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Scaling properties of the most frequently used classes of phase space functions
5 June 1995 PHYSICS LETTERS A EISEVIER
Physics LettersA 201(1995) 393-396
Scaling properties of the most frequently used classes of phase space functions D.M. Davidovi6, D. Lalovib, A.R. TanEiC Laboramy for TheoreticalE’hysics,Instituteof Nuclear Sciences “Vi&a”, P.O. Box 522, 11001 Belgrade,Yugoslavia
Received22 December 1994;revised manuscript received4 April 1995; acceptedfor publication 6 April 1995 Communicatedby J.P. Vigier
Abstract
We prove that in every class of frequently used phase space functions, except for the Husimi distributions, there always exist functions which do not remain in the same class after the change of variables (q, p) + (Aq, Ap), 0 < A < 1. We analyse some physical implications of these scaling properties.
In Ref. [l] we proved that the normalized function F(q, p) belongs to the class of Husimi distributions if, after the obvious change of variables in F(q, p), the expression F
i(P* -p2)
(PI -P2Y 4bfL2
2bh
becomes positive definite with respect to p1 and pz for some positive values of the parameter b. Using this criterion we recently proved [2] that for every scaling parameter which is smaller than unity, a scaled Husimi function remains in the class of Husimi functions. For completeness, we will first prove this fact here, in a different way. Let us examine expression (1) when F(q, p) = h2D(hq, hp), where D(q, p) is a Husimi function. We may write h2D
iA(PI
-PA
(PI
2bfi
-Pd2 4b&’
ih(p,
-Pz)
h2CP,
2bfi
-Pd2
4bti2
- (1 - A2) (PkifiF’2).
(2)
We will keep the parameter b fixed because we want to check whether the scaled function belongs to the class of Husimi functions with the same b. This is important because for example the well-known Q function for a given harmonic oscillator has this parameter fixed to the value mw. As Dfq, p) is a Husimi ~s~bution, the function in brackets is, according to (11, positive definite. Thus, the whole expression in (2) will be positive 03759601/95/$09.50
Q 1995 Elsevier Science B.V. All rights reserved
SSDI 0375-9601(95)00276-6
394
D.M. Davia!oviC et al. /Physics
Letters A 201 (1995) 393-396
definite when the second factor on its right-hand side is positive definite. When 0 < A < 1 this factor may be written in the form
-a(z+
q[
-p2j2]=/exp(-~2)S(x-y)ew[i(w-p2y)l
(a=(l-A2)/4M2>0,
dxdy
A=1/4a).
Now we have for any 1,5(p) E L2 Z=/exp[ =
/
-a(pr
-p2)2]JI(~1)JI*(P2)
dpr
exp( -Ax2) 6(x - y) exp[i( pr
02
~-P,Y)]~(P,)~*(P,)
dxdydp,
dp2.
Changing the order of integration and introducing the notation 4(x> for the Fourier transform of Z=/exp(-Ax2)8(x-y)$(x)$*(y)
dx dy=/exp(-Ax2)$(x)$*(x)
$(p)
we get
dx>O.
So, for 0 < A < 1 the second factor in (2) is certainly positive definite and due to this the function h2D(hq, hp) is a Husimi distribution. Let us now consider the following three-parametric phase space function introduced by us in Ref. 131,which will be called the ABC function (hereafter h = 1)
a2
a2
DAB&, d =exp AQ +Bap2 +C
WI,
(3)
P),
where W(q, p) is a Wigner function. For special choices of the parameters of the ABC function one can obtain various well-known phase space functions as its special cases [4]: (a) The Husimi function when A = l/4& B = fb, C = 0. (b) The P function when A = - l/4& B = - ab, C = 0. (c) The standard function when A = B = 0, C = ii. (d) The antistandard function when A = B = 0, C = - ii. (e) The O’Connel and Wang function when A = B = 0, C = $ib. (f) The Wigner function when A = B = C = 0. Taking into account the relation between the Husimi function D&q, p) and the Wigner function 141,we can write the ABC function in the form
a2 D,,,(q,
p) = exp
$
+ (B - $b)-$
+ Caqap
1D,h D, (Aq, P).
(4)
Using now the change of variables (q, p> + (Aq, Ap), 0 < A < 1 we obtain
%d hi,
AP)
AP) = ev
.
(5)
One concrete class of ABC functions is characterised by the concrete choice of the parameters A, B, C. We will now analyse the question whether it can happen that for every A and every fixed set of the parameters A, B, C, one can always find another function from the same class so that the following equality is fulfilled, A2%&q,
AP) =D,k(q,
P).
