Besides the number states and the coherent states the squeezed coherent states or, what is the same, the displaced squeezed vacuum states belong to the most interesting states in quantum optics for which, practically, all interesting parameters and quasiprobabilities may be calculated in closed exact way. The coherent states are the vacuum states displaced by a complex parameter in the phase space (for one mode). The higher number states with are the discrete excitations of the ground state of a harmonic oscillator and they also can be displaced and squeezed but this we do not consider in present article. All minimum uncertainty states belong to the squeezed coherent states and therefore some aspects of these states were already considered in the early years of the development of quantum mechanics although not under this name, for example, by Schrödinger  , Pauli  and Louisell  . The name “squeezed states” appeared in the eighties by Walls  and others and numerous articles and reviews are published since this time, e.g.,  -  and, e.g.,      .
In the narrow sense the squeezing operations form together with rotations in a plane (the two-dimensional phase plane) the Lie group with 3 real parameters. This Lie group possesses different realizations in quantum optics of a single mode and also a basic nontrivial realization in a two-mode system. We will deal with in this article a single mode where the basic operators of the Lie group are realized by quadratic combinations of the annihilation and creation operators of this mode but in Appendix A we represent in detail the basic relations for . Besides this, the Lie group may find application within a single mode also for the treatment of phase states and as mentioned possesses a basic realization in a two-mode system (e.g.,  ). Dynamical squeezing appears if the Hamiltonian or Liouvillean of a process is described by quadratic combinations of annihilation and creation operators.
The main purpose of this paper is the representation of the formalism of squeezing in two approaches concerning the complex squeezing parameter which we call non-unitary and unitary approach and the calculation of expectation values and of the basic quasiprobabilities for squeezed vacuum and squeezed coherent states. The squeezed coherent states are well appropriate to demonstrate some problems of the distinction of sub- and super-Poissonian photon statistics because the whole set of these states can be not assigned to only one of these two kinds of statistics and it requires substantial efforts to find out to which of these statistics it belongs in a special case. The cases when they are neither sub- nor super-Poissonian statistics may be very far from a Poisson statistics that can be seen by the distance parameter. This shows in an example of nonclassical states the problems of classification of statistics in quantum optics in this way and is discussed in Section 11.
2. Squeezed Vacuum States in Non-Unitary Approach and Their Photon Statistics
In this section we begin with the discussion of squeezed vacuum states in the non-unitary approach. For their definition we apply the non-unitary operator
with as a complex parameter onto the vacuum
state . As usual, denote a pair of annihilation and creation operators of a single boson mode with the commutation relations , (I unity operator) and they act onto the number states which are orthonormalized and complete
Now, we define the squeezed vacuum states in the non-unitary approach by
where the zero in the notation is arranged for the substitution by a complex displacement parameter in the later generalization to displaced vacuum states (see Figure 1 and from Section 7 on).
Since is not a unitary operator the right-hand side of (2) is not normalized and using the Taylor series we find the
The complex parameter is restricted in the non-unitary approach for normalizable states to but can be continued to non-normalizable states for . In the unitary approach (2) we apply a unitary operator
1Note that the operators with complex do not form a group that means the product of two such operators with different parameters is, in general, not an operator of this type but by a small extension one comes to the group of squeezing and rotation operators; see Appendix A.
to the vacuum state according to
The connection between the two parameters and is given by1
Figure 1. Position of ellipses of squeezed vacuum states in dependence on the phase of squeezing parameter . The drawn squeezing ellipses for are a contour of equal height of the Wigner quasiprobability . The mean value of the number operator N depends for the squeezed vacuum states only on the modulus of the squeezing parameter and is and the variance is that means and in our case. This shows that all squeezed vacuum states possess a super-Poissonian photon statistics. In the second picture we have shown a displacement of squeezed vacuum states in the quantum phase plane (see from Section 7 on). The squeezing parameter remains the same under displacements. The circle on the left figure corresponds to the vacuum state and on the right figure to a coherent state.
The parameter is stretched in comparison to and takes on the whole complex plane for normalizable states but and possess the same directions in the complex plane. It is easy to rewrite the formulae derived in the following from parameters to parameters using (5), for example
The complex parameter has often some advantages in comparison to the complex parameter concerning compactness of formulae but sometimes, e.g., in the dynamics to quadratic Hamiltonians in , the representation by is to prefer and in literature notations corresponds mostly to our . In Appendix A we consider the relations in detail.
The equivalence of the two approaches is given by the following general disentanglement of the unitary squeezing operator in (almost) normal ordering
The basic relations for squeezing operators between unitary and non-unitary approach were developed already earlier (e.g.,  ) for the group and we collect the most important relations in Appendix A.
From (2.3) follow the probabilities to the photon statistics of squeezed vacuum states
The sum over the are normalized according to
as one affirms from the Taylor series of the function . Only the
probabilities for even are non-vanishing and the probabilities are
strictly decreasing ( ) for increasing m.
