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 varies with time,
while the corresponding signal-idler photon number cross-correlation function is complex and
experiences an interference phenomenon driven by the interaction parameters.
The intensity operators and related polarization operators of the polarized
signal-idler photon pair in positive and negative helicity states are
identified as the appropriate operators specifying a conservation law and the
dynamical symmetry group (SU(1, 1))
of the parametric amplification process. The conservation of the mean positive
and negative helicity photon intensity difference and the purely real
positive-negative helicity intensity cross-correlation function correctly
account for the simultaneous production of polarized signal and idler photons
in positive and negative helicity states.
The fully quantized parametric amplification process treated in this paper consists of a pump photon of angular frequency ω, annihilation operator and creation operator interacting with a non-linear crystal to gener- ate two photons, the signal and idler, of angular frequencies, , annihilation operators, and crea- tion operators, , respectively. The model Hamiltonian for this process is obtained in the trilinear form [1] [2]
where g is a constant coupling parameter.
Transformation to the interaction frame through application of a transformation operator according to a general transformation law
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 (-mode) and idler (-mode) photons coupled by a sin- gle quantized pump photon (a-mode).
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 (,) in the form
which on substituting from Equation (2c) become
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 [3] . A summary of the procedure is presented in the next section to obtain exact analytical solutions for a fully quan- tized parametric amplification/down-conversion process.
2. The Matrix Method: Jaynes-Cummings Interaction
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 defined by
to obtain
where is a Hamiltonian matrix obtained as
Introducing the identity and the usual Pauli spin operators, , giving
the Hamiltonian matrix in Equation (4c) is expressed in the form
where the frequency difference and sum are defined by
An important point to note here is that, except for the factor in the interaction term specifying the parametric interaction mechanism, the Hamiltonian H in Equation (6a) takes the form of a Jaynes-Cummings interaction Hamiltonian [4] -[10] for a 2-level atom interacting with a quantized single-mode electromagnetic field. Since the interaction term of in Equation (6a) is anti-Hermitian according to
it may appropriately be referred to as an anti-Jaynes-Cummings Hamiltonian.
In this respect, the Hamiltonian in Equation (6a) is non-Hermitian, yet it generates the internal dynamics of a non-dissipative parametric amplification process governed by the original Hermitian Hamiltonian H in Eq- uation (1).
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 generates of the dynamics vector the signal-idler photon system by operating on the two-com- ponent operator defined in Equation (4a), which is now expressed in the appropriate form
after introducing the 2-dimensional Hilbert space basis vectors and defined as usual by
It is clear that takes exactly the form of a two-level atomic state vector as defined within the standard Jaynes-Cummings model in quantum optics [4] -[10] and in general quantum mechanics. In standard photon dy- namics, the basis vectors and are interpreted as the basic circular polarization state vectors. In partic- ular, using the Pauli matrix defined by
gives
which leads to the standard interpretation that is a positive helicity state vector, while is a negative helicity state vector for circularly polarized photons [11] .
The two-component vector is then specifically interpreted as a polarization operator vector for the coupled circularly polarized signal-idler photon pair. The component annihilation and creation operators and are interpreted as photon intensity operator amplitudes for signal and idler photons in positive and neg- ative helicity states and, respectively.
Using from Equation (7a) and its Hermitian conjugate obtained as
the total signal-idler photon intensity operator follows from the inner product according to
after introducing positive and negative helicity polarized signal-idler photon intensity operators, , re- spectively obtained as
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.
2.2. General Solution
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 from Equation (2b) according to
which is used in Equation (4b) to obtain the effective time evolution equation in the original frame in the form
where the Hamiltonian follows from the transformation in the form
This essentially reverses the general transformation law in Equation (2a) as expected. Substituting from Equation (6a) into Equation (8c) and applying standard algebraic relations gives the final form
The obvious interpretation is that in Equation (8d) is the full (anti)-Jaynes-Cummings Hamiltonian which generates the dynamics of a fully quantized parametric amplification process through its action on the signal-idler photon polarization operator vector according to Equation (8b). The anti-Hermitian property of its interaction term is responsible for the characteristic features of the amplification process. As observed earlier, the non-Hermitian nature of is not in any way associated with dissipation, since the (anti)-Jaynes-Cummings mode of interaction only characterizes the general dynamics generated by the original continuous variable operator Hermitian Hamiltonian H in Equation (1).