(6)
D.M. Davi~v~~
et al. /Physics
Letters A 201 &B.S~ 393-396
395
We have just shown that this is the case for the class of Husimi distributions. Now we shall prove that all the other classes of ABC functions are not invariant under scaling; i.e. equality (6) is not valid for every function from a given ABC class. Namely, as D,(hq, hp) in (5) is scaling invariant, then if the function on the left-hand side in (5) were scaling invariant for every A and every state described by the corresponding Husimi dis~ibution, it would be necessary that the following conditions are fulfilled,
(A--&)$=(A--$),(B-+b);=(~-;h),.;=c. The last conditions follow from the comparison of (5) and (4). However, these conditions may be satisfied only if A = 1, which is a trivial case, or when
A=-$
B=$b,
C=Q
and this is a class of Husimi distributions. So, in every class of the ABC functions, except the class of Husimi distributions, there always may be found at least one function that after scaling does not remain in the same class of unctions. Up to now this fact was only known for the class of Winger functions, for which this feature was shown by construction of a concrete example [Z]. In this regard the class of Husimi functions has a priviledged position among the other most frequently used phase space functions. The property of scaling invariance of the class of Husimi functions may be of special importance in nonlinear optics where the well-known Q function is widely used. This function is the special case of the Husimi function, when one chooses the parameter b so that b = mo. Hereafter, we will consider the physical implications of scaling invariance for the case of the Q function. It can be easily shown that when A decreases the average photon number in A2D(Ag, Ap) increases so that this transformation may represent physically some sort of photon pumping of the state. When the average number of photons becomes sufficiently large, the state becomes very close to the classical state. In mathematical formalism this reflects the following. The exact quantum mechanical average value for any physical quantity f(q, p) in a quantum state described by a Husimi dis~bution may be obtained by replacing the coordinate 9 and the momentum p in the function f(q, p) by the operators 4 and fi where [5]
so that all the average values in a state A2~(Aq, Ap) may be obtained by calculating f= j-f( 4, ri) h2D(Aq, AP)
dq
dp.
every differentiation introduces a multiplication by a small parameter A, it is obvious that for sufficiently small A the leading term in the expression for the average value is exactly the classical term 1 f(q, p)A2iXAq, Ap) dq dp. Let us note that the Wigner function is defined so that using it the average values are calculated by the formulae having an exactly classical structure, however the Wigner function cannot be considered as a true probability distribution because it is not non-negative. When used for the formulation of quantum mechanics in phase space the other standard phase space functions also suffer from some defect, for example, the O’Connell and Wang function is complex, while the Husk distribution in the general case does not give correct marginal dist~butions. Due to this, using these functions it is not possible to give the formulation of quantum mechanics in phase space which would be physically and intuitively transparent. The quantum mechanical Husimi distribution A2D(Aq, Ap) may be considered to behave classically for small A, because it is a normalized non-negative function and consequently a true probability distribution; and As
396
D.M. Davidovit et al. f Ptysics Letters A 201 (199S) 393-396
because the average values of all physical quantities in such a state may be obtained in the classical way so that the error introduced by such a procedure may be made negligible. From the classical state one can obtain the phase distribution in the standard way introducing the polar coordinates p, rp and integrating over p. The initial Husimi distribution and the scaled one, which for small h is classical, describe different physical states. However, it seems plausible to assume that the transformation (q, p) --) (hq, hp) does not change the distribution of the phase of the state so that both initial and transformed Husimi distributions have the same phase distribution. If it were so, the scaling transformation would be equivalent to the amplification of the quantum mechanical state by increasing the photon number by the phase insensitive amplifier. The ~ssibi~ty of ~pl~ing the state without changing its phase was discussed in Ref. [6]. If such an ~pli~cation were possible, the Husimi unction would be the most approp~ate phase space dist~bution for adequate and simple description of this process, because the other phase space functions are not scaling invariant.
References [l] [2] [3] [4] [S) [6]
D.M. DavidoviC and D. Lalovie, J. Phys. A 26 (1993) 5099. D.M. Davidovik, D. LaloviC and A.R. Tan&?, J. Phys. A 27 (1994) 8247. D. Lalovib, D.M. DavidoviC and N. BijediC, Phys. L&t. A 166 (1992) 99. R.F. O’Connell and L. Wang, Phys. Rev. A 31(1985) 1707. D. L.alovib, D.M. Davidovit aad N. BijediC, Phys. Rev. A 46 (1992) 1206. W. SchIeich, A. Bandilla and H. Paul, Phys. Rev. A 45 (1992) 6652.
