_{1}

^{*}

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
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 (1,1)
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**
.

Besides the number states | n 〉 and the coherent states | α 〉 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 〉 displaced by a complex parameter β in the phase space (for one mode). The higher number states | n 〉 with n ≠ 0 are the discrete excitations of the ground state | n = 0 〉 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 [

In the narrow sense the squeezing operations form together with rotations in a plane (the two-dimensional phase plane) the Lie group S U ( 1,1 ) 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 S U ( 1,1 ) are realized by quadratic combinations of the annihilation and creation operators ( a , a † ) of this mode but in Appendix A we represent in detail the basic relations for S U ( 1,1 ) . Besides this, the Lie group S U ( 1,1 ) 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., [

The main purpose of this paper is the representation of the formalism of S U ( 1,1 ) 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.

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

e x p ( − ζ 2 a † 2 ) with ζ as a complex parameter ( | ζ | < 1 ) onto the vacuum

state | 0 〉 . As usual, ( a , a † ) denote a pair of annihilation and creation operators of a single boson mode with the commutation relations [ a , a † ] ≡ a a † − a † a = I , (I unity operator) and they act onto the number states | n 〉 which are orthonormalized and complete

a | n 〉 = n | n − 1 〉 , a † | n 〉 = n + 1 | n + 1 〉 , N | n 〉 ≡ a † a | n 〉 = n | n 〉 ,

〈 m | n 〉 = δ m , n , ∑ n = 0 ∞ | n 〉 〈 n | = I , | n 〉 = a † n n ! | 0 〉 , ( n = 0 , 1 , 2 , ⋯ ) . (2.1)

Now, we define the squeezed vacuum states | 0, ζ 〉 in the non-unitary approach by

| 0 , ζ 〉 ∝ exp ( − ζ 2 a † 2 ) | 0 〉 = ∑ m = 0 ∞ ( − 1 ) m 2 m m ! ζ m a † 2 m | 0 〉 = ∑ m = 0 ∞ ( − 1 ) m ( 2 m ) ! 2 m m ! ζ m | 2 m 〉 , (2.2)

where the zero in the notation | 0, ζ 〉 is arranged for the substitution by a complex displacement parameter β in the later generalization to displaced vacuum states | β , ζ 〉 (see

Since e − ζ 2 a † 2 is not a unitary operator the right-hand side of (2) is not normalized and using the Taylor series ∑ m = 0 ∞ ( 2 m ) ! 2 2 m m ! 2 q m = 1 1 − q we find the

normalization factor

| 0 , ζ 〉 = ( 1 − ζ ζ * ) 1 4 exp ( − ζ 2 a † 2 ) | 0 〉 = ( 1 − ζ ζ * ) 1 4 ∑ m = 0 ∞ ( − 1 ) m ( 2 m ) ! 2 m m ! ζ m | 2 m 〉 , 〈 0 , ζ | 0 , ζ 〉 = 1 , | ζ | ≡ ζ ζ * < 1. (2.3)

The complex parameter ζ is restricted in the non-unitary approach for normalizable states to | ζ | < 1 but can be continued to non-normalizable states for | ζ | ≥ 1 . In the unitary approach (2) we apply a unitary operator

^{1}Note that the operators S ( ζ ′ * ,0, ζ ′ ) ≡ e x p ( ζ ′ * 2 a 2 − ζ ′ 2 a † 2 ) with complex ζ ′ 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 S U ( 1,1 ) group of squeezing and rotation operators; see Appendix A.

e x p ( ζ ′ * 2 a 2 − ζ ′ 2 a † 2 ) to the vacuum state | 0 〉 according to

| 0, ζ 〉 ≡ e x p { Arth | ζ | | ζ | ( ζ * 2 a 2 − ζ 2 a † 2 ) } | 0 〉 ≡ e x p ( ζ ′ * 2 a 2 − ζ ′ 2 a † 2 ) | 0 〉 . (2.4)

The connection between the two parameters ζ and ζ ′ is given by^{1}

ζ ′ = Arth ( | ζ | ) | ζ | ζ , ζ ′ * = Arth ( | ζ | ) | ζ | ζ * , | ζ ′ | = Arth | ζ | , ζ ′ | ζ ′ | = ζ | ζ | ,

ζ = th ( | ζ ′ | ) | ζ ′ | ζ ′ , ζ * = th ( | ζ ′ | ) | ζ ′ | ζ ′ * , | ζ | = th ( | ζ ′ | ) . (2.5)

The parameter ζ ′ is stretched in comparison to ζ and takes on the whole complex plane for normalizable states but ζ and ζ ′ possess the same directions in the complex plane. It is easy to rewrite the formulae derived in the following from parameters ( ζ , ζ * ) to parameters ( ζ ′ , ζ ′ * ) using (5), for example

1 1 − | ζ | 2 = ch 2 ( | ζ ′ | ) , | ζ | 2 1 − | ζ | 2 = sh 2 ( | ζ ′ | ) , 1 ± | ζ | 1 ∓ | ζ | = exp ( ± | ζ ′ | ) . (2.6)

The complex parameter ζ has often some advantages in comparison to the complex parameter ζ ′ concerning compactness of formulae but sometimes, e.g., in the dynamics to quadratic Hamiltonians in ( a , a † ) , the representation by ζ ′ is to prefer and ζ in literature notations | α , ζ 〉 corresponds mostly to our ζ ′ . 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

e x p { Arth | ζ | | ζ | ( ζ * 2 a 2 − ζ 2 a † 2 ) } = e x p ( − ζ 2 a † 2 ) ( 1 − ζ ζ * ) 1 4 ( a a † + a † a ) e x p ( ζ * 2 a 2 ) . (2.7)

The basic relations for squeezing operators between unitary and non-unitary approach were developed already earlier (e.g., [

From (2.3) follow the probabilities p n to the photon statistics of squeezed vacuum states

p n ≡ 〈 n | 0, ζ 〉 〈 0, ζ | n 〉 = { p 2 m = 1 − | ζ | 2 ( 2 m ) ! 2 2 m m ! 2 | ζ | 2 m = 1 − | ζ | 2 ( 2 m − 1 ) ! ! 2 m m ! | ζ | 2 m p 2 m + 1 = 0. (2.8)

The sum over the p n are normalized according to

∑ n = 0 ∞ p n = 1 − | ζ | 2 ∑ m = 0 ∞ ( 2 m ) ! 2 2 m m ! 2 | ζ | 2 m = 1 , (2.9)

as one affirms from the Taylor series of the function ( 1 − | ζ | 2 ) − 1 2 . Only the

probabilities for even n = 2 m are non-vanishing and the probabilities p 2 m are

strictly decreasing ( p 2 ( m + 1 ) ≤ p 2 m ) for increasing m.

From the commutation relations

e x p ( − ζ 2 a † 2 ) a e x p ( ζ 2 a † 2 ) = a − ζ 2 [ a † 2 , a ] + 1 2 ! ( ζ 2 ) 2 [ a † 2 , [ a † 2 , a ] ] − ⋯ = a + ζ a † , (2.10)

in connection with a | 0 〉 = 0 follows that the states | 0, ζ 〉 are eigenstates of the operator a + ζ a † to the eigenvalue zero

0 = exp ( − ζ 2 a † 2 ) a | 0 〉 = ( a + ζ a † ) exp ( − ζ 2 a † 2 ) | 0 〉 , (2.11)

that means

( a + ζ a † ) | 0, ζ 〉 = 0 , ⇔ a | 0, ζ 〉 = − ζ a † | 0, ζ 〉 . (2.12)

In representation by canonical operators ( Q , P ) this is equivalent to

( ( 1 + ζ ) Q + i ( 1 − ζ ) P ) | 0, ζ 〉 = 0, ⇔ P | 0, ζ 〉 = i 1 + ζ 1 − ζ Q | 0, ζ 〉 . (2.13)

Thus both the states a | 0, ζ 〉 and a † | 0, ζ 〉 as well as the states Q | 0, ζ 〉 and P | 0, ζ 〉 are linearly dependent. Furthermore, from (2.12) follows

a † k a | 0, ζ 〉 = − ζ a † k + 1 | 0, ζ 〉 , a l | 0, ζ 〉 = − ζ a l − 1 a † | 0, ζ 〉 . (2.14)

If one forms the scalar products of these relations by multiplication with 〈 0, ζ | one obtains the expectation values of a † k a ¯ and − ζ a † k + 1 ¯ and the expectation values a l ¯ and − ζ a l − 1 a † ¯ , respectively, with equality relations between them. By differentiations of (2.3) with respect to ζ and to ζ * in connection with (2.14) for k = 1 we find

N | 0, ζ 〉 = − ζ a † 2 | 0, ζ 〉 = 2 ( ζ ∂ ∂ ζ − ζ * ∂ ∂ ζ * ) | 0, ζ 〉 , (2.15)

as a further interesting relation which can also be written as eigenvalue equation for | 0, ζ 〉 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 | 0, ζ 〉 is the vacuum state | 0 〉 . It may be considered as a measure of nonclassicality of a state [

d ( | 0 , ζ 〉 〈 0 , ζ | , | 0 〉 〈 0 | ) = 2 ( 1 − 〈 0 | 0 , ζ 〉 〈 0 , ζ | 0 〉 ) = 2 ( 1 − 1 − | ζ | 2 ) = 2 ( 1 − 1 ch ( | ζ ′ | ) ) = | ζ | ( 1 + 1 8 | ζ | 2 + ⋯ ) = | ζ ′ | ( 1 − 5 24 | ζ ′ | 2 + ⋯ ) . (2.16)

It depends only on the modulus of ζ . For strong squeezing | ζ | → 1 this distance goes to 2 that is the largest distance for normalized states in Hilbert space and means orthogonality of the two states (

The wave functions of squeezed vacuum states are the scalar products 〈 q | 0, ζ 〉 and 〈 p | 0, ζ 〉 with the eigenstates | q 〉 and | p 〉 of the operators Q and P. Their number representations are

| q 〉 = 1 ( ℏ π ) 1 4 e x p ( − q 2 2 ℏ ) ∑ n = 0 ∞ 1 2 n n ! H n ( q ℏ ) | n 〉 ,

| p 〉 = 1 ( ℏ π ) 1 4 e x p ( − p 2 2 ℏ ) ∑ n = 0 ∞ i n 2 n n ! H n ( p ℏ ) | n 〉 , (3.1)

where H n ( z ) denotes the Hermite polynomials. They are not normalizable as it is well known and are only normalized by means of the delta function

〈 q | q ′ 〉 = δ ( q − q ′ ) , 〈 p | p ′ 〉 = δ ( p − p ′ ) , 〈 q | p 〉 = 1 2 ℏ π e x p ( i q p ℏ ) = 〈 p | q 〉 * . (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

〈 q | 0, ζ 〉 = ( 1 − ζ ζ * ℏ π ) 1 4 e x p ( − q 2 2 ℏ ) ∑ m = 0 ∞ ( − 1 ) m ζ m 2 2 m m ! H 2 m ( q 2 ℏ ) = ( 1 − ζ ζ * ℏ π ) 1 4 1 1 − ζ e x p ( − ( 1 + ζ ) q 2 ( 1 − ζ ) 2 ℏ ) ,

〈 p | 0, ζ 〉 = ( 1 − ζ ζ * ℏ π ) 1 4 e x p ( − p 2 2 ℏ ) ∑ m = 0 ∞ ζ m 2 2 m m ! H 2 m ( p 2 ℏ ) = ( 1 − ζ ζ * ℏ π ) 1 4 1 1 + ζ e x p ( − ( 1 − ζ ) p 2 ( 1 + ζ ) 2 ℏ ) . (3.3)

From this follows

W ( q ) ≡ 〈 q | 0, ζ 〉 〈 0, ζ | q 〉 = 1 − ζ ζ * ( 1 − ζ ) ( 1 − ζ * ) ℏ π e x p ( − ( 1 − ζ ζ * ) q 2 ( 1 − ζ ) ( 1 − ζ * ) ℏ ) ,

W ( p ) ≡ 〈 p | 0, ζ 〉 〈 0, ζ | p 〉 = 1 − ζ ζ * ( 1 + ζ ) ( 1 + ζ * ) ℏ π e x p ( − ( 1 − ζ ζ * ) p 2 ( 1 + ζ ) ( 1 + ζ * ) ℏ ) , (3.4)

with the normalization

∫ − ∞ + ∞ d q W ( q ) = ∫ − ∞ + ∞ d p W ( p ) = 1. (3.5)

The functions W ( q ) and W ( p ) are the Wigner quasiprobability W ( q , p ) integrated over one of the canonical variable p or q. The functions (3.4) remain invariant by interchanging the squeezing parameter ζ with its complex conjugate ζ ↔ ζ * . This shows in an example that, in general, a state (here | 0, ζ 〉 ) cannot uniquely be reconstructed from W ( q ) and W ( p ) alone.

