
Remarkably, entanglement properties of the state can be verified by intensity measurements independently performed on the two modes. In fact, with an ideal detection system, the inequality
reproduces exactly the entanglement condition in Eq. (7). Therefore, the amount of the violation of the separability boundary may be quantified by means of the parameter
corresponds to the boundary between separable and entangled states. For the PDC output we obtain
Maximally entangled states () thus correspond to the TWB (), whereas entanglement is degraded in the presence of thermal seeds. Notice, however, that if one of the two modes at the input is the vacuum, the state is always entangled irrespective of the intensity of the other seeds.
In Fig. 1 we show the nonclassicality regions in terms of the seeding, , , and downconversion, , mean photon numbers, i.e. the triples for which the parameters lie in the interval . As it is apparent from the plot, a hierarchy of nonclassicality concepts and thresholds naturally appears. The most stringent criterion of nonclassicality corresponds to negativity, followed by sub-shot-noise intensity correlations and then by entanglement.
We can express the thresholds for the appearance of nonclassicality as conditions on the mean number of photons resulting from the downconversion process
In other words, being negative-nonclassical is a sufficient condition to have sub-shot-noise intensity correlation. Moreover, either of the two (negativity and sub-shot-noise) is a sufficient condition for entanglement, i.e. for any value of and . Remarkably, the three nonclassicality conditions collapse into a single one when the seeding intensities are equal and differ by terms up to the second order in in the neighbourhood of this condition. It is already evident in Fig. 1, as well as in Fig. 2, where we show the three parameters as function of the seeding intensities for different values of , that the stronger is the spontaneous PDC (large ) the larger is the number of thermal photons that can be injected while preserving negativity and hence sub-shot-noise correlations and entanglement.
3 Effect of losses
In order to see whether the nonclassicality thresholds identified in the previous section may be investigated experimentally, one should take into account losses occurring during propagation, which generally degrade quantum features, and non-unit quantum efficiency in the detection stage, which may prevent the demonstration of nonclassicality. The two mechanisms may be subsumed by an overall loss factor [43, 44] using a beam splitter model [45, 46] in which the signal enters one port and the second one is left unexcited. Upon tracing out the second mode we describe the loss of photons during the propagation and the detection stage. In the following we assume equal transmission factor for the two channels and evaluate the nonclassicality parameters in the presence of losses.
Upon assuming that dark counts have been already subtracted, the positive operator-valued measure (POVM) of each detector is given by a Bernoullian convolution of the ideal number operator spectral measure. The moments of the distribution are evaluated by means of the operators
where . Of course, since are operatorial moments of a POVM, in general we have , with the first two moments given by
Upon inserting the above expressions in the nonclassicality parameters (i.e. replacing and by and respectively), we obtain that, for all of them, the inclusion of losses results in a simple re-scaling
In other words, the effect of losses is that of decreasing the amount of nonclassicality, whereas the thresholds for the quantum-classical transitions are left unaffected. This also means that the twin-beam still corresponds to the maximal violation of classicality condition independently of the kind of nonclassicality parameter we are taking into account. These are shown in Fig. 3, where the parameters for are compared with those in ideal condition for a fixed value of the PDC gain.
4 The multimode case
The (quantum) correlations introduced by the PDC process are intrinsically pairwise and thus no qualitative differences should be expected when considering the multimode case. On the other hand, the expression of the parameters does depend on the number of modes and thus it is worth explicitly addressing the multimode case . Besides, from the experimental point of view, this is a situation often encountered in travelling-wave PDC pumped by pulsed lasers.
The evolution operator for the multimode case can be rewritten in terms of the operators as thus emphasizing the pairwise structure. In our analysis we focus on the case in which all the modes are seeded with uncorrelated multi-mode thermal fields with mean photon number per mode
where . The density matrix at the output is thus given by
and the calculation for each pair of coupled modes is completely analogous to that performed in the first Section (see also ). The Heisenberg evolution of modes is
where and . The spontaneous PDC energy for each pair of modes is . In this case, the number of photons measured in each arm is , with (). The quantities relevant to our analysis are the mean photon values and the variances of the difference photocurrent , . Since correlations are only pairwise we have when and thus
Using this result, the extension to the multimode case for intensity correlations is straightforward, and the violation of the SNL in Eq. (1) can be rewritten as
If we assume that each mode of the seeding thermal fields in the arm () has the same mean photon number, , and that the parametric gain is the same for each pair of coupled mode, , the condition for the violation of the SNL in the multimode case is the same as for the single-mode seeds. The same is true in presence of losses, upon assuming equal transmission factor for the modes, as it can also easily seen by inspecting Eq. (21). On the other hand, for the negativity, as expressed by Eq. (5), the extension to the multimode case is not possible, since its derivation is explicitly based on the assumption of a single pair of downconverted modes [39, 40].
Finally, the separability/entanglement condition for the multimode thermally seeded PDC has already been analyzed , and it has been demonstrated that the separability properties of state may be checked by intensity measurements on the two arms, though not for a generic multimode field. An interesting case is when each mode of the seeding thermal fields in the arm () has the same mean number of photons, , and the parametric gain is the same for each pair of coupled modes, . In this case, the entanglement condition is still given by Eq. (7) and it is possible to reveal entanglement of the state by means of direct photon counting measurements on and arms exploiting the inequality
which is almost equal to Eq. (8) except for the second term where the number of modes appears. In fact, by starting from Eq. (21) and substituting the multimode expression of and , it can easily be proved that Eq. (22) leads to the entanglement condition in Eq. (7). As it has already been demonstrated  that the boundary between separability and entanglement is not modified by presence of losses, it is straightforward to prove that Eq. (22) still holds.
5 Conclusions and outlooks
In this paper we have addressed the quantum-classical transition for the radiation field in the continuous variable regime. We have analyzed in detail the pair of conjugated field modes obtained from parametric-downconversion and explicitly evaluated intensity correlations, negativity and entanglement for the system seeded by radiation in a thermal state. Our results have shown that a hierarchy of nonclassicality thresholds naturally emerges in terms of thermal and downconversion photon number and that the transition from quantum to classical regime may be tuned by controlling the seed intensities. The quantum/classical thresholds derived in this paper have two features that make them appealing for an experimental verification: i) they are not affected by losses, which only modify the amount of violation; ii) they can be verified by intensity measurements, without phase-dependent measurements and full state reconstruction. According to Fig. 1, in order to appreciate the differences among the criteria discussed above, the fields should contain a non-negligible number of photons coming both from the PDC process and from the seeds. We plan to generate such states by frequency-degenerate, noncollinear, travelling-wave PDC pumped by a high energy pulsed laser . In the experiment we should take advantage of the fact that can be reasonably high and bring the nonclassicality parameters to interesting regions. Tens of photons are expected from the process that may be measured by a pair of linear photodetectors with internal gain (photomultipliers or hybrid photodetectors) as described in . Besides, as an alternative to conventional crystals, a periodically-poled-non-linear waveguide medium and CW laser may be employed, aiming at the production of inherently single-mode (frequency) non-degenerate PDC light. Extension to the tripartite case [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59] is also in progress and results will be reported elsewhere.
M.G.A.P. thanks S. Olivares and M. Genoni for useful discussions. M.G. and I.P. D. thank M. Chekhova for useful discussions. This work has been supported by C.N.I.S.M., Regione Piemonte (E14), SanPaolo Foundation, Lagrange Project CRT Foundation.
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