Wigner Quasiprobability with an Application to Coherent Phase States

Starting from Wigner’s definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables ( ) , q p of phase space and using the known relation to the parity operator. One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics. For the coherent states we show that their Fourier transforms are again coherent states. We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries. It is not well suited for the calculation of expectation values. A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new. We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult. The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values. By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics. Both considered examples, the square well and the coherent phase states, belong to systems with ( ) 1,1 SU symmetry with the same index 1 2 k = of unitary irreducible representations.

One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics. For the coherent states we show that their Fourier transforms are again coherent states. We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries. It is not well suited for the calculation of expectation values. A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new. We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult. The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values. By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics. Both considered examples, the square well and the coherent phase states, belong to systems with

Introduction
Most representations of probability theory begin with the discussion of some examples where probabilities play a main role and introduce then axiomatically the probability as a positively semi-definite and "normalized" function over a set of events. The main purpose of this function is to allow us to calculate mean values and their variances or more generally expectation values for arbitrary functions over the set of events when the initial conditions or the prehistory of the events are not fully under control of the experimenter or observer. The results of such calculations are then true in the mean for great ensembles of "equal" events and of their dynamics which are made under the same uncontrolled or uncontrollable initial conditions. In Hamilton dynamics of a system of one degree of freedom a trajectory is fully determined by a pair ( ) , q p of canonical coordinate and momentum in two-dimensional phase space as initial condition and the probability function is given by a positive semi-definite function ( ) , F q p (called distribution or partition function) and from the dynamics of single trajectories, in principle, can be determined by the time evolution of the function ( ) , F q p (the lately discovered cases of possible chaotic dynamics destroy this assumption in some way). After foundation of probability theory in 17-th century mainly by Blaise Pascal and great contributions by Jakob B. 1654-1705 and Daniel (I.) Bernoulli 1700-1782 and by Huygens in 17-th century and by Laplace and Bayes in 18-th century a culminating point was reached by the axiomatic foundation which in the now finally accepted form was given in 1933 by Kolmogorov.
Quantum mechanics gave birth to a new kind of probabilities which are not definitely non-negative and which are called quasiprobabilities. The first and, for some reason, forever the most important such quasiprobability is the Wigner quasiprobability ( ) , W q p [1] (see also [2]), often called Wigner function. Due to substitution of the classical canonical variables ( ) tions and correspond to canonical transformations in classical theory [3].
Expectation values cannot only be calculated by probability or distribution functions ( ) , F q p but also from all of their transforms which can be uniquely inverted, for example, with the Fourier transform or with the Radon transform of ( ) , F q p or also from the wave functions. An important criterion for a probability function ( ) , F q p is that a displacement of the state in the phase plane displaces the arguments ( ) , q p in corresponding way. In Section 4 we calculate the Wigner quasiprobability for a particle in a square well with infinitely high impenetrable walls. This example provides a Wigner quasiprobability which for the spatial coordinate q vanishes outside the wall borders and therefore is not smooth (infinitely differentiable with continuous derivatives) at the wall borders. Due to the quadratic energy spectrum it possesses internal ( ) 1,1 SU -symmetry. Both properties are equivalent [4]. In Section 5 we make the transition from canonical operators to pairs of boson annihilation and creation operators (and, correspondingly, complex coordinates) that is important for quantum optics, e.g., [5]- [10]. The coherent states are here a very effective mean for the treatment of many problems but it is less known that the Fourier transformation of coherent states provides anew coherent states (Section 6 and Appendix A). A further effective mean for the short representation of many basic relations of quantum optics is the use of Laguerre 2D polynomials which we dealt with in some papers (see [11] and references therein) which is shortly touched in Section 7. From Section 8 on we prepare the calculation of the Wigner quasiprobability for coherent phase states and which similar to states in an infinitely high square well (Section 4) possess internal ( )

1,1 SU
-symmetry (Perelomov [12] [13] [14]). The difficulty is here that a very unorthodox entire function is involved (Appendix D). In the paper [15] the authors claim that they calculate the Wigner function for a phase state but they make this for a non-normalizable London phase state and the calculated function is properly speaking not a Wigner function and is non-normalizable (Sections [8][9][10][11]. Due to the presence of the mentioned unorthodox entire function which is difficult to manage the Wigner quasiprobability of a coherent phase state is less appropriate for the calculation of expectation values and variances. We discuss in addition in Sections 12-14 another possibility by means of Generalized Eulerian numbers, which we represented in a recent publication [16] (see also [17] [18]).