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

Q ¯ = 0 , P ¯ = 0 , ( Δ Q ) 2 ¯ = ( 1 − ζ ) ( 1 − ζ * ) 1 − ζ ζ * ℏ 2 , ( Δ P ) 2 ¯ = ( 1 + ζ ) ( 1 + ζ * ) 1 − ζ ζ * ℏ 2 . (3.6)

The product of their uncertainties ( Δ Q ) 2 ¯ and ( Δ P ) 2 ¯ (note inequality z 2 + z * 2 ≤ 2 z z * for arbitrary complex z = x + i y in contrast to x 2 + y 2 ≥ 2 x y for arbitrary real x and y)

( Δ Q ) 2 ¯ ( Δ P ) 2 ¯ − ( 1 − ζ 2 ) ( 1 − ζ * 2 ) ( 1 − ζ ζ * ) 2 ℏ 2 4 ≥ ℏ 2 4 . (3.7)

It depends on the phase χ of the complex squeezing parameter ζ = | ζ | e i χ that means on the position of the principal axes of squeezing in comparison to the axes of the canonical coordinates ( q , p ) (see

( Δ Q ) 2 ¯ + ( Δ P ) 2 ¯ = 1 + ζ ζ * 1 − ζ ζ * ℏ , (3.8)

does not depend on the phase of the squeezing parameter ζ . For real squeezing parameter ζ = ζ * we find for the uncertainties

( Δ Q ) 2 ¯ = 1 − ζ 1 + ζ ℏ 2 , ( Δ P ) 2 ¯ = 1 + ζ 1 − ζ ℏ 2 , ( ζ = ζ * ) , (3.9)

and their product is equal to ℏ 2 4 the minimal possible one. The principal axes

of the squeezing ellipses are then in direction of the coordinate axes ( q , p ) (see

addition the uncertainty correlation 1 2 Δ Q Δ P + Δ P Δ Q ¯ [

The uncertainty correlation 1 2 Q P + P Q ¯ arises in a natural way as non-

diagonal elements of the (symmetrical) uncertainty matrix S if we consider the cumulant expansion of the Wigner quasiprobability W ( q , p ) or the corresponding expansion of its Fourier transform W ˜ ( u , v ) [

S ≡ ( ( Δ Q ) 2 ¯ 1 2 Δ Q Δ P + Δ P Δ Q ¯ 1 2 Δ Q Δ P + Δ P Δ Q ¯ ( Δ P ) 2 ¯ ) = ( Δ a Δ a † + Δ a † Δ a ¯ + ( ( Δ a ) 2 ¯ + ( Δ a † ) 2 ¯ ) − i ( ( Δ a ) 2 ¯ − ( Δ a † ) 2 ¯ ) − i ( ( Δ a ) 2 ¯ − ( Δ a † ) 2 ¯ ) Δ a Δ a † + Δ a † Δ a ¯ − ( ( Δ a ) 2 ¯ + ( Δ a † ) 2 ¯ ) ) ℏ 2 . (3.10)

It is also called variance matrix [

〈 S 〉 = ( Δ Q ) 2 ¯ + ( Δ P ) 2 ¯ = ( Δ a Δ a † + Δ a † Δ a ¯ ) ℏ , (3.11)

is the uncertainty sum and the determinant of the matrix S denoted by [ S ] is essentially the uncertainty product but modified by the uncertainty correlations

[ S ] = ( Δ Q ) 2 ¯ ( Δ P ) 2 ¯ − 1 4 ( Δ Q Δ P + Δ P Δ Q ¯ ) 2 = { ( Δ a Δ a † + Δ a † Δ a ¯ ) 2 − 4 ( Δ a ) 2 ¯ ( Δ a † ) 2 ¯ } ℏ 2 4 ≥ ℏ 2 4 . (3.12)

The chain of inequalities which generalizes the basic uncertainty relation of quantum mechanics is ( [

ℏ 2 ≤ [ S ] ≤ ( Δ Q ) 2 ¯ ( Δ P ) 2 ¯ ≤ 〈 S 〉 . (3.13)

Both quantities 〈 S 〉 and [ S ] are invariant with respect to rotations and displacement of the states in the quantum phase plane and 〈 S 〉 is additionally invariant with respect to squeezing [

For squeezed vacuum states | 0, ζ 〉 we find using their number representation (2.2)

1 2 ( Δ Q Δ P + Δ P Δ Q ¯ ) = − i ( ( Δ a ) 2 ¯ − ( Δ a † ) 2 ¯ ) ℏ 2 = i ζ − ζ * 1 − ζ ζ * ℏ 2 , (3.14)

and therefore for the modified uncertainty product [ S ] in (3.12) using the explicit expressions (3.7) and (3.14)

[ S ] = ℏ 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 ( q , p ) shown in

By a rotation of the canonical coordinates ( q , p ) to new canonical coordinates ( q ′ , p ′ ) 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.

The Bargmann representation of a state is a representation by an analytic function which, in particular, leads immediately to the Husimi-Kano quasiprobability Q ( α , α * ) [

coherent states | | α 〉 ≡ exp ( α α * 2 ) | α 〉 = ∑ n = 0 ∞ α n n ! | n 〉 with arbitrary complex α .

For the squeezed vacuum state | 0, ζ 〉 this provides its Bargmann representations

e x p ( α α * 2 ) 〈 0, ζ | α 〉 = ( 1 − ζ ζ * ) 1 4 ∑ m = 0 ∞ ( − 1 ) m 2 m m ! ζ * m α 2 m = ( 1 − ζ ζ * ) 1 4 e x p ( − ζ * 2 α 2 ) = ( e x p ( α α * 2 ) 〈 α | 0, ζ 〉 ) * . (4.1)

From this one obtains the Husimi-Kano quasiprobability Q ( α , α * ) for density operator ρ = | 0, ζ 〉 〈 0, ζ |

Q ( α , α * ) ≡ 〈 α | 0, ζ 〉 〈 0, ζ | α 〉 π = 1 − ζ ζ * π e x p { − ( α α * + ζ * 2 α 2 + ζ 2 α * 2 ) } , ∫ i 2 d α ∧ d α * Q ( α , α * ) = 1. (4.2)

In representation by real canonical variables ( q , p ) this is

Q ( q , p ) = 1 − ζ ζ * 2 ℏ π e x p { − 2 ℏ ( ( 1 + ζ + ζ * 2 ) q 2 + ( 1 − ζ + ζ * 2 ) p 2 − i ( ζ − ζ * ) ) q p } , ∫ d q ∧ d p Q ( q , p ) = 1. (4.3)

There are different possibilities to calculate the Wigner quasprobability [

W ( α , α * ) = W 0 ( α + ζ α * 1 − ζ ζ * , α * + ζ * α 1 − ζ ζ * ) . (4.4)

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

W 0 ( α , α * ) = 2 π e x p ( − 2 α α * ) , ( ρ = | 0 〉 〈 0 | ) , (4.5)

we obtain

W ( α , α * ) = 2 π e x p { − 2 ( α + ζ α * ) ( α * + ζ * α ) 1 − ζ ζ * } = 2 π e x p { − 2 ( 1 + ζ ζ * ) α α * + ζ * α 2 + ζ α * 2 1 − ζ ζ * } , ∫ i 2 d α ∧ d α * W ( α , α * ) = 1. (4.6)

In representation by the real canonical variables ( q , p )

W ( q , p ) = 1 ℏ π e x p { − ( 1 + ζ ) ( 1 + ζ * ) q 2 + ( 1 − ζ ) ( 1 − ζ * ) p 2 − i ( ζ − ζ * ) 2 q p ( 1 − ζ ζ * ) ℏ } = 1 ℏ π e x p { − ( ( 1 + ζ ) q + i ( 1 − ζ ) p ) ( ( 1 + ζ * ) q − i ( 1 − ζ * ) p ) ( 1 − ζ ζ * ) ℏ } , ∫ d q ∧ d p W ( q , p ) = 1. (4.7)

Another often easy way to calculate the Wigner quasiprobability W ( α , α * ) is to make first the normal ordering of the operator involved in the representation (4.1) that leads to [

W ( α , α * ) = 〈 ρ e x p ( − a ∂ ∂ α − a † ∂ ∂ α * ) 〉 δ ( α , α * ) = 〈 ρ e x p ( − a † ∂ ∂ α * ) e x p ( − a ∂ ∂ α ) 〉 e x p ( 1 2 ∂ 2 ∂ α ∂ α * ) δ ( α , α * ) = 〈 ρ e x p ( − a † ∂ ∂ α * ) e x p ( − a ∂ ∂ α ) 〉 2 π e x p ( − 2 α α * ) . (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

e x p ( − a ∂ ∂ α ) to the state | 0 〉 and of e x p ( − a † ∂ ∂ α * ) from the right to 〈 0 |

reproduces them. In the real representation one may use the following equivalent definitions where the first corresponds to the definition given by Wigner [

W ( q , p ) = 1 ℏ π ∫ − ∞ + ∞ d x 〈 q − x | ρ | q + x 〉 e x p ( i 2 p x ℏ ) = 1 ℏ π ∫ − ∞ + ∞ d y 〈 p + y | ρ | p − y 〉 e x p ( i 2 q y ℏ ) . (4.9)

We checked (4.7) by these formulae using the wave functions 〈 q | 0, ζ 〉 and 〈 p | 0, ζ 〉 derived in (3.3).

Without presenting its detailed calculation let us give the more general quasiprobabilities F r ( α , α * ) with the parameter r and defined by [

F r ( α , α * ) = e x p ( r 2 ∂ 2 ∂ α ∂ α * ) W ( α , α * ) = 2 π r 2 e x p ( − 2 α α * r ) * W ( α , α * ) , (4.10)

Our result for the squeezed vacuum states | 0, ζ 〉 is

F r ( α , α * ) = 1 π 1 − ζ ζ * ( 1 + r 2 ) 2 − ( 1 − r 2 ) 2 ζ ζ * × e x p { − ( 1 + r 2 + 1 − r 2 ζ ζ * ) α α * + ζ * 2 α 2 + ζ 2 α * 2 ( 1 + r 2 ) 2 − ( 1 − r 2 ) 2 ζ ζ * } . (4.11)

For r = 0 one obtains the Wigner quasiprobability W ( α , α * ) , for r = + 1 the Husimi-Kano quasiprobability Q ( α , α * ) and for r = − 1 the Glauber- Sudarshan quasiprobability P ( α , α * ) . The Glauber-Sudarshan quasiprobability P ( α , α * ) for squeezed vacuum states is a singular generalized function and makes for ζ = 0 the transition to the delta function δ ( α , α * ) . The representation of (4.10) by real canonical variables ( q , p ) 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.

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 ( a , a † ) . We begin with the special expectation values a † k a k ¯ since from these expectation values one may calculate the expectation values N l ¯ of the number operator N.

Using the number representation (2.3) of the squeezed vacuum states | 0, ζ 〉 we proceed quickly for the expectation values a † k a k ¯ which depend only on | ζ | ≡ ζ ζ * to the following intermediate result

a † k a k ¯ ≡ 〈 0, ζ | a † k a k | 0, ζ 〉 = 1 − | ζ | 2 ∑ m = 0 ∞ ( 2 m ) ! 2 2 2 m m ! 2 ( 2 m − k ) ! | ζ | 2 m = 1 − | ζ | 2 | ζ | k ∂ k ∂ | ζ | k ∑ m = 0 ∞ ( 2 m ) ! 2 2 m m ! 2 | ζ | 2 m = 1 − | ζ | 2 | ζ | k ∂ k ∂ | ζ | k 1 1 − | ζ | 2 , (5.1)

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

( 1 − | ζ | 2 ) − 1 2 . This result can be expressed by the Ultraspherical polynomials P n ( α , α ) ( u ) as special case α = β of the Jacobi polynomials P n ( α , β ) ( u ) (e.g., [

[

a † k a k ¯ = k ! ( − 2 | ζ | 1 − | ζ | 2 ) k P k ( − 1 2 − k , − 1 2 − k ) ( | ζ | ) = k ! ( − i | ζ | 1 − | ζ | 2 ) k P k ( i | ζ | 1 − | ζ | 2 ) , (5.2)

where P n ( u ) ≡ P n ( 0,0 ) ( u ) are the Legendre polynomials as special case of the Jacobi polynomials P n ( α , β ) ( u ) . More explicitly this provides

a † k a k ¯ = k ! ( | ζ | 2 ( 1 − | ζ | 2 ) ) k ∑ j = 0 [ k 2 ] k ! j ! 2 ( k − 2 j ) ! ( 2 | ζ | ) k − 2 j ≥ 0. (5.3)

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

( | ζ | 1 − | ζ | 2 ) k multiplied by a polynomial in comparison to the infinite sums in (5.1).