From the commutation relations
in connection with follows that the states are eigenstates of the operator to the eigenvalue zero
In representation by canonical operators this is equivalent to
Thus both the states and as well as the states and are linearly dependent. Furthermore, from (2.12) follows
If one forms the scalar products of these relations by multiplication with one obtains the expectation values of and and the expectation values and , respectively, with equality relations between them. By differentiations of (2.3) with respect to and to in connection with (2.14) for we find
as a further interesting relation which can also be written as eigenvalue equation for to eigenvalue zero.
Another interesting characteristics of a state is its (Hilbert-Schmidt) distance to the nearest coherent state which in case of squeezed vacuum states is the vacuum state . It may be considered as a measure of nonclassicality of a state    . For this distance one finds (see (5) for and )
It depends only on the modulus of . For strong squeezing this distance goes to that is the largest distance for normalized states in Hilbert space and means orthogonality of the two states (Figure 2).
3. Wave Functions of Squeezed Vacuum States and Uncertainty Matrix
The wave functions of squeezed vacuum states are the scalar products and with the eigenstates and of the operators Q and P. Their number representations are
where denotes the Hermite polynomials. They are not normalizable as it is well known and are only normalized by means of the delta function
Using the number-state representation (3.3) of squeezed vacuum states we find by applying the first of the generating functions (3.3) for even Hermite polynomials
Figure 2. Distance of squeezed vacuum states to vacuum state in dependence on squeezing parameters and in non-unitary and in unitary approach. The figures show that for the parameter is stretched up to in comparison to . The maximal distance of two pure normalized states in Hilbert space is .
From this follows
with the normalization
The functions and are the Wigner quasiprobability integrated over one of the canonical variable p or q. The functions (3.4) remain invariant by interchanging the squeezing parameter with its complex conjugate . This shows in an example that, in general, a state (here ) cannot uniquely be reconstructed from and alone.
The functions and are two normalized Gaussian distribution with the expectation values
The product of their uncertainties and (note inequality for arbitrary complex in contrast to for arbitrary real x and y)
It depends on the phase of the complex squeezing parameter that means on the position of the principal axes of squeezing in comparison to the axes of the canonical coordinates (see Figure 1). In contrast, the sum of the uncertainties
does not depend on the phase of the squeezing parameter . For real squeezing parameter we find for the uncertainties
and their product is equal to the minimal possible one. The principal axes
of the squeezing ellipses are then in direction of the coordinate axes (see Figure 1). Clearly, it is not satisfying to consider squeezed vacuum states with real as minimum uncertainty states and such with complex which are only rotated in the phase plane (Figure 1), in general, not as minimum uncertainty states. The satisfying solution of this problem is to consider in
addition the uncertainty correlation   .
The uncertainty correlation arises in a natural way as non-
diagonal elements of the (symmetrical) uncertainty matrix if we consider the cumulant expansion of the Wigner quasiprobability or the corresponding expansion of its Fourier transform 
It is also called variance matrix  and is related to the covariance matrix  (3.p. 61). The trace of the uncertainty matrix denoted by
is the uncertainty sum and the determinant of the matrix denoted by is essentially the uncertainty product but modified by the uncertainty correlations
The chain of inequalities which generalizes the basic uncertainty relation of quantum mechanics is (   )
Both quantities and are invariant with respect to rotations and displacement of the states in the quantum phase plane and is additionally invariant with respect to squeezing  .
For squeezed vacuum states we find using their number representation (2.2)
and therefore for the modified uncertainty product in (3.12) using the explicit expressions (3.7) and (3.14)
This modified uncertainty product for squeezed vacuum states does no more depend on the position of the principal axes of squeezing ellipses in phase plane of canonical coordinates shown in Figure 1. It is the minimum possible one characterizing the states as minimum uncertainty states in this generalized sense.
By a rotation of the canonical coordinates to new canonical coordinates one may bring them in the position of the principal axes of the squeezing ellipses and since this is fully obvious we do not give the explicit transformation relations. However, this suggests that it is better not to exclude squeezed vacuum states with arbitrary positions of the squeezing ellipses from the minimum uncertainty states.
4. Bargmann Representation and Quasiprobabilities for Squeezed Vacuum States
The Bargmann representation of a state is a representation by an analytic function which, in particular, leads immediately to the Husimi-Kano quasiprobability  (chap. 7) and  . For this purpose we calculate the scalar product of a state with the analytic (but non-normalized)
coherent states with arbitrary complex .
