Volume 102A, number 1,2 PHYSICS LETTERS 30 April 1984

GENERAL SOLUTION FOR THE QND OBSERVABLES OF A GIVEN SYSTEM

Robert LYNCH

Physics Department, University o f Petroleum and Minerals, Dhahran, Saudi Arabia and Blackett Laboratory 1, Imperial College, London SI¢7 2BZ, UK

Received 15 November 1983 Revised manuscript received 16 February 1984

The general solution to finding the continuous QND observables of a given system is obtained. The method is also ex- tended to finding the QNDF observables of the system. An example is given, using the harmonic oscillator.

Recently I addressed the problem of obtaining all the continuous "quantum nondemoli t ion" (QND) ob-

servables * 1 of a given system. The solution [2] was obtained via classical canonical transformation theory.

Thus it could only be applied to systems with a classi- cal analog and suffered from factor ordering problems.

Guided by this approach, however, I have now found the general solution to the problem, which turns out to be quite simple in form. Before turning to the detailed argument it may prove useful to re- state the problem and briefly outline the previous ap- proach.

Suppose, then, that for a system with hamiltonian *2 HO( q, p, t) one is to decide whether a given operator A( t ) is a continuous QND observable. From the require- ment that a sequence of measurements of A( t ) (except for the first at time to, involving an unimportant [3] reduction of the state vector) be completely predict- able the condition follows that for all t

A( t ) = f [A( to ) ; t, t0l , (1)

where f is an arbitrary (but see below) real-valued func-

1 Sabbatical leave address 1983/84. ,1 For an excellent review of the subject, see ref. [1]. ,2 In this paper q, p, H0, etc., are operators. Classical quan-

tities carry the subscript "c". Calculations are carried out in the Heisenberg picture. For clarity the solution is given for a system with one degree of freedom; the generaliza- tion to n degrees of freedom is straightforward.

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tion. It is perhaps worth emphasizing that all continu- ous QND observables have this form.

Thus if one can find an A( to ) and a suitable func- tion f s u c h that (1) holds for all t, or prove that none

such exist, one can decide whether A( t ) is continuous QND or not. However such an approach is bound to be tedious.

For screening purposes it is far easier to use a neces-

sary condition which follows directly from (1), that if A( t ) is continuous QND then

[A(t), A(t ')] = 0 for all times t and t ' . (2)

However this is a sufficient condition if and only if A( to ) is not degenerate *3

It might turn out easier still to decide about A( t ) if one could find a method to generate all the continu- ous QND observables of the given system, and to see i fA( t ) is among them. Also, one might hope that such an investigation would reveal something profound about the nature of QND observables. These were the motivations for the work undertaken in ref. [2].

The nature of (1) indicates that such an investiga- tion will involve transformations among operators, at

this time apparently a somewhat impoverished and

*3 ifA(to ) is degenerate a similar relation to (1) can be ob- tained by including in f n further commuting operators B 1, B z ..... B n. A(t) is then a generalized (continuous) QND observable (see ref. [ 1 ], p. 365). The techniques developed above also have applications to generalized QND observables, but these will not be developed here.

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Volume 102A, number 1,2 PHYSICS LETTERS 30 April 1984

difficult subject [4] ,4 . Thus in ref. [2] a less ambi- tious and more limited program was at tempted, to find a solution based on classical canonical transfor- mation theory.

In this reformulation of the problem one seeks the

canonical transformation qc, Pc ~ Ac(qc, Pc, t) such that the condit ion (2), now regarded as a Poisson bracket expression, obtains. One result * s involves an arbitrary function A of the constants of the motion,

which are taken here to be the t = t o values Ofqc, Pc, i.e. qc0, Pc0' It is found that the most general "clas- sical" QND observable has the form

Ac = G[A(qc0, Pc0), t] , (3)

where G is also an arbitrary function. Reflecting further on the result (3) I have now

found the general solution to the problem. Instead of regarding (1) as a condition on continuous QND ob- servables, one regards it as a method of generating all such observables.

Hence allowing A( to) to range through all possible hermitian operators, and f t o range through all real- valued functions (with a restriction, given below), one obtains all possible QND observables. Thus, for exam-

ple, given A(to), the operator A(to) + (sin ¢o(t - to) ) × A2( t0) is QND.

With the benefit of this viewpoint one sees that QND observables are a generalization of constant oper- ators (in the Heisenberg picture), i.e. functions of con- stant operators with c-number time dependences.

At this point one might well stop, regarding the problem as solved. However, there is one practical dif- ficulty: all the operators so generated will be referred to time to, which may not be particularly convenient for calculations at time t.

This difficulty may be circumvented by the follow- ing technique. Solve (by whatever means) the equa- tions of mot ion for q, p

dq/dt = [q, H0] / ih , dp/dt = [p, Ho]/ih . (4a,b)

*4 In ref. [4] a quantum Hamil ton-Jacobi theory is con- structed. However, the theory is developed in terms of eigenvalues, and it is not clear that it encompasses true operator transformations.

*s This result was contributed by W.G. Unruh, acting as a referee for the paper.