The expectation values of symmetrically ordered power operators S { a † k a k } ¯ are connected with a † k a k ¯ by (see, e.g., Equation (7.6) in [

S { a † k a k } ¯ = ∑ l = 0 k k ! 2 l ! ( k − l ) ! 2 ( 1 2 ) l a † k − l a k − l ¯ . (5.4)

Inserting for a † k − l a k − l ¯ the result (5.3) one may transform the arising double sum by reordering to

S { a † k a k } ¯ = k ! ( 2 ( 1 − | ζ | 2 ) ) k ∑ j = 0 [ k 2 ] k ! j ! 2 ( k − 2 j ) ! | ζ | 2 j ( 1 + | ζ | 2 ) k − 2 j = k ! ( 2 ( 1 − | ζ | 2 ) ) k ∑ l = 0 k ( k ! l ! ( k − l ) ! ) 2 | ζ | 2 l . (5.5)

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

series expansion of ( 1 + | ζ | 2 ) k − 2 j 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

S { a † k a k } ¯ = k ! 2 k P k ( 1 + | ζ | 2 1 − | ζ | 2 ) = k ! ( 2 ( 1 − | ζ | 2 ) ) k P k ( 0, − 2 k − 1 ) ( 1 − 2 | ζ | 2 ) = k ! ( | ζ | 2 2 ( 1 − | ζ | 2 ) ) k P k ( 0, − 2 k − 1 ) ( 1 − 2 | ζ | 2 ) . (5.6)

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

the sequence S { a † k a k } ¯ .

The expectation values N k ¯ of powers of the number operator can be calculated from the expectation values a † k a k ¯ by the relation

N k ¯ = ∑ l = 0 k S ( k , l ) a † l a l ¯ , (5.7)

where S ( k , l ) 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 | ζ | 2

in the polynomial in the numerator over the denominator ( 1 − | ζ | 2 ) k in N k ¯

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

From the calculated expectation values we find in first order

a † a ¯ = N ¯ = | ζ | 2 1 − | ζ | 2 , S { a † a } ¯ = 1 + | ζ | 2 2 ( 1 − | ζ | 2 ) = N ¯ + 1 2 , (5.8)

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

a † 2 a 2 ¯ = | ζ | 2 ( 1 + 2 | ζ | 2 ) ( 1 − | ζ | 2 ) 2 = N ¯ ( 3 N ¯ + 1 ) , N 2 ¯ = | ζ | 2 ( 2 + | ζ | 2 ) ( 1 − | ζ | 2 ) 2 = N ¯ ( 3 N ¯ + 2 ) ,

S { a † 2 a 2 } ¯ = 1 + 4 | ζ | 2 + | ζ | 4 2 ( 1 − | ζ | 2 ) 2 = 3 N ¯ ( N ¯ + 1 ) + 1 2 . (5.9)

From this follows for squeezed vacuum states using | ζ | 2 = N ¯ N ¯ + 1 < 1

( Δ N ) 2 ¯ ≡ N 2 ¯ − ( N ¯ ) 2 = 2 | ζ | 2 ( 1 − | ζ | 2 ) 2 = 2 N ¯ ( N ¯ + 1 ) , (5.10)

that means the number uncertainty for squeezed vacuum states is larger than,

for example, for coherent states ( ( Δ N ) 2 ¯ = N ¯ ) and furthermore

a † 2 a 2 ¯ − ( a † a ¯ ) 2 = | ζ | 2 ( 1 + | ζ | 2 ) ( 1 − | ζ | 2 ) 2 = ( Δ N ) 2 ¯ − N ¯ = N ¯ ( 2 N ¯ + 1 ) ,

S { a † 2 a 2 } ¯ − ( S { a † a } ¯ ) 2 = 1 + 6 | ζ | 2 + | ζ | 4 4 ( 1 − | ζ | 2 ) 2 = 1 4 + 2 N ¯ ( N ¯ + 1 ) ≥ 1 4 . (5.11)

This shows that sub-Poissonian statistics a † 2 a 2 ¯ − ( a † a ¯ ) 2 < 0 does not exist for squeezed vacuum states and that S { a † 2 a 2 } ¯ − ( S { a † a } ¯ ) 2 ≥ 1 4 satisfies the general inequality for this quantity [

In this Section we calculate more general expectation values for the squeezed vacuum states of the form a † k a l ¯ = a † l a k ¯ * than in the preceding Section. They are only non-vanishing if the difference k − l 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 | 2 m 〉 Therefore, we now calculate separately the expectation values a † 2 k a 2 l ¯ and a † 2 k + 1 a 2 l + 1 ¯ . They depend on ζ and ζ * separately and, therefore, we use here the pair of variables ( ζ , ζ * ) for the representation of the results.

For a † 2 k a 2 l ¯ using (2.2) and doubling relations for the Gamma function applied to ( 2 x ) ! we arrive at the intermediate result

a † 2 k a 2 l ¯ ≡ 〈 0, ζ | a † 2 k a 2 l | 0, ζ 〉 = 1 − ζ ζ * π ( − 2 ) k + l ζ * k ζ l ∑ j = 0 ∞ ( j + k − 1 2 ) ! ( j + l − 1 2 ) ! j ! ( j − 1 2 ) ! ( ζ ζ * ) j , ( ( − 1 2 ) ! ≡ π ) . (6.1)

By comparison of this expression with explicit expressions for the Hyper- geometric function 2 F 1 ( α , β ; γ ; z ) we see that this is a polynomial case with the Jacobi polynomials P n ( α , β ) ( u ) involved. The main polynomial case of the Hypergeometric function is

2 F 1 ( − n , β ; γ ; z ) = k ! ( γ − 1 ) ! ( n + γ − 1 ) ! P n ( γ − 1, β − γ − n ) ( 1 − 2 z ) = n ! ( γ − 1 ) ! ( n + γ − 1 ) ! z n P n ( − β − n , β − γ − n ) ( 1 − 2 z ) = F 2 1 ( β , − n ; γ ; z ) , (6.2)

and the other possible polynomial case is ( see [

z n 2 F 1 ( − n 2 , − n − 1 2 ; α + 1 ; − 1 z 2 ) = n ! α ! ( n + α ) ! ( 1 + z 2 ) n P n ( α , α ) ( z 1 + z 2 ) . (6.3)

By transformation relations of the Hypergeometric function, in general, and in possible special cases we find the relation between the Jacobi polynomials

( 1 − z ) n P n ( α , β ) ( 1 + z 1 − z ) = P n ( α , − 2 n − α − β − 1 ) ( 1 − 2 z ) , (6.4)

and in special case if one of the upper indices is an integer

n ! ( n + β ) ! ( 2 u − 1 ) n P n ( m − n , β ) ( u ) = m ! ( m + β ) ! ( 2 u − 1 ) m P m ( n − m , β ) ( u ) . (6.5)

Together with the more trivial relation

P n ( α , β ) ( u ) = ( − 1 ) n P n ( β , α ) ( − u ) , (6.6)

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 k ↔ l

a † 2 k a 2 l ¯ = ( − 2 ) k + l k ! ( l − 1 2 ) ! ( − 1 2 ) ! ζ * k ζ l ( 1 − ζ ζ * ) l P k ( − 1 2 , l − k ) ( 1 + ζ ζ * 1 − ζ ζ * ) = ( − 2 ) k + l l ! ( k − 1 2 ) ! ( − 1 2 ) ! ζ * k ζ l ( 1 − ζ ζ * ) k P l ( − 1 2 , k − l ) ( 1 + ζ ζ * 1 − ζ ζ * ) , (6.7)

or alternatively

a † 2 k a 2 l ¯ = ( − 2 ) k + l k ! ( l − 1 2 ) ! ( − 1 2 ) ! ζ * k ζ l ( 1 − ζ ζ * ) k + l P k ( − 1 2 , − k − l − 1 2 ) ( 1 − 2 ζ ζ * ) = ( − 2 ) k + l l ! ( k − 1 2 ) ! ( − 1 2 ) ! ζ * k ζ l ( 1 − ζ ζ * ) k + l P l ( − 1 2 , − k − l − 1 2 ) ( 1 − 2 ζ ζ * ) . (6.8)

In case of the expectation values a † 2 k + 1 a 2 l + 1 ¯ the analogous intermediate result to (6.1) is

a † 2 k + 1 a 2 l + 1 ¯ = 1 − ζ ζ * π ( − 1 ) k + l 2 k + l + 1 ζ * k ζ l ∑ j = 1 ∞ ( j + k − 1 2 ) ! ( j + l − 1 2 ) ! ( j − 1 2 ) ! ( j − 1 ) ! ( ζ ζ * ) j . (6.9)

Using the relation to the Hypergeometric function and the Jacobi polynomials from this follows in analogy to (6.7)

a † 2 k + 1 a 2 l + 1 ¯ = ( − 2 ) k + l k ! ( l + 1 2 ) ! ( 1 2 ) ! ζ * k + 1 ζ l + 1 ( 1 − ζ ζ * ) l + 1 P k ( 1 2 , l − k ) ( 1 + ζ ζ * 1 − ζ ζ * ) = ( − 2 ) k + l l ! ( k + 1 2 ) ! ( 1 2 ) ! ζ * k + 1 ζ l + 1 ( 1 − ζ ζ * ) k + 1 P l ( 1 2 , k − l ) ( 1 + ζ ζ * 1 − ζ ζ * ) , (6.10)

or, alternatively, in analogy to (6.8)

a † 2 k + 1 a 2 l + 1 ¯ = ( − 2 ) k + l k ! ( l + 1 2 ) ! ( 1 2 ) ! ζ * k + 1 ζ l + 1 ( 1 − ζ ζ * ) k + l + 1 P k ( 1 2 , − k − l − 3 2 ) ( 1 − 2 ζ ζ * ) = ( − 2 ) k + l l ! ( k + 1 2 ) ! ( 1 2 ) ! ζ * k + 1 ζ l + 1 ( 1 − ζ ζ * ) k + l + 1 P l ( 1 2 , − k − l − 3 2 ) ( 1 − 2 ζ ζ * ) . (6.11)

We checked the special cases k = l 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 a † k a l ¯ in the special case k = l 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 k ≠ l and had to distinguish the cases of even k and l and of odd k and l and in the specialization k = l there are involved Jacobi polynomials which are different from that for k = l and it is not possible (or simple) to join these polynomials for k = l 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 W ( α , α * ) one may calculate basically the expectation values of symmetrically (Weyl) ordered operators, for example, by integration over the

function α α * for the expectation value of the operator 1 2 ( a a † + a † a ) = N + 1 2 I .

With the Husimi-Kano quasiprobability Q ( α , α * ) 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 a † k and of a l can be calculated with an arbitrary quasiprobability with the parameter r considered in Section 4.