The time evolution Equation (8b) can now be solved exactly by applying the usual procedure for solving the Jaynes-Cummings problem in quantum optics [9] [10] . Adding and subtracting in Equation (8e) and in- troducing frequency detuning δ defined by
puts in the form
where the operator and interaction Hamiltonian H are defined by
Using standard algebraic relations for, , , , , easily gives
Since both components and commute with, they are constants of the motion. The Hamil- tonian is thus time-independent, leading to a solution of the time evolution Equation (8b) through simple integration giving
where the initial polarization operator vector takes the form in Equation (7a). The general time evo- lution operator governing the dynamics under has been obtained as
after applying the commutation of and as in Equation (9d) and using
Substituting from Equation (10d) into Equation (10b) and using from Equation (9c) gives in the form
after considering that the identity commutes with the rest of the operators to effect a factorization as appropriate. Expanding the exponential operator in Equation (11a) and carrying out straightforward algebraic manipulation with details presented in [3] yields the final form
after introducing the time evolving pump photon interaction operators, , , defined by
Substituting from Equation (11b) into Equation (10a), using the initial from Equa- tion (7a), together with T(t) from Equation (10c) and applying
then reorganizing the result gives the final form
where the general time evolving signal photon annihilation operator and idler photon creation operator have been obtained in the form
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 in Equation (1).
2.3. Pump Photon Operator Action on Fock State <img src="htmlimages\16-7501380x\f1133ce7-8cf8-40fa-89af-be29ec03f5cc.png" width="40" height="45" />
The pump photon operators, , , , together with their Hermitian conjugates defined in the set of Equations (12a)-(12h) act on the pump photon Fock state vector according to [3]
which provide useful c-number variables, , , in explicit forms to describe features of the dy- namics. It is to be noted that, generate eigenvalue equations, while acts like an annihilation opera- tor and acts like a creation operator. Indeed, using Equations (15b) and (15d) shows that the operator prod- ucts and act like photon number operators according to
where
show that is easily obtained by setting n → n + 1 in Equation (15b), while is easily obtained by setting n → n − 1 in Equation (15d). These simple operations apply to the pump photon Fock state and indeed to vari- ous forms of the pump photon initial state vectors generally expressible as superpositions of Fock state vectors.
3. Fluctuation (Noise) State Vectors
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 of the pump, signal and idler photons is taken as the Fock state vector defined using usual notation in the product form
For an observable represented by an operator at any time t ≥ 0, the Q-fluctuation state vector is ob- tained as
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, , where j , k are suitable labels, which can be numbers or appropriate symbols, the cross-correlation functions are obtained according to
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.
4. Signal and Idler Photon Number Fluctuation State Vectors
This section investigates the suitability of the signal and idler photon number operators, as dy- namical operators for studying photon statistics in a fully quantized parametric amplification process. In this re- spect, the observable operator is the photon number. For signal and idler photons, ,. Photon number fluctuation state vectors are defined to evaluate photon number mean values, fluctuations and cross-correlation functions. These will reveal some new, i.e., unfamiliar, features originating from the full quan- tum treatment of the parametric amplification process generated by the trilinear Hamiltonian in Equation (1).
The signal and idler photon number operators and generally used in studying photon statistics in the parametric amplification process in quantum optics [1] [2] [12] -[15] are defined by
Setting, in Equation (16b), using the definitions from Equation (17a) and applying Eq- uations (14b), (14c) and their Hermitian conjugates, together with Equations (15a)-(15e), gives the signal and idler photon number fluctuation state vectors and, respectively in the final form
where
The photon number fluctuation state vectors in Equations (17b), (17c) are used in the general definitions in Equations (16c) and (16e) with, to obtain the mean signal and idler photon numbers and and their corresponding fluctuations in the Fock state in the final forms
Finally, using equations (17b)-(17c) in the general definition in Equations (16f)-(16h) with, , , , easily gives the signal-idler photon number cross-correlation functions and the corresponding fluctuation in the Fock state in the form
4.1. New Features in Signal-Idler Photon Statistics
Substituting, and from Equations (15a)-(15c) into Equation (18a), the mean signal and idler pho- ton numbers are expressed in the explicit form
+ (20b)
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 evolves over two different time scales specified by and. The physical consequence is that for values such that
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 Figure 1.
On the other hand, Equation (20b) shows that the mean idler photon number evolves over a single time scale specified only by so that for values (implies), the time evolution of the mean idler photon number is composed of simple oscillations as demonstrated in Figure 2.