The solutions to these equations can always be put in the form

q( t )=q(qo , Po, t), p ( t )=p(qo , Po, t ) , (5a,b)

where qo, PO are the Schr6dinger-picture operators at t = t 0. Invert these equations *6

qo = qo(q, P, t), PO = PO(q, P, t ) . (6a,b)

Plugging q0, P0 into A(to) then gives the desired form of the operator at time t. A more direct ap- proach is to choose any hermitian function A'(q0, P0) of q0 and Po- Using (6) this can be rewritten as a func- tion of q and p

A'(q0, P0) = A"(q, p, t ) . (7)

Choosing another real-valued function G ' , then

A(t) = G ' I A ' ( q 0, P0); t] (8)

is a QND observable. Varying A' and G ' one generates all possible QND observables * 7. Note the resemblance

between the classical result (3) and the quantum re- sult (8).

There is one restriction on the form of G' . From the QND condit ion (1) it is seen that a "hidden" re- quirement on G ' is

A(to) = A'(q0, P0) = G'[A'(qo, P0); to] , (9)

i.e. as t ~ t o the function G ' must reduce to its oper- ator argument. Thus a G ' of the form, say, cos[A'(q 0, po)( t - to)] is not allowed. This condition is related to degeneracy considerations , s and is obviously ab- sent in the classical case (3).

:~6

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#8

The inversion can always be carried out since there exists a unitary operator U (the inverse of the time development operator) such that q = UqoUt, p = UPoUt. I am in- debted to W.G. Unruh for pointing this out. This assumes that every hermitian operator corresponds to an observable, which is somewhat controversial. See the discussion in ref. [5], however. Obviously observ- ables without a classical analog, e.g. spin, will not be gen- erated. These may be handled by separately solving and inverting their time dependence. If G' were to reduce to a c-number at t = t o, for example, as with the cosine function, then every state would be an eigenstate at t o ; later measurements would then demolish a subset of these states. For a detailed discussion of the properties of the cosine operator see ref. [6]. I would like to thank G.J. Milburn for bringing this operator to my attention.

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Volume 102A, number 1,2 PHYSICS LETTERS 30 April 1984

After this somewhat complicated discussion it might be useful to consider an example, the harmonic oscillator, with HO( q, p) = p2 /2m + (mco2 /2 )q 2 . Solv-

ing (4) one finds (taking t O = 0)

q(t) = q0cos cot + (PO/mco)sin cot, (10a)

p(t) = - q o rnco sin cot + P0COS cot. (10b)

Inverting

q0 = q cos cot - (p/rnco)sin cot , (1 l a )

PO = qrnco sin cot + p cos cot . (1 l b )

Arbitrary QND observables can then be composed using (7) and (8).

For example X 1 = q0, X2 = PO/rnco, X2 + X2, ~-(X1X 2 + X 2 X 1)exp(cot), etc., are all QND observ- ables.

The technique can also be used to generate the QNDF observables of a system, although it is not quite as straightforward as the above.

As previously one solves for q(t), p(t), inverts, etc., only now with the hamiltonian

H = H 0 - qF( t ) , (12)

where F(t) is some (usually) unknown c-number func-

tion of time. Unfortunately the resulting q0, P0 gen- erally depend on F(t), which is awkward. With a bit of fiddling one can get around this difficulty, however. This is best seen by means of an example.

Consider, then, the forced harmonic oscillator. Carrying out the indicated prescription one obtains without too much difficulty

t 1

/ F ( t ' ) s i n cot 'dt ' , (13a) qo = X l + ~ " ~ 0

t

PO/mco= X2 -m~ f F(t')coscot'dt' . (13b) 0

As stated q0, for example, is an awkward QNDF observable. However, since q0 and X 1 differ only by a c-number function, they have common eigenfunc- tions, and X t is also QNDF. Alternatively, and more in keeping with the method, one may automatically generate X 1 as a QNDF observable by choosing the arbitrary function G ' to satisfy

t 1

f F(t ')sin cot'dt' . (14) G ' (q0 ; t) = q0 m ~ 0

Similarly X 2 can also be shown to be QNDF. On the other hand, one finds by this sort of argu-

ment that X 2 + X 2 is not a QNDF observable.

The author would like to thank the UPM Research Committee for supporting his sabbatical year during which this research was done, and also Professor T.W.B. Kibble and his group at Imperial College for making the author feel at home during that year.

References

[ 1 ] C.M. Caves, K.S. Thorne, R.W.P. Drever, V.D. Sandberg and M. Zimmerman, Rev. Mod. Phys. 52 (1980) 341.

[2] R. Lynch, Phys. Rev. Lett. 51 (1983) 1405. [3] R. Lynch, Phys. Lett. 87A (1982) 277; 92A (1982) 9. [4] R.A. Leacock and M.J. Padgett, Phys. Rev. Lett. 50

(1983) 3; Phys. Rev. 28D (1983) 2491. [5] W.E. Lamb Jr., Physics Today 22 (1969) 23. [6] G.J. Milburn and D.F. Walls, Phys. Rev. 28A (1983)

2065.

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