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 ( q , p ) ) 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 D ( β , β * ) to squeezed vacuum states | 0, ζ 〉 according to

| β , ζ 〉 ≡ D ( β , β * ) | 0, ζ 〉 = D ( β , β * ) S ( ζ ′ * ,0, ζ ′ ) | 0 〉 , ( ζ ′ , ζ ′ * ) = Arth ( | ζ | ) | ζ | ( ζ , ζ * ) , (7.1)

where S ( ζ ′ * ,0, ζ ′ ) 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 η = 0 and substituting ζ → − ζ ′

( S ( ζ ′ * ,0, ζ ′ ) ) † ( a , a † ) S ( ζ ′ * ,0, ζ ′ ) = S ( − ζ ′ * ,0, − ζ ′ ) ( a , a † ) ( S ( − ζ ′ * ,0, − ζ ′ ) ) † = ( a , a † ) ( 1 1 − | ζ | 2 , − ζ * 1 − | ζ | 2 − ζ 1 − | ζ | 2 , 1 1 − | ζ | 2 ) . (7.2)

These transformations of ( a , a † ) after transition to the Hermitean basis of operators ( Q , P ) are very similar to Special Lorentz transformations of ( x , c t ) with x one space coordinate and t the time and this is not incidental since it is for real parameter ζ the same one-parameter Lie group. From (7.2) follows

( S ( ζ ′ * ,0, ζ ′ ) ) † D ( β , β * ) S ( ζ ′ * ,0, ζ ′ ) = ( S ( ζ ′ * ,0, ζ ′ ) ) † e x p ( β a † − β * a ) S ( ζ ′ * ,0, ζ ′ ) = e x p ( β a † − ζ * a 1 − | ζ | 2 − β * a − ζ a † 1 − | ζ | 2 ) = e x p ( β + ζ β * 1 − | ζ | 2 a † − β * + ζ * β 1 − | ζ | 2 a ) = D ( β + ζ β * 1 − | ζ | 2 , β * + ζ * β 1 − | ζ | 2 ) . (7.3)

Applied to (7.1) we find (see also Schleich [

| β , ζ 〉 = D ( β , β * ) | 0, ζ 〉 = D ( β , β * ) S ( ζ ′ * ,0, ζ ′ ) | 0 〉 = S ( ζ ′ * ,0, ζ ′ ) D ( β + ζ β * 1 − | ζ | 2 , β * + ζ * β 1 − | ζ | 2 ) | 0 〉 = S ( ζ ′ * ,0, ζ ′ ) | β ′ ,0 〉 , ( β ′ , β ′ * ) ≡ ( β + ζ β * 1 − | ζ | 2 , β * + ζ * β 1 − | ζ | 2 ) , (7.4)

as alternative representation of squeezed vacuum states. This means that we make first a displacement of the vacuum state | 0 〉 to a coherent state | β ′ ,0 〉 with the changed displacement parameter β ′ and after this the squeezing of the coherent state with the same squeezing operator S ( ζ ′ * ,0, ζ ′ ) 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 ζ = 0 one obtains from | β , ζ 〉 the coherent states and for β = 0 the squeezed vacuum states

| β , 0 〉 = D ( β , β * ) | 0 , 0 〉 ≡ | β 〉 , | 0 , ζ 〉 = S ( ζ ′ * , 0 , ζ ′ ) | 0 , 0 〉 , | 0 , 0 〉 ≡ | 0 〉 . (7.5)

The squeezing operator is the stable part in these two alternative representations.

In generalization of the well-known eigenvalue equation of coherent states | β ,0 〉 ≡ | β 〉 the displaced squeezed vacuum states | β , ζ 〉 are right-hand eigenstates of the operator a + ζ a † to eigenvalues β + ζ β * according to

( a + ζ a † ) | β , ζ 〉 = ( β + ζ β * ) | β , ζ 〉 . (7.6)

This follows from the relation

( a + ζ a † ) e x p ( β a † − β * a ) | 0, ζ 〉 = e x p ( β a † − β * a ) ( a + β I + ζ ( a † + β * I ) ) | 0, ζ 〉 = ( β + ζ β * ) e x p ( β a † − β * a ) | 0, ζ 〉 , (7.7)

using that | 0, ζ 〉 are right-hand eigenstates of a + ζ a † 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

D ( β , β * ) = e x p ( − β β * 2 ) e x p ( β a † ) e x p ( − β * a ) , (7.8)

from the definition (7.1) follows

| β , ζ 〉 = e x p ( − β β * 2 ) ( 1 − ζ ζ * ) 1 4 e x p ( β a † ) e x p ( − β * a ) e x p ( − ζ 2 a † 2 ) | 0 〉 = ( 1 − ζ ζ * ) 1 4 e x p ( − β β * 2 ) e x p ( β a † ) e x p ( − ζ 2 ( a † − β * I ) 2 ) e x p ( − β * a ) | 0 〉 = ( 1 − ζ ζ * ) 1 4 e x p ( − β β * + ζ β * 2 2 ) e x p ( ( β + ζ β * ) a † − ζ 2 a † 2 ) | 0 〉 . (7.9)

If we now apply the generating function (D.1) for Hermite polynomials to the factor in front of | 0 〉 we obtain the following form of the representation of displaced squeezed vacuum states

| β , ζ 〉 = ( 1 − ζ ζ * ) 1 4 e x p ( − β β * + ζ β * 2 2 ) ∑ n = 0 ∞ 1 n ! ( 2 ζ 2 a † ) n H n ( β + ζ β * 2 ζ ) | 0 〉 , (7.10)

and using the generation of number states | n 〉 from the vacuum state | 0 〉 (see (2.1)) one finds the final form of the number-state representation^{2}

| β , ζ 〉 = ( 1 − ζ ζ * ) 1 4 e x p ( − ( β + ζ β * ) β * 2 ) ∑ n = 0 ∞ 1 n ! ( 2 ζ 2 ) n H n ( β + ζ β * 2 ζ ) | n 〉 . (7.11)

It is easy to see that for β = 0 using H 2 m ( 0 ) = ( − 1 ) m ( 2 m ) ! m ! , H 2 m + 1 ( 0 ) = 0 one obtains the number representation (2.3) of squeezed vacuum states | 0, ζ 〉 and for ζ = 0 using l i m z → ∞ 1 ( 2 z ) n H n ( z ) = 1 the number representation of coherent states | β 〉 .

From (7.10) one finds the probabilities p n of the photon statistics

p n ≡ 〈 n | β , ζ 〉 〈 β , ζ | n 〉 = 1 − ζ ζ * e x p ( − 2 β β * + ζ * β 2 + ζ β * 2 2 ) 1 n ! ( ζ ζ * 2 ) n × H n ( β + ζ β * 2 ζ ) H n ( β * + ζ * β 2 ζ * ) . (7.12)

^{2}In both parts with 2 ζ 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 | β , ζ 〉 are normalized and that the probabilities p n satisfy the necessary relation

∑ n = 0 ∞ p n = 1. (7.13)

In contrast to the photon distributions of coherent states | β 〉 ≡ | β ,0 〉 which depends only on | β | 2 ≡ β β * and of squeezed vacuum states | 0, ζ 〉 which depends only on | ζ | 2 ≡ ζ ζ * < 1 the photon distribution (7.12) depends in addition to the moduli also on the phases of ζ and β in the complex plane.

The nearest coherent state to the state | β , ζ 〉 is the state | β 〉 and for this distance d ( | β , ζ 〉 〈 β , ζ | , | β 〉 〈 β | ) one obtains the same value as on the right-hand side of (2.16). This means that it does not depend on β .

As important characteristics of the displaced squeezed vacuum states | β , ζ 〉 we now calculate their wave functions 〈 q | β , ζ 〉 and 〈 p | β , ζ 〉 in the eigenstates | q 〉 and | p 〉 of the canonical operators Q and P (in the usual standardizations

〈 q | q ′ 〉 = δ ( q − q ′ ) , ∫ − ∞ + ∞ d q | q 〉 〈 q | = I and similar for | p 〉 ). From the number

representation (7.11) of the states and the well-known number representation of | q 〉 and | p 〉 follows as the first step in the calculation

〈 q | β , ζ 〉 = ( 1 − ζ ζ * ℏ π ) 1 4 e x p ( − q 2 2 ℏ − β * ( β + ζ β * ) 2 ) × ∑ n = 0 ∞ ( ζ ) n 2 n n ! H n ( q ℏ ) H n ( β + ζ β * 2 ζ ) ,

〈 p | β , ζ 〉 = ( 1 − ζ ζ * ℏ π ) 1 4 e x p ( − p 2 2 ℏ − β * ( β + ζ β * ) 2 ) × ∑ n = 0 ∞ ( − i ζ ) n 2 n n ! H n ( p ℏ ) H n ( β + ζ β * 2 ζ ) . (8.1)

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 Q ¯ ≡ 〈 β , ζ | Q | β , ζ 〉 and P ¯ ≡ 〈 β , ζ | P | β , ζ 〉 of the canonical operators Q and P. For squeezed vacuum states | 0, ζ 〉 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 ( β , β * ) according to

Q ¯ = ℏ 2 ( β + β * ) , P ¯ = − i ℏ 2 ( β − β * ) . (8.2)

With these parameters the result of the evaluation of the sums in (8.1) can be represented in the form

〈 q | β , ζ 〉 = ( 1 − ζ ζ * ℏ π ) 1 4 1 1 − ζ e x p ( i P ¯ ( 2 q − Q ¯ ) 2 ℏ ) e x p { − ( 1 + ζ ) ( q − Q ¯ ) 2 2 ( 1 − ζ ) ℏ } ,

〈 p | β , ζ 〉 = ( 1 − ζ ζ * ℏ π ) 1 4 1 1 + ζ e x p ( − i Q ¯ ( 2 p − P ¯ ) 2 ℏ ) e x p { − ( 1 − ζ ) ( p − P ¯ ) 2 2 ( 1 + ζ ) ℏ } . (8.3)

One may make cross checks of these relations using the pair of Fourier transformations

〈 p | β , ζ 〉 = 1 2 ℏ π ∫ − ∞ + ∞ d q e x p ( − i p q ℏ ) 〈 q | β , ζ 〉 , 〈 q | β , ζ 〉 = 1 2 ℏ π ∫ − ∞ + ∞ d p e x p ( i q p ℏ ) 〈 p | β , ζ 〉 . (8.4)

From (8.3) one finds the Gaussian distributions

W ( q ) ≡ 〈 q | β , ζ 〉 〈 β , ζ | q 〉 = 1 − ζ ζ * ( 1 − ζ ) ( 1 − ζ * ) ℏ π e x p { − 1 − ζ ζ * ( 1 − ζ ) ( 1 − ζ * ) ( q − Q ¯ ) 2 ℏ } ,

W ( p ) ≡ 〈 p | β , ζ 〉 〈 β , ζ | p 〉 = 1 − ζ ζ * ( 1 + ζ ) ( 1 + ζ * ) ℏ π e x p { − 1 − ζ ζ * ( 1 + ζ ) ( 1 + ζ * ) ( p − P ¯ ) 2 ℏ } , (8.5)

with the normalization

∫ − ∞ + ∞ d q W ( q ) = ∫ − ∞ + ∞ d p W ( p ) = 1. (8.6)

The functions W ( q ) and W ( p ) are equal to the Wigner quasiprobability W ( q , p ) 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 ζ ↔ ζ * and the states are not uniquely reconstructible from W ( q ) and W ( p ) alone. In our case of (8.5) they are one-dimensional normalized Gaussian distributions around Q ¯ and P ¯ , respectively, with the variances of Q ¯ and P ¯

( Δ Q ) 2 ¯ = ( 1 − ζ ) ( 1 − ζ * ) 1 − ζ ζ * ℏ 2 , ( Δ P ) 2 ¯ = ( 1 + ζ ) ( 1 + ζ * ) 1 − ζ ζ * ℏ 2 , (8.7)

The uncertainty product (see also (3.7))

( Δ Q ) 2 ¯ ( Δ P ) 2 ¯ = ( 1 − ζ 2 ) ( 1 − ζ * 2 ) ( 1 − ζ ζ * ) 2 ℏ 2 4 ≥ ℏ 2 4 , (8.8)

is only for real squeezing parameter ζ but not for complex ζ the minimal possible one.

In case of real squeezing parameter ζ = ζ * one obtains from (8.3)

〈 q | β , ζ 〉 = 1 ( 2 ( Δ Q ) 2 ¯ π ) 1 4 e x p ( i P ¯ ( 2 q − Q ¯ ) 2 ℏ ) e x p { − ( q − Q ¯ ) 2 4 ( Δ Q ) 2 ¯ } ,

〈 p | β , ζ 〉 = 1 ( 2 ( Δ P ) 2 ¯ π ) 1 4 e x p ( − i Q ¯ ( 2 p − P ¯ ) 2 ℏ ) e x p { − ( p − P ¯ ) 2 4 ( Δ P ) 2 ¯ } , (8.9)

with the variances of Q and P

( Δ Q ) 2 ¯ = 1 − ζ 1 + ζ ℏ 2 , ( Δ P ) 2 ¯ = 1 + ζ 1 − ζ ℏ 2 , ( Δ Q ) 2 ¯ ( Δ P ) 2 ¯ = ℏ 2 4 , ( ζ = ζ * ) , (8.10)

As already explained in Section 3 only in case of real ζ = ζ * the squeezing axes coincide with the axes of ( q , p ) 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

− i ℏ ( ∂ ∂ q + q 2 ( Δ Q ) 2 ¯ ) 〈 q | β , ζ 〉 ≡ 〈 q | ( P − i ℏ 2 ( Δ Q ) 2 ¯ Q ) | β , ζ 〉 = ( P ¯ − i ℏ Q ¯ 2 ( Δ Q ) 2 ¯ ) 〈 q | β , ζ 〉 ,

i ℏ ( ∂ ∂ p + p 2 ( Δ P ) 2 ¯ ) 〈 p | β , ζ 〉 ≡ 〈 p | ( Q + i ℏ 2 ( Δ Q ) 2 ¯ P ) | β , ζ 〉 = ( Q ¯ + i ℏ P ¯ 2 ( Δ P ) 2 ¯ ) 〈 p | β , ζ 〉 , (8.11)

where we used 〈 q | P = − i ℏ ∂ ∂ q 〈 q | and 〈 p | Q = i ℏ ∂ ∂ p 〈 p | . The normalization of the states in (8.10) and the pure phase factors e x p ( ∓ i P ¯ Q ¯ 2 ℏ ) 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 | q 〉 and | p 〉 .

