Interpolation between phase space quantities with bifractional displacement operators

Physics Letters A 379 (2015) 255–260

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Physics Letters A www.elsevier.com/locate/pla

Interpolation between phase space quantities with bifractional displacement operators S. Agyo, C. Lei, A. Vourdas ∗ Department of Computing, University of Bradford, Bradford BD7 1DP, United Kingdom

a r t i c l e

i n f o

Article history: Received 18 September 2014 Received in revised form 17 November 2014 Accepted 18 November 2014 Available online 21 November 2014 Communicated by P.R. Holland Keywords: Fractional Fourier transforms Phase space methods Wigner functions

a b s t r a c t Bifractional displacement operators, are introduced by performing two fractional Fourier transforms on displacement operators. They are shown to be special cases of elements of the group G, that contains both displacements and squeezing transformations. Acting with them on the vacuum we get various classes of coherent states, which we call bifractional coherent states. They are special classes of squeezed states which can be used for interpolation between various quantities in phase space methods. Using them we introduce bifractional Wigner functions A (α , β; θα , θβ ), which are a two-dimensional continuum of functions, and reduce to Wigner and Weyl functions in special cases. We also introduce bifractional Q -functions, and bifractional P -functions. The physical meaning of these quantities is discussed. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Coherent states have been studied extensively in the literature for a long time [1–3]. They play a central role in phase space methods in quantum mechanics [4,5]. Various generalizations of coherent states have also been studied, especially in connection with groups like SU (2), SU (1, 1), etc. [6]. Two important operators in phase space methods, are the displacement operators and the parity operators [7–10]. They are related to each other through a two-dimensional Fourier transform. In this paper we replace the two Fourier transforms with two fractional Fourier transforms [11–14], and we get new unitary operators U (α , β; θα , θβ ) which we call bifractional displacement operators. Both displacement operators and parity operators, are special cases of these more general operators. We show that U (α , β; θα , θβ ) are elements of the group G = HW  SU (1, 1) that contains both displacements and squeezing transformations. The latter have been studied extensively in the literature, and in this paper we show that the operators U (α , β; θα , θβ ), which are introduced with an interpolation motivation, are elements of the group G (but the general element of G cannot always be written in the form U (α , β; θα , θβ )). Acting with the bifractional displacement operators on the vacuum, we get various classes of generalized coherent states (one class for each pair (θα , θβ )), which we call bifractional coherent states. They are squeezed states, and here we study them briefly

*

Corresponding author. E-mail address: [email protected] (A. Vourdas).

http://dx.doi.org/10.1016/j.physleta.2014.11.034 0375-9601/© 2014 Elsevier B.V. All rights reserved.

as a topic in its own right, because of their role in interpolations of different quantities in phase space methods. Using the bifractional displacement operators, we also introduce bifractional Wigner functions. Both the Wigner and Weyl functions are special cases of these more general functions. Examples of such functions are calculated numerically. Functions which interpolate between various phase space quantities, provide a deeper insight to current work on the phase space formalism. In Section 2 we review briefly fractional Fourier transforms, in order to define the notation. In Section 3 we introduce the bifractional displacement operators and show that they are elements of the group G. In Section 4 we introduce the bifractional coherent states. In Section 5 we use the bifractional displacement operators and the bifractional coherent states, to define bifractional Wigner functions and bifractional Q -functions. We conclude in Section 6 with a discussion of our results. 2. Fractional Fourier transform The fractional Fourier transform on a function f (x) is given by







f (x; θ) = F(θ) f (x) =

 Δ(x, y ; θ) =

1 + i cot θ

Δ(x, y ; θ) f ( y )dy 1/2

2π



× exp

−i (x2 + y 2 ) cot θ 2

+

ixy sin θ

 (1)

The fractional Fourier transform can be written in terms of the creation and annihilation operators a† , a as exp(i θ a† a) and in this

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S. Agyo et al. / Physics Letters A 379 (2015) 255–260

sense it is the time evolution operator of the oscillator, with respect to the ‘time variable’ θ . Below we present some special cases of the Δ(x, y ; θ):

The bifractional displacement operator is also given by

Δ(x, y ; 0) = δ(x − y )   π exp(ixy ) Δ x, y ; = 2 (2π )1/2 Δ(x, y ; π ) = δ(x + y )



U (α , β; θα , θβ ) = exp(i φ) exp i τ ( pˆ − tan θα xˆ + σ )2



(2)

In the case θ = π /2 the fractional Fourier transform reduces to the Fourier transform. In the case θ = π it is the parity transform (later we introduce the parity operator Π(0, 0)). For later use we give the formula



dy Δ(x, y ; θ1 )Δ( y , z; θ2 ) = Δ(x, z; θ1 + θ2 )

(3)

3. Bifractional displacement operators We consider the usual position and momentum operators xˆ , pˆ of the harmonic oscillator, and the displacement operators

√

√

D (α , β) = exp(i 2β xˆ − i 2α pˆ );

α , β ∈ R.

