ABSTRACT

After developing the concept of displaced squeezed vacuum states in the non- unitary approach and establishing the connection to the unitary approach we calculate their quasiprobabilities and expectation values $a^{†k}a$ in general form. Then we consider the displacement of the squeezed vacuum states and calculate their photon statistics and their quasiprobabilities. The expectation values of the displaced states are related to the expectation values of the undisplaced states and are calculated for some simplest cases which are sufficient to discuss their categorization as sub-Poissonian and super-Poissonian statistics. A large set of these states do not belong to sub- or to super-Poissonian states but are also not Poissonian states. We illustrate in examples their photon distributions. This shows that the notions of sub- and of super-Poissonian statistics and their use for the definition of nonclassicality of states are problematic. In Appendix A we present the most important relations for $SU\left(\mathrm{1,1}\right)$ treatment of squeezing and the disentanglement of their operators. Some initial members of sequences of expectation values for squeezed vacuum states are collected in Appendix E.

Keywords:

$SU\left(\mathrm{1,1}\right)$ Group of Squeezing and Rotation, Wigner Quasiprobability, Unitary Approach to Squeezing, Nonclassical States, Uncertainty Matrix, Distance of States, Jacobi, Ultraspherical, Legendre and Hermite Polynomials, Poisson Statistics

1. Introduction

Besides the number states $|n\rangle$ and the coherent states $|\alpha \rangle$ 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 $|n=0\rangle$ displaced by a complex parameter $\beta$ in the phase space (for one mode). The higher number states $|n\rangle$ with $n\ne 0$ are the discrete excitations of the ground state $|n=0\rangle$ 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 [1] , Pauli [2] and Louisell [3] . The name “squeezed states” appeared in the eighties by Walls [4] and others and numerous articles and reviews are published since this time, e.g., [5] - [14] and, e.g., [15] [16] [17] [18] [19] .

In the narrow sense the squeezing operations form together with rotations in a plane (the two-dimensional phase plane) the Lie group $SU\left(\mathrm{1,1}\right)$ 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 $SU\left(\mathrm{1,1}\right)$ are realized by quadratic combinations of the annihilation and creation operators $\left(a\mathrm{,}{a}^{†}\right)$ of this mode but in Appendix A we represent in detail the basic relations for $SU\left(\mathrm{1,1}\right)$ . Besides this, the Lie group $SU\left(\mathrm{1,1}\right)$ 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., [15] ). 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 $SU\left(\mathrm{1,1}\right)$ 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

$exp\left(-\frac{\zeta }{2}{a}^{†2}\right)$ with $\zeta$ as a complex parameter $\left(\left|\zeta \right|<1\right)$ onto the vacuum

state $|0\rangle$ . As usual, $\left(a\mathrm{,}{a}^{†}\right)$ denote a pair of annihilation and creation operators of a single boson mode with the commutation relations $\left[a\mathrm{,}{a}^{†}\right]\equiv a{a}^{†}-a^{†}a=I$ , (I unity operator) and they act onto the number states $|n\rangle$ which are orthonormalized and complete

$a|n\rangle =\sqrt{n}|n-1\rangle ,a^{†}|n\rangle =\sqrt{n+1}|n+1\rangle ,N|n\rangle \equiv a^{†}a|n\rangle =n|n\rangle ,$

$\langle m|n\rangle ={\delta }_{m,n},{\displaystyle \sum _{n=0}^{\infty }}|n\rangle \langle n|=I,|n\rangle =\frac{a^{†n}}{\sqrt{n!}}|0\rangle ,\left(n=0,1,2,\cdots \right).$ (2.1)

Now, we define the squeezed vacuum states $|\mathrm{0,}\zeta \rangle$ in the non-unitary approach by

$|0,\zeta \rangle \propto \exp \left(-\frac{\zeta }{2}{a}^{†2}\right)|0\rangle ={\displaystyle \underset{m=0}{\overset{\infty }{\sum }}}\frac{{\left(-1\right)}^m}{2^{m}m!}{\zeta }^m{a}^{†2m}|0\rangle ={\displaystyle \underset{m=0}{\overset{\infty }{\sum }}}\frac{{\left(-1\right)}^m\sqrt{\left(2m\right)!}}{2^{m}m!}{\zeta }^m|2m\rangle ,$ (2.2)

where the zero in the notation $|\mathrm{0,}\zeta \rangle$ is arranged for the substitution by a complex displacement parameter $\beta$ in the later generalization to displaced vacuum states $|\beta \mathrm{,}\zeta \rangle$ (see Figure 1 and from Section 7 on).

Since ${\text{e}}^{-\frac{\zeta }{2}{a}^{†2}}$ is not a unitary operator the right-hand side of (2) is not normalized and using the Taylor series ${\displaystyle {\sum }_{m=0}^{\infty }}\frac{\left(2m\right)!}{2^{2m}m{!}^2}{q}^m=\frac{1}{\sqrt{1-q}}$ we find the

normalization factor

$\begin{array}{l}|0,\zeta \rangle ={\left(1-\zeta {\zeta }^{*}\right)}^{\frac{1}{4}}\exp \left(-\frac{\zeta }{2}{a}^{†2}\right)|0\rangle \\ ={\left(1-\zeta {\zeta }^{*}\right)}^{\frac{1}{4}}{\displaystyle \sum _{m=0}^{\infty }}\frac{{\left(-1\right)}^m\sqrt{\left(2m\right)!}}{2^{m}m!}{\zeta }^m|2m\rangle ,\langle 0,\zeta |0,\zeta \rangle =1,\left|\zeta \right|\equiv \sqrt{\zeta {\zeta }^{*}}<1.\end{array}$ (2.3)

The complex parameter $\zeta$ is restricted in the non-unitary approach for normalizable states to $\left|\zeta \right|<1$ but can be continued to non-normalizable states for $\left|\zeta \right|\ge 1$ . In the unitary approach (2) we apply a unitary operator

¹Note that the operators $S\left({{\zeta }^{\prime }}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,0,}{\zeta }^{\prime }\right)\equiv exp\left(\frac{{{\zeta }^{\prime }}^{\mathrm{<mo>\; *\; </mo>}}}{2}{a}^2-\frac{{\zeta }^{\prime }}{2}{a}^{†2}\right)$ with complex ${\zeta }^{\prime }$ 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 $SU\left(\mathrm{1,1}\right)$ group of squeezing and rotation operators; see Appendix A.

