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Generalized uncertainty relation and correlated coherent states
Volume 79A, number 2,3
PHYSICS LETTERS
29 September 1980
GENERALIZED UNCERTAINTY RELATION AND CORRELATED COHERENT STATES V.V. DODONOV, E.V. KURMYSHEV and V.!. MAN’KO P.N. Lebedev Institute of Physics, Moscow, USSR Received 26 June 1980
A generalized Heisenberg-type uncertainty relation is obtained for two arbitrary operators both in the case of pure and of mixed states. As a rule equality is found to hold for pure quantum states only. New minimizing states called correlated coherent states, are constructed in explicit form, and their properties are studied.
The concept of coherent states introduced by Glauber [1] has proved to be fruitful in quantum physics. There is a large number of papers concerning various generalizations of these states [2—7] (we imply coherent states of the Heisenberg—Weyl group). All the states of the above mentioned papers are related to each other by means of canonical (unitary) transformations, but each type has specific features which might be useful for the solution of concrete physical problems. In this paper we consider the corparticular case of generalized Glauber states called related coherent states. These states minimize a more accurate uncertainty relation (UR) than that usually considered in quantum mechanics. The UR discussed below has been found for the first time by Schrodinger [8] and Robertson [9] for hermitian operators and pure states. We shall first of all generalize this relation for arbitrary operators and open quantum systems and then analyze it. Let A and B be arbitrary (in general nonhermitian) operators, and ,3 an arbitrary density matrix. Let us introduce the operator F aZ~A+ i~B,where a ÷ia 2 is an arbitrary complex number and z.\A (A>,(A)~Sp(15A). The operator F~FisHence, evidently hermitian and positive semidefinite. because the density matrix ~ is positive semidefinite too, the inequality Sp(,~k~F) ~ 0 should be valid. Taking into account the definition ofF one can rewrite this inequality as follows: —
(~A~Z~A (a 2
+
(L~IB~Z~B>
-
~ +
+
2~ ~~~AA)) )2
2 (~A~L~A
+
—
4(Z~A~~A) >2 +_~h~~A)] 4(~4+~4)
(1)
—‘
(It is assumed that all the mean values exist.) Omitting the case (z~sA+Z~A)= 0 to be discussed separately one finds from condition (1) the required relation: -
+ ~ [((~A~B
-
~[i((~A~B
(2)
+
It should be noted that relation (2) is a generalization of the Schwarz inequality for mixed states. As for hermitian operators, introducing the dispersions
2),
((~A)
=
= ~
one can 0A0B
where r 150
/
Sp(~F~F)
0B
=
(3) +
represent the UR (2) in the form 0AB GAUB(! — r2)~~ ([A, B])12 a.4B1(UA UB)V2
(4)
is the correlation coefficient.
Volume 79A, number 2,3
PHYSICS LETTERS
29 September 1980
This UR generalizes the result of refs. [8,9] for the most general case of mixed states. In some particular cases of mixed states this relation has been previously proved in ref. [10] (two-dimensional Hilbert space) and in ref. [11] (coordinate and momentum operators). As is seen from relation (1), the necessary and sufficient condition for the equality in (2) to hold is that Sp(,3F~F~)= 0, where the operator F~is obtained from F by setting
Let us consider the case when A and B are coordinate and momentum operators, respectively, and find the states, for which the equality in (4) holds at a fixed value of r. First of all one notes that these states are pure ones, because the operator Fis, in this case, a linear combination ofcoordinate and momentum; thus the equation Flp)= 0 has only one solution for each concrete F. However, the minimizing state is not uniquely determined here. We take for simplicity
a1
1, thushave [~,j3] It is clear that new minimizing= states to =bei.eigenstates of anthe operator of the formd = ~ei~ + ne~ñ,where ~,i~,p, ~,1i are real numbers. These states, denoted by a, r), are some generahzation of Glauber’s coherent states [1]: ala, r), = a a, r) (a a complex number). The peculiarity of the discussed states, as compared with the ones of refs. [2—7],is that the taken correlation coefficient r 0. It should bebelongtothe emphasized Hilbert that thespace, eigenstates theif operatora if andof only sin (i/i p) >0; this is equivalent to the condition [a, à~J= 2~sin(ili p) >0 (by definition ~,~ >0). Further, keeping in mind the condition mentioned above, oneeasily finds r = —cos(i,li p) and the equality Gpaq(l r2) = 1/4 to hold true for all states Ia, r). If we set [a, â~]= 1 and ~i = 77/2, the following family of operators depending on the parameter i~ = ~ is obtained:
a2
—i((L~A~I~B ~Ap/2~A~~A>
=
—
—((~A~B + ~~A)/2~A~A>.