The Wigner Quasiprobability in Different Representations and Their Equivalence
In quantum mechanics it is not possible to find a probability function and which is the best compromise for such a function in quantum theory and where q are the eigenstates of the operator Q to eigenvalues q. By Fourier transformation according to one finds the analogous formula by the wave functions p ψ in momentum representation ( ) ( ) In this transformation we used the scalar products and the completeness relations The definitions (2.4) and (2.6) are fully equivalent to the following definition showing explicitly the symmetry between the canonical variables ( ) where we made the substitution 0 0 , q q q q q q of the integration variables. In a last step using the completeness of the states p according to (2.8) and their scalar products q p Thus the relation (2.10) in connection with (2.11) proves the equivalence of (2.9) to (2.4).
We now give a representation of the Wigner quasiprobability by the displaced parity operator. With respect to the complete sets of eigenstates q or p of the Hermitean operators Q or P the parity operator Π acts according to and the parity operator Π itself can be represented by the following equivalent and furthermore The product of two displacement operators is also a displacement operator (group property) multiplied by a phase factor Klauder and Sudarshan [6] (see also Klauder and according to definition (2.20) where we used the property 2 I Π = given in (2.13).

( )
, W q p is up to a factor 1 π  the expectation value of the displaced parity operator Π in the following sense From this follows taking into account that and it can now be seen that ( ) , W q p in (2.23) is identical with the Wigner definition (2.4).

Further Properties of the Wigner Quasiprobability
We refer now further properties of the Wigner quasiprobability (e.g., [5] and taking into account 1 =  one has proved (3.4). It can be also derived from (2.4). In comparison to the bound (3.4) a classical probability function of two real variables ( ) , q p apart from being non-negative must not possess such a bound.
From ( or if we apply the product formula (2.18) for displacement operators For the trace over the product of two density operators 1  and 2  corresponding to two Wigner quasiprobabilities   In case of 1 one has a pure state that requires the equality sign in (3.11) for the integral over the squared Wigner quasiprobability that can be used as indicator for a pure state.

The Wigner Quasiprobability for a Particle in a Square Well with Infinitely High Impenetrable Walls
As an example, we consider the stationary Schrödinger equation for a particle in a square well with width a symmetrically to the coordinate origin 0 q = and with infinitely high impenetrable walls (e.g., [22] as it is well known, are ( ( ) ( ) ( ) ( )       q pa n m a n m q pa n m a q pa n m a n m q pa n m a For the eigenstates n ψ of the Hamilton operator to this system this leads to ( ) They are illustrated in Figure 2.
We emphasize here that the calculation of expectation values from the Wigner quasiprobability is by no means in every case the simplest way. Often it is much simpler to calculate them from the wave functions in position or momentum representation. However, the principal possibility to calculate them from the Wigner quasiprobability shows the way of correspondences between classical and quantum mechanics including the transition from the last to the classical phase space by approximations.
A Hamilton system of one degree of freedom with a general quadratic energy In a general quadratic energy spectrum which satisfy the commutation relations The Casimir operator C as invariant of the considered irreducible representation is Thus the considered system corresponds to the index 1 2 k = of the irreducible representation. The completeness relation is here The Hamilton operator H of the system can be represented by In addition we introduce the operators E − and E + by Their commutation relations with the operator These relations may be considered as a possible equivalent to the quantum-mechanical commutation relations for classical action j and phase ϕ with the classical to quantum correspondences ( ) Hamilton operator H and action operator J are here not generally proportional as it is the case for a harmonic oscillator and, therefore, is omnipresent in quantum optics. Due to The operators E − and E + are not unitary operators and it is not possible to determine a Hermitean phase operator from them by transition to the Loga- This is related to the fact that in classical theory the transition from canonical variables ( ) For systems such as here a quantum-mechanical particle in a potential well or,