Pauli in his hand-book article [

The displaced squeezed vacuum states or squeezed coherent states | β , ζ 〉 are the most general pure states with Gaussian distributions.

The Bargmann representation of states is the representation by an analytic function of α obtain by forming the scalar product of the state with the

analytic but non-normalized coherent state | | α 〉 ≡ ∑ n = 0 ∞ α n n ! | 0 〉 (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

exp ( α α * 2 ) 〈 0 , ζ | α 〉 = ( 1 − ζ ζ * ) 1 4 exp ( − β β * 2 + α β * − ζ * 2 ( α − β ) 2 ) = ( exp ( α α * 2 ) 〈 α | 0 , ζ 〉 ) * . (9.1)

The Bargmann representation of a state contains the full information about the state. From (9.1) one finds the Husimi-Kano quasiprobability

Q ( α , α * ) ≡ 〈 α | β , ζ 〉 〈 β , ζ | α 〉 π = 1 − ζ ζ * π e x p { − ( ( α − β ) ( α * − β * ) + ζ * 2 ( α − β ) 2 + ζ 2 ( α * − β * ) 2 ) } , ∫ i 2 d α ∧ d α * Q ( α , α * ) = 1. (9.2)

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 W ( α , α * ) of a displaced state can be obtained from Wigner quasiprobability W 0 ( α , α * ) of the corresponding undisplaced state by a simple argument displacement ( α , α * ) → ( α − β , α * − β * ) . This is also true for other quasiprobabilities such as for example the Husimi-Kano quasiprobability Q ( α , α * ) as we saw and for the Glauber-Sudarshan quasiprobability P ( α , α * ) . In this way one obtains from (4.6) for the Wigner quasiprobability of displaced squeezed vacuum states

W ( α , α * ) = 2 π e x p { − 2 ( α − β + ζ ( α * − β * ) ) ( α * − β * + ζ ( α − β ) ) 1 − ζ ζ * } , ∫ i 2 d α ∧ d α * W ( α , α * ) = 1. (9.3)

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

( β + ζ β * 1 − ζ ζ * , β * + ζ * β 1 − ζ ζ * ) that provides the Wigner quasiprobability

W 0 ( α , α * ) = 2 π exp { − 2 ( α − β + ζ β * 1 − ζ ζ * ) ( α * − β * + ζ * β 1 − ζ ζ * ) } . (9.4)

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 ( α , α * ) → ( α + ζ α * 1 − ζ ζ * , α * + ζ * α 1 − ζ ζ * ) (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 ( q , p ) can be obtained from (4.7) by the following substitution of the canonical variables ( q , p )

( q , p ) → ( q − Q ¯ , p − P ¯ ) , ( a ¯ , a ¯ * ) ≡ ( Q ¯ + i P ¯ 2 ℏ , Q ¯ − i P ¯ 2 ℏ ) = ( β , β * ) , ( a † ¯ = a ¯ * ) . (9.5)

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.

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 D ( β , β * ) one finds applied to displaced squeezed vacuum states

( a † k a l ¯ ) β ≡ 〈 β , ζ | a † k a l | β , ζ 〉 = 〈 0, ζ | ( D ( β , β * ) † a † k a l D ( β , β * ) | 0, ζ 〉 = 〈 0, ζ | ( a † + β * I ) k ( D ( β , β * ) ) † D ( β , β * ) ︸ = I ( a + β I ) l | 0, ζ 〉 (10.1)

that after Taylor series expansion of the binomials can be written

( a † k a l ¯ ) β = ∑ i = 0 k ∑ j = 0 l k ! l ! i ! ( k − i ) ! j ! ( l − j ) ! β * i β j ( a † k − i a l − j ¯ ) 0 , (10.2)

where index 0 at expectation values means the expectation values before the displacement and with index β after the displacement with the complex parameter β .

For squeezed vacuum states the expectation values ( a † k a l ¯ ) 0 are only non-

vanishing if the difference | k − l | is an even number. Taking this into account we find for displaced squeezed vacuum states the expectation values of the operators a and a †

( a ¯ ) β = β ( I ¯ ) 0 = β , ( a † ¯ ) β = β * ( I ¯ ) 0 = β * , (10.3)

and for the expectation values of the operators a 2 and a † 2

( a 2 ¯ ) β = ( a 2 ¯ ) 0 + β 2 ( I ¯ ) 0 = − ζ 1 − ζ ζ * + β 2 ,

( a † 2 ¯ ) β = ( a † 2 ¯ ) 0 + β * 2 ( I ¯ ) 0 = − ζ * 1 − ζ ζ * + β * 2 . (10.4)

Next we calculate expectation values of operators which are important for the photon statistics. The expectation value of the number operator N = a † a depends only on the squared moduli | ζ | 2 ≡ ζ ζ * and | β | 2 ≡ β β * of the complex ζ and β and we find

( N ¯ ) β = ( a † a ¯ ) β = ( a † a ¯ ) 0 + β β * ( I ¯ ) 0 = ζ ζ * 1 − ζ ζ * + β β * = | ζ | 2 1 − | ζ | 2 + | β | 2 . (10.5)

For the discussion of sub- and super-Poissonian statistics of displaced squeezed vacuum states we need in addition the expectation value of the operator a † 2 a 2 . It does not only depend on the moduli of ζ and β but also on their phases and from (10.2) we find

( a † 2 a 2 ¯ ) β = ( a † 2 a 2 ¯ ) 0 + 4 β β * ( a † a ¯ ) 0 + ( β β * ) 2 ( I ¯ ) 0 + β 2 ( a † 2 ¯ ) 0 + β * 2 ( a 2 ¯ ) 0 = ζ ζ * ( 1 + 2 ζ ζ * ) ( 1 − ζ ζ * ) 2 + 4 β β * ζ ζ * 1 − ζ ζ * + ( β β * ) 2 − β 2 ζ * 1 − ζ ζ * − β * 2 ζ 1 − ζ ζ * . (10.6)

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 N 2 = a † 2 a 2 + a † a

( N 2 ¯ ) β = ζ ζ * ( 2 + ζ ζ * ) ( 1 − ζ ζ * ) 2 + β β * 1 + 3 ζ ζ * 1 − ζ ζ * + ( β β * ) 2 − β 2 ζ * 1 − ζ ζ * − β * 2 ζ 1 − ζ ζ * , (10.7)

and for the variance of the number operator

( ( Δ N ) 2 ¯ ) β = ( N 2 ¯ ) β − ( N ¯ ) β 2 = 2 ζ ζ * ( 1 − ζ ζ * ) 2 + β β * 1 + ζ ζ * 1 − ζ ζ * − β 2 ζ * 1 − ζ ζ * − β * 2 ζ 1 − ζ ζ * = 2 ζ ζ * ( 1 − ζ ζ * ) 2 + ( β − ζ β * ) ( β * − ζ * β ) 1 − ζ ζ * ≥ 0. (10.8)

Finally we calculate the expectation value which plays a role for the definition of sub- and super-Poissonian statistics by its sign

( a † 2 a 2 ¯ ) β − ( a † a ¯ ) β 2 = ( ( Δ N ) 2 ¯ ) β − ( N ¯ ) β = ζ ζ * ( 1 + ζ ζ * ) ( 1 − ζ ζ * ) 2 + 2 β β * ζ ζ * 1 − ζ ζ * − β 2 ζ * 1 − ζ ζ * − β * 2 ζ 1 − ζ ζ * = ζ ζ * ( 1 + ζ ζ * ) ( 1 − ζ ζ * ) 2 − β β * + ( β − ζ β * ) ( β * − ζ * β ) 1 − ζ ζ * . (10.9)

The expectation values (10.6)-(10.9) depend not only on the moduli of β and ζ but also on their phases e i φ and e i χ . For comparison with (10.9) one finds for the corresponding symmetrically (Weyl) ordered quantity

( S { a † 2 a 2 } ¯ ) β − ( S { a † a } ¯ ) β 2 = ( a † 2 a 2 ¯ ) β − ( a † a ¯ ) β 2 + ( a † a ¯ ) β + 1 4 = 2 ζ ζ * ( 1 − ζ ζ * ) 2 + ( β − ζ β * ) ( β * − ζ * β ) 1 − ζ ζ * + 1 4 ≥ 1 4 . (10.10)

This quantity is greater than or equal to 1 / 4 in every case but depends also on the phases of β and ζ .

The sign of the quantity (10.9) was taken by Mandel to define of sub- and super-Poissonian statistics as follows [

a † 2 a 2 ¯ − ( a † a ¯ ) 2 = ( Δ N ) 2 ¯ − N ¯ < 0 , ( sub-Poissonian ) ,

a † 2 a 2 ¯ − ( a † a ¯ ) 2 = ( Δ N ) 2 ¯ − N ¯ > 0 , ( super-Poissonian ) . (11.1)

We now investigate the photon statistics of this quantity for displaced squeezed vacuum states starting from (10.9). First we denote the phases of β and ζ in the the complex plane as follows

β = | β | e i φ , ζ = | ζ | e i χ . (11.2)

The mean value N ¯ of the number operator N according to (10.5) does not depend on the angle between β and ζ and if we change only the angle φ of β leaving | β | and ζ constant the mean value N ¯ remains constant and the squeezing ellipses change in their position in comparison to the displacement β . This is shown in

From (10.9) and from definition (11.1) follows as condition for sub- Poissonian statistics of squeezed coherent states | β , ζ 〉

| ζ | 2 ( 1 + | ζ | 2 ) ( 1 − | ζ | 2 ) 2 − 2 | β | 2 | ζ | ( c o s ( χ 0 ) − | ζ | ) 1 − | ζ | 2 < 0, ( χ 0 ≡ χ − 2 φ ) , (11.3)

or resolved to | β | 2 for cos ( χ 0 ) − | ζ | > 0 (pay attention that in case of cos ( χ 0 ) − | ζ | < 0 the inequality (11.3) cannot be satisfied or changes “>” into “<” if divided by cos ( χ 0 ) − | ζ | < 0 !)

| β | 2 > | ζ | ( 1 + | ζ | 2 ) 2 ( 1 − | ζ | 2 ) ( cos ( χ 0 ) − | ζ | ) > 0 , ( cos ( χ 0 ) > | ζ | ) . (11.4)

For possible sub-Poissonian statistics it is necessary that cos ( χ 0 ) > | ζ | whereas in case of cos ( χ 0 ) < | ζ | we have super-Poissonian statistics. For χ 0 = 0 or cos ( χ 0 ) − | ζ | = 1 − | ζ | > 0 that means for squeezing in direction of the displacement parameter β (amplitude squeezing, see

cos ( χ 0 ) − | ζ | > 0 , (11.5)

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 | β | by the expectation value N ¯ of the number operator since it is then easier to compare the results with coherent states with the same N ¯ .