=
Any density matrix can be represented in the form p,~In )(n ,
=
n 5mn , p,~~‘0, = I (mln> ‘ Therefore, we have in the case under consideration .
-
—
—
-
~p~(nlfr~frIn)~0
Sp(~FF~)=
-
—
—
As far as the coefficients p,~are nonnegative one concludes from the preceding relation that the minimizing density matrices, for which the equality in (2) holds, have the form
~ !rl<1. (8) (1—r) Their normalized eigenfunctions, called by us correlated coherent states, have the form:
Pn>O, ~p~=l, (6) n n where the state vectors I ~ are different orthonormal solutions of the equation
ar,~[1
~lp)
(xla,r,
~*=~PnnXPnI,
0.
(7)
If eq. (7) has one solution only, then the minimum is approached in pure states. As far as the condition Sp(,~F~F~) = 0 is necessary and sufficient, and the eigenstates of the operator F coincide with those of the operator 13F (i3 a complex number), the most general procedure of constructing the minimizing state is as follows. One finds all solutions of the equation F I p) = 0 and chooses those, which satisfy eqs. (5). The states selected in this way are the minimizing ones. It should be noted that the variational technique for determining those normalizable pure states, which give a minimum for the uncertainty product of two noncommuting hermitian operators, has been developedin ref. [12].
77
(2ir?72)~/4ex~~_~ [i
4~ ax +
—
‘
—
(1
—
r
) (9)
—
~(a2 ÷al2)
It is interesting to note that states of the type (9) have already been studied by Kennard [13], who has also proved exactly the relation ~q ~p ~‘h/2. Recently similar states were also considered in ref. [7], but without connection with relation (4). If we set r = 0 and 77 = l/V’~,the states (9) become the usual Glauber coherent states. The analytic properties of the states (9) (expansion of the unity operator, etc.) are just as these of other coherent states. For example, 151
Volume 79A, number 2,3
PHYSICS LETTERS
the expansion of the state I a, r, ~) in powers of a gives us the eigenstates of the number operator: 772,772
+
4772(1
2(l_r2)h/2 ~
[
Xexpl—-----—4772 1
—
(1
—
r2)h/2
(11)
)
r_____
~
=
~
~ = V’~77~ — [2(1 —
r2)I 1/2
(12) i.e. by changing the dimension of the coordinate and by introducing a new generalized momentum. The time-dependent canonical transformation
2)112,
~=~
= —r/(l
—
r
(13)
.
leads to the equation of the oscillator with friction: 2) = 0 (14) Q + 2yQ + Q/(l r Therefore, the states (9) concern the problem of the damped quantum oscillator [7,151. The states (9) have the most clear form in Wigner’s representation: -
-
—
~
be satisfied by any operator. We consider the density main result of the present paper and the meaning of the introduced correlated coherent states toabout mation be as the follows. system, If one one has canno only additional say that inforUpUq
I H~(f).
A direct calculation of the correlation factor Rn in the state (11) leads us to the result ~R 0= r independent of n. It should be pointed out that the system described by the hamiltonian (10) is in fact the usual harmonic oscillator with the equation of motion ~ + = o. The operator (10)is reduced the hamiltonian of the osdillator(Q2 +~2)/2by the to canonical transformation =
(16)
can be a quantum Wigner function, if and only if ~ > 1/4, because this condition, coinciding with (4), leads to the relation ff W2(q, p) dp dq ( 1 being equivalent to the condition Sp(~32)~ 1, which has to
2 77772)_1/4(2nn!)_11
—
W=~/2exp(_2_2+~~~)
with the corresponding eigenvalues n = 0, 1, 2 These normalized eigenfiinctions in coordinate representation are, as usual, expressed by means of Hermite polynomials: (xln, r, s~)=(2
mal distribution to give the dispersion uq = 772, ~ 2)] 1 and the correlation coefficient r. = [4772(1 r We note that the most general normal distribution (here z~= ~ u,~q) —
(10)
r2)
29 September 1980
p)
>)~2/4 (we
restrict our discussion to coordinate and
momentum operators. If one knows the correlation coefficient r = apq/(opuq)1/2 not to be equal to zero (there is additional inftrmation about the system), the lower limit of the dispersion clear to have 2/[4(lproduct r2)].is In other words, athelarger value: UpUq > h quantum fluctuations in a state with r * 0, generally speaking, are larger than in the state with r = 0. This means in reality that the probability of tunnel penetration in a state with r *0 can be much larger than in that with r = 0. We believe, in connection with the above discussion, the following problem to be of interest: to generalize the UR (4) for the case of the higher correlation momentsM an inequality of the form u~UB 1, > h2/4f(M ,M,~,i.e. to obtain 1 Ma). —
...