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 in the form
This result contradicts the expected conservation of the mean signal-idler photon number difference,
, generally obtained in the parametric/semi-classical approximation and inferred through the com- mutation of the corresponding number difference operator with the bilinear or trilinear Hamiltonian () of the parametric amplification process [1] [2] [13] -[16] . The time variation of
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 does not account for the experimentally or nu- merically observed simultaneous production of signal and idler photon pairs in a parametric amplification process. This observation may also be supported by the fact that the differing photon number fluctuations in Equations (18b)-(18c) mean that the measurements of signal and idler photon numbers involve different quan- tum noise levels, which can cause delays in the counts of either signal or idler photons. The important pheno- menon of simultaneous production of signal and idler photons is addressed through photon polarization state dynamics in the next section.
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 being the complex conjugate of. According to the definitions of the c-number variables, , in Equations (15a), (15c) and (17d), the complex form of the number cross-correlation functions is an off-resonance feature associated with non-zero values of the frequency detuning parameter k, i.e., is complex for, otherwise, the signal-idler photon number cross-correlation functions are real under the resonance condition,. Calculations performed under the resonance condition do not reveal the complex form of the signal-idler photon number cross-correlation functions.
The average signal-idler photon cross-correlation function denoted here by is obtained as
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 and defined by
The amplitudes and phase angles (,), (,) and (,) are obtained using Equations (15a), (15c) and (17d), respectively, in the form
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, Equ- ation (22b) shows that the signal-idler photon number cross-correlation function is characterized by an interfe- rence phenomenon controlled by interaction parameters. It is an interesting interference phenomenon characte- rized by two generally unequal phase differences and. The interference phenomenon in the off-reson- ance signal-idler photon number dynamics may also have some constructive or destructive effect on the ex- pected simultaneous production of signal and idler photons.
4.2. Recalling the Parametric/Semi-Classical Approximation
An important observation is that the time variation of the mean signal-idler photon number difference and the complex form yielding the interference phenomenon in the signal-idler photon number cross-correlation function are both quantum effects characterizing the dynamics of a fully quantized parametric amplification process. These effects do not appear in the parametric/semi-classical approximation.
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 [1] [16]
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) [1] .
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 and are not well-behaved and are therefore not suitable dynamical operators for study- ing photon statistics in a fully quantized parametric amplification process. The non-conservation of the mean number difference is inconsistent with the commutation relation, which shows that the number operator difference is a constant of the motion, while the number cross-correlation function is generally complex and experiences an interference phenomenon. In addition, comparing the re- sults obtained within the full quantum treatment in Equations (19a), (20e) with the results obtained within the parametric/semi-classical approximation in Equations (22j), (22k) as appropriate reveals contradictions between number-based photon statistics determined in the fully quantized and in the parametric/semi-classical approxi- mation models of the parametric amplification process. Polarized signal-idler photon intensity operators are identified to be the well-behaved dynamical operators for studying photon statistics in parametric amplification/ down-conversion processes in the next section.
5. The Polarization Operators and Symmetry Group of the Parametric Amplification Process
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 derivable from the polarization operator vector and its Hermitian conjugate ac- cording to
where
and are the usual Pauli spin operators defined according to
Using Equations (23c), (23d) in Equation (23a) provides the desired polarization operators in the form
where and are the positive and negative helicity intensity operators defined earlier in Equation (7g).
Application of the usual commutation relations between the annihilation and creation operators provides the algebraic relations
which constitute the Lie algebra of the symmetry group. By Equation (24d), commutes with all the other operators and may be identified with the identity operator in algebra. The set of polarization operators defined in Equations (24a), (24b) therefore constitute the continuous variable genera- tors of the symmetry group covering the dynamics of the non-degenerate parametric amplification process. The polarization operators thus specify the Lie group as the dynamical symmetry group of the parametric amplification/down-conversion process. The polarization states of the coupled signal-idler photon pairs are therefore appropriately specified by points on the two-sheet hyperboloid of the manifold.
Using Equations (24a), (24b) in Equation (1) puts the Hamiltonian H in the form
where the angular frequency difference and sum are as defined earlier in Equation (6b). An inconse- quential constant term in has been ignored. Application of Equation (24d) gives the important result
which leads to the conclusion that the polarization operator is a constant of the motion specifying a con- servation law governing the dynamics of the coupled signal-idler photon pairs in the fully quantized parametric amplification process. The polarization operator will be treated as a well-behaved dynamical operator if its mean value is conserved to maintain consistency with the result in Equation (24f). This is established below.