Transition to Complex Conjugate Coordinates in Quantum Optics and Coherent States
The following well-known considerations serve at once for the further introduction of our notations. The Hamilton operator H ′ to a one-dimensional harmonic oscillator of frequency ω in quantum mechanics of a (charged) particle in a potential ( ) , V q p k q ∂ = ∂ its second derivative), and with kinetic energy ( ) (m mass) can be represented by which is the usual form for a member (mode) of the sum of harmonic oscillators into which a bounded electromagnetic field in quantum optics can be decomposed. Canonical coordinate and momentum ( ) , Q P are then in "symmetric way" related to the electric and magnetic field and have nothing to do with "coordinate" and moment of the mode as wave packet in free space.
We introduce now the pair of boson annihilation and creation operators ( ) † , a a and corresponding complex conjugate variable ( ) From this results the representation of the Wigner quasiprobability by the plus the substitution of ( ) ( ) The transition from (5.2) (or equivalently from (5.6)) to (5.1) and inversely is a special squeezing transformation with real squeezing parameter The eigenstates n (number states) of the number operator N to eigenvalues N n n n n a n n n a n n n n The vacuum state 0 as ground state of the Hamilton operator H in (5.6) for a single mode of the electromagnetic field is in quantum optics considered as the genuine vacuum to this single mode in contrast to possible squeezed vacua but, apparently, in agreement with experiments.
Coherent states α are the displaced states (or excitations) of the vacuum state 0 as follows The displacement operator The coherent states α are the right-hand eigenstates of the operator a to complex eigenvalues α and they are mutually non-orthogonal and (over-)complete. This is well known [5] [6] and, e.g., [7], and is written here for convenience. It is, however, less known that the two-dimensional Fourier transformation of coherent states provides again coherent states in the following sense with specialization 0 β = to the vacuum state 0 0 β = ≡ (vacuum state 0 possesses even parity, i.e., providing the parity operator Π as "sum" (integral) over all weighted elements of the Heisenberg-Weyl group.
A general group relation for arbitrary operators A which includes the (over-)completeness of coherent states is the following ( A is trace of A) It is written in [8] in another but equivalent form (Equation  and for 0, 0 m n ≠ = the following series of special forms ( ) For finite groups G with N elements g G ∈ and dimension s n of the irreducible representation If we sum over an operator A in its matrix representation , that is analogous to the form (5.14). This operator relation can be multiplied by arbitrary states ψ and we obtain different forms of linear dependence between the coherent states.
The parity operator Π using the completeness of the coherent states (5.17) can be represented now by ( ) Using the action of the displacement operators from this relation follows where , j α are displaced number states j . Thus the displaced parity operator is the Fourier transformation of the operator π α β α β

The Wigner Quasiprobability in Complex Conjugate Coordinates and Related Quasiprobabilities of Quantum Optics
The In its structure this is up to a constant factor a two-dimensional Fourier  5 We mention the identity The Wigner quasiprobability in representation by the variables ( ) As another possible basic definition of the (normalized) Wigner quasiprobability in complex conjugate coordinates ( ) * , α α follows from (2.9) using where the delta "functions" in representation by real coordinates and complex conjugate coordinates are related by ( ) ( ) ( ) . Using (6.1) and taking into account that  is an arbitrary density operator from (6.5) follows the pure operator relation It is normalized as follows by partial integration This is in agreement with where we used (7.9) but the arising geometric series is only conditionally convergent.
In anti normal ordering of the operator involved in the definition (6.5) one finds ( ) .
One may introduce in this way by ( ) * , Q α α the coherent-state quasiprobability also called Husimi-Kano quasiprobability which is connected with the Wigner quasiprobability as follows ("*" is notation for convolution of two func- The normalization of ( ) * , Q α α follows immediately using partial integra- The coherent-state quasiprobability ( ) due to definition (6.11). It is easy to show that similarly to the Wigner quasiprobability a displacement of the state In the process of "desmoothing" the singular zero points of ( )   can be considered at once as the "most classical" quantum-optical states and a measure of nonclassicality of an arbitrary state may be defined as the nearest distance to such a state (in case of pure states the nearest distance to a coherent state [24] [25]). , P α α which is then not specific for its quantum-optical origin that may be one key to a deeper understanding of this function which seems to be absent now. It has to be noted that for the transition 0 →  one has primarily to make the transition from variables ( ) * , α α to real canonical variables ( )