For the squeezed vacuum state this provides its Bargmann representations
From this one obtains the Husimi-Kano quasiprobability for density operator
In representation by real canonical variables this is
There are different possibilities to calculate the Wigner quasprobability   . In most convenient way the Wigner quasiprobability of an arbitrary squeezed state can be calculated from the Wigner quasiprobability of the corresponding non-squeezed state using the relation (4.9) of Appendix B. The parameters for our non-unitary approach (4.4) are derived in (4.18) where we have to set and have
Taking into account the well-known Wigner quasiprobability for the vacuum state
In representation by the real canonical variables
Another often easy way to calculate the Wigner quasiprobability is to make first the normal ordering of the operator involved in the representation (4.1) that leads to 
For example, one obtains then immediately from it the already used Wigner quasiprobability for the vacuum state (4.5) since the application of the operator
to the state and of from the right to
reproduces them. In the real representation one may use the following equivalent definitions where the first corresponds to the definition given by Wigner   for pure states (written here with density operator and with Dirac’s notations for states)
We checked (4.7) by these formulae using the wave functions and derived in (3.3).
Without presenting its detailed calculation let us give the more general quasiprobabilities with the parameter r and defined by  (‘*’ means convolution)
Our result for the squeezed vacuum states is
For one obtains the Wigner quasiprobability , for the Husimi-Kano quasiprobability and for the Glauber- Sudarshan quasiprobability . The Glauber-Sudarshan quasiprobability for squeezed vacuum states is a singular generalized function and makes for the transition to the delta function . The representation of (4.10) by real canonical variables is easily to make. Furthermore, with some calculation one may bring all quasiprobabilities in representation by real variable to principal axes form that, however, we do not demonstrate here.
5. Expectation Values of Powers of Number Operator and Related Ordered Operators for Squeezed Vacuum States
We now calculate expectation values for the squeezed vacuum states. In particular interesting are the expectation values of ordered powers of the annihilation and creation operators . We begin with the special expectation values since from these expectation values one may calculate the expectation values of the number operator N.
Using the number representation (2.3) of the squeezed vacuum states we proceed quickly for the expectation values which depend only on to the following intermediate result
where we used that the last sum on the right-hand side is the Taylor series of
. This result can be expressed by the Ultraspherical polynomials as special case of the Jacobi polynomials (e.g., 
 ) in two essentially different ways. We represent this in a slightly more general form as needed here in Appendix C. The result specialized for (5.1) is
where are the Legendre polynomials as special case of the Jacobi polynomials . More explicitly this provides
Remarkable in these transformations is that we could split an essential factor
multiplied by a polynomial in comparison to the infinite sums in (5.1).
The expectation values of symmetrically ordered power operators are connected with by (see, e.g., Equation (7.6) in  )
Inserting for the result (5.3) one may transform the arising double sum by reordering to
The transition from the first line to the second line is possible after Taylor
series expansion of and applying then Vandermonde’s convolution
identity which provides a particularly interesting representation with the squared binomial coefficients involved. One may also directly make the transition to representations of the result by the Jacobi polynomials and by its special case of Legendre polynomials using their explicit representations and transformation formulae as follows
In Appendix E we give some first members of the explicit representation of
the sequence .
The expectation values of powers of the number operator can be calculated from the expectation values by the relation
where denotes the Stirling numbers of second kind. We could not find up to now a closed representation of the coefficients in front of powers of
in the polynomial in the numerator over the denominator in
and we give explicit results in Appendix E up to .
From the calculated expectation values we find in first order
and in second order (see also Appendix E for more and higher-order expectation values)
From this follows for squeezed vacuum states using
that means the number uncertainty for squeezed vacuum states is larger than,
for example, for coherent states ( ) and furthermore
This shows that sub-Poissonian statistics does not exist for squeezed vacuum states and that satisfies the general inequality for this quantity  .
6. Further Expectation Values for Squeezed Vacuum States
In this Section we calculate more general expectation values for the squeezed vacuum states of the form than in the preceding Section. They are only non-vanishing if the difference is an even number 2m. This follows from symmetry considerations of the squeezed vacuum states or from their number-state representation which contains only even number states Therefore, we now calculate separately the expectation values and . They depend on and separately and, therefore, we use here the pair of variables for the representation of the results.
For using (2.2) and doubling relations for the Gamma function applied to we arrive at the intermediate result
By comparison of this expression with explicit expressions for the Hyper- geometric function we see that this is a polynomial case with the Jacobi polynomials involved. The main polynomial case of the Hypergeometric function is
and the other possible polynomial case is ( see  Section 5 there)
By transformation relations of the Hypergeometric function, in general, and in possible special cases we find the relation between the Jacobi polynomials
and in special case if one of the upper indices is an integer
Together with the more trivial relation
this provides many possibilities to represent our final results.
Our final result admits the following two equivalent representations showing some symmetry with respect to interchanging
In case of the expectation values the analogous intermediate result to (6.1) is
Using the relation to the Hypergeometric function and the Jacobi polynomials from this follows in analogy to (6.7)
or, alternatively, in analogy to (6.8)
We checked the special cases in comparison with the representation (5.2) by the Legendre polynomials. We checked too that the right-hand sides of this formula gives for all the four different representations the same result.