$exp\left(\frac{{{\zeta }^{\prime }}^{\mathrm{<mo>\; *\; </mo>}}}{2}{a}^2-\frac{{\zeta }^{\prime }}{2}{a}^{†2}\right)$ to the vacuum state $|0\rangle$ according to

$|\mathrm{0,}\zeta \rangle \equiv exp\left\{\frac{\text{Arth}\left|\zeta \right|}{\left|\zeta \right|}\left(\frac{{\zeta }^{\mathrm{<mo>*</mo>}}}{2}{a}^2-\frac{\zeta }{2}{a}^{†2}\right)\right\}|0\rangle \equiv exp\left(\frac{{{\zeta }^{\prime }}^{\mathrm{<mo>*</mo>}}}{2}{a}^2-\frac{{\zeta }^{\prime }}{2}{a}^{†2}\right)|0\rangle .$ (2.4)

The connection between the two parameters $\zeta$ and ${\zeta }^{\prime }$ is given by¹

${\zeta }^{\prime }=\frac{\text{Arth}\left(\left|\zeta \right|\right)}{\left|\zeta \right|}\zeta \mathrm{,}{{\zeta }^{\prime }}^{\mathrm{<mo>\; *\; </mo>}}=\frac{\text{Arth}\left(\left|\zeta \right|\right)}{\left|\zeta \right|}{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}\left|{\zeta }^{\prime }\right|=\text{Arth}\left|\zeta \right|\mathrm{,}\frac{{\zeta }^{\prime }}{\left|{\zeta }^{\prime }\right|}=\frac{\zeta }{\left|\zeta \right|}\mathrm{,}$

$\zeta =\frac{\text{th}\left(\left|{\zeta }^{\prime }\right|\right)}{\left|{\zeta }^{\prime }\right|}{\zeta }^{\prime },{\zeta }^{*}=\frac{\text{th}\left(\left|{\zeta }^{\prime }\right|\right)}{\left|{\zeta }^{\prime }\right|}{{\zeta }^{\prime }}^{*},\left|\zeta \right|=\text{th}\left(\left|{\zeta }^{\prime }\right|\right).$ (2.5)

The parameter ${\zeta }^{\prime }$ is stretched in comparison to $\zeta$ and takes on the whole complex plane for normalizable states but $\zeta$ and ${\zeta }^{\prime }$ possess the same directions in the complex plane. It is easy to rewrite the formulae derived in the following from parameters $\left(\zeta \mathrm{,}{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)$ to parameters $\left({\zeta }^{\prime }\mathrm{,}{{\zeta }^{\prime }}^{\mathrm{<mo>\; *\; </mo>}}\right)$ using (5), for example

$\frac{1}{1-{\left|\zeta \right|}^2}={\text{ch}}^2\left(\left|{\zeta }^{\prime }\right|\right),\frac{{\left|\zeta \right|}^2}{1-{\left|\zeta \right|}^2}={\text{sh}}^2\left(\left|{\zeta }^{\prime }\right|\right),\sqrt{\frac{1\pm \left|\zeta \right|}{1\mp \left|\zeta \right|}}=\exp \left(\pm \left|{\zeta }^{\prime }\right|\right).$ (2.6)

The complex parameter $\zeta$ has often some advantages in comparison to the complex parameter ${\zeta }^{\prime }$ concerning compactness of formulae but sometimes, e.g., in the dynamics to quadratic Hamiltonians in $\left(a\mathrm{,}{a}^{†}\right)$ , the representation by ${\zeta }^{\prime }$ is to prefer and $\zeta$ in literature notations $|\alpha \mathrm{,}\zeta \rangle$ corresponds mostly to our ${\zeta }^{\prime }$ . 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

$\begin{array}{l}exp\left\{\frac{\text{Arth}\left|\zeta \right|}{\left|\zeta \right|}\left(\frac{{\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{2}{a}^2-\frac{\zeta }{2}{a}^{†2}\right)\right\}\\ =exp\left(-\frac{\zeta }{2}{a}^{†2}\right){\left(1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\frac{1}{4}\left(a{a}^{†}+a^{†}a\right)}exp\left(\frac{{\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{2}{a}^2\right)\mathrm{.}\end{array}$ (2.7)

The basic relations for squeezing operators between unitary and non-unitary approach were developed already earlier (e.g., [18] ) for the $SU\left(\mathrm{1,1}\right)$ group and we collect the most important relations in Appendix A.

From (2.3) follow the probabilities $p_n$ to the photon statistics of squeezed vacuum states

$\begin{array}{c}{p}_n\equiv \langle n|\mathrm{0,}\zeta \rangle \langle \mathrm{0,}\zeta |n\rangle \\ =\{\begin{array}{l}{p}_{2m}=\sqrt{1-{\left|\zeta \right|}^2}\frac{\left(2m\right)!}{2^{2m}m{!}^2}{\left|\zeta \right|}^{2m}=\sqrt{1-{\left|\zeta \right|}^2}\frac{\left(2m-1\right)!!}{2^{m}m!}{\left|\zeta \right|}^{2m}\hfill \\ p_{2m+1}=0.\hfill \end{array}\end{array}$ (2.8)

The sum over the $p_n$ are normalized according to

${\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{p}_n=\sqrt{1-{\left|\zeta \right|}^2}{\displaystyle \underset{m=0}{\overset{\infty }{\sum }}}\frac{\left(2m\right)!}{2^{2m}m{!}^2}{\left|\zeta \right|}^{2m}=1,$ (2.9)

as one affirms from the Taylor series of the function ${\left(1-{\left|\zeta \right|}^2\right)}^{-\frac{1}{2}}$ . Only the

probabilities for even $n=2m$ are non-vanishing and the probabilities $p_{2m}$ are

strictly decreasing ( $p_{2\left(m+1\right)}\le p_{2m}$ ) for increasing m.

