Operators of the phase. Fundamentals

ANNALS

OF PHYSICS

209, 479-505

(1991)

Operators

of the Phase. Fundamentals

JANOS BERGOU* AND BERTHOLD-GEORG ENGLERT+

Hungarian

Academy

Central Research Institute for Physics, of Sciences, P.O. Box 49, H-1525 Budapest, Received

August

Hungary

10, 1990

The asymptotic form of the Wigner function to a quantum operator is used to introduce a clear concept of the classical limit that is more than just a figure of speech. Quantum analogs of classical functions of the phase (or amplitude) variable in phase space are then identified. We study natural extrapolation procedures from the classical into the quantum regime, one of which is based upon writing the ladder operators as a product, in which one factor is a unitary quantum analog of the classical phase factor e@. The spectral decompositions of such unitary operators supply complete, orthogonal sets of states, each of which can be associated with a definite phase. 0 1991 Academic Press, Inc.

I. INTRODUCTION Terms like “phase uncertainty,” “phase distribution,” “phase diffusion,” and the like hold a prominent place in quantum optics, although-so we think-their meaning is clear only in the classical regime. A proper definition in the quantum domain is still missing. This is not as simple a matter as one might presume naively, because such notions, though well established in the classical limit, are not easily extrapolated into the realm of the quantum world. 1.a. Ladder Operators It is common knowledge [ 1 ] these days that a unitary polar decomposition of the ladder operators a, at of the (one-mode-) photon field is impossible [2]. The operator A defined by a=dm~A=Afi, (1)

or A=,/m-‘a, * Permanent address: New York, NY 10021. + Permanent address: Germany.

Department Sektion

A+=at,/z-’ of Physics,

Physik,

Universitlt

Hunter

College

Munchen,

(2) of the City Am Coulombwall

University

of New

1, D-8046

York,

Garching,

479 0003-4916/91

$7.50

Copyright Q 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.

480

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AND

ENGLERT

is not unitary, since AA+ = 1,

ATA = 1 - lO)(Ol # 1,

(3)

where In), here for n=O, stands for the nth number state,

ata In) = In) n (4 a=&?

(n+l),

atIn>=In+l>Jn+l,

(4)

with n = 0, 1, 2, ... . If instead of focusing on one degree of freedom, one considers a photon field with infinitely many modes, it is possible to have states beyond the Fock space, such as states with a spatially constant photon density. Then there is no limit to the number of photons that one can take out of the field. Accordingly, normalized ladder operators exist which are truly unitary-and consequently define a phase observable-because they are not embarrassed by a ground state as is the situation, witnessed in (3) met by A and At. For more detail on this development consult Refs. [3,4]. We shall, however, in this paper stick to the one-mode problem. Recently the attempts of Pegg and Barnett [IS, 63 got a lot of publicity [7]. Their recipe amounts to replacing the normalized ladder operators (2) by the cyclic permutators A,=

IN) ei(N+l)‘PyOI

+ ; In- l)(nl, n=I

‘4k=

IO) e-‘(N+‘)yN(

+ 5 In)(n“==l

(5)

11,

where cp,, is an arbitrary constant that is presently irrelevant. In the subspace spanned by In), n = 0, .... N, these are, indeed, unitary-not so in the entire state space. However, in the limit N-, co, one comes essentially back to the Susskind-Glogower [2] operators (2); and as long as N stays finite, the cyclic Pegg-Barnett operators are no operators of the phase at all, if one accepts the criterion formulated in Section 1I.b. The insistence that the limit N+ cc is performed only after everything else is said and done is of no help in our opinion. For, does the injunction, to pick N sufficiently large, depending on the state of the physical system, not signify that the operators (5) themselves are state dependent? For those who, as we do, answer yes, does this not make havoc of the linearity of the “operators” (5)? We shall have more to say about the Pegg-Barnett approach at the end of Section 1II.a. We shall offer our resolution in Section 111.~. It consists of replacing (1) by a = URfa,

at = RfU-‘/&

(6)

OPERATORS

OF THE

PHASE.

481

FUNDAMENTALS

where, in the classical limit, the amplitude R is indistinguishable from ,/?%& and the unitary U approaches the normalized ladder operator A of (2). Of course, the factorization (6) is not unique; there is a choice among various Us and corresponding Rs. Consequently, the hermitian phase operator 4, introduced via U = e+,

(7)

also is not unique. This is as it should be. The royal road from the classical principality to the quanta1 kingdom is a fiction; there are always many routes to choose from. Whatever choice is made, the eigenstates Id’) of U, U 14’) = I@) e”‘, are complete

(c)‘l U=e’“’

(++‘I,

(8)

EI&>(@ =1 s,2n)

and orthogonal (f/Y 1$4”) = 27r S(qY- If’),

(10)

so that we are justified in calling (4’ 1p 14’) a phase probability density for any density operator p. Another generally accepted statement [8] says that a bounded selfadjoint “phase operator” @, canonically conjugate to aTa in the sense that the commutation relation [ata, CD] = i

(11)

holds, does not exist. This is not quite true. It has been shown by Garrison and Wong [9] that (11) can be satisfied on a dense set of the Hilbert space of state vectors. More about this in Section III.a, where we discuss the Garrison-Wong operator in some detail. It turns out, however, that insisting upon (11) is hardly a practical attitude, because approximating even simple physical states by members of said dense set is a very awkward business. (Naturally, our Q of (7) does not obey (11) identically, but (11) becomes essentially true in the classical limit, in the sense that the unitary operator of (6) is such that [U,ata]=U+u

(12)

with a quanta1 correction u that vanishes in the classical limit.) 1.b. Phase Space Geometry The notions of amplitude and phase of a harmonic oscillator are familiar in classical mechanics. One can regard them as polar coordinates r’, cp’ in the phase space, whose Cartesian coordinates are the position q’ and the momentum p’, q’ + ip’ = r’e’v’.