Π(α , β) = D

α β





Π(0, 0) D

,

2 2

†

α β

Π(α , β) =

D

2π

 =





,

2 2

(5)





α  , β  exp i β α  − β  α dα  dβ  

dα  dβ  Δ β, α  ;

π 2

  

π Δ α , −β  ; D α, β 



1/2

U (α , β; θα , θβ ) = cos(θα − θβ )



2



dα  dβ  Δ β, α  ; θβ



α , β;

π π 2

,

2

(7)

2 sin θβ



α 2 cot θα + β 2 cot θβ +

α2 sin 2θα

(9)

.

The proof is complex and is based on the integration of Eq. (7). In the integration we are careful with the ordering of the operators. We recall here that the operators xˆ 2 , pˆ 2 , xˆ pˆ + pˆ xˆ , xˆ , pˆ , 1, form a closed structure under commutation. Therefore the

(10)

form a group G which is the semidirect product of the Heisenberg Weyl group HW of displacements, by the SU (1, 1) group of squeezing transformations: G = HW  SU (1, 1). The operators U (α , β; θα , θβ ) depend on four parameters, and they are special cases of the operators T (a1 , a2 , a3 , a4 , a5 , a6 ). But clearly the general element T (a1 , a2 , a3 , a4 , a5 , a6 ) which depends on six parameters cannot always be written as U (α , β; θα , θβ ) which depends on four parameters.

Given a pair (θα , θβ ) we introduce the following set of ‘bifractional coherent states’:



C (θα , θβ ) = |α , β; θα , θβ  = U (α , β; θα , θβ )|0

(11)

     α , β; π , π = U α , β; π , π |0 = D (α , β)|0.  2 2 2 2

   π π = |α , β |−β, α ; 0, 0 = α , β; , 2

(14)

2

Since the bifractional operator is unitary, the coherent states in the set C (θα , θβ ) are Glauber coherent states with respect to the operators

†



†

b† (θα , θβ ) = U (0, 0; θα , θβ )a† U (0, 0; θα , θβ ) . (8)

(13)

Therefore in these special cases we get the standard (Glauber) coherent states, and



= Π(α , β)

(12)

In the special cases that θα = θβ = π2 we get

b(θα , θβ ) = U (0, 0; θα , θβ )a U (0, 0; θα , θβ )



U (α , β; π , π ) = D (−β, α )

2

β

|α , β; 0, 0 = U (α , β; 0, 0)|0 = D (β, −α )|0.

U (α , β; 0, 0) = D (β, −α )



2

φ = − (θα + θβ ) −

2 cos θα

−√

In the special cases that θα = θβ = 0 we get

Here we replaced the two Fourier transforms in Eq. (6) with two fractional Fourier transforms. We note that the two fractional Fourier transforms, use the variables α  , β  which are related to position and momentum and are dual to each other. In this sense our two-dimensional fractional Fourier transform is not a straightforward generalization of a one-dimensional fractional Fourier transform. The prefactor | cos(θα −θβ )|1/2 in Eq. (7) is important for unitarity. The fact that this prefactor cannot be factorized as a function of θα times a function of θβ , is related to the fact that the variables α  , β  are dual quantum variables. Also for θα − θβ = π /2, the prefactor is zero and the integral diverges, and in numerical work below, we do not go near this point. The following are special cases:

U

1

1

α

4. Bifractional coherent states



× Δ α , −β  ; θα D α  , β 

† U (α , β; θα , θβ ) = U (−α , −β; −θα , −θβ )

σ=√

;

cos θα

(6)

A generalization of the displaced parity operator is the following operator which we call bifractional displacement operator. The bifractional displacement operator is a unitary operator and defined as



cos θα sin θβ cos(θα − θβ )

−i



2α xˆ

  = exp a1 xˆ 2 + a2 pˆ 2 + a3 (ˆx pˆ + pˆ xˆ ) + a4 xˆ + a5 pˆ + a6 1

It is related with the displacement operator through the twodimensional Fourier transform (e.g., [7–10])



τ=

cot θα

√

T (a1 , a2 , a3 , a4 , a5 , a6 )



= D (α , β)Π(0, 0);  Π(0, 0) = dx|x−x|.