From the commutation relations

$\begin{array}{l}exp\left(-\frac{\zeta }{2}{a}^{†2}\right)aexp\left(\frac{\zeta }{2}{a}^{†2}\right)\\ =a-\frac{\zeta }{2}\left[a^{†2}\mathrm{,}a\right]+\frac{1}{\mathrm{2\; <mo>\; !\; </mo>}}{\left(\frac{\zeta }{2}\right)}^2\left[a^{†2}\mathrm{,}\left[a^{†2}\mathrm{,}a\right]\right]-\cdots =a+\zeta a^{†}\mathrm{,}\end{array}$ (2.10)

in connection with $a|0\rangle =0$ follows that the states $|\mathrm{0,}\zeta \rangle$ are eigenstates of the operator $a+\zeta a^{†}$ to the eigenvalue zero

$0=\exp \left(-\frac{\zeta }{2}{a}^{†2}\right)a|0\rangle =\left(a+\zeta a^{†}\right)\exp \left(-\frac{\zeta }{2}{a}^{†2}\right)|0\rangle ,$ (2.11)

that means

$\left(a+\zeta a^{†}\right)|\mathrm{0,}\zeta \rangle =0,\iff a|\mathrm{0,}\zeta \rangle =-\zeta a^{†}|\mathrm{0,}\zeta \rangle .$ (2.12)

In representation by canonical operators $\left(Q\mathrm{,}P\right)$ this is equivalent to

$\left(\left(1+\zeta \right)Q+\text{i}\left(1-\zeta \right)P\right)|\mathrm{0,}\zeta \rangle =\mathrm{0,}\iff P|\mathrm{0,}\zeta \rangle =\text{i}\frac{1+\zeta }{1-\zeta }Q|\mathrm{0,}\zeta \rangle \mathrm{.}$ (2.13)

Thus both the states $a|\mathrm{0,}\zeta \rangle$ and $a^{†}|\mathrm{0,}\zeta \rangle$ as well as the states $Q|\mathrm{0,}\zeta \rangle$ and $P|\mathrm{0,}\zeta \rangle$ are linearly dependent. Furthermore, from (2.12) follows

$a^{†k}a|\mathrm{0,}\zeta \rangle =-\zeta a^{†k+1}|\mathrm{0,}\zeta \rangle \mathrm{,}{a}^l|\mathrm{0,}\zeta \rangle =-\zeta a^{l-1}{a}^{†}|\mathrm{0,}\zeta \rangle \mathrm{.}$ (2.14)

If one forms the scalar products of these relations by multiplication with $\langle \mathrm{0,}\zeta |$ one obtains the expectation values of $\overline{a^{†k}a}$ and $-\zeta \overline{a^{†k+1}}$ and the expectation values $\overline{a^l}$ and $-\zeta \overline{a^{l-1}{a}^{†}}$ , respectively, with equality relations between them. By differentiations of (2.3) with respect to $\zeta$ and to ${\zeta }^{\mathrm{<mo>\; *\; </mo>}}$ in connection with (2.14) for $k=1$ we find

$N|\mathrm{0,}\zeta \rangle =-\zeta a^{†2}|\mathrm{0,}\zeta \rangle =2\left(\zeta \frac{\partial }{\partial \zeta }-{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\frac{\partial }{\partial {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}\right)|\mathrm{0,}\zeta \rangle \mathrm{,}$ (2.15)

as a further interesting relation which can also be written as eigenvalue equation for $|\mathrm{0,}\zeta \rangle$ 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 $|\mathrm{0,}\zeta \rangle$ is the vacuum state $|0\rangle$ . It may be considered as a measure of nonclassicality of a state [20] [21] [22] . For this distance one finds (see (5) for $\zeta$ and ${\zeta }^{\prime }$ )

$\begin{array}{c}d\left(|0,\zeta \rangle \langle 0,\zeta |,|0\rangle \langle 0|\right)=\sqrt{2\left(1-\langle 0|0,\zeta \rangle \langle 0,\zeta |0\rangle \right)}\\ =\sqrt{2\left(1-\sqrt{1-{\left|\zeta \right|}^2}\right)}=\sqrt{2\left(1-\frac{1}{\text{ch}\left(\left|{\zeta }^{\prime }\right|\right)}\right)}\\ =\left|\zeta \right|\left(1+\frac{1}{8}{\left|\zeta \right|}^2+\cdots \right)=\left|{\zeta }^{\prime }\right|\left(1-\frac{5}{24}{\left|{\zeta }^{\prime }\right|}^2+\cdots \right).\end{array}$ (2.16)

It depends only on the modulus of $\zeta$ . For strong squeezing $\left|\zeta \right|\to 1$ this distance goes to $\sqrt{2}$ 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 $\langle q|\mathrm{0,}\zeta \rangle$ and $\langle p|\mathrm{0,}\zeta \rangle$ with the eigenstates $|q\rangle$ and $|p\rangle$ of the operators Q and P. Their number representations are

$|q\rangle =\frac{1}{{\left(\hslash \text{π}\right)}^{\frac{1}{4}}}exp\left(-\frac{q^2}{2\hslash }\right){\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}\frac{1}{\sqrt{2^{n}n\mathrm{<mo>\; !\; </mo>}}}{\text{H}}_n\left(\frac{q}{\sqrt{\hslash }}\right)|n\rangle \mathrm{,}$

$|p\rangle =\frac{1}{{\left(\hslash \text{π}\right)}^{\frac{1}{4}}}exp\left(-\frac{p^2}{2\hslash }\right){\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}\frac{{\text{i}}^n}{\sqrt{2^{n}n\mathrm{<mo>\; !\; </mo>}}}{\text{H}}_n\left(\frac{p}{\sqrt{\hslash }}\right)|n\rangle \mathrm{,}$ (3.1)

where ${\text{H}}_n\left(z\right)$ denotes the Hermite polynomials. They are not normalizable as it is well known and are only normalized by means of the delta function

$\begin{array}{l}\langle q|q^{\prime }\rangle =\delta \left(q-q^{\prime }\right)\mathrm{,}\langle p|p^{\prime }\rangle =\delta \left(p-p^{\prime }\right)\mathrm{,}\\ \langle q|p\rangle =\frac{1}{\sqrt{2\hslash \text{π}}}exp\left(\text{i}\frac{qp}{\hslash }\right)={\langle p|q\rangle }^{\mathrm{<mo>\; *\; </mo>}}\mathrm{.}\end{array}$ (3.2)

