Mean photon number

Figure 4. Conditional nonclassical state generation. Conditional (f f (.4,9) as a function of the number of detected Stokes photons. Diamonds show experimentally measured values, which are calculated from the two arms of the anti-Stokes beam-splitter via g (A.S ) = (AS AS-2)/ AS ) AS-2) (see Fig. 1 C). The measured mean photons number on the Stokes and anti-Stokes channels were fis = 1.06 and has = 0.36 respectively. The solid line shows the result of a theoretical model including background and loss on both the Stokes and anti-Stokes channels. The overall detection efficiency (a) and number of background photons (hbg ) used in the model were as = 0.35, n G = 0.27 (qas = 0.1, rdfs = 0.12) on the Stokes (anti-Stokes) channel, and were estimated from experimental measurements. For these measurements an optically-pumped 87Rb cell was used to filter the Stokes photons from the write laser. The dotted line represents < ns (AS) corrected for loss and background on the anti-Stokes channel, obtained by setting the anti-Stokes channel loss and background to zero in this model. Inset measured mean anti-Stokes number n s conditioned on the Stokes photon number ns- The solid line represents n s as predicted by the model.

Table 1. Scaling for the anti-Stokes pulse Q-parameter and Fock state fidelity F. n refers to the mean number of excitations in the rubidium cell, nfG is the mean photon number of background counts in the write channel ( we assume they are mainly due to leak of the write drive and so follow Poisson statistics), as is the Stokes detection efficiency and ns is the number of Stokes photons on which we condition. The mean number of atomic excitations is calculated via (nsw) = Tr (pas nsw), similarly (n2sw = Tr (pas h2sw). The subscript T (P) refers to thermal (Poisson) photon statistics of the unconditional Stokes light.

Here the mean photons number of subharmonic modes are represented as the sum n = rid + Sn of the semiclassical and quantum parts. Straightforward, but complicated analytical calculations (see details in [Kryuchkyan 2004]) show that 5n —> —0.125 in the limit E —> 00, which leads to the asymptotic value Kuin = 0.75 < 1. Therefore, as the analysis shows, allowing for quantum fluctuations of arbitrary level, CV entanglement is always achieved in the NOPO... 

We shall study the solution of stochastic equations in the semiclassical treatment, neglecting the noise terms, for mean photon numbers rij and phases ipj of the modes (rij = ay (3j, ipj = /(ij)/ li) for time-intervals exceeding... 

We present the final results for the case of harmonic modulation with modulation amplitude /(f) = f + fi cos(5t + ), assuming, without loss of generality, / > 0, f > 0 and 4 0. In this case the mean photon number... 

First, we shall study the steady-state solution of the stochastic Eqs. (10a) to (lOd) in the semiclassical treatment, ignoring the noise terms for the mean photon numbers rijo and the phases decay rates and the detunings do not depend on the polarization (71 = 72 = 7, Ai = A2 = A). 

Generally, the second-harmonic generation is described by the quantum state (127) and we use this state in our further calculations. Classical solutions discussed earlier, ua x) = sech x and = tanh x, indicated that the amplitudes of the two modes are monotonic functions of time and that eventually all the energy from the fundamental mode will be transferred into the second-harmonic mode, assuming that there was no second-harmonic signal initially. It is well known [20,48], however, that the quantum solution has oscillatory character and does not allow for the complete power transfer. Using the state (127) we find that the mean photon numbers evolve in time according to the formulas... 

Figure 15. Maximum efficiency of energy transfer in the degenerate downconversion versus the initial mean photon number of the pump mode.

We see that the total energy can be found without any knowledge of the Bogoliubov coefficients. However, these coefficients are necessary, if one wants to know the distribution of the energy or the mean photon numbers over the modes. To solve the infinite set of equations (33), we introduce the generating function... 

The function (151) can be simplified in the long-time limit t> 1, when the average number of created photons, /L n h fa (V +U)/2 exceeds 1. Then the mean-square fluctuation of the photon number has the same order of magnitude as the mean photon number itself, s/2 Jf, and the most significant part of the spectmm corresponds to the values n > 1. Using the Laplace-Heine asymptotical formula for the Legendre polynomial [283]... 

First, for simplicity, we assume that the mean photon numbers of chaotic photons in both modes are the same (wch,a) = ( <, ) = (wch) and the initial coherent amplitudes 7 (l are real. We find... 

As the second example, we analyze another special case of the field (72), evolving in a way opposite the field evolution under initial condition (74). Let the signal mode is initially coherent (with real amplitude aHo), whereas the idler mode is chaotic (with the mean photon number (nch,i>))- We obtain the following... 

The time dependence of Nj(x) in Eq. (104) resembles that of the pump field, if DpjL characteristic time of the change of pump-field intensity). This means that one-photon multimode Fock-state fields with a given mean-photon-number time dependence can be generated for suitably chosen pump-field-intensity profiles. For instance, if the pump field consists of two femtosecond pulses of the same duration and one has no chirp whereas the other one is highly chirped, the overall pump field as well as Nj(t) have a peaked structure (see Fig. 17). 

Rate equations expressed in terms of mean photon number and quantization volume are still not directly applicable to experiment. Moreover, since the quantization volume is no more than a theoretical artifact, it must invariably cancel out in any final result. However, the ratio of these two quantities, which represents a mean photon density, is directly related to the mean irradiance, and the relationship may be derived as follows. Consider a quantization volume represented by a small cube of space of side length l and volume V through which the incident beam passes by definition, this cube contains on average q photons of circular frequency m, and its energy content is qhatk (see Fig. 4). For... 

In these equations the position of the molecule is described by the vector R the wavevectors of the two beams of modes r2 and are k2 and k3 respectively, with ( 2) and (q3) the corresponding mean photon numbers (mode occupancies) and is a unit vector describing the polarization state of mode rn. In deriving Eqs. (120) and (121), the state vectors describing the radiation fields have been assumed to be coherent laser states, and so, for example, (<72) = (oc n a(2 ), where a ) is the coherent state representing mode 2 and h is the number operator a similar expression may be written for (<73). Also, the molecular parameters apparent in Eqs. (120) and (121) are the components of the transition dipole, p °, and the index-symmetric second-order molecular transition tensor,... 

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