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Eigenstates of squeeze operators do not exist
ISApril 1991
PHYSICS LETTERS A
Volume 154, number 7.8
Eigenstates of squeeze operators do not exist Arkadiusz Orlowski Department
of MathematicalStatistics
ul. Rakowiecka
Z&/30. 02-528
and Experimentation,
Warsaw Agricultural
University,
Warsaw, Poland
Received 27 November 1990; accepted for publication 12 February 199 I Communicated by J.P. Vigier
Nonexistence of normalized eigenstates for the SU ( 1, I ) Lie group displacement operator is proven. The proof is independent of the Bargmann index so it provides a generalization and unification of some previous results. Other applications of the developed method are also indicated.
2. Calculations
1. Introduction The single-mode and two-mode squeeze operators are very important and useful in the description and generation of squeezed states of light [ l-31. It is often essential to recognize that they are special cases of the SU ( 1, 1) Lie group displacement operator [ 41 D(C)=exp(zK+ =exp(tK+)
-ICK_) ewb(l-
lt12)K~1exp( -5*K- 1,
where e;=z tgh ( 1z[ ) / I z( and the generators of the SU ( I, I ) Lie algebra satisfy the following commutation relations, [K_,K+J=ZK,,
(1)
D(W~-‘(O=P~, where p denotes an arbitrary Making use of the relations
D(W+D-‘((I= D(t)K_D-l(t)=
D(t)K,D-l(t)=
complex
K, +r2K_
parameter.
-2pCK,
l-IQ’
’
‘2K+
:_:,;‘” ,
-yC+
-9CK-
[K3,KJ=kKT.
Also the new multi-mode squeeze operators [ 51 can be written in this general form. Various realizations are distinguished by different values of their Bargmann index [ 61, In this paper I show that normalized eigenstates of the above operator do not exist. The proof is independent of the Bargmann index so it provides a generalization and unification of some previous results (the nonexistence of proper eigenstates for singlemode, two-mode and recently multi-mode squeeze operators has been separately proven in ref. [ 71 and ref. [ 5 ] respectively ).
03759601/91/$
Let Z=c, K, +c2K_ +c,K3 be a complex linear combination of generators. Next let us consider the following system of linear algebraic equations for the coefficients c,,
I-
+ ( I+ It;12W3 ItI2
7
we find an explicit form of system ( I ), [1-p(1-l~12)]c,+~2c2--rc3=o, ~‘2~,+[~-~(~-l~12)~~2-c*~3=~,
-2gC~,-2&~+[l-p+(l+p)]Q~]c~=O. The nonzero (nonunique) solution of this system exists iff the corresponding determinant vanishes. Thisispossibleforp=l orP=(l+]Q)/(lT]Q). Now let us assume that a normalized eigenstate of D(t) exists. Then due to the unitarity of this operator we have
03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
319
Volume
154. number 7.8
PHYSICS
is real. We also have
D(r)~l~(r))=D(r)~D-‘(r)D(r)lY(~)) =GD(O
Iv(r) > =Pexp[iy(t;)
Takingp=(l~le()/(lTIt(() c2 =c~ = 1 we immediately K, +K_
+
‘(‘-‘) l-(l?lrlY
PI Y(t) > .
and (forsimplicity) obtain that the state
K:,
>
IAO>,
(which can be normalized) is also an eigenstate D(c) with the appropriate eigenvalue equal to
l+lrl m
15 April 1991
A
SU( 1, I ) Lie group displacement operator do not exist. The proof is independent of the Bargmann index of the representation so it contains as special cases the results mentioned earlier. Let us note that this method can be immediately applied to the SU (2 ) Lie group with similar result. However for the Heisenberg-Weyl group of the harmonic oscillator (with the well known displacement operator D(a) = exp(cua+ - (~*a) and [a, a+ ] = I ) the equivalent of system ( 1) has nonzero solutions only for p= I.
D(r)ly(r))=exp[iy(t;)lly(r)). where ~(0
LETTERS
of
exp[i7(8 1 .
References [ I ] C.M.
Caves and B.L. Schumaker.
Phys. Rev. A 31
( 1985 )
3068.
[ 21 This is obviously in contradiction with the unitarity of D(r). Hence normalized eigenstates of this operator do not exist.
B.L. Schumaker
and C.M.
Caves, Phys. Rev. A 31 (1985)
3093. [3] B.L. Schumaker,
[ 4 ] K.
Wbdkiewicz
Phys. Rep. 135 (1986)
317.
and J.H. Eberly, J. Opt. Sot. Am. B 2
( I985 )
458; A. Orlowski
and K. W&Ikiewicz,
J. Mod. Opt. 37
3. Remarks
[S] X.
Ma and W. Rhodes, Phys. Rev. A 41
[6] V. Bargmann, Ann. Math. 48 (1947)
I have shown that normalized
320
( 1990)
295.
eigenstates
of the
[ 7 ] X.
( 1990)
4625.
568.
Ma and W. Rhodes, Nuovo Cimento B 104 ( 1989)
159.