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

$\begin{array}{c}\langle q|\mathrm{0,}\zeta \rangle ={\left(\frac{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{\hslash \text{π}}\right)}^{\frac{1}{4}}exp\left(-\frac{q^2}{2\hslash }\right){\displaystyle \underset{m=0}{\overset{\infty }{\sum }}}\frac{{\left(-1\right)}^m{\zeta }^m}{2^{2m}m\mathrm{<mo>\; !\; </mo>}}{\text{H}}_{2m}\left(\frac{q}{\sqrt{2\hslash }}\right)\\ ={\left(\frac{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{\hslash \text{π}}\right)}^{\frac{1}{4}}\frac{1}{\sqrt{1-\zeta }}exp\left(-\frac{\left(1+\zeta \right)q^2}{\left(1-\zeta \right)2\hslash }\right)\mathrm{,}\end{array}$

$\begin{array}{c}\langle p|\mathrm{0,}\zeta \rangle ={\left(\frac{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{\hslash \text{π}}\right)}^{\frac{1}{4}}exp\left(-\frac{p^2}{2\hslash }\right){\displaystyle \underset{m=0}{\overset{\infty }{\sum }}}\frac{{\zeta }^m}{2^{2m}m\mathrm{<mo>\; !\; </mo>}}{\text{H}}_{2m}\left(\frac{p}{\sqrt{2\hslash }}\right)\\ ={\left(\frac{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{\hslash \text{π}}\right)}^{\frac{1}{4}}\frac{1}{\sqrt{1+\zeta }}exp\left(-\frac{\left(1-\zeta \right)p^2}{\left(1+\zeta \right)2\hslash }\right)\mathrm{.}\end{array}$ (3.3)

From this follows

$W\left(q\right)\equiv \langle q|\mathrm{0,}\zeta \rangle \langle \mathrm{0,}\zeta |q\rangle =\sqrt{\frac{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{\left(1-\zeta \right)\left(1-{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)\hslash \text{π}}}exp\left(-\frac{\left(1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)q^2}{\left(1-\zeta \right)\left(1-{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)\hslash }\right)\mathrm{,}$

$W\left(p\right)\equiv \langle p|\mathrm{0,}\zeta \rangle \langle \mathrm{0,}\zeta |p\rangle =\sqrt{\frac{1-\zeta {\zeta }^{\mathrm{<mo>*</mo>}}}{\left(1+\zeta \right)\left(1+{\zeta }^{\mathrm{<mo>*</mo>}}\right)\hslash \text{π}}}exp\left(-\frac{\left(1-\zeta {\zeta }^{\mathrm{<mo>*</mo>}}\right)p^2}{\left(1+\zeta \right)\left(1+{\zeta }^{\mathrm{<mo>*</mo>}}\right)\hslash }\right)\mathrm{,}$ (3.4)

with the normalization

${\displaystyle {\int }_{-\infty }^{+\infty }}\text{d}qW\left(q\right)={\displaystyle {\int }_{-\infty }^{+\infty }}\text{d}pW\left(p\right)=1.$ (3.5)

The functions $W\left(q\right)$ and $W\left(p\right)$ are the Wigner quasiprobability $W\left(q\mathrm{,}p\right)$ integrated over one of the canonical variable p or q. The functions (3.4) remain invariant by interchanging the squeezing parameter $\zeta$ with its complex conjugate $\zeta \leftrightarrow {\zeta }^{\mathrm{<mo>\; *\; </mo>}}$ . This shows in an example that, in general, a state (here $|\mathrm{0,}\zeta \rangle$ ) cannot uniquely be reconstructed from $W\left(q\right)$ and $W\left(p\right)$ alone.

The functions $W\left(q\right)$ and $W\left(p\right)$ are two normalized Gaussian distribution with the expectation values

$\overline{Q}=0,\overline{P}=0,\overline{{\left(\Delta Q\right)}^2}=\frac{\left(1-\zeta \right)\left(1-{\zeta }^{*}\right)}{1-\zeta {\zeta }^{*}}\frac{\hslash }{2},\overline{{\left(\Delta P\right)}^2}=\frac{\left(1+\zeta \right)\left(1+{\zeta }^{*}\right)}{1-\zeta {\zeta }^{*}}\frac{\hslash }{2}.$ (3.6)

The product of their uncertainties $\overline{{\left(\Delta Q\right)}^2}$ and $\overline{{\left(\Delta P\right)}^2}$ (note inequality $z^2+z^{\mathrm{<mo>\; *\; </mo>2}}\le 2z{z}^{\mathrm{<mo>\; *\; </mo>}}$ for arbitrary complex $z=x+\text{i}y$ in contrast to $x^2+y^2\ge 2xy$ for arbitrary real x and y)

$\overline{{\left(\Delta Q\right)}^2}\overline{{\left(\Delta P\right)}^2}-\frac{\left(1-{\zeta }^2\right)\left(1-{\zeta }^{\mathrm{<mo>\; *\; </mo>2}}\right)}{{\left(1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)}^2}\frac{{\hslash }^2}{4}\ge \frac{{\hslash }^2}{4}\mathrm{.}$ (3.7)

It depends on the phase $\chi$ of the complex squeezing parameter $\zeta =\left|\zeta \right|{\text{e}}^{\text{i}\chi }$ that means on the position of the principal axes of squeezing in comparison to the axes of the canonical coordinates $\left(q\mathrm{,}p\right)$ (see Figure 1). In contrast, the sum of the uncertainties

$\overline{{\left(\Delta Q\right)}^2}+\overline{{\left(\Delta P\right)}^2}=\frac{1+\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}\hslash \mathrm{,}$ (3.8)

does not depend on the phase of the squeezing parameter $\zeta$ . For real squeezing parameter $\zeta ={\zeta }^{*}$ we find for the uncertainties

$\overline{{\left(\Delta Q\right)}^2}=\frac{1-\zeta }{1+\zeta }\frac{\hslash }{2}\mathrm{,}\overline{{\left(\Delta P\right)}^2}=\frac{1+\zeta }{1-\zeta }\frac{\hslash }{2}\mathrm{,}\left(\zeta ={\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)\mathrm{,}$ (3.9)

and their product is equal to $\frac{{\hslash }^2}{4}$ the minimal possible one. The principal axes

of the squeezing ellipses are then in direction of the coordinate axes $\left(q\mathrm{,}p\right)$ (see Figure 1). Clearly, it is not satisfying to consider squeezed vacuum states $|\mathrm{0,}\zeta \rangle$ with real $\zeta$ as minimum uncertainty states and such with complex $\zeta$ 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 $\frac{1}{2}\overline{\Delta Q\Delta P+\Delta P\Delta Q}$ [23] [24] .

