_{1}

^{*}

An exact
quantum treatment reveals that signal and idler photon number operators are not
well-behaved dynamical operators for studying photon statistics in parametric
amplification/down-conversion processes. Contrary to expectations, the mean
signal-idler photon number difference

The fully quantized parametric amplification process treated in this paper consists of a pump photon of angular frequency ω, annihilation operator

where g is a constant coupling parameter.

Transformation to the interaction frame through application of a transformation operator

where the transformation operator takes the form

puts the Hamiltonian H from Equation (1) into an interaction form

which specifically describes the interaction of signal (

The time evolution equations governing the dynamics of signal and idler photons are obtained through Hei- senberg’s equations for the annihilation and creation operator pair (

which on substituting

The general procedure for obtaining exact analytical solutions to these equations has been developed by the present author in a recent paper on fully quantized parametric oscillation/frequency-conversion process [

The coupled time evolution Equations (3b), (3c) are expressed in an appropriate matrix form by introducing a two-component signal-idler photon annihilation-creation operator column matrix

to obtain

where

Introducing the

the Hamiltonian matrix in Equation (4c) is expressed in the form

where the frequency difference

An important point to note here is that, except for the factor

it may appropriately be referred to as an anti-Jaynes-Cummings Hamiltonian.

In this respect, the Hamiltonian

To gain complete understanding of the 2-level Jaynes-Cummings mode of interaction in the fully quantized pa- rametric amplification process, consider that according to Equation (4b), the Jaynes-Cummings interaction Ha- miltonian

after introducing the 2-dimensional Hilbert space basis vectors

It is clear that

gives

which leads to the standard interpretation that

The two-component vector

Using

the total signal-idler photon intensity operator

after introducing positive and negative helicity polarized signal-idler photon intensity operators

It follows from Equations (4b), (6a) and (7a) that the underlying dynamics in a fully quantized parametric amplification process is a two-state dynamics characterized by the time evolution of circular polarization state vectors of the coupled signal-idler photon pair generated by a Jaynes-Cummings mode of interaction. This leads to the general interpretation that the coupled signal-idler photon pair constitutes a composite circularly polarized two-state system specified by the positive and negative helicity states interacting with a single-mode quantized pump field equivalent to an (anti)-Jaynes-Cummings model for a single two-level atom. A comprehensive pres- entation of photon polarization state dynamics in linear and nonlinear quantum optics is currently under prepara- tion in a more elaborate paper by the present author.

A complete picture of the Jaynes-Cummings mode of interaction is obtained by transforming Equation (4b) back to the original frame by applying the inverse operator

which is used in Equation (4b) to obtain the effective time evolution equation in the original frame in the form

where the Hamiltonian

This essentially reverses the general transformation law in Equation (2a) as expected. Substituting

The obvious interpretation is that

The time evolution Equation (8b) can now be solved exactly by applying the usual procedure for solving the Jaynes-Cummings problem in quantum optics [

puts

where the operator

Using standard algebraic relations for

Since both components

where the initial polarization operator vector

after applying the commutation of

Substituting

after considering that the identity

after introducing the time evolving pump photon interaction operators

with Hermitian conjugates easily obtained as

Substituting

then reorganizing the result gives the final form

where the general time evolving signal photon annihilation operator

These are the desired exact analytical solutions determined within the Heisenberg picture, where they can be used in the calculation of mean values, fluctuations and cross-correlation functions of various physical quantities which characterize the dynamics of a fully quantized parametric amplification process generated by the trilinear Hamiltonian

The pump photon operators

which provide useful c-number variables

where

show that

State vectors which describe the fluctuations of various physical quantities (observables) are here referred to as fluctuation state vectors in the sense that their occupation numbers (or inner products) define the fluctuations. The interpretation of fluctuations as noise may also lead to a corresponding reference as noise state vectors. The fluctuation (noise) state vectors are generated through repeated application of annihilation (fundamental) or cre- ation (dual) operators on the appropriate initial state vectors. The action of the annihilation or creation operators causes de-excitation or excitation, which generally produce fluctuations or noise during measurements. In gen- eral, repeated applications of annihilation and creation operators in appropriate order are equivalent to opera- tions with corresponding observable operators formed as products of the annihilation and creation operators.

In this study, the initial state vector

For an observable represented by an operator

The Q-mean value is obtained as

while the Q-second moment is obtained as the occupation number (or inner product) according to

The Q-fluctuation is obtained using equations (16c)-(16d) in the general form

For two observables represented by operators

while the cross-correlation fluctuations defined by

are obtained according to

These results are general enough to apply to various operators and their Hermitian conjugates.

This section investigates the suitability of the signal and idler photon number operators

The signal and idler photon number operators

Setting

where

The photon number fluctuation state vectors in Equations (17b), (17c) are used in the general definitions in Equations (16c) and (16e) with

Finally, using equations (17b)-(17c) in the general definition in Equations (16f)-(16h) with

Substituting

+

which display some new, i.e., unfamiliar, features of signal-idler photon statistics appearing in the dynamics of a fully quantized parametric amplification process.

According to Equation (20a), the mean signal photon number

with

the time evolution becomes oscillatory and the beating of competing oscillations over the two time scales gene- rates fractional revivals in the mean signal photon number as demonstrated in

On the other hand, Equation (20b) shows that the mean idler photon number

These results show that signal and idler photon numbers have different modes of behavior, which may also account for the difference in number fluctuations in Equations (18b) and (18c).