We express now the displacement parameter β in (10.9) by the expectation value N ¯ of the number operator N using (10.5)

| β | 2 = N ¯ − | ζ | 2 1 − | ζ | 2 ≥ 0 , (11.6)

where we have omitted the index β in ( N ¯ ) β since we use it in the following also for coherent states. If we do so then we obtain from (10.9) with abbreviation χ 0 ≡ χ − 2 φ

( a † 2 a 2 ¯ ) β − ( a † a ¯ ) β 2 = | ζ | 2 ( 1 + | ζ | 2 ) ( 1 + | ζ | 2 ) 2 − 2 ( N ¯ − | ζ | 2 1 − | ζ | 2 ) | ζ | ( c o s ( χ 0 ) − | ζ | ) 1 − | ζ | 2 = | ζ | ( 1 − | ζ | 2 ) 2 { | ζ | ( 1 + | ζ | 2 ) − 2 ( N ¯ − ( N ¯ + 1 ) | ζ | 2 ) ( c o s ( χ 0 ) − | ζ | ) } . (11.7)

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 | ζ | in dependence on N ¯ and on the angle χ 0 as follows

| ζ | 3 − 2 ( N ¯ + 1 ) c o s ( χ 0 ) 2 N ¯ + 1 | ζ | 2 − | ζ | + 2 N ¯ c o s ( χ 0 ) 2 N ¯ + 1 = 0. (11.8)

As a third-order equation for | ζ | with real coefficients in dependence on N ¯ and χ 0 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 | ζ | < 1 . The results are presented in

which one has sub-Poissonian statistics if β possesses the same or the opposite direction of ζ (amplitude squeezing). Then by rotating the phase e i χ of ζ from χ = 0 (amplitude squeezing) up to χ = π (phase squeezing) leaving constant the modulus | ζ | (see

of 1 2 ( a a † + a † a ) 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., p n 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 ( ( Δ N ) 2 ¯ − N ¯ = 0 ) and this can be considered in dependence on arbitrary N ¯ . 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 [

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 | 1 〉 and then to its displacement since

the operator e x p ( − ζ * 2 a 2 ) in its disentanglement (2.7)) acts on | 1 〉 by reproducing it (case k = 3 4 in (A.4)). This becomes more complicated if we

apply it to the number states | n 〉 with n ≥ 2 and to extend the theory to these cases.

Wünsche, A. (2017) Squeezed Coherent States in Non-Unitary Approach and Relation to Sub- and Super-Poissonian Statistics. Advances in Pure Mathematics, 7, 706-757. https://doi.org/10.4236/apm.2017.712044

Squeezing of states in the narrow sense is connected with the Lie group S U ( 1,1 ) ≅ S p ( 2, ℝ ) ≅ S L ( 2, ℝ ) with 3 real parameters (or one complex and one real in our representation) of squeezing operators [

We define operators S ( ξ , η , ζ ) with 3 complex parameters which are operators in S L ( 2, ℂ ) as follows

S ( ξ , η , ζ ) ≡ e x p { ξ K − + i 2 η K 0 − ζ K + } (A.1)

where ( K − , K 0 , K + ) are three abstract operators of the Lie algebra s l ( 2, ℂ ) to the Lie group S L ( 2, ℂ ) satisfying the commutation relation [

[ K − , K + ] = 2 K 0 , [ K 0 , K − ] = − K − , [ K 0 , K + ] = + K + , (A.2)

with Casimir operator C

C ≡ K 0 2 − K 1 2 − K 2 2 = K 0 2 − 1 2 ( K − K + + K + K − ) , [ C , K i ] = 0. (A.3)

One basic discrete realization of these operators in quantum optics of a single mode is

K − ≡ 1 2 a 2 = 1 4 ℏ ( Q 2 − P 2 + i ( Q P + P Q ) ) ,

K 0 ≡ 1 4 ( a a † + a † a ) = 1 4 ℏ ( Q 2 + P 2 ) ,

K + ≡ 1 2 a † 2 = 1 4 ℏ ( Q 2 − P 2 − i ( Q P + P Q ) ) ,

C ≡ k ( k − 1 ) I = − 3 16 , k = 1 4 , ( or k = 3 4 ) , (A.4)

where ( a , a † ) is a pair of boson annihilation and creation operators (A.1) and k = 3 4 belongs to squeezed number states | 1 〉 (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 ( a , a † ) are particularly convenient in combination with the realization (A.4). Squeezing with S U ( 1,1 ) operators within two modes and genuinely different from single-mode squeezing is also possible [

The transition from S L ( 2, ℂ ) to S L ( 2, ℝ ) ≅ S U ( 1,1 ) can be made by specializing the parameters ( ξ , η , ζ ) in (A.1) according to

S ( ζ * , η = η * , ζ ) = e x p { ζ * 2 a 2 + i η 2 ( a a † + a † a ) − ζ 2 a † 2 } = ( S ( − ζ * , − η = − η * , − ζ ) ) † (A.5)

with complex parameter ζ and real parameter η . The operators S ( ζ * , η = η * , ζ ) are then unitary operators S − 1 = S † in the infinite-dimensional unitary representation in Hilbert space and we have ( K − † , K 0 † , K + † ) = ( K + , K 0 , K − ) . Instead of ( K − , K + ) one may introduce Hermitean operators ( K 1 , K 2 ) by K ∓ ≡ K 1 ∓ i K 2 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 S L ( 2, ℂ ) by calculating the matrix with elements ( κ , λ , μ , ν ) in the following relation

S ( ξ , η , ζ ) ( a , a † ) ( S ( ξ , η , ζ ) ) − 1 = ( a , a † ) ( κ λ μ ν ) = ( κ a + μ a † , λ a + ν a † ) . (A.6)

This step is essentially a Bogolyubov transformation. It can be represented by (e.g., [

( κ λ μ ν ) = ( ch ( ε ) − i η sh ( ε ) ε ξ sh ( ε ) ε ζ sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ) . ε ≡ ξ ζ − η 2 . (A.7)

For ε 2 > 0 , ( ε real) we call this transformation squeezing-like and for ε 2 < 0 , ( ε imaginary) rotation-like. In last case it is better to write the hyperbolic functions by trigonometric functions. The special case ε = 0

( κ λ μ ν ) = ( 1 − i η , ξ ζ , 1 + i η ) , ξ ζ = η 2 , (A.8)

could be called cone-like. The matrices (A.7) are unimodular

| ( κ λ μ ν ) | = κ ν − λ μ = 1 , (A.9)

but in general, not unitary.

The inversion of (A.7) which is unique can be simply written

ξ = λ Arsh ( ϑ ) ϑ , η = i κ − ν 2 Arsh ( ϑ ) ϑ , ζ = μ Arsh ( ϑ ) ϑ , (A.10)

with the following relations between the abbreviations ε and ϑ

ε = Arch ( κ + ν 2 ) = Arsh ( ( κ + ν 2 ) 2 − 1 ) = Arsh ( ( κ − ν 2 ) 2 + λ μ ) ≡ Arsh ( ϑ ) ,

ϑ ≡ sh ( ε ) = ( κ + ν 2 ) 2 − 1 = ( κ − ν 2 ) 2 + λ μ . (11)

From Arch ( z ) the sign of Arsh ( z 2 − 1 ) does not follow uniquely but Arsh ( z 2 − 1 ) z 2 − 1 is then uniquely determined if one chooses the same sign of z 2 − 1 in numerator and denominator.

We now write down some special correspondences between the operators S ( ξ , η , ζ ) and its two-dimensional matrices in the fundamental representation

S ( ξ ,0,0 ) = e x p ( ξ K − ) → e x p ( ξ 2 a 2 ) , ⇔ ( 1, ξ 0, 1 ) ,

S ( 0, η ,0 ) = e x p ( i η 2 K 0 ) → e x p { i η 2 ( a a † + a † a ) } ⇔ ( e − i η , 0 0, e i η ) ,

S ( 0,0, ζ ) = e x p ( − ζ K + ) → e x p ( − ζ 2 a † 2 ) , ⇔ ( 1, 0 ζ , 1 ) . (A.12)

The triangular operators S ( ξ ,0,0 ) and S ( 0,0, ζ ) and their correspondent two-dimensional matrices form a group for themselves and in the same way the operators S ( 0, η ,0 ) together with its matrices. However, the operators S ( ξ ,0, ζ ) do not form a group its extension to a group needs all operators S ( ξ , η , ζ ) .

We may decompose the matrices in (A.7) into products of special matrices, in particular, in the following for us important ways

( κ λ μ ν ) = ( 1 0 μ κ 1 ) ( 1 λ κ 0 1 ) ( κ 0 0 1 κ ) = ( 1 0 μ κ 1 ) ( κ 0 0 1 κ ) ( 1 λ κ 0 1 ) = ( κ 0 0 1 κ ) ( 1 0 μ κ 1 ) ( 1 λ κ 0 1 )

= ( 1 ν 0 0 ν ) ( 1 λ ν 0 1 ) ( 1 0 μ ν 1 ) = ( 1 λ ν 0 1 ) ( 1 ν 0 0 ν ) ( 1 0 μ ν 1 ) = ( 1 λ ν 0 1 ) ( 1 0 μ ν 1 ) ( 1 ν 0 0 ν ) , κ ν − λ μ = 1. (A.13)

As consequences we obtain the following disentanglements of the general operator S ( ξ , η , ζ ) into products of partial operators

S ( ξ , η , ζ ) ≡ e x p ( ξ K − + i 2 η K 0 − ζ K + ) = e x p ( − μ κ K + ) e x p ( λ κ K − ) e x p ( − 2 l o g ( κ ) K 0 ) = e x p ( − μ κ K + ) e x p ( − 2 l o g ( κ ) K 0 ) e x p ( λ κ K − )

= exp ( − 2 log ( κ ) K 0 ) exp ( − μ κ K + ) exp ( λ κ K − ) = exp ( 2 log ( ν ) K 0 ) exp ( λ ν K − ) exp ( − μ ν K + ) = exp ( λ ν K − ) exp ( 2 log ( ν ) K 0 ) exp ( − μ ν K + ) = exp ( λ ν K − ) exp ( − μ ν K + ) exp ( 2 log ( ν ) K 0 ) . (A.14)

For the operators ( K − , K 0 , K + ) we may insert the realization (A.4) of S U ( 1,1 ) but every other realization of S L ( 2, ℂ ) is also appropriate.

In the special case η = 0 of operators S ( ξ , η , ζ ) the matrices (A.7) specialize

S ( ξ ,0, ζ ) ⇔ ( κ λ μ ν ) = ( ch ( ξ ζ ) ξ sh ( ξ ζ ) ξ ζ ζ sh ( ξ ζ ) ξ ζ ch ( ξ ζ ) ) , (A.15)

that leads to the disentanglement relations

S ( ξ , 0 , ζ ) = exp ( ξ K − − ζ K + ) = exp ( − ζ th ( ξ ζ ) ξ ζ K + ) exp ( ξ sh ( 2 ξ ζ ) 2 ξ ζ K − ) ( ch ( ξ ζ ) ) − 2 K 0 = exp ( − ζ th ( ξ ζ ) ξ ζ K + ) ( ch ( ξ ζ ) ) − 2 K 0 exp ( ξ th ( ξ ζ ) ξ ζ K − )

= ( ch ( ξ ζ ) ) − 2 K 0 exp ( − ζ sh ( 2 ξ ζ ) 2 ξ ζ K + ) exp ( ξ th ( ξ ζ ) ξ ζ K − ) = ( ch ( ξ ζ ) ) 2 K 0 exp ( ξ sh ( 2 ξ ζ ) 2 ξ ζ K − ) exp ( − ζ th ( ξ ζ ) ξ ζ K + ) = exp ( ξ th ( ξ ζ ) ξ ζ K − ) ( ch ( ξ ζ ) ) 2 K 0 exp ( − ζ th ( ξ ζ ) ξ ζ K + ) = exp ( ξ th ( ξ ζ ) ξ ζ K − ) exp ( − ζ sh ( 2 ξ ζ ) 2 ξ ζ K + ) ( ch ( ξ ζ ) ) 2 K 0 , (A.16)

where again we may insert the realization (A.4). In addition to the operators K − and K + we have involved then on the right-hand side also the operator K 0 .

If we make in (A.15) the transition to the variables in the non-unitary approach

S ( ξ ,0, ζ ) → S ( Arth ( ξ ζ ) ξ ζ ξ ,0, Arth ( ξ ζ ) ξ ζ ζ ) , (A.17)

then the matrix in (A.15) makes the transition to

( κ , λ μ , ν ) → ( 1 1 − ξ ζ ξ 1 − ξ ζ ζ 1 − ξ ζ 1 1 − ξ ζ ) , (A.18)

as it is easily to see.