1

× exp i

xˆ 2



(4)

The displaced parity operator is given by



3.1. U (α , β; θα , θβ ) as special elements of the group G of squeezing and displacement transformations

(15)

But they have novel non-trivial properties with respect to a, a† .

S. Agyo et al. / Physics Letters A 379 (2015) 255–260

257

The coherent states in the set C (θα , θβ ) satisfy the resolution of the identity



1

dαdβ|α , β; θα , θβ α , β; θα , θβ | = 1

2π

(16)

From this follows that an arbitrary state | g  can be written as

 |g =

dαdβ|α , β; θα , θβ  g (α , β; θα , θβ ); 1

g (α , β; θα , θβ ) =

2π

α , β; θα , θβ | g 

(17)

We note that the

 B( z) = g α , β;

π π ,

2



2

 exp

1 2



2

α +β

2



;

z = α + iβ

(18)

is a Bargmann function with respect to the Glauber coherent states |α , β; π2 , π2 . We next consider the special case θβ = 0 and calculate the Bargmann functions B ( z; θα , 0) [15,16,10] for the coherent states in the set C (θα , 0).



B( z; θα , 0) = | cos θα |1/2 exp Az2 + B z + Γ A=−

Fig. 1. The uncertainty σ pp , the g (2) and the average number of photons n as a function of θα (in rads), for the coherent states |2, 2; θα , 0.

4.1. Statistical properties of the coherent states |α , β; θα , 0 For the coherent states |α , β; θα , 0 (with θβ = 0) we have calculated the wavefunction f (x) using the relation



f (x) = π −3/4 exp −

1 2(1 + i cot θα )

B =β+

Γ =−





2

2

β 2 + α2



(19)

4

−1/2

(ˆx + i pˆ );

f (x) = | cos θα |1/2 π −1/4

a =2

−1/2

(ˆx − i pˆ )

(20)

 

r

2

b = w 1 − |a|2

 

e −i φ

1 + 2A

κ x2 + 23/2 Bx + λ



2 + 4A (24)

x





1 2





2



∗

dxxn  f (x)

=





pn =

dx f (x) (−i ∂x )n f (x)

 (xp + px) − x p ;

   ∗ i (xp + px) = − + dx f (x) x(−i ∂x ) f (x) 2

(25)

and similarly for Δ p. For squeezed states, the Robertson–Schrödinger relation holds:

(21)

1

2 σxx σ pp − σxp = .

If in Eq. (21) we replace a with 2 A, b with B, and c with Γ , we get Eq. (19). The coherent states in the set C (θα + φα , θβ + φβ ) are related to the coherent states in the set C (θα , θβ ) through the fractional Fourier transform

(26)

4

We have checked that this is the case. We also found analytically that

√ x = β 2;

 2 x

1

= 2β 2 + ; 2

1

σxx = . 2

(27)

It is seen that these quantities do not depend on α , θα (in the case θβ = 0 considered in this section). The σ pp has been calculated numerically for the case α = β = 2. In Fig. 1 we plot the σ pp as a function of θα . Also in this example

|α , β; θα + φα , θβ + φβ  | cos(θα + φα − θβ − φβ )|1/2 | cos(θα − θβ )|1/2 



  × dα  dβ  Δ β, β  ; φβ Δ α , α  ; φα α  , β  ; θα , θβ .

1 2

1/2

1 1 c = − a ∗ w 2 − | w |2 . 2 2

 n

σxx = x2 − x2 ;

σxp =

2

 exp

We then calculated the uncertainties (all integrals are moments of Gaussian functions):



 

1/2

1

κ = 2 A − 1; λ = 2Γ + 4 A Γ − B .