Wigner Quasiprobability in Number-State Representation
In this Section we derive the number representation of the displaced number states and formulae for the Wigner quasiprobability of arbitrary states in expansions of the number states. For symmetries in the formulae it is favorable to use the Laguerre 2D polynomials with the special and limiting cases Another basic relation for the displaced number states follows from (5.14) in- In special case of the Wigner quasiprobabilities for the number states n n ρ = one finds from (7.12) [5] The Wigner quasiprobability ( ) , W q p for the first 6 number states ( ) , 0,1, , 5 n n =  is illustrated in Figure 3. In principle, it is known [5] but we illustrate it for easy comparison with the eigenstates of the Hamiltonian for an impenetrable square well discussed in Section 4.

Coherent Phase States with London Phase States as Their Limiting Case and Quasi-Distribution of Phase
In this and in the following Sections we discuss coherent phase states and calculate their Wigner quasiprobability.
We consider now the normalizable right-hand and left-hand eigenstates ε of the operator E − to eigenvalues ε or * ε , respectively, according to which are well known (e.g., [29]) and are = that we will not discuss here.
The London phase states can be used to define for arbitrary (pure or mixed) states with density operator  a normalized 2π-periodic phase distribution Since the London phase states are non-orthogonal it is not a genuine probability but only a "one-dimensional" quasiprobability (however, of other kind than the Wigner quasiprobability since it is positively semi-definite) defined over the unit circle The centered distribution for coherent phase states (i.e., for T n x (e.g., [32], chap. 10.11., Equations (2) and (29)). The Fourier series of the more general distribution  (e.g., Vladimirov [33], chap. 2, § 9, Equation (35)

Coherent Phase States as SU(1,1) Coherent States
Up to now we developed the elementary theory of coherent phase states ε .
Less elementary is that the coherent phase states are ( ) is the same as for the system of a particle in a square well with infinitely high impenetrable walls (see Section 4).
The generation of the coherent phase states ε from the vacuum state 0 by a unitary operator can now be written

Wave Functions and Further Characteristics of Coherent Phase States
We begin with the Bargmann (-Segal) representation of states (e.g., [8]) which is the scalar product of the considered state with analytic but non-normalized coherent states This function which is difficult to deal with plays a role, practically, in all representations of the coherent phase states ε [36] [37] 7 . Although the Taylor series (10.2) represents an entire function which, therefore, converges in the 7 Besides a lecture about Hermite and Laguerre 2D polynomials at a Conference in Patras (Greece) I presented there in the Section "Open Problems" a short remark about the very unorthodox entire function (10.2) and the difficulties of its treatment and it was admitted and desired to publish from this one page [37]. Much later, I got an email from Skorokhodov (see Appendix D) with some information, in particular, about the zeros of this function (35 pairs with high precision). It is a pity that with my present computer I was no more able up to now to open again the appended file.
The operator