The next considerations are the effort to split the general operator S ( ξ , η , ζ ) into products of a proper squeezing operator S ( ξ ′ ,0, ζ ′ ) and a proper rotation operator S ( 0, η ′ ,0 ) . The general two-dimensional matrix (A.7) can be split also in the following two ways

( κ , λ μ , ν ) = ( κ ν λ κ ν μ ν κ κ ν ) ( κ ν 0 0 ν κ ) = ( κ ν 0 0 ν κ ) ( κ ν λ ν κ μ κ ν κ ν ) . (A.19)

This corresponds explicitly to the two possibilities ( ε ≡ ξ ζ − η 2 )

= ( κ λ μ ν ) = ( ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 ξ sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ζ sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 ) × ( ch ( ε ) − i η sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε 0 0 ch ( ε ) + i η sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε )

= ( ch ( ε ) − i η sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε 0 0 ch ( ε ) + i η sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ) × ( ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 ξ sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ζ sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 ) . (A.20)

and if we introduce abbreviations

( κ λ μ ν ) = ( ch ( ξ ′ ζ ′ ) ξ ′ sh ( ζ ′ ζ ′ ) ξ ′ ζ ′ ζ ′ sh ( ζ ′ ζ ′ ) ξ ′ ζ ′ ch ( ξ ′ ζ ′ ) ) ( e − i η 0 0 0 e i η 0 ) = ( e − i η 0 0 0 e i η 0 ) ( ch ( ξ ″ ζ ″ ) ξ ″ sh ( ζ ″ ζ ″ ) ξ ″ ζ ″ ζ ″ sh ( ζ ″ ζ ″ ) ξ ″ ζ ″ c h ( ξ ″ ζ ″ ) ) . (A.21)

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

( e − i η 0 0 0 e i η 0 ) , ⇔ S ( 0, η 0 ,0 ) = e i η 0 2 ( a a † + a † a ) . (A.22)

The first possibility of the splitting in a squeezing and a rotation part is

( κ λ μ ν ) = ( ch ( ε ) − i η sh ( ε ) ε ξ sh ( ε ) ε ζ sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ) = ( ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 ξ sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ζ sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 )

× ( ch ( ε ) − i η sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε 0 0 ch ( ε ) + i η sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ) = ( ch ( ξ ′ ζ ′ ) , ξ ′ sh ( ζ ′ ζ ′ ) ξ ′ ζ ′ ζ ′ sh ( ζ ′ ζ ′ ) ξ ′ ζ ′ , ch ( ξ ′ ζ ′ ) ) ( e − i η 0 , 0 0, e i η 0 ) = ( κ ν κ ν λ ν κ μ κ ν ) ( κ ν 0 0 ν κ ) , (A.23)

with correspondences

ch ( ξ ′ ζ ′ ) = ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 = 1 + ξ ζ sh 2 ( ε ) ε 2 ,

sh ( ξ ′ ζ ′ ) = ξ ζ sh ( ε ) ε , th ( ξ ′ ζ ′ ) = ξ ζ sh ( ε ) 1 + ξ ζ sh 2 ( ε ) ,

sh ( ξ ′ ζ ′ ) ξ ′ ζ ′ = ξ ζ sh ( ε ) ε Arsh ( ξ ζ sh ( ε ) ε ) ,

e ± i η 0 = ch ( ε ) ± i η sh ( ε ) ε ch ( ε ) ∓ i η sh ( ε ) ε = ch ( ε ) ± i η sh ( ε ) ε ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 = ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 ch ( ε ) ∓ i η sh ( ε ) ε ,

tg ( η 0 ) = η th ( ε ) ε , c o s ( η 0 ) = ch ( ε ) ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 , s i n ( η 0 ) = η sh ( ε ) ε ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2

ξ ′ = Arsh ( ξ ζ sh ( ε ) ε ) ξ ζ sh ( ε ) ε ξ e − i η , ζ ′ = Arsh ( ξ ζ sh ( ε ) ε ) ξ ζ sh ( ε ) ε ζ e i η . (A.24)

The parameters in the factorized matrices are ( κ , λ , μ , ν are such as in (A.7))

S ( ξ , η , ζ ) = S ( Arsh ( λ μ ) λ μ κ ν λ ,0, Arsh ( λ μ ) λ μ ν κ μ ) S ( 0, iArsh ( 1 2 ( κ ν − ν κ ) ) ,0 ) = S ( Arsh ( ξ ζ s h ( ε ) ε ) ξ ζ ch ( ε ) − i η sh ( ε ) ε ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 ξ ,0, Arsh ( ξ ζ sh ( ε ) ε ) ξ ζ ch ( ε ) + i η sh ( ε ) ε ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 ζ ) ⋅ S ( 0, iArsh ( − i η sh ( ε ) ε 2 + ξ ζ sh 2 ( ε ) ) ,0 ) . (A.25)

The second possibility with interchanged order of the splitting in a squeezing and a rotation part is

( κ λ μ ν ) = ( ch ( ε ) − i η sh ( ε ) ε ξ sh ( ε ) ε ζ sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ) = ( ch ( ε ) − i η sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε 0 0 ch ( ε ) + i η sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ) × ( ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 ξ sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ζ sh ( ε ) ε ch ( ε ) − i η sh ( ε ) ε ch ( ε ) + i η sh ( ε ) ε ch 2 ( ε ) + η 2 sh 2 ( ε ) ε 2 ) = ( e − i η 0 0 0 e i η 0 ) ( ch ( ξ ″ ζ ″ ) ξ ″ sh ( ζ ″ ζ ″ ) ξ ″ ζ ″ ζ ″ sh ( ζ ″ ζ ″ ) ξ ″ ζ ″ c h ( ξ ″ ζ ″ ) )

= ( κ ν 0 0 ν κ ) ( κ ν ν κ λ κ ν μ κ ν ) . (A.26)

The diagonal matrix of a rotation with parameter η 0 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 S ( ξ ′ ,0, ζ ′ ) which are equivalent to the general operators S ( ξ , η , ζ ) after splitting rotation factors. However, these special squeezing operators alone do not form a group.

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 ( α , α * ) in the following form [

W ( α , α * ) = 〈 ρ e x p ( − a ∂ ∂ α − a † ∂ ∂ α * ) 〉 δ ( α , α * ) , (B.1)

where ρ is the density operator of the state. First we investigate the displace- ment of a state with the density operator ρ 0 and the Wigner quasiprobability W 0 ( α , α * ) according to

ρ = D ( β , β * ) ρ 0 ( D ( β , β * ) ) † . (B.2)

Then one finds for W ( α , α * )

W ( α , α * ) = 〈 D ( β , β * ) ρ 0 ( D ( β , β * ) ) † exp ( − a ∂ ∂ α − a † ∂ ∂ α * ) 〉 δ ( α , α * ) = 〈 ρ 0 ( D ( β , β * ) ) † exp ( − a ∂ ∂ α − a † ∂ ∂ α * ) D ( β , β * ) 〉 δ ( α , α * ) = 〈 ρ 0 ( D ( β , β * ) ) † exp ( − ( a + β I ) ∂ ∂ α − ( a † + β * I ) ∂ ∂ α * ) 〉 δ ( α , α * ) = 〈 ρ 0 exp ( − a ∂ ∂ α − a † ∂ ∂ α * ) 〉 exp ( − β ∂ ∂ α − β * ∂ ∂ α * ) δ ( α , α * ) = 〈 ρ 0 exp ( − a ∂ ∂ α − a † ∂ ∂ α * ) 〉 δ ( α − β , α * − β * ) . (B.3)

Therefore we obtained

W ( α , α * ) = W 0 ( α − β , α * − β * ) . (B.4)

This means that the Wigner quasiprobability W ( α , α * ) for the displaced state is equal to the Wigner quasiprobability W 0 ( α , α * ) for the undisplaced state with displaced arguments corresponding to the displacement parameters ( β , β * ) . This displacement property is one of the minimal requirements for a phase-space function F ( α , α * ) in quantum theory to be called a quasiprobability.

In widely analogous way one may treat the squeezing of a state with density operator ρ 0 according to

ρ = S ( ζ * , η = η * , ζ ) ρ 0 ( S ( ζ * , η = η * , ζ ) ) † . (B.5)

Instead of the unitary squeezing operators S ( ζ * , η = η * , ζ ) we will calculate the transformations first a little more general with the operators S ( ξ , η , ζ ) and at the end we will specialize the result to ξ = ζ * and η = η * . Therefore, we

have first to substitute ( S ( ζ * , η = η * , ζ ) ) † → ( S ( ξ , η , ζ ) ) − 1 = S ( − ξ , − η , − ζ ) and

find

W ( α , α * ) = 〈 S ( ξ , η , ζ ) ρ 0 ( S ( ξ , η , ζ ) ) − 1 exp ( − a ∂ ∂ α − a † ∂ ∂ α * ) 〉 δ ( α , α * ) = 〈 ρ 0 ( S ( ξ , η , ζ ) ) − 1 exp ( − a ∂ ∂ α − a † ∂ ∂ α * ) S ( ξ , η , ζ ) 〉 δ ( α , α * ) = 〈 ρ 0 exp ( − ( ν a − μ a † ) ∂ ∂ α − ( − λ a + κ a † ) ∂ ∂ α * ) 〉 δ ( α , α * )

= 〈 ρ 0 exp ( − a ( ν ∂ ∂ α − λ ∂ ∂ α * ) − a † ( − μ ∂ ∂ α + κ ∂ ∂ α * ) ) 〉 δ ( α , α * ) = 〈 ρ 0 exp ( − a ∂ ∂ γ − a † ∂ ∂ γ ′ ) 〉 δ ( ν γ − μ γ ′ , − λ γ + κ γ ′ ) = 〈 ρ 0 exp ( − a ∂ ∂ γ − a † ∂ ∂ γ ′ ) 〉 δ ( γ , γ ′ ) . (B.6)

Formally this can be written

W ( α , α * ) ≡ W 0 ( γ , γ ′ ) , (B.7)

with the matrix elements in explicit form given in (A.7). This corresponds to the transformation

γ = κ α + μ α * , α = ν γ − μ γ ′ , ∂ ∂ γ = ν ∂ ∂ α − λ ∂ ∂ α * , γ ′ = λ α + ν α * , α * = − λ γ + κ γ ′ , ∂ ∂ γ ′ = − μ ∂ ∂ α + κ ∂ ∂ α * . (B.8)

In the last step in (B.6) was used that the two-dimensional delta function δ ( γ , γ ′ ) 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 S ( ζ * , η = η * , ζ ) we can substitute κ → ν * , λ → μ * and γ ′ → γ * and (B.7) using (B.8) can be specialized to [

W ( α , α * ) = W 0 ( ν * α + μ α * , μ * α + ν α * ) ≡ W 0 ( α ′ , α ′ * ) . (B.9)

We mention that a displacement relation of the form (B.4) is true for all quasiprobabilities (e.g., Q ( α , α * ) and P ( α , α * ) ) 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 − a ∂ ∂ α − a † ∂ ∂ α * as a whole in

the exponent of only one exponential function.

The transformation of the variables ( α ′ , α ′ * ) ↔ ( α , α * ) possesses the property

d α ∧ d α * = ( ν * d α ′ + μ d α ′ * ) ∧ ( μ * d α + ν d α ′ * ) = ( ν ν * − μ μ * ) d α ′ ∧ d α ′ * = d α ′ ∧ d α ′ * , (B.10)

and therefore

d q ∧ d p 2 ℏ = i 2 d α ∧ d α * = i 2 d α ′ ∧ d α ′ * = d q ′ ∧ d p ′ 2 ℏ . (B.11)

This means that each area element of the Wigner quasiprobability W 0 ( α , α * ) is mapped in a new area element of W ( α , α * ) 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

( a b c d ) with a d − b c = 1 can be given by extension of coordinates ( x , y ) to

their own complex planes according to

δ ( a x + b y ) δ ( c x + d y ) = δ ( ( a d − b c ) d x ) δ ( d ( c d x + y ) ) = δ ( x d ) δ ( d ( c d x + y ) ) = δ ( x d ) δ ( d y ) = δ ( x ) δ ( y ) , ( a d − b c = 1 ) . (B.12)

This is then also correct for the representation of the two-dimensional delta function by complex variables.

We derive here an identity between Jacobi polynomials which we used in Section 5 in specialized form.

Starting from the right-hand side of (5.1) we make a transformation of the following more general expression

( 1 − y 2 ) α + 1 2 ∂ n ∂ y n 1 ( 1 − y 2 ) α + 1 2 = ( ∂ ∂ y + ( 2 α + 1 ) y 1 − y 2 ) n = ( ∂ ∂ y − α + 1 2 1 − y − α + 1 2 1 + y ) n = ∑ j = 0 n ( − 2 1 − y 2 ) j P j ( − α − 1 2 − j , − α − 1 2 − j ) ( y ) n ! ( n − j ) ! ∂ n − j ∂ n − j . (C.1)

This is an operator identity which in the last line is written in the entangled form derived in [

( ∂ ∂ y − α ′ 1 − y + β ′ 1 + y ) n = ∑ j = 0 j ( − 2 1 − y 2 ) j P j ( α ′ − j , β ′ − j ) ( y ) n ! ( n − j ) ! ∂ n − j ∂ y n − j , (C.2)

and can be proved by complete induction. With P n ( α , β ) ( u ) are denoted the Jacobi polynomials.