(13)

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(Both q’ and p’ are understood to be dimensionless variables referring to the natural scales associated with a harmonic oscillator. Here and in the sequel primes are used to distinguish numbers from operators symbolized by the same letter.) One way of making contact with these classical concepts is to replace the left-hand side of (13) by the corresponding quantum mechanical expectation value, so that amplitude and phase variables for a quantum oscillator would be introduced by (q)

+ i(p)

= r’f?’

(14)

or with the ladder operator a = (4 + ip)lJi

(15)

(a) =i.lew. a

(16)

more compactly

It is then the result of natural geometrical uncertainties of the rotated coordinates,

considerations,

see Fig. 1, to use the

Q(4) = q cos 40’ + P sin cp’ = @?
(17)

P(cp’)=pcoscp’-qsincp’=i(at(a)-a(at))/J2o(a),

as measures for the spreads 6r in amplitude

and 6~ in phase:

dr=6Q=((Q2)-

(Q>‘)“*, (18)

tan@=$fiP=

FIG. 1. Phase and amplitude, space which has the expectation

(P’)“‘/(Q),

and their uncertainties, as natural values (q), (p) as basic Cartesian

geometrical coordinates.

concepts

in the phase

OPERATORS

OF THE

PHASE.

483

FUNDAMENTALS

where (Q) = r’ and (P) = 0 have entered. An immediate definitions is

consequence of these

an uncertainty relation for amplitude and phase. In terms of expectation values referring to ladder operators, equivalently

Eqs. (18) read

/c-b\’ p) =~[(;,:;;~;‘+~+~]-1,

(tan6q)2=i

(ad+da> 4 [ (at>(a)

-- (d2>

-(a>2

(at)’

1

(20)

’

For example, if the system is in an eigenstate of the ladder operators (a “coherent” state), then (a2> = 2, (ata)



=

= (au?) - 1 = (at)(u)

2Y = irf2,

(21)

with the consequences (22)

and in (19) the equal sign would hold. In (16) and (20) everything is well determined in terms of expectation values of familiar operators, and naturally for (a) = 0 one does, in this approach, not attempt to speak of a phase of the system. No doubt, this direction is worth exploring, but it does have its drawbacks. The principle one is that cp’ = (i/2) log ((at)/(u)) (symbolically) is not itself the expectation value of an one. Indeed, the concept of amplitude and operator, neither is r’ = Jm phase of (16) is of rather poor content. A glimpse at the uncertainty ellipse of Fig. 2 suffices to justify questions like: what is the probability for finding the phase in a given range of values? One partial answer is based on a refinement of the geometrical considerations hinted at in Figs. 1 and 2. This approximation method, developed by Schleich and collaborators [ 10, 111, makes extensive use of areas in phase space and, in particular, their overlaps. Whereas this program has been astoundingly successful, its results are, by construction, reliable only in the semiclassical regime. The final answer to the question just posed can only be given after a quantum analog of (13) is found. Proceeding from (6) one then arrives at (8)-( lo), from where on sailing is straight ahead.

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ENGLERT

each state of a one-dimensional system one can associate an uncerkzinty ellipse in phase space. The coordinates of the center are the expectation values (q) and (p). distance from the center to the point on the circumfence in the direction cp’ is R(q’) = + n/2), where 6Q(rp’) is the uncertainty of the operator Q(cp’) = q cos cp’+ p sin cp’, 26Q,,, SQ,,/SQ(cp’ and 46Q,i,. and SQ,., , SQ,,, are the two extreme values of 6Q((p’). The lengths of the axes are 46Q,,, The area of the ellipse, 4n SQ,,, 8Qmin, is never less then 2n. It equals 2n for minimum uncertainty states. Under linear canonical transformations-translations, rotations, scale changes in phase spacethe uncertainty ellipse is transformed accordingly. FIG.

the The

2.

AND

With

(q), (p)

I.c. Phase Space Functions Another way of making contact with the classical concepts of (13) employs the language of phase space functions to talk about operators. In quantum optics [ 123 it is popular to express the density operator p with the aid of its P-function pp (provided that it exists), (23) Here we encounter the coherent states, the eigenstates of the ladder operators, a la’) = la’) a’,

(at’1 at = at’(at’l,

if

la’) = (a?‘[+

at’ = a’*,

for which we use the normalization (at’

1a”) = exp(at’a”),

(25)

which implies

(at’1 a = &


at la’) = &

la’).

(26)

OPERATORS

OF THE

PHASE.

485

FUNDAMENTALS

In (23) it is understood that a’ = .A- (q’ + ip’), Yh

af’ = -L fi

(q’ - ip’),

(da’) = dq’ dp’

(27)

or a’ = -

y~e”p’,

ati =-

3i

yle-i4+,

(da’) = dr’r’ dcp’,

(28)

3i

reflecting the choice between Cartesian or polar coordinates in phase space. For later reference we remark, incidentally, that in the completeness relation

5

(d0 la’> (at’1 = 1 2n (at’1 a’)

(29)

neither (27) nor the equivalent (28) is mandatory. It is only required that the two independent complex variables a’ and at’ are integrated along orthogonal contours [13]. Thus a’ = q’,

at’ = -ip’,

(da’) = dq’ dp’

(30)

is also a permissible parametrization. Generally speaking, the integration contours must be chosen such that the resulting numerical integral exists. The counter part to the P-function in (23) is the Q-function [ 121, which for any operator F(a(at, a) is given by FQ(ay,d)=

(a~‘lFla’)/(ayla’).