[1] D. R.J.Stoler, Glauber, Phys. 131 (1963) [2] Phys. Rev.Rev. 1D (1970) 3217.2766. [3] E.Y.C. Lu, Lett. Nuovo Cimento 2 (1971) 1241. [4] l.A. Malkin, VI. Man’ko and D.A. Trifonov, J. Math. Phys. 14 (1973) 576. Phys. Rev. A13 (1976) 2226. [6] V. Canivell and P. Seglar, Phys. Rev. D15 (1977) 1050. [7] A.D. Jannussis, G.N. Brodimas and L.C. Papaloucas, Phys.
[51 H.P. Yuen,
Lett. 71A (1979) 301. [81 E. Schrodinger, Sitzungsber. Preuss. Akad. Wiss. (Berlin, 1930) p. 296
f d~(q + ~ ~Ia, r, 77Xa, r, ~ 277 J —~—
—
~ ~)e~
[9] H.P. Robertson, Phys. Rev. 35 (1930) 667A; 46(1934)
794.
(15) = ~~exp[_2772f2
q
—
2n2(l where ,i~p
152
—
(‘b), c~ q
—
+ —
r2)
2r
(1
—
r2)h/2
(ci). This is the classical nor-
[10] [11] [12] [13] [14]
W. Band and J.L. Park, Found. Phys. 1(1970)133. D. Stoler and S. Newman, Phys. Lett. 38A (1972) 433. R. Jackiw, J. Math. Phys. 9 (1968) 339. E.H. Kennard, Z. Phys. 44 (1927) 326. E.H. Kerner, Can. J. Phys. 36 (1958) 371.
[15] V.V. 550. Dodonov and V.1. Man’ko, Phys. Rev. A20 (1979)
PHYSICS LETTERS
29 September 1980
GENERALIZED UNCERTAINTY RELATION AND CORRELATED COHERENT STATES V.V. DODONOV, E.V. KURMYSHEV and V.!. MAN’KO P.N. Lebedev Institute of Physics, Moscow, USSR Received 26 June 1980
A generalized Heisenberg-type uncertainty relation is obtained for two arbitrary operators both in the case of pure and of mixed states. As a rule equality is found to hold for pure quantum states only. New minimizing states called correlated coherent states, are constructed in explicit form, and their properties are studied.
The concept of coherent states introduced by Glauber [1] has proved to be fruitful in quantum physics. There is a large number of papers concerning various generalizations of these states [2—7] (we imply coherent states of the Heisenberg—Weyl group). All the states of the above mentioned papers are related to each other by means of canonical (unitary) transformations, but each type has specific features which might be useful for the solution of concrete physical problems. In this paper we consider the corparticular case of generalized Glauber states called related coherent states. These states minimize a more accurate uncertainty relation (UR) than that usually considered in quantum mechanics. The UR discussed below has been found for the first time by Schrodinger [8] and Robertson [9] for hermitian operators and pure states. We shall first of all generalize this relation for arbitrary operators and open quantum systems and then analyze it. Let A and B be arbitrary (in general nonhermitian) operators, and ,3 an arbitrary density matrix. Let us introduce the operator F aZ~A+ i~B,where a ÷ia 2 is an arbitrary complex number and z.\A (A>,(A)~Sp(15A). The operator F~FisHence, evidently hermitian and positive semidefinite. because the density matrix ~ is positive semidefinite too, the inequality Sp(,~k~F) ~ 0 should be valid. Taking into account the definition ofF one can rewrite this inequality as follows: —
(~A~Z~A (a 2
+
(L~IB~Z~B>
-
~ +
+
2~ ~~~AA)) )2
2 (~A~L~A
+
—
4(Z~A~~A) >2 +_~h~~A)] 4(~4+~4)
(1)
—‘
(It is assumed that all the mean values exist.) Omitting the case (z~sA+Z~A)= 0 to be discussed separately one finds from condition (1) the required relation: -
+ ~ [((~A~B
-
~[i((~A~B
(2)
+
It should be noted that relation (2) is a generalization of the Schwarz inequality for mixed states. As for hermitian operators, introducing the dispersions
2),
((~A)
=
= ~
one can 0A0B
where r 150
/
Sp(~F~F)
0B
=
(3) +
represent the UR (2) in the form 0AB GAUB(! — r2)~~ ([A, B])12 a.4B1(UA UB)V2
(4)
is the correlation coefficient.