Calculation of the Wigner Quasiprobability of a Coherent Phase State
For the calculation of the Wigner quasiprobability This is in number representation equivalent to the formula (see (7.12) and After different trials we found that 0.98 ε = was for us the maximally acceptable value concerning the calculation time. Trials were also made to suppress the visibility of the oscillations at the borders due to truncation of the involved series. We needed approximately 40 hours calculation time for one such picture (we did not apply any function of exponential type for acceleration of the convergence). In crass opposition to this was the pure calculation time for the given squeezed coherent state with some similarity to the Wigner quasiprobability of the coherent phase state which after programming according to the known formula was approximately of the order of a very few minutes.  Let us make a few remarks about the article [15]. According to its title, in this article the Wigner function of a phase state is calculated and some related functions are represented graphically. As phase state the authors use the London phase states ϕ given here in (8.7) which are non-normalizable (see (8.9)). They declare ρ ϕ ϕ = as the density operator but it is ϕ ϕ = ∞ and the obtained "Wigner function" is also non-normalizable and, therefore, is not a genuine Wigner quasiprobability. Expectation values cannot be calculated with such a function. One may think that the calculated function plays, at least, an auxiliary role for the calculation of interesting properties of normalizable phase states i e ϕ ε ε = for 1 ε < defined in (8.4) from which the London phase states (8.7) follow as their limiting case 1 ε = without the normalization factor 2 1 ε − . However, we did not find a relation where they may play an auxiliary role. The authors [15] made their London phase states finite by truncating the sum over n in (8.7) with certain max n n = but then the scalar product of these changed states is max 2π n and is not normalized to the value equal to 1 and the results for the Wigner function are badly defined. In case of our normalizable coherent phase states ε the truncation is made "in natural way" by the powers ( ) , 1 n ε ε < for n → ∞ which is a very slowly convergent procedure and which caused our extremely long calculation times. We have to underline the following. To get use from the calculated functions in [15] (or its generalization for ϕ ϕ′ ≠ ) as an auxiliary quantity for calculations, for example, of expectation values it has to be given, at least, in general analytic form. However, due to the involved unorthodox entire function ( ) f z this is very difficult to achieve. This also concerns our calculations of a genuine example of the Wigner quasiprobability for coherent phase states ε in Figure 5.
This may only serve as a certain illustration but for the calculation of expectation values from the Wigner quasiprobability ( ) , W q p this is inappropriate. Some expectation values for the coherent phase states ε can be effectively calculated in other way that we demonstrate in the next Sections.

Expectation Values of Functions of the Number Operator N for Coherent Phase States
The , E k l denotes the Eulerian numbers [17] [18] and also [16] with some of the first given in Table 1.
In particular  We use this result in Section 13 in the calculation of ( )

Expectation Values for Canonical Operators Q and P and Their Variances for Coherent Phase States
In this Section we discuss the calculation of expectation values of the canonical operators Q and P and of simple functions of these operators for coherent phase states ε .
We begin with the calculation of the expectation value of the annihilation operator a for which we find from (8.4) the following series  , The numbers

( )
, E k l µ are generalized Eulerian numbers for which we found the general formula We see that the sum ( The splitting of the functions ( ) is very important for the acceleration of the convergence of series in the evaluation, in particular, for 1 z < in the neighborhood of 1 z  and for approximate solutions it is then necessary to take into account only a few number of terms of the series 1 1 , satisfying for 1 ε ≤ the inequality ( ) In the following we need also the square of the expression (13.9) for which we find the first sum terms of the expansion As an interesting observation we remark that in contrast to the expectation values Q and P in (13.13)  , E k l that their squares possess the mentioned property).
For the expectation values of the canonical operators Q and P we find ( ) For their squares we calculate and for their product We come now to the expectation values of the operators 2 Q and 2 P for which we find using (13.7) ( )

Uncertainties, Uncertainty Matrix and Uncertainty Correlations
The uncertainties of the expectation values of the canonical operators emphasizing their transformation and invariance properties are well summarized in the following symmetric uncertainty matrix S (e.g., [39]) We find the following few first terms of expansions of common partial ex-  The uncertainty sum is the trace S of the uncertainty matrix S and we find ( ) ( ) In the same way as the uncertainty sum S it depends only on the modulus of ε but not on its angle. If we apply the binomial formula to the two quadratic terms in (14.7) then the last two terms cancel but we do not come in this way to an essentially shorter representation. We come back to this below. In the li- The most expressions which we may calculate from Q and P are not invariant with respect to the phase of the complex parameter ε . For example, the mean values Q and P depend on the phase of ε . If we choose * , ε ε ε = = (14.9) then we find A. Wünsche For the corresponding variances we now obtain the formally asymmetric ex- As known, in general, the minimal uncertainty product for a given state is only obtained under special positions of the axes ( ) , q p of the phase space but the uncertainty product taking into account the uncertainty correlations provide it in every case. The inequality between the square of the uncertainty product taking into account the uncertainty correlation and the half uncertainty sum is the inequality between geometric and arithmetic mean. For imaginary parameter ε instead of real one the roles of the axes are interchanged.
The uncertainties for coherent phase states ε are graphically represented in Figure 6.

Conclusions
We derived in this article formulae for the calculation of the Wigner quasipro- Finally, we mention that the Wigner quasiprobability possesses also something which is to feel as a very aesthetical aspect.