The second very important feature arising as a consequence of the different time evolution patterns in Equa- tions (20a)-(20b) is the unexpected time evolution of the mean signal-idler photon number difference

This result contradicts the expected conservation of the mean signal-idler photon number difference,

Signal number in (20a) over scaled time τ = gt, n = 3, k = 5, ,

Idler number in (20b) over scaled time τ = gt, n = 3, k = 5, ,

emerges here as a new feature of the dynamics of a fully quantized parametric amplification process. The imme- diate physical implication is that the time varying

The third important feature follows from the results in Equations (19a)-(19b), which show that the signal-idler photon number cross-correlation functions are generally complex, with

The average signal-idler photon cross-correlation function denoted here by

which on substituting Equation (19a) and its complex conjugate in accordance with Equation (19b) becomes

It is interesting to introduce the polar forms

to express Equation (21b) in the convenient form

after introducing interaction parameter dependent phase differences

The amplitudes and phase angles (

It is clear that the phase angles vanish under the resonance condition k = 0 and the number cross-correlation function as presented in Equations (22b)-(22c) becomes real. But, for off-resonance dynamics with

An important observation is that the time variation of the mean signal-idler photon number difference

In the parametric approximation or semi-classical model where the pump photon is considered to be generated by a high intensity laser field, the mean pump photon number n is obtained from a c-number field amplitude α of very large magnitude according to

from which follows

Using Equation (22h) in Equations (18a) and (19a) gives the mean photon numbers, number difference and cross-correlation function in the parametric approximation/semi-classical model in the familiar forms [

Using Equation (22h) in Equations (18b)-(18c) and comparing the results with Equation (22k) gives the num- ber fluctuations in the parametric/semiclassical approximation in the form

These results show that, in the parametric approximation or the semi-classical model, the mean signal-idler photon number difference is conserved according to Equations (22i), (22j) and the number cross-correlation function is real according to Equation (22k). The simultaneous production of signal and idler photons predicted within the parametric/semi-classical approximation is based on Equations (22j) and (22l) [

The above results confirm that the complex forms and the time variation obtained in Equations (19a) and (20e), respectively, emerge entirely as quantum effects in the fully quantized parametric amplification process.

The main conclusion drawn from the investigation in this section is that the signal and idler photon number operators

This section establishes that the appropriate dynamical operators which characterize the dynamics and specify the dynamical symmetry group of the parametric amplification process are the polarization operators

where

and

Using Equations (23c), (23d) in Equation (23a) provides the desired polarization operators in the form

where

Application of the usual commutation relations between the annihilation and creation operators

which constitute the Lie algebra of the

Using Equations (24a), (24b) in Equation (1) puts the Hamiltonian H in the form

where the angular frequency difference

which leads to the conclusion that the polarization operator

It is necessary to study the behavior of the intensity operators which constitute the polarization operators

The time evolving positive and negative helicity intensity fluctuation state vectors

Using the intensity fluctuation state vectors from Equations (25b), (25c) according to the general definitions in Equations (16c) and (16e) with

The positive and negative helicity intensity cross-correlation functions and cross-correlation fluctuations are obtained using Equations (25b)-(25c) and the general definitions in Equations (16f) and (16h) with

Comparing Equations (26b) and (26d) gives the important result

Two important features characterizing the coupled signal-idler photon polarization state dynamics follow from the results in Equations (26a) and (26e). The results in Equation (26a) give the mean positive and negative helic- ity intensity difference as

which on using

provides the conservation law

where

The two results in Equations (27c) and (27d) show that in the fully quantized parametric amplification process, polarized signal and idler photons in the positive and negative helicity states are produced simultaneously.

Using the results and general definitions obtained above as appropriate give the polarization fluctuation state vectors

where

take the final form (noting

The mean values of the time evolving polarization operators in the Fock state are easily obtained using the general definitions as

where

Using Equations (15a)-(15d) in Equation (26a) and substituting the results into Equation (29a) gives

It is clear from Equation (29c) that the mean value

The result in Equation (29d) shows that

The fluctuations in

after applying the definitions of fluctuations in the intensities and cross-correlation functions obtained earlier. Application of Equation (26e) in Equations (30a), (30b) gives the final results

Noting that

the second moments

where Equations (31c), (31d) are noted to define some commonly calculated cross-correlation functions ex- pressed in normal or anti-normal order [

The fluctuations follow easily in the form

Finally, the definitions of

The time evolution of the fluctuations

This paper reveals that the internal dynamics of a fully quantized parametric amplification process is characte- rized by an (anti)-Jaynes-Cummings mode of interaction governing the time evolution of the polarization state vectors of the coupled signal-idler photon pair. The exact analytical solutions obtained in the paper have suc- cessfully revealed some uniquely quantum mechanical effects such as the unexpected time variation (non-con- servation) of the signal-idler photon number inversion (

Polarization fluctuations in (32d) over scaled time τ = gt, n = 3, k = 5, ,

parametric amplification processes. The conservation of the positive-negative helicity intensity inversion and the purely real cross-correlation function directly specify the simultaneous production of positive and negative he- licity polarized signal and idler photons. The dynamics of the fully quantized parametric amplification process has also been shown to be characterized by the fundamental quantum mechanical phenomenon of fractional re- vivals when the pump, signal and idler photons are in the initial Fock state. General collapses and revivals (not explicitly included in this paper) emerge when the pump photon is taken in an initial coherent state, as elabo- rated in [

In general, the exact analytical expressions obtained in this paper provide opportunities for in-depth studies of the statistical properties of fully quantized parametric amplification/down-conversion processes. The concept of fluctuation state vectors, which arises from the evaluation of expectation values of time evolving operators within the Heisenberg picture, constitutes an exact quantum state engineering process [

I thank my colleague Dr. B. O. Ndinya for valuable support in preparation of diagrams. This work was fully supported by Maseno University.