There is yet an interesting mathematical aspect. In (5.2) Section 5 we calculated the expectation values in the special case by formulae which involve Jacobi polynomials (or their special case Legendre polynomials) without distinction of even and odd k, whereas in present Section we calculated the more general cases and had to distinguish the cases of even k and l and of odd k and l and in the specialization there are involved Jacobi polynomials which are different from that for and it is not possible (or simple) to join these polynomials for to one common formula. The manifold of different transformation relations for Jacobi polynomials is very astonishing (see Appendix C and  ).
As alternative to the calculation of expectation values by the number representation of squeezed vacuum states one may calculate them from the quasiprobabilities that, however, is also not very simple. With the Wigner quasiprobability one may calculate basically the expectation values of symmetrically (Weyl) ordered operators, for example, by integration over the
function for the expectation value of the operator .
With the Husimi-Kano quasiprobability one may calculate basically the expectation values of anti-normally operators which one has then to transform to the more interesting expectation values of normally ordered operators. The expectation values of and of can be calculated with an arbitrary quasiprobability with the parameter r considered in Section 4.
7. Displaced Squeezed Vacuum States or Squeezed Coherent States in Non-Unitary Approach
As a generalization of squeezed vacuum states we derive here shortly their representation in the basis of number state and discuss a very interesting aspect. It is difficult to deal with squeezing in full generality and one may find in literature many approaches which are special ones (squeezing only in directions of coordinate axes ) or with absent calculation of basic functions connected with them.
We define displaced squeezed vacuum states in the non-unitary approach by applying the displacement operator to squeezed vacuum states according to
where is the squeezing operator in unitary approach. We may change the order of operations of displacement and squeezing where the squeezing operator remains the stable part and the displacement operator has to be change. The basic relations for this provides the fundamental representation (A.7) setting there and substituting
These transformations of after transition to the Hermitean basis of operators are very similar to Special Lorentz transformations of with one space coordinate and t the time and this is not incidental since it is for real parameter the same one-parameter Lie group. From (7.2) follows
Applied to (7.1) we find (see also Schleich  , p. 125)
as alternative representation of squeezed vacuum states. This means that we make first a displacement of the vacuum state to a coherent state with the changed displacement parameter and after this the squeezing of the coherent state with the same squeezing operator as in the first variant. Therefore, displaced squeezed vacuum states are fully equivalent to squeezed coherent states with the changed displacement parameters as seen from (7.1) and (7.4). For one obtains from the coherent states and for the squeezed vacuum states
The squeezing operator is the stable part in these two alternative representations. Figure 3 shows schematically the displacement of squeezed vacuum states under fixed complex squeezing parameters in different directions of the complex phase plane described by the complex parameter ( in the four particular pictures).
In generalization of the well-known eigenvalue equation of coherent states the displaced squeezed vacuum states are right-hand eigenstates of the operator to eigenvalues according to
This follows from the relation
using that are right-hand eigenstates of to eigenvalue zero (see (2.12)).
We now derive the number representation of displaced squeezed vacuum states. Using the normally order form (2.7) of the squeezing operator and the following normally ordered form of the displacement operator
from the definition (7.1) follows
Figure 3. Squeezing ellipses in relation to the displacement parameter of squeezed coherent states (schematically). The mean value of the number operator N depends for squeezed coherent states only on the modulus of the squeezing parameter left constant here and on the modulus of the complex displacement parameter and, therefore, is the same on the shown circles in all four partial figures. The case of squeezing in direction of the displacement parameter is also called “amplitude squeezing” (in q-direction in first figure) and in opposite direction to the displacement parameter “phase squeezing” (in p-direction in first figure).
If we now apply the generating function (D.1) for Hermite polynomials to the factor in front of we obtain the following form of the representation of displaced squeezed vacuum states
and using the generation of number states from the vacuum state (see (2.1)) one finds the final form of the number-state representation2
It is easy to see that for using one obtains the number representation (2.3) of squeezed vacuum states and for using the number representation of coherent states .
From (7.10) one finds the probabilities of the photon statistics
2In both parts with one has to choose the same sign of the root but which sign does not matter.
By means of the generating function (D.2) for products of two Hermite polynomials it can be affirmed that the states are normalized and that the probabilities satisfy the necessary relation
In contrast to the photon distributions of coherent states which depends only on and of squeezed vacuum states which depends only on the photon distribution (7.12) depends in addition to the moduli also on the phases of and in the complex plane.
The nearest coherent state to the state is the state and for this distance one obtains the same value as on the right-hand side of (2.16). This means that it does not depend on .