Volume 79A, number 2,3
PHYSICS LETTERS
29 September 1980
This UR generalizes the result of refs. [8,9] for the most general case of mixed states. In some particular cases of mixed states this relation has been previously proved in ref. [10] (two-dimensional Hilbert space) and in ref. [11] (coordinate and momentum operators). As is seen from relation (1), the necessary and sufficient condition for the equality in (2) to hold is that Sp(,3F~F~)= 0, where the operator F~is obtained from F by setting
Let us consider the case when A and B are coordinate and momentum operators, respectively, and find the states, for which the equality in (4) holds at a fixed value of r. First of all one notes that these states are pure ones, because the operator Fis, in this case, a linear combination ofcoordinate and momentum; thus the equation Flp)= 0 has only one solution for each concrete F. However, the minimizing state is not uniquely determined here. We take for simplicity
a1
1, thushave [~,j3] It is clear that new minimizing= states to =bei.eigenstates of anthe operator of the formd = ~ei~ + ne~ñ,where ~,i~,p, ~,1i are real numbers. These states, denoted by a, r), are some generahzation of Glauber’s coherent states [1]: ala, r), = a a, r) (a a complex number). The peculiarity of the discussed states, as compared with the ones of refs. [2—7],is that the taken correlation coefficient r 0. It should bebelongtothe emphasized Hilbert that thespace, eigenstates theif operatora if andof only sin (i/i p) >0; this is equivalent to the condition [a, à~J= 2~sin(ili p) >0 (by definition ~,~ >0). Further, keeping in mind the condition mentioned above, oneeasily finds r = —cos(i,li p) and the equality Gpaq(l r2) = 1/4 to hold true for all states Ia, r). If we set [a, â~]= 1 and ~i = 77/2, the following family of operators depending on the parameter i~ = ~ is obtained:
a2
—i((L~A~I~B ~Ap/2~A~~A>
=
—
—((~A~B + ~~A)/2~A~A>.
=
Any density matrix can be represented in the form p,~In )(n ,
=
n 5mn , p,~~‘0, = I (mln> ‘ Therefore, we have in the case under consideration .
-
—
—
-
~p~(nlfr~frIn)~0
Sp(~FF~)=
-
—
—
As far as the coefficients p,~are nonnegative one concludes from the preceding relation that the minimizing density matrices, for which the equality in (2) holds, have the form
~ !rl<1. (8) (1—r) Their normalized eigenfunctions, called by us correlated coherent states, have the form:
Pn>O, ~p~=l, (6) n n where the state vectors I ~ are different orthonormal solutions of the equation
ar,~[1
~lp)
(xla,r,
~*=~PnnXPnI,
0.
(7)
If eq. (7) has one solution only, then the minimum is approached in pure states. As far as the condition Sp(,~F~F~) = 0 is necessary and sufficient, and the eigenstates of the operator F coincide with those of the operator 13F (i3 a complex number), the most general procedure of constructing the minimizing state is as follows. One finds all solutions of the equation F I p) = 0 and chooses those, which satisfy eqs. (5). The states selected in this way are the minimizing ones. It should be noted that the variational technique for determining those normalizable pure states, which give a minimum for the uncertainty product of two noncommuting hermitian operators, has been developedin ref. [12].
77
(2ir?72)~/4ex~~_~ [i
4~ ax +
—
‘
—
(1
—
r
) (9)
—
~(a2 ÷al2)
It is interesting to note that states of the type (9) have already been studied by Kennard [13], who has also proved exactly the relation ~q ~p ~‘h/2. Recently similar states were also considered in ref. [7], but without connection with relation (4). If we set r = 0 and 77 = l/V’~,the states (9) become the usual Glauber coherent states. The analytic properties of the states (9) (expansion of the unity operator, etc.) are just as these of other coherent states. For example, 151
Volume 79A, number 2,3
PHYSICS LETTERS
the expansion of the state I a, r, ~) in powers of a gives us the eigenstates of the number operator: 772,772
+
4772(1
2(l_r2)h/2 ~
[
Xexpl—-----—4772 1
—
(1
—
r2)h/2
(11)
)
r_____
~
=
~
~ = V’~77~ — [2(1 —
r2)I 1/2
(12) i.e. by changing the dimension of the coordinate and by introducing a new generalized momentum. The time-dependent canonical transformation
2)112,
~=~
= —r/(l
—
r
(13)
.