σ pp = p 2 −  p 2 ;

 

1/4 1 2 B( z) = 1 − |a|2 exp az + bz + c



2

4

†

which is [10]

a = − tanh



and the Bargmann function of Eq. (19). We found

  2 1

1 | w ; r , θ = exp − re −i φ a† + re i φ a2 exp wa† − w ∗ a |0

=

√



dp B (x + ip ) 2; θα , 0 exp − p 2

(23)

sin θα (1 + i cot θα )

1





α

This result should be compared and contrasted with the Bargmann function for the squeezed states

a=2

x2

(22)

1

1

2

4

2 σxp = σ pp −

(28)

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S. Agyo et al. / Physics Letters A 379 (2015) 255–260

Using a Taylor expansion we have expressed the Bargmann function B ( z; θα , 0) of Eq. (19), with α = β = 2, as

B( z; θα , 0) =

∞  an zn √ n! n =0

(29)

Numerically we have the series at 30. We have found  truncated that in this case |an |2 = 0.999. We have then calculated the quantities 30  ν  n = nν |an |2 ;

g (2 ) =

n =0

n2  − n n2

(30)

In Fig. 1 we show n and g (2) as a function of θα . It is seen that for θα < 0.8 we have antibunching (g (2) < 1). 5. Interpolations between phase space quantities In the first part of this section, we introduce a two-dimensional continuum of functions A (α , β; θα , θβ ), which we call bifractional Wigner functions, and which have the Weyl and Wigner functions, as special cases. In the second part, we introduce bifractional Q -functions, and bifractional P -functions.

Fig. 2. [ A (α , β; 0, 0)] (Weyl function) for the state of Eq. (37) with β0 = 0. The arrows indicate the autoparts (A) and cross-parts (C).

α0 = 2 and

Fig. 3. A (α , β; π2 , π2 ) (Wigner function) for the state of Eq. (37) with β0 = 0. The arrows indicate the autoparts (A) and cross-parts (C).

α0 = 2 and

5.1. Bifractional Wigner functions For a trace class operator Θ , the Weyl function is given by

   W (α , β|Θ) = Tr Θ D (α , β)

(31)

and the Wigner function is given by





W (α , β|Θ) = Tr ΘΠ(α , β)

(32)

From Eq. (6) it follows that the Wigner and Weyl functions are related through the two-dimensional Fourier transform (e.g., [17, 18]):

W (α , β|Θ) =

1 2π







  W α  , β  Θ exp i β α  − β  α dα  dβ  (33)

Using Eq. (7) we define the bifractional Wigner function





A (α , β; θα , θβ |Θ) = Tr Θ U (α , β; θα , θβ )

1/2  = cos(θα − θβ )

  |s = N |α0 , β0  + |−α0 , −β0 



Δ β, α  ; θβ

    ×Δ α , −β  ; θα  W α , β Θ dα dβ

As an example we consider as Θ the density matrix ρ = |ss| where |s is the following superposition of two Glauber coherent states:

(34)

(36)

where N is a normalization factor. This is the even coherent state introduced in Ref. [20]. In this case

If it is obvious which operator we use, we omit Θ in the notation. A (α , β; θα , θβ |Θ) includes both the Wigner and Weyl functions as special cases:

ρ = ρ A + ρC ;

A (α , β; 0, 0|Θ) =  W (β, −α |Θ)

ρC = N 2 |α0 , β0 −α0 , −β0 | + |−α0 , −β0 α0 , β0 |

  π π A α , β; , Θ = W (α , β|Θ) 

2

2

A (α , β; π , π |Θ) =  W (−β, α |Θ)

(35)

We note that [19] has considered a single fractional Fourier transform between Wigner and Weyl functions and this gives generalized Wigner functions that depend on one angle. Here we use a double fractional Fourier transform and we get the bifractional Wigner functions that depend on two angles. As explained earlier, our two-dimensional fractional Fourier transform is not a straightforward generalization of a one-dimensional fractional Fourier transform.