8. Wave Functions of Displaced Squeezed Vacuum States or Squeezed Coherent States
As important characteristics of the displaced squeezed vacuum states we now calculate their wave functions and in the eigenstates and of the canonical operators Q and P (in the usual standardizations
, and similar for ). From the number
representation (7.11) of the states and the well-known number representation of and follows as the first step in the calculation
The infinite sums can be calculated in closed form using the generating function (D.2) for the product of two Hermite polynomials. We represent the result by the mean values and of the canonical operators Q and P. For squeezed vacuum states these mean values vanish already due to the symmetry and for the displaced squeezed vacuum states they are simply connected with the complex displacement parameter according to
With these parameters the result of the evaluation of the sums in (8.1) can be represented in the form
One may make cross checks of these relations using the pair of Fourier transformations
From (8.3) one finds the Gaussian distributions
with the normalization
The functions and are equal to the Wigner quasiprobability integrated over one of the canonical variables p or q and nothing speaks as known against an interpretation of genuine one-dimensional probability densities. They remain invariant with interchanging and the states are not uniquely reconstructible from and alone. In our case of (8.5) they are one-dimensional normalized Gaussian distributions around and , respectively, with the variances of and
The uncertainty product (see also (3.7))
is only for real squeezing parameter but not for complex the minimal possible one.
In case of real squeezing parameter one obtains from (8.3)
with the variances of Q and P
As already explained in Section 3 only in case of real the squeezing axes coincide with the axes of and the uncertainty product becomes the minimal one but taking into account the uncertainty correlation the Gaussian states with other positions of the squeezing axes can be included into the minimum uncertainty states (see Section 3).
By differentiation of relations (8.9) with respect to variables q and p follows
where we used and . The normalization of the states in (8.10) and the pure phase factors do not follow from
the differential equations (8.11) and must be determined where the mentioned phase factors must be present to get full agreement with the usual definitions of the phases of the states and .
Pauli in his hand-book article  (pp. 20, 21) derives the first of the differential equations (8.11), however, with the right-hand side equal to zero that means he does not take into account the possible displacements and that corresponds to squeezed vacuum states (displaced squeezed vacuum states were not known or discussed at that time but Schrödinger considered already coherent states although not under this name in  ). More general and complete are the derivations in the monographs of Louisell  (p. 50, Equation (1.12.23)) and of Leonhardt  (p. 32, Equation (2.85)). The derivations of Pauli are the usual ones in the theory of elements (states) and operators in a Hilbert space known already at that time. Starting from the axiom of positive definiteness of the norm (square root of scalar product of an element with itself) of non-zero elements they consider the superpositions of two arbitrary states and from their norms they derive first the Cauchy-Bunyakovski-Schwarz inequality and second from the vanishing of the superpositions of the two states that they have to be linearly dependent  . In the derivations of uncertainty relations this is applied to two states which are generated from one state by applying two different operators to this one state (here of and ). For the minimum of the norm (equal to zero) this superposition has to be linearly dependent as seen in (8.11). This leads then to the superposition of the two states and (see also (2.13)) with real and in which Pauli in foresight of the result introduced instead of a more indifferent real parameter . Thus the result is connected with the basic assumption that all states in quantum optics may be considered as states with positive norm in a Hilbert space.
The displaced squeezed vacuum states or squeezed coherent states are the most general pure states with Gaussian distributions.
9. Bargmann Representation and Wigner Quasiprobability for Displaced Squeezed Vacuum States
The Bargmann representation of states is the representation by an analytic function of obtain by forming the scalar product of the state with the
analytic but non-normalized coherent state (see Section 4).
From the number representation of displaced squeezed vacuum states (7.11) using the generating function for Hermite polynomials (9.1) we find
The Bargmann representation of a state contains the full information about the state. From (9.1) one finds the Husimi-Kano quasiprobability
This is the corresponding Husimi-Kano quasiprobability (4.2) for squeezed vacuum states with argument displacement.
In Appendix B it is shown by a very simple transformation that the Wigner quasiprobability of a displaced state can be obtained from Wigner quasiprobability of the corresponding undisplaced state by a simple argument displacement . This is also true for other quasiprobabilities such as for example the Husimi-Kano quasiprobability as we saw and for the Glauber-Sudarshan quasiprobability . In this way one obtains from (4.6) for the Wigner quasiprobability of displaced squeezed vacuum states
As discussed in Section 7 as alternative we may first make a displacement of the vacuum state to a coherent state with the displacement parameter
that provides the Wigner quasiprobability
If we now make the squeezing of coherent state with the same squeezing operator as in the other variant (see (7.4)) we have to transform the arguments
in (9.4) according to (see also (4.4)) and we
obtain again the Wigner quasiprobability (9.3) that affirms the inter- changeability of squeezing and displacement according to (7.4).
The Wigner quasiprobability for displaced squeezed vacuum states in the representation by canonical variables can be obtained from (4.7) by the following substitution of the canonical variables
We do not write down this. In analogous way we may find the other quasiprobabilities in the representation by canonical variables.
The given quasiprobabilities can be used for the calculation of expectation values for displaced squeezed vacuum states or squeezed coherent states but in next Section we present an alternative for such calculations.