The uncertainty correlation $\frac{1}{2}\overline{QP+PQ}$ arises in a natural way as non-

diagonal elements of the (symmetrical) uncertainty matrix $\text{S}$ if we consider the cumulant expansion of the Wigner quasiprobability $W\left(q\mathrm{,}p\right)$ or the corresponding expansion of its Fourier transform $\tilde{W}\left(u\mathrm{,}v\right)$ [24]

$\begin{array}{l}\text{S}\equiv \left(\begin{array}{cc}\overline{{\left(\Delta Q\right)}^2}& \frac{1}{2}\overline{\Delta Q\Delta P+\Delta P\Delta Q}\\ \frac{1}{2}\overline{\Delta Q\Delta P+\Delta P\Delta Q}& \overline{{\left(\Delta P\right)}^2}\end{array}\right)\\ =\left(\begin{array}{cc}\overline{\Delta a\Delta a^{†}+\Delta a^{†}\Delta a}+\left(\overline{{\left(\Delta a\right)}^2}+\overline{{\left(\Delta a^{†}\right)}^2}\right)& -\text{i}\left(\overline{{\left(\Delta a\right)}^2}-\overline{{\left(\Delta a^{†}\right)}^2}\right)\\ -\text{i}\left(\overline{{\left(\Delta a\right)}^2}-\overline{{\left(\Delta a^{†}\right)}^2}\right)& \overline{\Delta a\Delta a^{†}+\Delta a^{†}\Delta a}-\left(\overline{{\left(\Delta a\right)}^2}+\overline{{\left(\Delta a^{†}\right)}^2}\right)\end{array}\right)\frac{\hslash }{2}\mathrm{.}\end{array}$ (3.10)

It is also called variance matrix [23] and is related to the covariance matrix [10] (3.p. 61). The trace of the uncertainty matrix $\text{S}$ denoted by $\langle \text{S}\rangle$

$\langle \text{S}\rangle =\overline{{\left(\Delta Q\right)}^2}+\overline{{\left(\Delta P\right)}^2}=\left(\overline{\Delta a\Delta a^{†}+\Delta a^{†}\Delta a}\right)\hslash \mathrm{,}$ (3.11)

is the uncertainty sum and the determinant of the matrix $\text{S}$ denoted by $\left[\text{S}\right]$ is essentially the uncertainty product but modified by the uncertainty correlations

$\begin{array}{c}\left[\text{S}\right]=\overline{{\left(\Delta Q\right)}^2}\overline{{\left(\Delta P\right)}^2}-\frac{1}{4}{\left(\overline{\Delta Q\Delta P+\Delta P\Delta Q}\right)}^2\\ =\left\{{\left(\overline{\Delta a\Delta a^{†}+\Delta a^{†}\Delta a}\right)}^2-4\overline{{\left(\Delta a\right)}^2}\overline{{\left(\Delta a^{†}\right)}^2}\right\}\frac{{\hslash }^2}{4}\ge \frac{{\hslash }^2}{4}\mathrm{.}\end{array}$ (3.12)

The chain of inequalities which generalizes the basic uncertainty relation of quantum mechanics is ( [18] [24] )

$\frac{\hslash }{2}\le \sqrt{\left[\text{S}\right]}\le \sqrt{\overline{{\left(\Delta Q\right)}^2}\overline{{\left(\Delta P\right)}^2}}\le \langle \text{S}\rangle \mathrm{.}$ (3.13)

Both quantities $\langle \text{S}\rangle$ and $\left[\text{S}\right]$ are invariant with respect to rotations and displacement of the states in the quantum phase plane and $\langle \text{S}\rangle$ is additionally invariant with respect to squeezing [24] .

For squeezed vacuum states $|\mathrm{0,}\zeta \rangle$ we find using their number representation (2.2)

$\frac{1}{2}\left(\overline{\Delta Q\Delta P+\Delta P\Delta Q}\right)=-\text{i}\left(\overline{{\left(\Delta a\right)}^2}-\overline{{\left(\Delta a^{†}\right)}^2}\right)\frac{\hslash }{2}=\text{i}\frac{\zeta -{\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}\frac{\hslash }{2}\mathrm{,}$ (3.14)

and therefore for the modified uncertainty product $\left[\text{S}\right]$ in (3.12) using the explicit expressions (3.7) and (3.14)

$\left[\text{S}\right]=\frac{{\hslash }^2}{4}.$ (3.15)

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 $\left(q\mathrm{,}p\right)$ 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 $\left(q\mathrm{,}p\right)$ to new canonical coordinates $\left(q^{\prime }\mathrm{,}{p}^{\prime }\right)$ 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 $Q\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ [25] (chap. 7) and [26] . For this purpose we calculate the scalar product of a state with the analytic (but non-normalized)

coherent states $||\alpha \rangle \equiv \exp \left(\frac{\alpha {\alpha }^{*}}{2}\right)|\alpha \rangle ={\displaystyle {\sum }_{n=0}^{\infty }}\frac{{\alpha }^n}{\sqrt{n!}}|n\rangle$ with arbitrary complex $\alpha$ .