Physics LettersA 201(1995) 393-396
Scaling properties of the most frequently used classes of phase space functions D.M. Davidovi6, D. Lalovib, A.R. TanEiC Laboramy for TheoreticalE’hysics,Instituteof Nuclear Sciences “Vi&a”, P.O. Box 522, 11001 Belgrade,Yugoslavia
Received22 December 1994;revised manuscript received4 April 1995; acceptedfor publication 6 April 1995 Communicatedby J.P. Vigier
Abstract
We prove that in every class of frequently used phase space functions, except for the Husimi distributions, there always exist functions which do not remain in the same class after the change of variables (q, p) + (Aq, Ap), 0 < A < 1. We analyse some physical implications of these scaling properties.
In Ref. [l] we proved that the normalized function F(q, p) belongs to the class of Husimi distributions if, after the obvious change of variables in F(q, p), the expression F
i(P* -p2)
(PI -P2Y 4bfL2
2bh
becomes positive definite with respect to p1 and pz for some positive values of the parameter b. Using this criterion we recently proved [2] that for every scaling parameter which is smaller than unity, a scaled Husimi function remains in the class of Husimi functions. For completeness, we will first prove this fact here, in a different way. Let us examine expression (1) when F(q, p) = h2D(hq, hp), where D(q, p) is a Husimi function. We may write h2D
iA(PI
-PA
(PI
2bfi
-Pd2 4b&’
ih(p,
-Pz)
h2CP,
2bfi
-Pd2
4bti2
- (1 - A2) (PkifiF’2).
(2)
We will keep the parameter b fixed because we want to check whether the scaled function belongs to the class of Husimi functions with the same b. This is important because for example the well-known Q function for a given harmonic oscillator has this parameter fixed to the value mw. As Dfq, p) is a Husimi ~s~bution, the function in brackets is, according to (11, positive definite. Thus, the whole expression in (2) will be positive 03759601/95/$09.50
Q 1995 Elsevier Science B.V. All rights reserved
SSDI 0375-9601(95)00276-6
394
D.M. Davia!oviC et al. /Physics
Letters A 201 (1995) 393-396
definite when the second factor on its right-hand side is positive definite. When 0 < A < 1 this factor may be written in the form
-a(z+
q[
-p2j2]=/exp(-~2)S(x-y)ew[i(w-p2y)l
(a=(l-A2)/4M2>0,
dxdy
A=1/4a).
Now we have for any 1,5(p) E L2 Z=/exp[ =
/
-a(pr
-p2)2]JI(~1)JI*(P2)
dpr
exp( -Ax2) 6(x - y) exp[i( pr
02
~-P,Y)]~(P,)~*(P,)
dxdydp,
dp2.
Changing the order of integration and introducing the notation 4(x> for the Fourier transform of Z=/exp(-Ax2)8(x-y)$(x)$*(y)
dx dy=/exp(-Ax2)$(x)$*(x)
$(p)
we get
dx>O.
So, for 0 < A < 1 the second factor in (2) is certainly positive definite and due to this the function h2D(hq, hp) is a Husimi distribution. Let us now consider the following three-parametric phase space function introduced by us in Ref. 131,which will be called the ABC function (hereafter h = 1)
a2
a2
DAB&, d =exp AQ +Bap2 +C
WI,
(3)
P),
where W(q, p) is a Wigner function. For special choices of the parameters of the ABC function one can obtain various well-known phase space functions as its special cases [4]: (a) The Husimi function when A = l/4& B = fb, C = 0. (b) The P function when A = - l/4& B = - ab, C = 0. (c) The standard function when A = B = 0, C = ii. (d) The antistandard function when A = B = 0, C = - ii. (e) The O’Connel and Wang function when A = B = 0, C = $ib. (f) The Wigner function when A = B = C = 0. Taking into account the relation between the Husimi function D&q, p) and the Wigner function 141,we can write the ABC function in the form
a2 D,,,(q,
p) = exp
$
+ (B - $b)-$
+ Caqap
1D,h D, (Aq, P).
(4)
Using now the change of variables (q, p> + (Aq, Ap), 0 < A < 1 we obtain
%d hi,
AP)
AP) = ev
.
(5)
One concrete class of ABC functions is characterised by the concrete choice of the parameters A, B, C. We will now analyse the question whether it can happen that for every A and every fixed set of the parameters A, B, C, one can always find another function from the same class so that the following equality is fulfilled, A2%&q,
AP) =D,k(q,
P).