10. Calculation of Expectation Values for Displaced States from the Expectation Values of the Undisplaced States
An alternative for the calculation of expectation values of ordered powers of the annihilation and creation operator for displaced states from that for the undisplaced states is the following possibility presented here for normal ordering. Using the unitary displacement operator one finds applied to displaced squeezed vacuum states
that after Taylor series expansion of the binomials can be written
where index 0 at expectation values means the expectation values before the displacement and with index after the displacement with the complex parameter .
For squeezed vacuum states the expectation values are only non-
vanishing if the difference is an even number. Taking this into account we find for displaced squeezed vacuum states the expectation values of the operators and
and for the expectation values of the operators and
Next we calculate expectation values of operators which are important for the photon statistics. The expectation value of the number operator depends only on the squared moduli and of the complex and and we find
For the discussion of sub- and super-Poissonian statistics of displaced squeezed vacuum states we need in addition the expectation value of the operator . It does not only depend on the moduli of and but also on their phases and from (10.2) we find
The expectation values for the squeezed vacuum states can be taken from Appendix E where they are collected. From this follows for the expectation value of the operator
and for the variance of the number operator
Finally we calculate the expectation value which plays a role for the definition of sub- and super-Poissonian statistics by its sign
The expectation values (10.6)-(10.9) depend not only on the moduli of and but also on their phases and . For comparison with (10.9) one finds for the corresponding symmetrically (Weyl) ordered quantity
This quantity is greater than or equal to in every case but depends also on the phases of and .
11. Sub- and Super-Poissonian Photon Statistics for Displaced Squeezed Vacuum States or Squeezed Coherent States
The sign of the quantity (10.9) was taken by Mandel to define of sub- and super-Poissonian statistics as follows 
We now investigate the photon statistics of this quantity for displaced squeezed vacuum states starting from (10.9). First we denote the phases of and in the the complex plane as follows
The mean value of the number operator N according to (10.5) does not depend on the angle between and and if we change only the angle of leaving and constant the mean value remains constant and the squeezing ellipses change in their position in comparison to the displacement . This is shown in Figure 2. In the following we show that the photon statistics may change from sub- to super-Poissonian statistics if the displacement moves around the circles shown schematically in Figure 2 and if is sufficiently large and somewhere between this has to be a position where it possesses the same values of and as the Poisson statistics of coherent states with the same .
From (10.9) and from definition (11.1) follows as condition for sub- Poissonian statistics of squeezed coherent states
or resolved to for (pay attention that in case of the inequality (11.3) cannot be satisfied or changes “>” into “<” if divided by !)
For possible sub-Poissonian statistics it is necessary that whereas in case of we have super-Poissonian statistics. For or that means for squeezing in direction of the displacement parameter (amplitude squeezing, see Figure 3) the necessary condition for sub-Poissonian statistics is satisfied but according to (11.4) the modulus of the displacement parameter has to be greater than a minimal value defined by the equality sign in (11.4). For or the left-hand side of (11.3) is positive and we have super-Poissonian statistics. Thus one of the necessary conditions for sub-Poissonian statistics is (see Figure 4, inner circle)
with the limiting curve by substitution of “>” by the equality sign “=” in (11.5). In the other necessary condition (11.4) it is better to substitute the displacement parameter by the expectation value of the number operator since it is then easier to compare the results with coherent states with the same .
We express now the displacement parameter in (10.9) by the expectation value of the number operator N using (10.5)
where we have omitted the index in since we use it in the following also for coherent states. If we do so then we obtain from (10.9) with abbreviation
Figure 4. Polar diagram of sub- and super-Poissonian statistics in dependence on the complex squeezing parameter in non-unitary approach and with displacement parameter with fixed (in positive axis direction) and therefore . Sub-Poissonian statistics is only possible within the inner circle . Within this circle the ellipse-like curves with constant separate regions of sub-Poissonian statistics from regions of super-Poissonian statistics where sub-Poissonian statistics is within the ellipse-like inner regions to the crescent-like outer regions with super-Poissonian statistics. The center ( arbitrary) corresponds to coherent states of arbitrary . Constant (circles around coordinate origin) correspond for fixed to constant touching the separatrices with the same at . Using the squeezing parameter in the unitary approach (see (5) and (6)) this scheme would take on the whole complex plane and, in particular, for parameters with one has a large stretching of corresponding parameters with and the inner circle becomes .
Setting this expression equal to zero one obtains an equation for states which belong neither to sub- nor to super-Poissonian statistics. This is a third-order equation for in dependence on and on the angle as follows
As a third-order equation for with real coefficients in dependence on and it may possess, in principal, three real or one real and two complex conjugate solutions but for our purpose the real solutions have to be positive ones and have to be restricted to . The results are presented in Figure 4 as polar diagram of . The ellipse-like regions of the inner smaller circle (11.5) in Figure 4 belong to sub-Poissonian and the outer crescent-like regions to super-Poissonian statistics in dependence on shown as separatrices.