For the squeezed vacuum state $|\mathrm{0,}\zeta \rangle$ this provides its Bargmann representations

$\begin{array}{c}exp\left(\frac{\alpha {\alpha }^{\mathrm{<mo>*</mo>}}}{2}\right)\langle \mathrm{0,}\zeta |\alpha \rangle ={\left(1-\zeta {\zeta }^{\mathrm{<mo>*</mo>}}\right)}^{\frac{1}{4}}{\displaystyle \sum _{m=0}^{\infty }}\frac{{\left(-1\right)}^m}{2^{m}m\mathrm{<mo>!</mo>}}{\zeta }^{\mathrm{<mo>*</mo>}m}{\alpha }^{2m}\\ ={\left(1-\zeta {\zeta }^{\mathrm{<mo>*</mo>}}\right)}^{\frac{1}{4}}exp\left(-\frac{{\zeta }^{\mathrm{<mo>*</mo>}}}{2}{\alpha }^2\right)={\left(exp\left(\frac{\alpha {\alpha }^{\mathrm{<mo>*</mo>}}}{2}\right)\langle \alpha |\mathrm{0,}\zeta \rangle \right)}^{\mathrm{<mo>*</mo>}}\mathrm{.}\end{array}$ (4.1)

From this one obtains the Husimi-Kano quasiprobability $Q\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ for density operator $\rho =|\mathrm{0,}\zeta \rangle \langle \mathrm{0,}\zeta |$

$\begin{array}{l}Q\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>*</mo>}}\right)\equiv \frac{\langle \alpha |\mathrm{0,}\zeta \rangle \langle \mathrm{0,}\zeta |\alpha \rangle }{\text{π}}=\frac{\sqrt{1-\zeta {\zeta }^{\mathrm{<mo>*</mo>}}}}{\text{π}}exp\left\{-\left(\alpha {\alpha }^{\mathrm{<mo>*</mo>}}+\frac{{\zeta }^{\mathrm{<mo>*</mo>}}}{2}{\alpha }^2+\frac{\zeta }{2}{\alpha }^{\mathrm{<mo>*</mo>2}}\right)\right\}\mathrm{,}\\ {\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>*</mo>}}Q\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>*</mo>}}\right)=1.\end{array}$ (4.2)

In representation by real canonical variables $\left(q\mathrm{,}p\right)$ this is

$\begin{array}{l}Q\left(q\mathrm{,}p\right)=\frac{\sqrt{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}}{2\hslash \text{π}}exp\left\{-\frac{2}{\hslash }\left(\left(1+\frac{\zeta +{\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{2}\right)q^2+\left(1-\frac{\zeta +{\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{2}\right)p^2-\text{i}\left(\zeta -{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)qp\right\}\mathrm{,}\\ {\displaystyle \int }\text{d}q\wedge \text{d}pQ\left(q\mathrm{,}p\right)=1.\end{array}$ (4.3)

There are different possibilities to calculate the Wigner quasprobability [27] [28] . In most convenient way the Wigner quasiprobability $W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ of an arbitrary squeezed state can be calculated from the Wigner quasiprobability $W_0\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ 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 $\xi ={\zeta }^{*}$ and have

$W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>*</mo>}}\right)=W_0\left(\frac{\alpha +\zeta {\alpha }^{\mathrm{<mo>*</mo>}}}{\sqrt{1-\zeta {\zeta }^{\mathrm{<mo>*</mo>}}}}\mathrm{,}\frac{{\alpha }^{\mathrm{<mo>*</mo>}}+{\zeta }^{\mathrm{<mo>*</mo>}}\alpha }{\sqrt{1-\zeta {\zeta }^{\mathrm{<mo>*</mo>}}}}\right)\mathrm{.}$ (4.4)

Taking into account the well-known Wigner quasiprobability for the vacuum state

$W_0\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=\frac{2}{\text{π}}exp\left(-2\alpha {\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\mathrm{,}\left(\rho =|0\rangle \langle 0|\right)\mathrm{,}$ (4.5)

we obtain

$\begin{array}{l}W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=\frac{2}{\text{π}}exp\left\{-2\frac{\left(\alpha +\zeta {\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\left({\alpha }^{\mathrm{<mo>\; *\; </mo>}}+{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\alpha \right)}{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}\right\}\\ =\frac{2}{\text{π}}exp\left\{-2\frac{\left(1+\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)\alpha {\alpha }^{\mathrm{<mo>\; *\; </mo>}}+{\zeta }^{\mathrm{<mo>\; *\; </mo>}}{\alpha }^2+\zeta {\alpha }^{\mathrm{<mo>\; *\; </mo>2}}}{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}\right\}\mathrm{,}{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=1.\end{array}$ (4.6)

In representation by the real canonical variables $\left(q\mathrm{,}p\right)$

$\begin{array}{l}W\left(q\mathrm{,}p\right)\\ =\frac{1}{\hslash \text{π}}exp\left\{-\frac{\left(1+\zeta \right)\left(1+{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)q^2+\left(1-\zeta \right)\left(1-{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)p^2-\text{i}\left(\zeta -{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)2qp}{\left(1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)\hslash }\right\}\\ =\frac{1}{\hslash \text{π}}exp\left\{-\frac{\left(\left(1+\zeta \right)q+\text{i}\left(1-\zeta \right)p\right)\left(\left(1+{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)q-\text{i}\left(1-{\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)p\right)}{\left(1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)\hslash }\right\}\mathrm{,}\\ {\displaystyle \int }\text{d}q\wedge \text{d}pW\left(q\mathrm{,}p\right)=1.\end{array}$ (4.7)

Another often easy way to calculate the Wigner quasiprobability $W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ is to make first the normal ordering of the operator involved in the representation (4.1) that leads to [22]

$\begin{array}{c}W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>*</mo>}}\right)=\langle \rho exp\left(-a\frac{\partial }{\partial \alpha }-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>*</mo>}}}\right)\rangle \delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>*</mo>}}\right)\\ =\langle \rho exp\left(-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>*</mo>}}}\right)exp\left(-a\frac{\partial }{\partial \alpha }\right)\rangle exp\left(\frac{1}{2}\frac{{\partial }^2}{\partial \alpha \partial {\alpha }^{\mathrm{<mo>*</mo>}}}\right)\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>*</mo>}}\right)\\ =\langle \rho exp\left(-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>*</mo>}}}\right)exp\left(-a\frac{\partial }{\partial \alpha }\right)\rangle \frac{2}{\text{π}}exp\left(-2\alpha {\alpha }^{\mathrm{<mo>*</mo>}}\right)\mathrm{.}\end{array}$ (4.8)