(31)

In view of (25) and (26) it is an everywhere analytical function in both of its complex arguments. Obviously one has Fo(at’, a’) =F(at’, a’) if F(at, a) is normally ordered, :F(at, a): = F(at, a), all ats to the left of all as. Combining (23) with (31) one obtains

(32)

upon using the polar parametrization (28) explicitly. Since both pP and F, are periodic functions of cp’, the cp’ integration covers any interval of length 271. Now, Eq. (32) invites the computation of [14, 151 (33)

486

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ENGLERT

and calling it the “expectation value of the normally ordered :@: operator.” (The specific choice of the range 0 ... 27c is, of course, arbitrary, but some choice is necessary.) The “phase uncertainty” 6~ would then be evaluated according to 1 -+ (6V)2 = 4

(:(p2:)-

(:(p:)2,

The term 1/(4(ata)) is put in by hand to account for the difference between the alleged operators :cp2: and (:(P:)~. Its particular value ensures that (34) agrees with (20) if p projects to a coherent state with large ( (a)[. Why does one have to resort to such measures? Can one not calculate (:(P:)~ and see how it differs from :(p2:? No, because there are no such operators :cp: and :(p2:. In fact, there are no operators at = af*, independent of r’ (except, of all, whose Q-function is, for at’ = r’e -iv’/* course, if the operator is a multiple of the identity). We supply a proof of this assertion in the Appendix. In general terms, the analyticity of the right-hand side in (31) restricts Q-functions rather severely. In particular, the natural attempt [16] :cp:= ; (log at -log

a)

must fail, because log z is not analytic in the entire complex z-plane. There are, of course, Q-functions which are independent of r’ for r’ $1. So we can regard (:cp”:) as a semiclassical approximation to the expectation value of some operator. Which one? When leaving this question consciously unanswered, it is all right to use (33) and the like in the classical regime, where it has led to enlightening insights. Let us only mention the well-known connection between the laser line-width and the time derivative of ( :(p2:), via the coefficient of the (a/acp’)* term in the Fokker-Planck equation obeyed by pp(r’, cp’, t) [ 121. When, however, one becomes interested in applying these concepts in the quantum regime or to highly nonclassical states of the system-such as squeezed states, for which there is no P-function-then it becomes an obligation to answer that question and have well-defined operators associated with such phase and amplitude properties. II. OPERATORS OF THE PHASE, OR AMPLITUDE 1I.a. The Classical Limit In Eq. (32) we have one particular example of evaluating quantum mechanical traces with the aid of phase space functions, here the P-function for one factor of the product Fp, the Q-function for the other one. Since not all operators possess a P-function, one cannot use (32) generally. It is more appropriate to employ Wigner functions, Fw and pw, in terms of which the analog of (32) reads (F)=tr(Fp)=jq

Fdat’,

a’) p&at’,

a’).

OPERATORS

OF THE PHASE.

487

FUNDAMENTALS

We use the normalization of Ref. [ 173, so that an operator F(at, a) is related to its Wigner function Fw(at’, a’) by

and Fw(at’, a’) = tr(F(at, = tr(F(at

a) :2e-2’a+--u+“(o~o’):} + at’, u + a’) 2( - l)“+“}.

(38)

(Incidentally, the latter version implies immediately that the Wigner function pw of a density operator p is restricted by - 2 < pw < 2, in contrast to the Q-function, which obeys 0 < pa d 1 on the diagonal.) Note that in (36) two Wigner functions occur, whereas in (32) one meets one P- and one Q-function. This is just one manifestation of the high symmetry of the Wigner function. Another consequence of the symmetry properties is that the Wigner function provides a perfectly unbiased formulation of operator equations in terms of the phase space functions, as discussed in Ref. [ 171. In fact, when using Wigner functions, one is as close to classical physics as one can possibly be. If the system is in a classical state, pw is essentially nonzero only for a range of uf’, a’ values with (very) large magnitudes. Upon using the polar parametrization (28) in (36)

(F) = Jam dr’r’{c2n) $ Fw(r’,$1 p&r’, cp’),

(39)

one observes that for a classical pw only (very) large values of r’ contribute. Under these circumstances it is permissible to replace Fw(r’, cp’) by its large-r’ asymptotic form. For all operators of interest this is a periodic function of rp’, f(cp’) = f(cp’ + 27r), multiplied by a power of r‘. Therefore, we propose to call clim F = r’“‘( cp’)

(40)

the classical limit of F if ,!Fa r’+“Fw(r’,

cp’) =f($).

(41)

It is of considerable practical importance that one can substitute the Q-function F, for Fw in (41). This is possible because they are related by FJut',

u')

(da”)

= 1 271

FJut",

.")

2e-2'"t'--at")(u'-

0,

(42)

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where-owing to the Gaussian exponential factor--only at”, a” values in the vicinity of at’, a’ contribute, which implies that the asymptotic forms of Fe and Fw do not differ. Usually, the Q-function (31) is more easily computed than the Wigner function (38). The latter can then be obtained by the inverse of (42), (du”)

Fdut’, 4 = 1 F

.,,).

2e2’“t’-“t”““‘~““)Fg(utN,

For most FQ, the standard parametrizations (27), (28) are not applicable here, and (30), for instance, has to be used. The resulting explicit form of (43) is &-“fP”F@f’

f’&t’,+j~

- @“, u’ + f),

(44)

where one should note that the arguments of FQ are not complex conjugates of each other. Let us illustrate this concept of the classical limit of an operator in a few examples. First, consider the normally ordered powers utjuk with i, k = 0, 1, 2, ... . Their Q- and Wigner functions are

.

‘k-jL!k-j’(2ut’u’)

fork > j

1 (45)

~I~~~:lk!!aat’j~~~~j-k,(2ut’u’)lork~jj k

=

ei(k-i)s’

(-2)-jj! (.-2)-k

(r’/JT)k-j~~-j)(r’2)fork~j k! (r-‘/J’?)-”

Lppk’(r”)

fork < j ’

for which ,!hrnm

(,.-j-k(utjuk)Q)

=

)irnm

(r)-J-k(utjuk),)

= 2-(i+kW&(k-j)rp’,

(46)

where the asymptotic form of the generalized Laguerre polynomials, L’*‘(x) g (-x)“/n! n

for

1x1$1

(47)

has been used. Thus we find the classical limit jtk

&(k ~ A P’ = ut’jufk,

(48)

OPERATORS

OF THE

PHASE.