ρ A = N 2 |α0 , β0 α0 , β0 | + |−α0 , −β0 −α0 , −β0 |

(37)

where ρ A and ρC are the ‘auto-part’ and the ‘cross-part’ of the density matrix. Inserting this in Eq. (34) we get the ‘auto-part’ and the ‘cross-part’ of the A (α , β; θα , θβ ), shown with A and C in the figures. In Figs. 2, 3, 4, we plot the [ A (α , β; 0, 0)] (Weyl function), A (α , β; π2 , π2 ) (Wigner function), and [ A (α , β; π4 , 0)] correspondingly, for α0 = 2 and β0 = 0. Here denotes the real part of the function. In the Weyl function of Fig. 2 the auto-terms are in the middle and the cross-terms are in the ‘wings’, while in the Wigner function of Fig. 3, the cross-terms are in the middle and the autoterms are in the wings. As we go from the Wigner (Fig. 3) to the Weyl function, the cross terms split in two parts in the beta

S. Agyo et al. / Physics Letters A 379 (2015) 255–260

Fig. 4. [ A (α , β; π4 , 0)] for the state of Eq. (37) with indicate the autoparts (A) and cross-parts (C).

α0 = 2 and β0 = 0. The arrows

259

Fig. 5. The bifractional Q -function Q (α , β; 0, 0) for the state of Eq. (37) with 1.2 and β0 = 0.

α0 =

Fig. 6. The bifractional Q -function Q (α , β; π4 , 0) for the state of Eq. (37) with 1.2 and β0 = 0.

α0 =

direction (see the intermediate function in Fig. 4), and the autoterms move from the wings to the center (see the Weyl function in Fig. 2). 5.2. Bifractional Q -functions and bifractional P -functions q-functions (or Husimi functions) for a trace class operator Θ , are defined as

q(α , β|Θ) = α , β|Θ|α , β

(38)

We define bifractional Q -functions with respect to the bifractional coherent states as follows:

Q (α , β; θα , θβ |Θ) = α , β; θα , θβ |Θ|α , β; θα , θβ 

(39)

Using the resolution of the identity in Eq. (16) we get

1



dαdβ Q (α , β; θα , θβ |Θ) = Tr Θ.

2π

6. Discussion

(40)

Clearly

Q (α , β; 0, 0|Θ) = q(β, −α |Θ);



Q

α , β;

π π 2

,

2

 Θ = q(α , β|Θ)

(41)

The bifractional Q -functions are generalizations of the q-functions. We also introduce the bifractional P -function P (α , β; θα , θβ |Θ) with respect to the bifractional coherent states, as

Θ=

1

π



dα dβ P (α , β; θα , θβ |Θ)|α , β; θα , θβ α , β; θα , θβ | (42)

In the case θα = θβ = π2 we get the ordinary P -function P (α , β). The P (α , β; θα , θβ |Θ) is given by

P (α , β; θ, φ|Θ) 

  1 = exp |α |2 + |β|2 dγ dδ exp 2i (β γ − α δ)

π



× exp |γ |2 + |δ|2 −γ , −δ; θ, φ|Θ|γ , δ; θ, φ

(43)

The proof of this is the same as in [21]. This is because the overlap γ , δ; θα , θβ |α , β; θα , θβ  does not depend on θα , θβ . In Figs. 5, 6 we present the Q (α , β; 0, 0|Θ) and Q (α , β; π4 , 0|Θ), correspondingly, for the state of Eq. (37) with α0 = 1.2 and β0 = 0. The results show the rotation of the autopart of Q (α , β; θα , 0|Θ), as θα changes.

The parity and displacement operators are important tools in phase space methods. The parity operators are a two-dimensional Fourier transform of the displacement operators (Eq. (6)). We have replaced the two Fourier transforms, with two fractional Fourier transforms, and we got the bifractional displacement operators U (α , β; θα , θβ ). We stressed the importance for unitarity, of the prefactor | cos(θα − θβ )|1/2 in these operators. We have shown that the U (α , β; θα , θβ ) are special cases of the squeezing operators acting on displacement operators, which are used here for interpolation purposes. Acting with the bifractional displacement operators on the vacuum, we get the bifractional coherent states, which are special cases of squeezed states. We have studied the uncertainties and statistical properties of these states. They can be used to define generalized phase space quantities. For example we have used them to define generalizations of the Q -functions. Using the bifractional displacement operators we introduced the bifractional Wigner functions in Eq. (34). Both the Wigner and Weyl functions are special cases of this more general function. Examples of these functions have been given in Figs. 2, 3, 4. We have also given an example of the bifractional Q -functions, in Figs. 5, 6. The work provides a deeper insight into the phase space methods. References [1] J.R. Klauder, B.-S. Skagerstam, Coherent States, World Scientific, Singapore, 1985. [2] A. Perelomov, Generalized Coherent States and Their Applications, Springer, Berlin, 1986.

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