(6)
D.M. Davi~v~~
et al. /Physics
Letters A 201 &B.S~ 393-396
395
We have just shown that this is the case for the class of Husimi distributions. Now we shall prove that all the other classes of ABC functions are not invariant under scaling; i.e. equality (6) is not valid for every function from a given ABC class. Namely, as D,(hq, hp) in (5) is scaling invariant, then if the function on the left-hand side in (5) were scaling invariant for every A and every state described by the corresponding Husimi dis~ibution, it would be necessary that the following conditions are fulfilled,
(A--&)$=(A--$),(B-+b);=(~-;h),.;=c. The last conditions follow from the comparison of (5) and (4). However, these conditions may be satisfied only if A = 1, which is a trivial case, or when
A=-$
B=$b,
C=Q
and this is a class of Husimi distributions. So, in every class of the ABC functions, except the class of Husimi distributions, there always may be found at least one function that after scaling does not remain in the same class of unctions. Up to now this fact was only known for the class of Winger functions, for which this feature was shown by construction of a concrete example [Z]. In this regard the class of Husimi functions has a priviledged position among the other most frequently used phase space functions. The property of scaling invariance of the class of Husimi functions may be of special importance in nonlinear optics where the well-known Q function is widely used. This function is the special case of the Husimi function, when one chooses the parameter b so that b = mo. Hereafter, we will consider the physical implications of scaling invariance for the case of the Q function. It can be easily shown that when A decreases the average photon number in A2D(Ag, Ap) increases so that this transformation may represent physically some sort of photon pumping of the state. When the average number of photons becomes sufficiently large, the state becomes very close to the classical state. In mathematical formalism this reflects the following. The exact quantum mechanical average value for any physical quantity f(q, p) in a quantum state described by a Husimi dis~bution may be obtained by replacing the coordinate 9 and the momentum p in the function f(q, p) by the operators 4 and fi where [5]
so that all the average values in a state A2~(Aq, Ap) may be obtained by calculating f= j-f( 4, ri) h2D(Aq, AP)
dq
dp.
every differentiation introduces a multiplication by a small parameter A, it is obvious that for sufficiently small A the leading term in the expression for the average value is exactly the classical term 1 f(q, p)A2iXAq, Ap) dq dp. Let us note that the Wigner function is defined so that using it the average values are calculated by the formulae having an exactly classical structure, however the Wigner function cannot be considered as a true probability distribution because it is not non-negative. When used for the formulation of quantum mechanics in phase space the other standard phase space functions also suffer from some defect, for example, the O’Connell and Wang function is complex, while the Husk distribution in the general case does not give correct marginal dist~butions. Due to this, using these functions it is not possible to give the formulation of quantum mechanics in phase space which would be physically and intuitively transparent. The quantum mechanical Husimi distribution A2D(Aq, Ap) may be considered to behave classically for small A, because it is a normalized non-negative function and consequently a true probability distribution; and As
396
D.M. Davidovit et al. f Ptysics Letters A 201 (199S) 393-396
because the average values of all physical quantities in such a state may be obtained in the classical way so that the error introduced by such a procedure may be made negligible. From the classical state one can obtain the phase distribution in the standard way introducing the polar coordinates p, rp and integrating over p. The initial Husimi distribution and the scaled one, which for small h is classical, describe different physical states. However, it seems plausible to assume that the transformation (q, p) --) (hq, hp) does not change the distribution of the phase of the state so that both initial and transformed Husimi distributions have the same phase distribution. If it were so, the scaling transformation would be equivalent to the amplification of the quantum mechanical state by increasing the photon number by the phase insensitive amplifier. The ~ssibi~ty of ~pl~ing the state without changing its phase was discussed in Ref. [6]. If such an ~pli~cation were possible, the Husimi unction would be the most approp~ate phase space dist~bution for adequate and simple description of this process, because the other phase space functions are not scaling invariant.
References [l] [2] [3] [4] [S) [6]
D.M. DavidoviC and D. Lalovie, J. Phys. A 26 (1993) 5099. D.M. Davidovik, D. LaloviC and A.R. Tan&?, J. Phys. A 27 (1994) 8247. D. Lalovib, D.M. DavidoviC and N. BijediC, Phys. L&t. A 166 (1992) 99. R.F. O’Connell and L. Wang, Phys. Rev. A 31(1985) 1707. D. L.alovib, D.M. Davidovit aad N. BijediC, Phys. Rev. A 46 (1992) 1206. W. SchIeich, A. Bandilla and H. Paul, Phys. Rev. A 45 (1992) 6652.