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

$exp\left(-a\frac{\partial }{\partial \alpha }\right)$ to the state $|0\rangle$ and of $exp\left(-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)$ from the right to $\langle 0|$

reproduces them. In the real representation one may use the following equivalent definitions where the first corresponds to the definition given by Wigner [27] [28] for pure states (written here with density operator $\rho$ and with Dirac’s notations for states)

$\begin{array}{c}W\left(q\mathrm{,}p\right)=\frac{1}{\hslash \text{π}}{\displaystyle {\int }_{-\infty }^{+\infty }}\text{d}x\langle q-x|\rho |q+x\rangle exp\left(\text{i}\frac{2px}{\hslash }\right)\\ =\frac{1}{\hslash \text{π}}{\displaystyle {\int }_{-\infty }^{+\infty }}\text{d}y\langle p+y|\rho \mathrm{<mo>\; |\; </mo>}p-y\rangle exp\left(\text{i}\frac{2qy}{\hslash }\right)\mathrm{.}\end{array}$ (4.9)

We checked (4.7) by these formulae using the wave functions $\langle q|\mathrm{0,}\zeta \rangle$ and $\langle p|\mathrm{0,}\zeta \rangle$ derived in (3.3).

Without presenting its detailed calculation let us give the more general quasiprobabilities $F_r\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ with the parameter r and defined by [29] (‘*’ means convolution)

$\begin{array}{l}{F}_r\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\mathrm{<mo>\; =\; </mo>}exp\left(\frac{r}{2}\frac{{\partial }^2}{\partial \alpha \partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\\ =\frac{2}{\text{π}\sqrt{r^2}}exp\left(-\frac{2\alpha {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}{r}\right)\mathrm{<mo>\; *\; </mo>}W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\mathrm{,}\end{array}$ (4.10)

Our result for the squeezed vacuum states $|\mathrm{0,}\zeta \rangle$ is

$\begin{array}{c}{F}_r\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=\frac{1}{\text{π}}\sqrt{\frac{1-\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{{\left(\frac{1+r}{2}\right)}^2-{\left(\frac{1-r}{2}\right)}^2\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}}\\ \times exp\left\{-\frac{\left(\frac{1+r}{2}+\frac{1-r}{2}\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}\right)\alpha {\alpha }^{\mathrm{<mo>\; *\; </mo>}}+\frac{{\zeta }^{\mathrm{<mo>\; *\; </mo>}}}{2}{\alpha }^2+\frac{\zeta }{2}{\alpha }^{\mathrm{<mo>\; *\; </mo>2}}}{{\left(\frac{1+r}{2}\right)}^2-{\left(\frac{1-r}{2}\right)}^2\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}\right\}\mathrm{.}\end{array}$ (4.11)

For $r=0$ one obtains the Wigner quasiprobability $W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ , for $r=+1$ the Husimi-Kano quasiprobability $Q\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ and for $r=-1$ the Glauber- Sudarshan quasiprobability $P\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ . The Glauber-Sudarshan quasiprobability $P\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ for squeezed vacuum states is a singular generalized function and makes for $\zeta =0$ the transition to the delta function $\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ . The representation of (4.10) by real canonical variables $\left(q\mathrm{,}p\right)$ 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 $\left(a\mathrm{,}{a}^{†}\right)$ . We begin with the special expectation values $\overline{a^{†k}{a}^k}$ since from these expectation values one may calculate the expectation values $\overline{N^l}$ of the number operator N.

Using the number representation (2.3) of the squeezed vacuum states $|\mathrm{0,}\zeta \rangle$ we proceed quickly for the expectation values $\overline{a^{†k}{a}^k}$ which depend only on $\left|\zeta \right|\equiv \sqrt{\zeta {\zeta }^{\mathrm{<mo>\; *\; </mo>}}}$ to the following intermediate result

$\begin{array}{c}\overline{a^{†k}{a}^k}\equiv \langle \mathrm{0,}\zeta |a^{†k}{a}^k|\mathrm{0,}\zeta \rangle =\sqrt{1-{\left|\zeta \right|}^2}{\displaystyle \underset{m\mathrm{<mo>\; =\; </mo>0}}{\overset{\infty }{\sum }}}\frac{\left(2m\right){\mathrm{<mo>\; !\; </mo>}}^2}{2^{2m}m{\mathrm{<mo>\; !\; </mo>}}^2\left(2m-k\right)\mathrm{<mo>\; !\; </mo>}}{\left|\zeta \right|}^{2m}\\ =\sqrt{1-{\left|\zeta \right|}^2}{\left|\zeta \right|}^k\frac{{\partial }^k}{\partial {\left|\zeta \right|}^k}{\displaystyle \underset{m\mathrm{<mo>\; =\; </mo>0}}{\overset{\infty }{\sum }}}\frac{\left(2m\right)\mathrm{<mo>\; !\; </mo>}}{2^{2m}m{\mathrm{<mo>\; !\; </mo>}}^2}{\left|\zeta \right|}^{2m}\\ =\sqrt{1-{\left|\zeta \right|}^2}{\left|\zeta \right|}^k\frac{{\partial }^k}{\partial {\left|\zeta \right|}^k}\frac{1}{\sqrt{1-{\left|\zeta \right|}^2}}\mathrm{,}\end{array}$ (5.1)

where we used that the last sum on the right-hand side is the Taylor series of

${\left(1-{\left|\zeta \right|}^2\right)}^{-\frac{1}{2}}$ . This result can be expressed by the Ultraspherical polynomials ${\text{P}}_n^{\left(\alpha \mathrm{,}\alpha \right)}\left(u\right)$ as special case $\alpha =\beta$ of the Jacobi polynomials ${\text{P}}_n^{\left(\alpha \mathrm{,}\beta \right)}\left(u\right)$ (e.g., [30]