PHYSICS LETTERS A
Volume 154, number 7.8
Eigenstates of squeeze operators do not exist Arkadiusz Orlowski Department
of MathematicalStatistics
ul. Rakowiecka
Z&/30. 02-528
and Experimentation,
Warsaw Agricultural
University,
Warsaw, Poland
Received 27 November 1990; accepted for publication 12 February 199 I Communicated by J.P. Vigier
Nonexistence of normalized eigenstates for the SU ( 1, I ) Lie group displacement operator is proven. The proof is independent of the Bargmann index so it provides a generalization and unification of some previous results. Other applications of the developed method are also indicated.
2. Calculations
1. Introduction The single-mode and two-mode squeeze operators are very important and useful in the description and generation of squeezed states of light [ l-31. It is often essential to recognize that they are special cases of the SU ( 1, 1) Lie group displacement operator [ 41 D(C)=exp(zK+ =exp(tK+)
-ICK_) ewb(l-
lt12)K~1exp( -5*K- 1,
where e;=z tgh ( 1z[ ) / I z( and the generators of the SU ( I, I ) Lie algebra satisfy the following commutation relations, [K_,K+J=ZK,,
(1)
D(W~-‘(O=P~, where p denotes an arbitrary Making use of the relations
D(W+D-‘((I= D(t)K_D-l(t)=
D(t)K,D-l(t)=
complex
K, +r2K_
parameter.
-2pCK,
l-IQ’
’
‘2K+
:_:,;‘” ,
-yC+
-9CK-
[K3,KJ=kKT.
Also the new multi-mode squeeze operators [ 51 can be written in this general form. Various realizations are distinguished by different values of their Bargmann index [ 61, In this paper I show that normalized eigenstates of the above operator do not exist. The proof is independent of the Bargmann index so it provides a generalization and unification of some previous results (the nonexistence of proper eigenstates for singlemode, two-mode and recently multi-mode squeeze operators has been separately proven in ref. [ 71 and ref. [ 5 ] respectively ).
03759601/91/$
Let Z=c, K, +c2K_ +c,K3 be a complex linear combination of generators. Next let us consider the following system of linear algebraic equations for the coefficients c,,
I-
+ ( I+ It;12W3 ItI2
7
we find an explicit form of system ( I ), [1-p(1-l~12)]c,+~2c2--rc3=o, ~‘2~,+[~-~(~-l~12)~~2-c*~3=~,
-2gC~,-2&~+[l-p+(l+p)]Q~]c~=O. The nonzero (nonunique) solution of this system exists iff the corresponding determinant vanishes. Thisispossibleforp=l orP=(l+]Q)/(lT]Q). Now let us assume that a normalized eigenstate of D(t) exists. Then due to the unitarity of this operator we have
03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
319
Volume
154. number 7.8
PHYSICS
is real. We also have
D(r)~l~(r))=D(r)~D-‘(r)D(r)lY(~)) =GD(O
Iv(r) > =Pexp[iy(t;)
Takingp=(l~le()/(lTIt(() c2 =c~ = 1 we immediately K, +K_
+
‘(‘-‘) l-(l?lrlY
PI Y(t) > .
and (forsimplicity) obtain that the state
K:,
>
IAO>,
(which can be normalized) is also an eigenstate D(c) with the appropriate eigenvalue equal to
l+lrl m
15 April 1991
A
SU( 1, I ) Lie group displacement operator do not exist. The proof is independent of the Bargmann index of the representation so it contains as special cases the results mentioned earlier. Let us note that this method can be immediately applied to the SU (2 ) Lie group with similar result. However for the Heisenberg-Weyl group of the harmonic oscillator (with the well known displacement operator D(a) = exp(cua+ - (~*a) and [a, a+ ] = I ) the equivalent of system ( 1) has nonzero solutions only for p= I.
D(r)ly(r))=exp[iy(t;)lly(r)). where ~(0
LETTERS
of
exp[i7(8 1 .
References [ I ] C.M.
Caves and B.L. Schumaker.
Phys. Rev. A 31
( 1985 )
3068.
[ 21 This is obviously in contradiction with the unitarity of D(r). Hence normalized eigenstates of this operator do not exist.
B.L. Schumaker
and C.M.
Caves, Phys. Rev. A 31 (1985)
3093. [3] B.L. Schumaker,
[ 4 ] K.
Wbdkiewicz
Phys. Rep. 135 (1986)
317.
and J.H. Eberly, J. Opt. Sot. Am. B 2
( I985 )
458; A. Orlowski
and K. W&Ikiewicz,
J. Mod. Opt. 37
3. Remarks
[S] X.
Ma and W. Rhodes, Phys. Rev. A 41
[6] V. Bargmann, Ann. Math. 48 (1947)
I have shown that normalized
320
( 1990)
295.
eigenstates
of the
[ 7 ] X.
( 1990)
4625.
568.
Ma and W. Rhodes, Nuovo Cimento B 104 ( 1989)
159.