Figure 5 shows that between sub- and super-Poissonian statistics may lie statistics which are very far from a Poisson statistics, in case of even maximally far (distance d to nearest coherent states goes to ). Alternatively to Figure 5 one may demonstrate the existence of displaced squeezed vacuum states with also in the following way. One begins with an arbitrarily chosen displacement parameter and looks according to condition (11.4) for displacement parameters above the minimal possible one for
Figure 5. Probabilities for squeezed coherent states (joined points) with and such as for a coherent state (orange points) with and nearest distance d to coherent states with or . Additionally, the displacement parameter is given. The last partial figure with small squeezing parameter is in its photon statistics already “visibly” near to a coherent state with but the nearest distance to a coherent state (with ) is not yet very small and shows that this measure is very sensible. All four partial figures belong to states with photon statistics which is neither sub- nor super-Poissonian but also not a Poisson statistics. For , (but, in principle, arbitrary) one has exactly a coherent state (see also Figure 4, center of diagram).
which one has sub-Poissonian statistics if possesses the same or the opposite direction of (amplitude squeezing). Then by rotating the phase of from (amplitude squeezing) up to (phase squeezing) leaving constant the modulus (see Figure 1, right partial figure) the photon statistics makes the transition from sub- to super-Poissonian statistics and one comes unavoidable to a value of the phase where but which does not belong to Poisson statistics. The same effect one has if we rotate the displacement parameter leaving constant the squeezing parameter in modulus and phase (see Figure 3). If the time evolution of a squeezed state is determined by a Hamiltonian H which is a linear combination of and
of then there may appear a complicated picture of changing with
time from sub- to super-Poissonian statistics or from amplitude to phase squeezing since then also the modulus of the squeezing parameter changes with time.
Clearly, one may make the division of photon statistics in sub- and super- Poissonian ones but both categories are very inhomogeneous concerning the comprised states and the set of states which are neither sub- nor super- Poissonian ones is also very large and inhomogeneous and the prefixes “sub” and “super” are here problematic. There are hardly to expect clear differences and correlations in experiments with states of both statistics or, moreover, even qualitatively different behavior. In general, a photon statistics is determined by a countable infinite number of parameters (e.g., or moments of the distribution) and for states which belong neither to sub- nor to super-Poissonian statistics only one from this countable infinite set is fixed ( ) and this can be considered in dependence on arbitrary . Therefore, also the classification of states with sub-Poissonian photon statistics as nonclassical states is highly problematic. In the same way it is also problematic to define the states with no regions of negativity of the Wigner quasiprobability as the classical states since then the set of these states is too large and inhomogeneous to be useful for comparative purpose (all squeezed coherent states belong then to them). Better seems to be for this purpose to use the nearest distance to a coherent state as quantitative measure but this measure cannot change its sign and is in every case positive or zero   . In this definition it is evident that the category of states with large distance to the nearest coherent states is very inhomogeneous and large.
A main purpose of this article was to discuss the distinction of cases of sub- and super-Poissonian statistics within the displaced squeezed vacuum states where the non-unitary approach is preferable. For this case it was necessary to calculate the expectation values of powers of the number operator for these states. We have chosen for this purpose mainly its calculation from the number-state representation and posed this in a more general connection to the calculation of properties of these states. For squeezed coherent states, practically, all interesting parameters can be calculated in exact and not very difficult way and, therefore, this category of states is very suited to demonstrate in examples more principal definitions for all states.
The means developed in this article can be applied without substantial changes to squeezing of the number state and then to its displacement since
the operator in its disentanglement (2.7)) acts on by reproducing it (case in (A.4)). This becomes more complicated if we
apply it to the number states with and to extend the theory to these cases.
Squeezing Operators and Their Disentanglement
Squeezing of states in the narrow sense is connected with the Lie group with 3 real parameters (or one complex and one real in our representation) of squeezing operators    . Its complex extension contains 3 complex parameters (or 6 real ones) and comprises also the group in the right specialization. In this Appendix we give a short but general and systematic representation of the action of squeezing operators in quantum optics. We cannot take care without making too clumsy notations in some cases to the notations in the main text but this concerns mostly the notations of the squeezing parameters with and without prime (or in non-unitary and unitary approaches) and one must pay attention to this when going from formulae of this Appendix to that of the main text.
We define operators with 3 complex parameters which are operators in as follows
where are three abstract operators of the Lie algebra to the Lie group satisfying the commutation relation    
with Casimir operator C
One basic discrete realization of these operators in quantum optics of a single mode is
where is a pair of boson annihilation and creation operators (A.1) and belongs to squeezed number states (and their displacement) not
dealt with in this article. Each realization of the commutation relations (A.2) is appropriate for the following derivations but for the two-dimensional fundamental representation it is important to know two basic operators for which are particularly convenient in combination with the realization (A.4). Squeezing with operators within two modes and genuinely different from single-mode squeezing is also possible   . A further realization of for a single mode is connected with coherent phase states.