FUNDAMENTALS

489

as could have been anticipated. As an immediate consequence, we note that a product containing j factors of at and k factors of a, in any order, has the same classical limit as atjak, clim(at i1ak1ati2akz . a-fjnakn) = at’ja’k

= c]im(at jak)

(49)

with j=j,+j,+ ... +j,, k=k,+kz+ ... + k,, because the process of normal ordering introduces terms that grow less rapidly than r’J+k as r’ + co. Equation (49) states that the order of operators in a product has no significance in the classical regime. In particular, both ata and (ata + aat)/ = (q2 + p2)/2 = ata + i have the same classical limit. Second, consider the normalized ladder operators of (2). Here we have

= a’

exp( - aT’a’( 1 - P-“))

(50)

and A, = (AL)* =a, jm dyexp(-2at’a’ taWr2/2)). (cosh( y2/2))’ -m J;; In both integrals there is only a contribution AQ Aw

(51)

from y g 0 if at’s’ = r”/2 % 1, so that

g*=eiw’

for

r’B1.

1

Thus, the classical limit is a pure phase factor, clim A = CT@‘,

clim At = e ~ @‘.

(53)

And in analogy to (49) one more generally has clim(AtjlAkl

. ..At/.~k.)=~i(k-j)rp’

(54)

withj=j, + ... +j,, k=k,+ ... +k,. Third, consider the parity operator ( - l)“+“, which is at the heart of the Wigner function, as emphasized in (38). Here we have

(( - l)“+“)w = 27c&at’ + a’) G(i(at’ -a’)) = 7t 6(r’ cos fp’) 6(r’ sin cp’), 595/209/2-I6

= 71d(q’) 6(p’) (55)

490

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which imply clim( - 1 )Ota = 0.

(56)

clim(( - l)n+o)* = clim 1 = 1,

(57)

clim( ( - 1)a+a)z # (clim( - 1)af@)2

(58)

Since, of course,

we observe that and learn that the algebraic properties of the clim operation are different from the familiar ones of the standard lim. Fourth, to disprove the conjecture-suggested by (56), (57)-that lclim F21 2 lclim FI 2, consider the operator F=

l+(-l)~+~+l-(-l)a+n (

ata+

1

a 2

> Jz’

(59)

for which

so that climI;=a-clima jiThis F is, therefore, classically indistinguishable the squared operator F2=

(61)

JiJ.

from CZ/J% On the other hand, for

l+(-l)~+~+l-(-l)ff+a a@+2 ( ata+ 1

a2 > T’

(62)

we have (F2)e

= (2a-f’s’ - 1 + e-2”+‘“‘)/(2at’2)

(F2)w

= e*@‘.

= e2i@(r’2 - 1 + e-“2)/rf2,

(63)

OPERATORS

OF THE

PHASE.

491

FUNDAMENTALS

Thus clim F* = e*@’ = clim A*.

(64)

In the classical regime, F* ?z A*, whereas F 2 a/& Here lclim F*I < lclim FI *, indeed. The classical limit (40) holds surprises in stock and requires extreme care, in particular, when one deals with operators whose Wigner function possesses singularities as in (55), (60), and to a lesser extent (63). 1I.b. Phasors and Amplors With this concept of the classical limit at hand, we now introduce the notion of phasors, short for: operators of the phase; and amplors, short for: operators of the amplitude. An operator F is a phasor if clim F depends only on the phase variable cp’, not on the amplitude variable r’. Likewise, F is an ampler if clim F depends only on r’, not on cp’. For example, the normalized ladder operators of (2) are phasors, and any operator function of ata is an amplor. In Eq. (1) the ladder operators are thus written as products of a phasor and an amplor. Naturally, if clim F is a constant, then F is both a phasor and an amplor; and example is provided by (- l)“+“, see (56). Amplors are rather familiar objects, so we focus our attention on phasors. It is an immediate consequence of their definition that sums of phasors are phasors, too. And since all functions of cp’ are necessarily periodic, climF=f(cp’)=f(cp’+2n)=

fkeik+",

f k=-s

the possible quantum F=

analogs F of the classical f(cp') are all of the form f

fkE(k)=j. 12nl$$f(q')

k=-w

f

E(kle-ikw

k=-m

where the defining property of the phasor basis Eck’, k = 0, f 1, f 2, .... is clim E(k) = &b

(67)

There is, of course, nothing unique about the phasors Eck’, which is to say that to each classical f(cp') there is a plethora of quantum analogs F. For instance, we already have two candidates for Ec2), namely A2 and the F* of (62). What one chooses for the Eck’ will be subject to the appliction in mind. Although not mandatory, it is certainly natural to have EC-k’=

E(k)+

(68)

for all k=O, 1,2 ,..., so that, in particular, E (‘) is hermitian, associated with a real f(cp'). Another desirable property is clim( E (kl)E(kz) . . . Eckn’) = e i(kl+kz+

.-- +k,)cp’

and a hermitian

F is

2

(69)

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which says, as in (49) and (54), that the order of factors in a product does not matter in the classical regime. If (69) is obeyed, then the classical limit of a function g(F) of the phasor F is clim g(F) = g(clim F).

(70)

In the sequel we shall only consider such Efk)s for which (68) and (69) hold. The phase operator CDis, as a matter of definition, such that clim@=cp’-27r

[cp’-cpo 1 ~

27c

’

(71)

where the square brackets in bold face [ ... ] symbolize the integer part of their contents. The constant cpO is arbitrary; it determines the range of q’ in which clim @ = cp’, namely, cp,,< cp’ < ‘p,, + 271. With the identity

we find, according to (65), (66), Q=(~~+~)E(O)+

f

$(e-‘*“nE(*‘-eikU)OE(-k)).

k=l

In view of (68) this is a hermitian

operator, and because of (69) we have

clim ei@= e’q’.