5.1. Intensity Fluctuation State Vectors
It is necessary to study the behavior of the intensity operators which constitute the polarization operators and according to Equation (24a). The definitions of the time evolving positive and negative helicity inten- sity operators and for the coupled polarized signal-idler photon pairs follow from Equation (7g) in the form
The time evolving positive and negative helicity intensity fluctuation state vectors and, re- spectively are easily obtained using Equations (14b), (14c), (25a) and the general definition in Equation (16b) with, in the final form (noting,)
Using the intensity fluctuation state vectors from Equations (25b), (25c) according to the general definitions in Equations (16c) and (16e) with, , the mean values and corresponding fluctuations of the positive and negative helicity intensities, in the Fock state are obtained in the final forms
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, , , in the final form
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 from Equations (15a)-(15d) to obtain
provides the conservation law
where, are the respective initial mean positive and negative helicity intensities. The result in Equation (26e) gives the cross-correlation coefficient
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.
5.2. Polarization Fluctuation State Vectors
Using the results and general definitions obtained above as appropriate give the polarization fluctuation state vectors, in the form
where and are already obtained in Equations (25b) and (25c), respectively, while the fluctua- tion state vectors and defined by
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 and are the mean positive and negative helicity intensities obtained earlier in Equation (26a).
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 evolves in time over two different time scales specified by and, which for develops into an oscillatory evolution cha- racterized by fractional revivals due to the beating of oscillations over the two time scales. This behavior takes exactly the form demonstrated in Figure 1.
The result in Equation (29d) shows that, which according to the definition in Equation (23a) for, also defines the mean intensity inversion between the positive and negative helicity states, is conserved in time. This is in complete agreement with the earlier result obtained in Equation (24f) establishing that the pola- rization operator is a constant of the motion specifying a conservation law governing the dynamics of a fully quantized parametric amplification process. This consistency leads to identification of the polarization op- erator (or generator) as defined in Equation (24a) as the appropriate well-behaved dynamical operator for specifying the conservation law governing the dynamics of a fully quantized parametric amplifica- tion process.
Fluctuations of the Polarization Operators
The fluctuations in and are easily obtained in the form
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, and cross-correlation functions, are defined by
where Equations (31c), (31d) are noted to define some commonly calculated cross-correlation functions ex- pressed in normal or anti-normal order [1] . Using Equations (28c), (28d) gives
The fluctuations follow easily in the form
Finally, the definitions of, in Equation (24b) easily gives the corresponding fluctuation state vectors, mean values and fluctuations in the respective forms (recalling Equation (31a)).
The time evolution of the fluctuations obtained using Equations (31f), (31g) in Eq- uation (32f) is characterized by fractional revivals as demonstrated in Figure 3.
6. Conclusions
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 () and the complex form as well as the interference phenomenon characterizing the signal-idler number cross-correlation function under off-reson- ance dynamics. The physical implication is that the signal and idler photon number operators are not well-be- haved operators for studying photon statistics in a fully quantized parametric amplification process. In particular, the time variation of the number inversion and the interference phenomenon mean that the observed simultane- ous production of signal and idler photons is not specified by the signal-idler photon number cross-correlations as generally believed. It has been established here that the dynamics is characterized by the polarization states of the coupled signal-idler photon pairs. Polarized signal-idler photon intensity operators and the related polariza- tion operators, which specify the dynamical symmetry group () of the system, have been identified as the well-behaved operators for describing photon statistics in the fully quantized (as well as the semi-classical)
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 [3] .
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 [17] and the correspond- ing density matrices can be used to study the entanglement properties of the fully quantized parametric amplifi- cation process. The positive and negative helicity intensity operators, together with the polarization operators, are useful in the construction of intelligent state vectors on an SU(1, 1) manifold [18] . In particular, polarization operators are currently employed in characterization and tomography of photon polarization states, mostly under parametric interactions [19] -[22] . These are important for applications in the design and implementation of quantum information processing, quantum computation, quantum tomography and other quantum-based preci- sion technologies.
Acknowledgements
I thank my colleague Dr. B. O. Ndinya for valuable support in preparation of diagrams. This work was fully supported by Maseno University.
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in (20a) over scaled time τ = gt, n = 3, k = 5, 
,
in (20b) over scaled time τ = gt, n = 3, k = 5, 
,
in (32d) over scaled time τ = gt, n = 3, k = 5, 
,