The transition from to can be made by specializing the parameters in (A.1) according to
with complex parameter and real parameter . The operators are then unitary operators in the infinite-dimensional unitary representation in Hilbert space and we have . Instead of one may introduce Hermitean operators by but we do not write down all relations for these new operators that is easy to make and is diligent work.
We now consider the generation of the two-dimensional fundamental representation of the group by calculating the matrix with elements in the following relation
This step is essentially a Bogolyubov transformation. It can be represented by (e.g.,  )
For , ( real) we call this transformation squeezing-like and for , ( imaginary) rotation-like. In last case it is better to write the hyperbolic functions by trigonometric functions. The special case
could be called cone-like. The matrices (A.7) are unimodular
but in general, not unitary.
The inversion of (A.7) which is unique can be simply written
with the following relations between the abbreviations and
From the sign of does not follow uniquely but is then uniquely determined if one chooses the same sign of in numerator and denominator.
We now write down some special correspondences between the operators and its two-dimensional matrices in the fundamental representation
The triangular operators and and their correspondent two-dimensional matrices form a group for themselves and in the same way the operators together with its matrices. However, the operators do not form a group its extension to a group needs all operators .
We may decompose the matrices in (A.7) into products of special matrices, in particular, in the following for us important ways
As consequences we obtain the following disentanglements of the general operator into products of partial operators
For the operators we may insert the realization (A.4) of but every other realization of is also appropriate.
In the special case of operators the matrices (A.7) specialize
that leads to the disentanglement relations
where again we may insert the realization (A.4). In addition to the operators and we have involved then on the right-hand side also the operator .
If we make in (A.15) the transition to the variables in the non-unitary approach
then the matrix in (A.15) makes the transition to
as it is easily to see.
The next considerations are the effort to split the general operator into products of a proper squeezing operator and a proper rotation operator . The general two-dimensional matrix (A.7) can be split also in the following two ways
This corresponds explicitly to the two possibilities ( )
and if we introduce abbreviations
The rotation operators are the same in both cases and the squeezing operators distinguish themselves only by phase factors in the non-diagonal matrix elements. We have
The first possibility of the splitting in a squeezing and a rotation part is
The parameters in the factorized matrices are ( are such as in (A.7))
The second possibility with interchanged order of the splitting in a squeezing and a rotation part is
The diagonal matrix of a rotation with parameter is the stable part in the two factorizations (A.23) and (A.26). These considerations show that without disadvantage for the generality we may use the special squeezing operators which are equivalent to the general operators after splitting rotation factors. However, these special squeezing operators alone do not form a group.
Influence of Displacement and Squeezing of States onto the Wigner Quasi-Probability
The displacement of states and the squeezing make transformations of the variables in the Wigner quasiprobability which can be given in a general form. For the derivation we use the representation of the Wigner quasiprobability by complex variables in the following form   which is equivalent to the definition given by Wigner  
where is the density operator of the state. First we investigate the displace- ment of a state with the density operator and the Wigner quasiprobability according to
Then one finds for
Therefore we obtained
This means that the Wigner quasiprobability for the displaced state is equal to the Wigner quasiprobability for the undisplaced state with displaced arguments corresponding to the displacement parameters . This displacement property is one of the minimal requirements for a phase-space function in quantum theory to be called a quasiprobability.
In widely analogous way one may treat the squeezing of a state with density operator according to
Instead of the unitary squeezing operators we will calculate the transformations first a little more general with the operators and at the end we will specialize the result to and . Therefore, we
have first to substitute and
Formally this can be written
with the matrix elements in explicit form given in (A.7). This corresponds to the transformation
In the last step in (B.6) was used that the two-dimensional delta function is invariant with respect to a (complex) unimodular transformation of the variables. This can be proved, for example, by transition to real variables and transformation to principal axes (see below).
In the special case of unitary transformations with the operator we can substitute and and (B.7) using (B.8) can be specialized to 
We mention that a displacement relation of the form (B.4) is true for all quasiprobabilities (e.g., and ) whereas for squeezing of states the form (B.9) is only true for the Wigner quasiprobability. The reason is
that we do not have in other cases the operator as a whole in
the exponent of only one exponential function.
The transformation of the variables possesses the property
This means that each area element of the Wigner quasiprobability is mapped in a new area element of of the same area with preservation of the topology but without preservation of angles such as for transformations in classical mechanics. Therefore, for example, if the Wigner quasiprobability possesses regions of negativity (squeezed coherent states do not possess such regions) then after the transformation the area of negativity remains the same as before the transformation. The similar property is true for displacements and is here obvious.
A simple proof of the invariance of the two-dimensional delta function under (in general, complex) unimodular transformations of the variables by a matrix
with can be given by extension of coordinates to
their own complex planes according to
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