(74)

This implies that all complex numbers of unit modulus are eigenvalues of the unitary exp(i@), so that the spectrum of the hermitian @ consists of the whole range cpo. . . cpo+ 2n. Further, upon recalling that, for any operator F, the Wigner function of the commutator with tits is given by

we observe that clim [ ata, F] = i -

a chm .

h’

F.

(76)

Thus clim[atu,

ECk)] = -ke”@

(77)

OPERATORS

OF THE

PHASE.

493

FUNDAMENTALS

and clim j [ata, @] = l-271

f k=

6(cp’-cp,-2nk),

(78)

-cc

quite becoming for the phase operator.

III. PHASOR BASES 1II.a. Ladder Phasor Basis We have already met one set of phasors that can be used as ECk’s, the powers the normalized ladder operators of (2). Indeed,

3

for

k>O,

for

k = 0,

for

k
of

(79)

satisfies all the criteria (67), (68), and (69). This is a very natural choice for the phasor basis ECk’ and deserves detailed consideration. The eigenstates of A and At, A IA’) = IA’) A’,

(A”1 if

(A+‘1 = (A’)’

At = At’(At’l, (80)

At’=A’*

are related to the eigenstates of a and at, see (24) and (25), by IA’) = dm

la’)

(At’1 = (at’1 ,/m Consequently,

with

a’ = A’,

with

at’ = At’.

(81)

we have (1 -At’A”)-I,

(At’IA”)=

;+n

f k=

i

b(a-2nk)+;cot(cx,2),

if

IA+‘A”I

if

&‘A”

if

IAt’A”I

< 1, = e’*,

(82)

-cc

00,

> 1,

which restricts A’ and At’ to complex numbers whose modulus does not exceed unity. The periphery of this range ([A’[ = 1, IAt’1 = 1) is of particular interest. Upon introducing 19’) = (A’ = e’q’),

(cp’l = Iqp’)t=

(Ai’=e-i’P’I,

(83)

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one easily verifies that

for all j, k =O, 1, 2, ... . Therefore the quantum analog F to the classical f(cp’) produced by the phasor basis (79) with the aid of (66) is

very reminiscent of the P-function expansion (23). Since the number-state matrix elements (n )FI m ) depend on the difference n - m only, not on n and m individually, (86)

these Fs are the Toeplitz

operators. The corresponding

phase operator @ is then

=

(87)

This is the phase operator of Garrison and Wong [9] (except for a change in sign and the particular choice cpO= - rc has not been made). To evaluate its commutator with ata we employ

(4 ata = i$

(cp’l,

ata Iv’) = -i$

1~‘)

(88)

with the consequence

to find

f Cata, f’l = jc2nj$14) for any F of the form (85). The Garrison-Wong

y

<@I

(90)

phase operator (87) thus obeys (91)

One the dense set of states I ), characterized by (cp’ = cpOI ) = (At’ = e-‘q” I ) = 0, Eq. (11) is obeyed, indeed. Unfortunately, however, when approximating even

OPERATORS

OF THE

PHASE.

495

FUNDAMENTALS

simple physical states in this dense set, one recognizes rather undesirable properties. Consider, for instance, a number state In), whose At’ wave function is (A+‘In)=A+‘“.

(92)

To approximate In) by (n; E) (with, say, 0 < E< 1) from said dense set, we need to find wave functions @,;,(A +‘) = (A +’ 1n; E), analytic in At’ for IAt’l < 1, such that, for all values of the parameter E, for

$n;e(A+‘) = 0

A+’ = epiqO, (93)

and as ~-1. One possibility

(94)

is (95)

for which 4k:e(A+‘) =

1 +s 1 -A+‘eirpo A+‘“, - 2

d- 1- &tfeivo

so that (93) and (94) hold by construction.


(96)

Further, we find l+&l+(l-22E)P

1- E2eia ’

(97)

which equals unity in the limit E --, 1. However, this does not happen for each order of CIindividually, since f$ (

k (,;Ele’a(~+o-n)(,;&)lol=o >

=(n;&I(ata-n)kIn;&) 1,

for

k = 0,

!-$-f(l-~)~(-&~~)~~‘(&)~,

for

k=l,2,...,

(98)

supplying, for k = 1 and k = 2, (n;EI(ata--)(n;&)=---

1 2(1+&)

1 4

(99)

496

BERGOU

AND

ENGLERT

and

1 +E2 2(1+&)2(1-4+03.

(n;&I(a~a-n)*(n;E)=

(100)

Whereas it is possible, for a In; E) different from (95), to achieve a null limit in (99), the limit in (100) will never be finite. Thus, when number states are approximated in said dense set of state vectors, the better the approximation is, the larger the spread of ata. Can one live with that? Hardly. In the Pegg-Barnett approach [5] that proceeds from (5) one eventually arrives at an alleged phase operator sN which represents a twofold approximation to the Garrison-Wong operator, inasmuch as the integral of (87) is not only replaced by a sum,

(101) but also the 1~‘) are projected to the subspace spanned by IO), II),

.... IN),

14;) = 5 In) e’“cn, fl=O

(102)

2TI 4h=cPo+N+mm’

m = 0, 1, .... N.

These Iqi) are the eigenstates of A, [see Eq. (5)], with eigenvalues ei@h. In addition, all the number states IN+ 1 ), IN+ 2), ... are eigenstates of A, with the common eigenvalue zero. Thus the spectrum of AN is distinctly different from that of A which, as we recall, consists of all A’ with IA’1 $ 1. Nevertheless, in the limit N --) co the Pegg-Barnett $‘N approaches the Garrison-Wong @. This limit is a delicate matter. Consider the unitary operator exp(icr&,,) for which

as is demonstrated by evaluating the (nl, In’) matrix elements on both sides. Note that the right-hand side here is not a unitary operator, save for CI= 0. In particular, we have for a= fl, exp(isN) = A,+

z

In)(nl

+ A,

n=N+l

exp(-i$N)=Af,+

2

(104) In)(nl

+A+,

n=N+l

and the terms of order GLand CI* supply lim N-CC

*N = @,

li?,

s,‘,= jqo+2n$ rpo

Icp’) cp
#G2.