The operator identity (C.1) can be applied to arbitrary functions of the variable y. If we apply it to the function f ( y ) = 1 we find

( 1 − y 2 ) α + 1 2 ∂ n ∂ y n 1 ( 1 − y 2 ) α + 1 2 1 = ∑ j = 0 n ( − 2 1 − y 2 ) j P j ( − α − 1 2 − j , − α − 1 2 − j ) ( y ) n ! ( n − j ) ! ∂ n − j ∂ y n − j 1 = n ! ( − 2 1 − y 2 ) n P n ( − α − 1 2 − n , − α − 1 2 − n ) ( y ) . (C.3)

Up to this point it may be considered as a transformation of the (Rodrigues- kind) definition of specialized Jacobi polynomials given by Szegö [

The special case of Jacobi polynomials with equal upper indices P n ( β , β ) ( y ) is called Ultraspherical polynomials and they are related to the Gegenbauer polynomials C n ν ( y ) in the following simple way

P n ( β , β ) ( y ) = ( 2 β ) ! ( n + β ) ! β ! ( n + 2 β ) ! C n β + 1 2 ( y ) . (C.4)

They possess the following for us interesting expansions (Equations (4.5) and (5.2) in [

P n ( β , β ) ( y ) = 2 2 β ( n + β ) ! ( n + 2 β ) ! ∑ k = 0 [ n 2 ] ( − 1 ) k ( n + β − k − 1 2 ) ! k ! ( n − 2 k ) ! ( − 1 2 ) ! ( 2 y ) n − 2 k = ∑ j = 0 [ n 2 ] ( − 1 ) j ( n + β ) ! j ! ( n − 2 j ) ! ( j + β ) ! 2 2 j ( 1 − y 2 ) j y n − 2 j . (C.5)

Substituting in (C.5) β = − α − 1 2 − n and using transformation relations for

the factorials one finds the following representation of the Ultraspherical polynomials involved in (C.3)

P n ( − α − 1 2 − n , − α − 1 2 − n ) ( y ) = ( − 1 ) n α ! ( n + 2 α ) ! 2 2 n ( 2 α ) ! ( n + α ) ! ∑ k = 0 [ n 2 ] ( n + α ) ! k ! ( n − 2 k ) ! ( k + α ) ! ( 2 y ) n − 2 k . (C.6)

In [

( 2 1 + x 2 ) n P n ( α , α ) ( x 1 + x 2 ) = ∑ k = 0 [ n 2 ] ( − 1 ) k ( n + α ) ! k ! ( n − 2 k ) ! ( k + α ) ! ( 2 x ) n − 2 k , (C.7)

or after the substitution x = i y

( − i 2 1 − y 2 ) n P n ( α , α ) ( i y 1 − y 2 ) = ∑ k = 0 [ n 2 ] ( n + α ) ! k ! ( n − 2 k ) ! ( k + α ) ! ( 2 y ) n − 2 k . (C.8)

An identity for these polynomials and their explicit representation is

( − i 2 1 − y 2 ) n P n ( α , α ) ( i y 1 − y 2 ) = ( − 1 ) n 2 2 n ( 2 α ) ! ( n + α ) ! α ! ( n + 2 α ) ! P n ( − α − 1 2 − n , − α − 1 2 − n ) ( y ) . (C.9)

These relations were applied in Section 5 to get the representation of a † k a k ¯ by known polynomials.

For easy use we collect here the most important generating functions for Hermite polynomials. The basic generating function for Hermite polynomials is

∑ n = 0 ∞ t n n ! H n ( z ) = e x p ( 2 t z − t 2 ) . (D.1)

The basic generating function for products of two Hermite polynomials with different arguments is the Mehler formula [

∑ n = 0 ∞ 1 n ! ( t 2 ) n H n ( x ) H n ( y ) = 1 ( 1 − t 2 ) 1 2 e x p ( 2 t x y − t 2 ( x 2 + y 2 ) 1 − t 2 ) . (D.2)

By setting y = 0 and using H 2 m ( 0 ) = ( − 1 ) m ( 2 m ) ! m ! and H 2 m + 1 ( 0 ) = 0 one

obtains for even Hermite polynomials and by its differentiation with respect to variable z for odd Hermite polynomials the following non-trivial generating functions

∑ m = 0 ∞ ( − 1 ) m m ! ( t 2 ) 2 m H 2 m ( z ) = 1 ( 1 − t 2 ) 1 2 e x p ( − ( t z ) 2 1 − t 2 ) ,

∑ m = 0 ∞ ( − 1 ) m m ! ( t 2 ) 2 m + 1 H 2 m + 1 ( z ) = t z ( 1 − t 2 ) 3 2 e x p ( − ( t z ) 2 1 − t 2 ) . (D.3)

The generating functions for even and odd Hermite polynomials which may be easily derived from (D.1) are different from the generating functions (D.3).

We compile here expectation values for ordered operators calculated from the formulae derived in the main text of Section 5.

For a few initial numbers of expectation values a † k a k ¯ we calculated by (5.2) and (5.3) or from (6.7)-(6.11)

a † 0 a 0 ¯ = 1,

a † 1 a 1 ¯ = | ζ | 2 1 − | ζ | 2 ,

a † 2 a 2 ¯ = | ζ | 2 ( 1 + 2 | ζ | 2 ) ( 1 − | ζ | 2 ) 2 ,

a † 3 a 3 ¯ = 3 | ζ | 4 ( 3 + 2 | ζ | 2 ) ( 1 − | ζ | 2 ) 3 ,

a † 4 a 4 ¯ = 3 | ζ | 4 ( 3 + 24 | ζ | 2 + 8 | ζ | 4 ) ( 1 − | ζ | 2 ) 4 ,

a † 5 a 5 ¯ = 15 | ζ | 6 ( 15 + 40 | ζ | 2 + 8 | ζ | 4 ) ( 1 − | ζ | 2 ) 5 . (E.1)

The expectation values of symmetrically (Weyl) ordered operators calculated according to (5.4) are

S { a † 0 a 0 } ¯ = 1,

S { a † 1 a 1 } ¯ = 1 2 1 + | ζ | 2 1 − | ζ | 2 ,

S { a † 2 a 2 } ¯ = 1 2 1 + 4 | ζ | 2 + | ζ | 4 ( 1 − | ζ | 2 ) 2 ,

S { a † 3 a 3 } ¯ = 3 4 1 + 9 | ζ | 2 + 9 | ζ | 4 + | ζ | 6 ( 1 − | ζ | 2 ) 3 ,

S { a † 4 a 4 } ¯ = 3 2 1 + 16 | ζ | 2 + 36 | ζ | 4 + 16 | ζ | 6 + | ζ | 8 ( 1 − | ζ | 2 ) 4 ,

S { a † 5 a 5 } ¯ = 15 4 1 + 25 | ζ | 2 + 100 | ζ | 4 + 100 | ζ | 6 + 25 | ζ | 8 + | ζ | 10 ( 1 − | ζ | 2 ) 5 . (E.2)

The coefficients in front of powers of | ζ | in the numerators can be obtained forming the squares of the numbers in the Pascal triangle and, therefore, the polynomials in the numerator are palindromic ones.

For the corresponding expectation values of powers N l ¯ of the number operator N we found from (5.7)

N 0 ¯ = 1 ,

N 1 ¯ = | ζ | 2 1 − | ζ | 2 , ⇔ | ζ | 2 = N ¯ 1 + N ¯ < 1 ,

N 2 ¯ = | ζ | 2 ( 2 + | ζ | 2 ) ( 1 − | ζ | 2 ) 2 , ⇒ ( Δ N ) 2 ¯ = 2 | ζ | 2 ( 1 − | ζ | 2 ) 2 = 2 N ¯ ( 1 + N ¯ ) ,

N 3 ¯ = | ζ | 2 ( 4 + 10 | ζ | 2 + | ζ | 4 ) ( 1 − | ζ | 2 ) 3 , ⇒ ( Δ N ) 3 ¯ = 4 | ζ | 2 ( 1 + | ζ | 2 ) ( 1 − | ζ | 2 ) 3 = 4 N ¯ ( 1 + N ¯ ) ( 1 + 2 N ¯ ) ,

N 4 ¯ = | ζ | 2 ( 8 + 60 | ζ | 2 + 36 | ζ | 4 + | ζ | 6 ) ( 1 − | ζ | 2 ) 4 , ⇒ ( Δ N ) 4 ¯ = 4 | ζ | 2 ( 2 + 11 | ζ | 2 + 2 | ζ | 4 ) ( 1 − | ζ | 2 ) 4 ,

N 5 ¯ = | ζ | 2 ( 16 + 296 | ζ | 2 + 516 | ζ | 4 + 116 | ζ | 6 + | ζ | 8 ) ( 1 − | ζ | 2 ) 5 , ⇒ ( Δ N ) 5 ¯ = 16 | ζ | 2 ( 1 + 16 | ζ | 2 + 16 | ζ | 4 + | ζ | 6 ) ( 1 − | ζ | 2 ) 5 . (E.3)

The photon distribution p n of squeezed vacuum states (2.8) is highly asymmetric to the mean value N ¯ with a longer tail to higher values of n and, therefore, ( Δ N ) k ¯ is nonnegative for all odd k (in addition to all even k for which this is trivial).

We give yet the initial members of the sequences of expectation values a 2 l ¯ and a † a 2 l − 1 ¯ for squeezed vacuum states calculated by computer from the more general formulae in Section 6. We found

a 0 ¯ = 1, a 2 ¯ = − ζ 1 − ζ ζ * , a † a ¯ = ζ * ζ 1 − ζ ζ * , a 4 ¯ = 3 ζ 2 ( 1 − ζ ζ * ) 2 , a † a 3 ¯ = − 3 ζ * ζ 2 ( 1 − ζ ζ * ) 2 ,

a 6 ¯ = − 15 ζ 3 ( 1 − ζ ζ * ) 3 , a † a 5 ¯ = 15 ζ * ζ 3 ( 1 − ζ ζ * ) 3 , a 8 ¯ = 105 ζ 4 ( 1 − ζ ζ * ) 4 , a † a 7 ¯ = − 105 ζ * ζ 4 ( 1 − ζ ζ * ) 4 , a 10 ¯ = − 945 ζ 5 ( 1 − ζ ζ * ) 5 , a † a 9 ¯ = 945 ζ * ζ 5 ( 1 − ζ ζ * ) 5 . (E.4)

The expectation values a † 2 k ¯ and a † 2 k − 1 a ¯ follow from (E.4) using the general relation a † k a l ¯ = a † l a k ¯ * . The obvious relation seen from (E.4)

a † a 2 l − 1 ¯ ≡ 〈 0, ζ | a † a 2 l − 1 | 0, ζ 〉 = − ζ * 〈 0, ζ | a 2 l | 0, ζ 〉 ≡ − ζ * a 2 l ¯ , a 2 l ¯ = ( − 1 ) l ( 2 l − 1 ) ! ! ζ l ( 1 − ζ ζ * ) l , (E.5)

results from the eigenvalue Equation (2.12) written for the left-hand squeezed vacuum states 〈 0, ζ | as follows

〈 0 , ζ | ( a † + ζ * a ) = 0. (E.6)

We calculated some of the members in (E.4) in independent alternative ways and did not find contradictions.

A pair of boson annihilation and creation operators ( a , a † ) is connected with

the Hermitean canonical operators ( Q , P ) by ( ℏ ≡ h 2 π with h Planck’s action

quantum)

( a , a † ) = ( Q + i P 2 ℏ , Q − i P 2 ℏ ) , ( Q , P ) = ℏ 2 ( a + a † , − i ( a − a † ) ) , [ a , a † ] = I , [ Q , P ] = i ℏ I . (1)

Corresponding pairs of complex conjugate variables ( α , α * ) and real canonical variables ( q , p ) are related by

( α , α * ) = ( q + i p 2 ℏ , q − i p 2 ℏ ) , ( q , p ) = ℏ 2 ( α + α * , − i ( α − α * ) ) , i 2 d α ∧ d α * = d q ∧ d p 2 ℏ . (2)

Furthermore: [ A , B ] denotes the commutator of operators A and B and 〈 A 〉 the trace of an operator A (usually in Hilbert space) and [ A ] the determinant of two-dimensional operators A. Expectation values of operators to pure states | ψ 〉 or density operators ρ are denoted by overlining, i.e. A ¯ ≡ 〈 ψ | A | ψ 〉 = 〈 ρ A 〉 .