[31] ) 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

$\begin{array}{c}\overline{a^{†k}{a}^k}=k\mathrm{<mo>\; !\; </mo>}{\left(-\frac{2\left|\zeta \right|}{1-{\left|\zeta \right|}^2}\right)}^k{\text{P}}_k^{\left(-\frac{1}{2}-k\mathrm{,}-\frac{1}{2}-k\right)}\left(\left|\zeta \right|\right)\\ =k\mathrm{<mo>\; !\; </mo>}{\left(-\frac{\text{i}\left|\zeta \right|}{\sqrt{1-{\left|\zeta \right|}^2}}\right)}^k{\text{P}}_k\left(\frac{\text{i}\left|\zeta \right|}{\sqrt{1-{\left|\zeta \right|}^2}}\right)\mathrm{,}\end{array}$ (5.2)

where ${\text{P}}_n\left(u\right)\equiv {\text{P}}_n^{\left(\mathrm{0,0}\right)}\left(u\right)$ are the Legendre polynomials as special case of the Jacobi polynomials ${\text{P}}_n^{\left(\alpha \mathrm{,}\beta \right)}\left(u\right)$ . More explicitly this provides

$\overline{a^{†k}{a}^k}=k!{\left(\frac{\left|\zeta \right|}{2\left(1-{\left|\zeta \right|}^2\right)}\right)}^k{\displaystyle \underset{j=0}{\overset{\left[\frac{k}{2}\right]}{\sum }}}\frac{k!}{j{!}^2\left(k-2j\right)!}{\left(2\left|\zeta \right|\right)}^{k-2j}\ge 0.$ (5.3)

Remarkable in these transformations is that we could split an essential factor

${\left(\frac{\left|\zeta \right|}{1-{\left|\zeta \right|}^2}\right)}^k$ multiplied by a polynomial in comparison to the infinite sums in (5.1).

The expectation values of symmetrically ordered power operators $\overline{\mathcal{S}\left\{a^{†k}{a}^k\right\}}$ are connected with $\overline{a^{†k}{a}^k}$ by (see, e.g., Equation (7.6) in [22] )

$\overline{\mathcal{S}\left\{a^{†k}{a}^k\right\}}={\displaystyle \underset{l=0}{\overset{k}{\sum }}}\frac{k{!}^2}{l!\left(k-l\right){!}^2}{\left(\frac{1}{2}\right)}^l\overline{a^{†k-l}{a}^{k-l}}\mathrm{.}$ (5.4)

Inserting for $\overline{a^{†k-l}{a}^{k-l}}$ the result (5.3) one may transform the arising double sum by reordering to

$\begin{array}{c}\overline{\mathcal{S}\left\{a^{†k}{a}^k\right\}}=\frac{k!}{{\left(2\left(1-{\left|\zeta \right|}^2\right)\right)}^k}{\displaystyle \underset{j=0}{\overset{\left[\frac{k}{2}\right]}{\sum }}}\frac{k!}{j{!}^2\left(k-2j\right)!}{\left|\zeta \right|}^{2j}{\left(1+{\left|\zeta \right|}^2\right)}^{k-2j}\\ =\frac{k!}{{\left(2\left(1-{\left|\zeta \right|}^2\right)\right)}^k}{\displaystyle \underset{l=0}{\overset{k}{\sum }}}{\left(\frac{k!}{l!\left(k-l\right)!}\right)}^2{\left|\zeta \right|}^{2l}\mathrm{.}\end{array}$ (5.5)

The transition from the first line to the second line is possible after Taylor

series expansion of ${\left(1+{\left|\zeta \right|}^2\right)}^{k-2j}$ 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

$\begin{array}{c}\overline{\mathcal{S}\left\{a^{†k}{a}^k\right\}}=\frac{k!}{2^k}{\text{P}}_k\left(\frac{1+{\left|\zeta \right|}^2}{1-{\left|\zeta \right|}^2}\right)=\frac{k!}{{\left(2\left(1-{\left|\zeta \right|}^2\right)\right)}^k}{\text{P}}_k^{\left(\mathrm{0,}-2k-1\right)}\left(1-2{\left|\zeta \right|}^2\right)\\ =k!{\left(\frac{{\left|\zeta \right|}^2}{2\left(1-{\left|\zeta \right|}^2\right)}\right)}^k{\text{P}}_k^{\left(\mathrm{0,}-2k-1\right)}\left(1-\frac{2}{{\left|\zeta \right|}^2}\right)\mathrm{.}\end{array}$ (5.6)

In Appendix E we give some first members of the explicit representation of

the sequence $\overline{\mathcal{S}\left\{a^{†k}{a}^k\right\}}$ .

The expectation values $\overline{N^k}$ of powers of the number operator can be calculated from the expectation values $\overline{a^{†k}{a}^k}$ by the relation

$\overline{N^k}={\displaystyle \underset{l=0}{\overset{k}{\sum }}}\text{S}\left(k\mathrm{,}l\right)\overline{a^{†l}{a}^l}\mathrm{,}$ (5.7)

where $\text{S}\left(k\mathrm{,}l\right)$ 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 ${\left|\zeta \right|}^2$

in the polynomial in the numerator over the denominator ${\left(1-{\left|\zeta \right|}^2\right)}^k$ in $\overline{N^k}$

and we give explicit results in Appendix E up to $k=5$ .

From the calculated expectation values we find in first order

$\overline{a^{†}a}=\bar{N}=\frac{{\left|\zeta \right|}^2}{1-{\left|\zeta \right|}^2},\overline{\mathcal{S}\left\{a^{†}a\right\}}=\frac{1+{\left|\zeta \right|}^2}{2\left(1-{\left|\zeta \right|}^2\right)}=\bar{N}+\frac{1}{2},$ (5.8)

and in second order (see also Appendix E for more and higher-order expectation values)

$\overline{a^{†2}{a}^2}=\frac{{\left|\zeta \right|}^2\left(1+2{\left|\zeta \right|}^2\right)}{{\left(1-{\left|\zeta \right|}^2\right)}^2}=\bar{N}\left(3\bar{N}+1\right)\mathrm{,}\overline{N^2}=\frac{{\left|\zeta \right|}^2\left(2+{\left|\zeta \right|}^2\right)}{{\left(1-{\left|\zeta \right|}^2\right)}^2}=\bar{N}\left(3\bar{N}+2\right)\mathrm{,}$