(105)

OPERATORS

OF THE

PHASE.

497

FUNDAMENTALS

The latter illustrates the general observation that, as a rule, lim,, cof($,,,) # f(limN - ocsN) = f(Q). We are here facing the obvious fact that the integral version of (87) is not the spectral decomposition of the Garrison-Wong @. Therefore, the quantity (cp’ 1p 1cp’) cannot be identified with the phase probability density of the system specified by the density operator p. Indeed, since (cp’ 19”) # 0 for cp’ - 9” # 0, f27c, +47c, .. . [see (82)] such an interpretation is unjustified a priori. This, we think, constitutes a very serious objection to the Pegg-Barnett procedure, because after the limit N+ cc is eventually performed, the resulting “continuous phase probability distribution” [6] is precisely (cp’ ( p 1cp’ ). Now, as soon as (106)

is accepted as the expectation value of the phase, one is dealing with the Garrison-Wong operator (87). Then the spread &D in phase is determined by (cm)*=

(CD’) - (CD)*,

(107)

where (@*) is not obtained by integrating (p”(‘p’I p I cp’). For example, if p projectstoanumberstate,wehave(cp’~p~cp’)=~(n~cp’)~*=l,sothat (@)=cp0+7c. Whereas this is anticipated, the standard preconception that in a number state all phases are equally probable, which feeds the expectation S@ = x/d, agrees with the actual situation only for sufficiently large values of n. Indeed, we find (108)

so that 6@ is short of the naive z/& by 29, 6.2, 3.4, 2.4, ... % for n = 0, 2, 4, 6, .... respectively. The true phase probability density (CD’ 1p I @’ ) can be computed by employing the At wave function of the eigenstate I@‘) to the Garrison-Wong operator @. It is given by [9] (A+‘1 @‘) = f

A+‘“(nl@‘)

fl=O

w+*a d$1+eisA+’ 9-Q’ I I> G 1 -p/j+’

log -

?I

’

(109)

where cpod @’ < cpo+ 271 is understood. The (n I @‘) can be found recursively by expanding (A” I@’ ) in powers of A “. A plot showing l(nl @‘)I2 for n=O, 1,4, 5 is contained in Ref. [9]. 595/209/2-I7

498

BERGOU

AND

ENGLERT

1II.b. Wigner Phasor Basis Another natural phasor basis ECk) consists of the operators whose Wigner functions are eikrp’, not just in the limit of (41), but for all r’( >O). With (28) in (37) these are

00 dV’ E(k) = -2ata s drlrle-‘,= s -G eikrp’ .ze

exp(JZ r’(ate@

0

For k > 0, explicit equivalent Eck)= Ecpk)’ =

+ ae-‘“‘)):.

(110)

(2n)

expressions, obtained by evaluating the integrals, are

:e-ata(ata)(1-k)‘2

(Zckp1,,2(ata)+Zck+1,,2(a‘fa)):

ak

(111)

([e.y-k)! =( [a’“:‘]

(112)

I k;l)Jfir

The modified Bessel functions in the normally ordered version (111) do not lend themselves to further simplification; the afa, a-ordered expression (112) involves the integer part of (ata + 1)/2, so that it distinguishes between odd and even values of ata, E’k’

= EC-M+

=

(k > 0), which-not

1

ata-

ata , (3 (jzk >

,

( >

1 + ( - l)“+” (.i.:k2

1 - ( - 1)Ota 2 + ‘, ata+k 2 ! 2 (

2

.

,

> .!

(113)

unexpectedly-reproduces akf(ata)

= f(ata

(62) for k = 2. The identity + k) ak,

(114)

valid for any operator function of ata, enables us to present (113) compactly: E’k’ = EC-k)+ = BakB-1 = (EW)V for k = 0, 2, 4, ... (115) and E(k)=E(-kl+=Bak~-l=(E(2))(k-~N2E(1) for k= 1, 3, 5, .... (116) wherein B(ata), = 2ata’2 ([qq-;)!, B(afa)=

2a+J2

[ 1 ata 2

!.

(117)

OPERATORS

OF THE

PHASE.

499

FUNDAMENTALS

Rather unfortunately, one cannot use (115) for odd k values as well because, for example, the k = 1 operator BuB-’ is the F of (59), about which we know that it is not a phasor. The phasor basis (110) obeys (67) and (68) as a matter of definition, and (69) also holds. We leave its verification to the reader. The definition (110) immediately implies that here the quantum analog F to the classicalf(cp’), produced with the aid of (66), is the operator whose Wigner function equalsf(cp’),

F= ~J-2”t”

,g R

f(cp’) exp($

J(ate@’ + ae-‘“‘)):.

(118)

The method of associating with a classical function f(at’, a’) that operator whose Wigner function equals f(at’, a’) is a quantization procedure that is both well known, powerful, and-because of the symmetry properties of the Wigner function--also highly justified. Equation (118) is an application of this method. Unfortunately, the phasor basis (110) is rather impractical because these EC” have intransparent commutation relations. In contrast to the set (79), the ECk) of (110) do not have common eigenstates for one sign of k. Of course, for k > 0 they do possess right eigenstates and for k < 0, left ones. For example, it is easily verified that lo,> =exp (2jd2)

IO>, (119)

are eigenstates of EC” (and therefore also of EC4), EC6), E@), ...) with eigenvalue p. For 1~1< 1 they are normalizable, (o,~o,~)=(1-~*$)“2, (1,11.S>=(1-~*/L’)-3’2,

(120)

(0,l l,.)=O. (Incidentally, the 10,) are called “squeezed vacuum” states in quantum optics.) We found these eigenstates of little use, mainly because nothing analogous to (84) exists here. Indeed, even the completeness relation for the states (119) is a surprisingly complicated expression. 111.~. Unitary Phusor Buses Proceeding from any phasor basis with the properties (68) and (69) we can construct a corresponding hermitian phase operator (73). Then Eck’ = exp(ik@)

(121)

500

BERGOU

AND

ENGLERT

defines another phasor basis which, in general, is different from the one chosen initially. For this new phasor basis (121), the quantum analog F to the classical f(cp’) is then simply F= f(Q) because the sum in (66) is here ~=~~exp(ik(a--p))=2n

f k=

6(@-cp’-271k).