leads to the equation of the oscillator with friction: 2) = 0 (14) Q + 2yQ + Q/(l r Therefore, the states (9) concern the problem of the damped quantum oscillator [7,151. The states (9) have the most clear form in Wigner’s representation: -
-
—
~
be satisfied by any operator. We consider the density main result of the present paper and the meaning of the introduced correlated coherent states toabout mation be as the follows. system, If one one has canno only additional say that inforUpUq
I H~(f).
A direct calculation of the correlation factor Rn in the state (11) leads us to the result ~R 0= r independent of n. It should be pointed out that the system described by the hamiltonian (10) is in fact the usual harmonic oscillator with the equation of motion ~ + = o. The operator (10)is reduced the hamiltonian of the osdillator(Q2 +~2)/2by the to canonical transformation =
(16)
can be a quantum Wigner function, if and only if ~ > 1/4, because this condition, coinciding with (4), leads to the relation ff W2(q, p) dp dq ( 1 being equivalent to the condition Sp(~32)~ 1, which has to
2 77772)_1/4(2nn!)_11
—
W=~/2exp(_2_2+~~~)
with the corresponding eigenvalues n = 0, 1, 2 These normalized eigenfiinctions in coordinate representation are, as usual, expressed by means of Hermite polynomials: (xln, r, s~)=(2
mal distribution to give the dispersion uq = 772, ~ 2)] 1 and the correlation coefficient r. = [4772(1 r We note that the most general normal distribution (here z~= ~ u,~q) —
(10)
r2)
29 September 1980
p)
>)~2/4 (we
restrict our discussion to coordinate and
momentum operators. If one knows the correlation coefficient r = apq/(opuq)1/2 not to be equal to zero (there is additional inftrmation about the system), the lower limit of the dispersion clear to have 2/[4(lproduct r2)].is In other words, athelarger value: UpUq > h quantum fluctuations in a state with r * 0, generally speaking, are larger than in the state with r = 0. This means in reality that the probability of tunnel penetration in a state with r *0 can be much larger than in that with r = 0. We believe, in connection with the above discussion, the following problem to be of interest: to generalize the UR (4) for the case of the higher correlation momentsM an inequality of the form u~UB 1, > h2/4f(M ,M,~,i.e. to obtain 1 Ma). —
...
[1] D. R.J.Stoler, Glauber, Phys. 131 (1963) [2] Phys. Rev.Rev. 1D (1970) 3217.2766. [3] E.Y.C. Lu, Lett. Nuovo Cimento 2 (1971) 1241. [4] l.A. Malkin, VI. Man’ko and D.A. Trifonov, J. Math. Phys. 14 (1973) 576. Phys. Rev. A13 (1976) 2226. [6] V. Canivell and P. Seglar, Phys. Rev. D15 (1977) 1050. [7] A.D. Jannussis, G.N. Brodimas and L.C. Papaloucas, Phys.
[51 H.P. Yuen,
Lett. 71A (1979) 301. [81 E. Schrodinger, Sitzungsber. Preuss. Akad. Wiss. (Berlin, 1930) p. 296
f d~(q + ~ ~Ia, r, 77Xa, r, ~ 277 J —~—
—
~ ~)e~
[9] H.P. Robertson, Phys. Rev. 35 (1930) 667A; 46(1934)
794.
(15) = ~~exp[_2772f2
q
—
2n2(l where ,i~p
152
—
(‘b), c~ q
—
+ —
r2)
2r
(1
—
r2)h/2
(ci). This is the classical nor-
[10] [11] [12] [13] [14]
W. Band and J.L. Park, Found. Phys. 1(1970)133. D. Stoler and S. Newman, Phys. Lett. 38A (1972) 433. R. Jackiw, J. Math. Phys. 9 (1968) 339. E.H. Kennard, Z. Phys. 44 (1927) 326. E.H. Kerner, Can. J. Phys. 36 (1958) 371.
[15] V.V. 550. Dodonov and V.1. Man’ko, Phys. Rev. A20 (1979)