(122)

-00

Further, the spectral decomposition of @ supplies a complete set of orthogonal states, each of which can be justifiably associated with a definite phase. No doubt, unitary phasor bases like (121) are potentially very useful. Since exponentiating @s constructed according to (73) is not done easily, it is advantageous to regard the unitary exp(iQi) as the fundamental quantity. We shall, therefore, now consider phasor bases of the form E’k’ = ,yk

(123)

with uu+=

u+u=

1

(124)

and, of course, clim U = e@‘.

(125)

Conditions (68) and (69) are obeyed automatically. As mentioned in the Introduction, the eigenstates of U-for which we shall use the notational conventions of Eqs. (8t(lO)-are not only complete but also orthogonal, in marked contrast to the sets of states (83) and (119). The operators @ of (73) and 4 of (7) are here related by (126)

if one chooses qpoG 4’ < cp,,+ 27~ for the range of eigenvalues of 4, then @ = 4. In any event, exp(i@) equals U and does not define, via (121) a new phasor basis. To each unitary phasor U, there are corresponding amplors R, Rt, R=$

U+a,

R+=&~U

(127)

with clim R = clim Rt = r’ that allow factorizations

(128)

of the ladder operators a, at and the number operator

ata: .=i

UR, Jz

+&+U+

ata = k RtR. fi

’

(129)

OPERATORS

Obviously,

OF THE

PHASE.

501

FUNDAMENTALS

these are the quantum analogs of the classical factorizations 1 .I climu=~Pr’, climai=-$r’eP”‘, climuta=kr”.

(130)

In contrast to (l), where a, at are written as products of a hermitian amplor and non-unitary phasors, we have non-hermitian amplors and unitary phasors in (129). The advantage of (129) over (1) is that the spectral decomposition of U immediately supplies states with a definite phase, whereas the route that starts from (1) leads to the phase operator (87) whose eigenstates are then to be found. Let us now study in some detail a class of unitary phasors of the structure u== f(+z)

A- f

A+kgk(apz).

(131)

k=O

The requirements f(n) -+ 1,

g,(n I+ 0,

as n-co,

ensure that (125) is obeyed. Upon evaluating number-state (124) we find that this U is indeed unitary, provided that (i)

klfo Igk(o)12= 1;

(ii)

l.m)12+

f

Ig,(n+

(132) matrix

elements of

1)12= 1;

k=O n

(iii)

If(n)

1

Igdn--k)12=1;

(133)

k=O

k=O

tv)

fb)

g:(n-m)=

for n = 0, 1, 2, . .. in (ii)-(v), the reader to verify that

-f k=O

gkb+

l)

&?+m+I(n--m);

m = 1, 2, 3, ... in (iv), and m = 0, 1, .... n in (v). We invite

m=~,

(134)

.eJn)= /G

is a solution (one of many, of course) of (132) and ( 133). By rotations in phase space, generated by atu, we can produce different, though equivalent, unitary phasors U,, closely related to the one of (131): u,

=

eiaeiaatoUe

-irate

(135) = f(atu)

A- f k=O

Atkgk(utu)

eick+ IJa.

502

BERGOU

AND

ENGLERT

The quanta1 correction u of Eq. (12) is then available through differentiation, .=i;

Uala=o=

f

(k+ l)Atkg,(.t,),

k=O

for which clim u = 0

(137)

is true, indeed. Before turning to the eigenstates of U, we mention-for the sake of completeness-that the amplor R that comes with the phasor (131) is explicitly given by R=,/%&

f*(ata-l)-

F

&(@&,h(‘++k+

(138)

l)Atk+‘,

k=O

where f ( - 1) = 0 is understood. The classical limit (128) is easily read off. The number-state amplitudes (n I@) of the eigenstates 14’) of U obey the recursion f(n)(n

+ 1 I&) = @‘(n 14’) t

(139)

which is a consequence of (8). If f(n) =O, then (i), (ii), and (iii) of (133) imply g,(m)=0 if On, so that U does not couple the subspace spanned by IO), 11), .... In) to the rest of the Hilbert space. In this situation, the spectrum of U contains discrete eigenvalues in addition to the continuum ei4’ (with any interval of 2n for 4’); the corresponding eigenstates, n + 1 in number, span that subspace. In view of (132), f(n) = 0 can at most occur for a finite number of n values, and since the continuous part of the spectrum is all that we are interested in, we shall dispose of this possibility for ‘good and assume f(n) #O for all n=O, 1,2, ... . Equation (139) tells us that (n 14’) equals (0 14’) times an nth degree polynomial in e”‘, (njqY)=

(01qY)

i

(140)

ci,,,eim6’.

t?l=O

Starting from Q~=

1, the coefficients CC,,, are recursively determined a

&m-l++

t k=m

&-k(k)ak,m

? >

by (141)

OPERATORS

OF THE

PHASE.

503

FUNDAMENTALS

where it is understood that CC,,, = 0 unless 0
ci n,n =

[

n k=O

we obtain

1

f(k)

‘7

n-1 u n.n-

1=

G.,

an,,-2

=

%,,

,r,

(142)

go(k)3

n-1

n-2

I--L

]Tl

go(j)

c k=O

go(k)

+

un.n

jFo

m

g1(A,

and so on. Thus the (n 14’) can be computed successively, as soon as (OlqY) is known. It is permissible to choose (0 14’) positive because only I(0 1#‘)I 2 matters. Now (143)

so that I(O(fj’)12=

f

eCik4’(Ol UklO)

(144)

k=--41

with, for instance, (o~u”~o)=(o~o)=l, (01 UlO)=

(01 u-‘lo)*=

-g,(O),

(01 WO)= (01 ~-21~>*=~~o~~~~2-~f(~~ Sl(O), ~~I~31~~=~~I~-31~~*=-~~~~~~~3+~f~~~~o~~~g1~~~ +f(O) go(l) s1(O)-f(O)f(l) g2(0)

(145)

and so forth. More generally, one can check that the recursion (139) is consistent with s(2A,g

(nlq5’)

ei’4’(f$‘lm)=

(nl U’(m)

(146)

for all integers I and all non-negative integers n, m. After inserting both (140) and (144) into (147)

we obtain (01 Un+l(0)=

(01 U-n-1l0)* =-

1 Ii@ %+ I,n+ I m=O

n+l,m>

(148)

504

BERGOU AND ENGLERT

which supplements (141). So one can compute both the coefficients CL,,, and the vacuum expectation values (0 ) U” )0) recursively, thereby determining all (n 14’) with the desired accuracy. More about this and other applications that require numerical work on another occasion. APPENDIX

Suppose one knew about an operator F that its Q-function depend on at’s’ for at’ = a’*. In other words, (at’1 Flu’) =e”*‘~(@)

for

FQ(at’, a’) does not

a-/-’= a’* = r’epi~‘/J5

holds for all r’> 0. The Fourier coefficients of the periodic f(cp’ + 271) can be evaluated according to

(149)

function f(cp’) =

which must be true for all j, k = 0, 1, 2, ... and all r’ > 0. The identity

(for r’ = fi this is (84)) enables us to express the trace in (150) in terms of the number-state matrix elements of F,

=e

-r’2~2(rr/~)lj-kl

nt,

(nlFln+j-k)(r’2/2)“/Jn!

(n+

for

jak,

for

k>j.

j-k)!

(152)

X nzo

The right-hand

(n-j+klFln)(r'2/2)"lJn!

(n-j+ky

side must not depend on r’, which implies (153)

for allj,k,n=0,1,2 ,.... Consequently, f(q’) multiple of the identity.

is a constant, and F is a numerical

505

OPERATORS OF THE PHASE. FUNDAMENTALS ACKNOWLEDGMENT

BGE expresses his gratitude for the kind hospitality experienced at the Central Research Institute for Physics in Budapest. Note added

in proo$

A generalization of (110) is

which agrees with (110) for I = 2. For 1.= 1, one has ELk) = elka’ and the limit I + co would correspond to Elki = eik+” if there were such a thing (but there is not, as stated in the Appendix). The spread 6@ of thz related phase operator @, constructed according to (73) in the n =0 number state, is given by (6@)‘=iarcsin(i)

[rr+arcsin(S)]

for 1< 2 and is infinite for I z 2. Thus @ is unbounded for I > 2 which indicates that condition (69) is violated. For 1,= 2 (Wigner phasor basis) the spread is An, larger than the naive rt/,,/?. And, by choosing a A value sufliciently small, 6@ can be made as small as one likes-quite contrary to the notion that number states do not have a preferred phase. [There is, of course, no physical preference, because the parameter qpois perfectly arbitrary and so is (0) = ‘pa + a.]

REFERENCES 1. P. CARRUTHERSAND M. M. NIETO, Rev. Mod. Phys. 40 (1968), 411 summarize the common knowledge of 20 years ago. 2. L. SUSSKINDAND J. GLCKXWER, Physics 1 (1964) 49. 3. F. ROCCA AND M. SIRUGUE,Commun. Math. Phys. 34 (1973) 111. 4. R. HONEGGERAND A. RIECKERS,Publ. Res. Inst. Math. Sri. 26 (1990) 391. 5. D. T. PEGG AND S. M. BARNETT, Europhys. Letf. 6 (1988). 483. 6. D. T. PEGG AND S. M. BARNETT, Phys. Rev. A 39 (1989) 1665. 7. See, for example, R. LOUDON, Phys. World 2 (1989), 16; or A. SCHENZLE,Phys. BI. 45 (1989). 84. 8. W. H. LOUISELL, Phys. Left. 7 (1963). 60. 9. J. C. GARRISONAND J. WONG, J. Math. Phys. 11 (1970). 2242; see also A. GALINDO, Letl. Math. Phys. 8 (1984), 495; 9 (1985). 263. 10. W. SCHLEICH,“Interference in Phase Space,” Ludwig-Maximilian-Universitat, Munchen, 1988. 11. W. SCHLEICH, R. J. HOROWICZ, AND S. VARRO, Phys. Rev. A 40 (1989) 7405, deal with the Pegg-Barnett “phase probability distribution.” This paper contains many references of historical relevance. 12. M. SARGENTIII, M. 0. SCULLY, AND W. E. LAMB, JR., “Laser Physics,” Addison-Wesley, Reading, MA, 1974 is a standard reference. 13. J. SCHWINGER,“Quantum Kinematics and Dynamics,” p. 141, Benjamin, New York, 1970. 14. M. 0. SCULLY, K. WI~DKIEWICZ, M. S. ZUBAIRY, J. BERGOU, NING Lu, AND J. MEYER-TER-VEHN, Phys. Rev. Left.

60 (1988),

1832.

15. J. BERGOU, M. ORSZAG. M. 0. SCULLY. AND K. W~DKIEWICZ, 16. Ref. [12], p. 337. 17. B.-G. ENGLERT, J. Phys. A: Math. Gen. 22 (1989), 625.

Phys. Rev. A 39 (1989).

5136.