Probability distributions for the phase difference A. Luis and L. L. Sa´nchez-Soto Departamento de Optica, Facultad de Ciencias Fı´sicas, Universidad Complutense, 28040 Madrid, Spain ~Received 30 November 1994; revised manuscript received 21 June 1995! In this work we analyze the quantum phase properties of pairs of electromagnetic field modes. Since phases differing by 2p are physically indistinguishable, we propose a general procedure to obtain the correct mod(2p) probability distributions for the phase difference. This allows us to investigate the properties of a number of phase approaches. This procedure provides deeper insight into the quantum nature of the phase difference. We relate this problem to the representation of nonbijective canonical transformations in quantum mechanics. PACS number~s!: 42.50.Dv, 03.65.2w I. INTRODUCTION The problem of quantum phase fluctuations of optical fields has a long history and has provoked many discussions @1#. There have been many attempts to properly introduce a satisfactory description of phase and significant progress has been achieved in the last fews years in clarifying the status of the quantum phase operator~for latest reviews see Refs. @2,3#!. Most of this previous work has been devoted to the prop- erties of the phase for a single-mode field or, equivalently, for a single-harmonic oscillator. The more relevant conclu- sion is that there is no such phase operator, at least verifying simultaneously a polar decomposition, self-adjointness, and adequate commutation relations. This has allowed the intro- duction of several approaches depending on which of those criteria should be fulfilled@4#. Although the definition of the absolute phase is, alone, an interesting problem, from a practical point of view an abso- lute phase has no meaning. Since in real measurements we are always forced to deal with the difference with respect to a reference phase, the phase difference should be the funda- mental quantity in describing the optical phase. It is worth emphasizing that the absence of a proper phase operator for a single mode is usually ascribed to the semiboundedness of the eigenvalue spectrum of the number operator. However, the conjugate variable to the phase difference is the number difference, that is not bounded from below. So, it is reason- able to expect that the phase difference will be free of the problems arising in the one-mode case. Taking this into account, two different ways to describe the phase difference emerge. One focuses on the phase- difference variable itself, without any previous assumption about the absolute phases. We have pursued this issue in previous works@5–7#, showing that the polar decomposition corresponding to the phase difference has a unitary solution, in contrast with the polar decomposition for the absolute phase. In fact, the solution has interesting commutation rela- tions with the number difference@8#. The other way to proceed is to describe the phase differ- ence in terms of previously introduced phase operators for the two systems. This approach is faced with an interesting difficulty. Due to its periodic character, adding and subtract- ing phases must be done carefully. The eigenvalue spectra of the sum and difference operators have widths of 4p, and this is not compatible with the idea that the phase must be 2p periodic. Thus there should be a way to cast the phase sum and difference into the 2p range. Such a casting procedure was proposed by Barnett and Pegg@9#. We should emphasize that although the 4p and 2p prob- ability distributions are both valid, they give different values for the variances. The former explicitly reveals the existence of correlations between single-mode phases, while the latter is easier to interpret because in it the phase sum or difference is a single-valued variable. Our aim here is to obtain a casting procedure more suited for a clear analysis of its implications and a comparison with the phase-difference operator arising directly from the polar decomposition. The plan of this paper is as follows. In Sec. II we first consider the enlightening subject of the angle difference. In this case we have the same kind of problems linked to the periodicity, but with the advantage of having an angle opera- tor. This fact makes the special behavior of the transforma- tion to the angle sum and difference more transparent and allows an easier translation to the very similar phase prob- lem. It also shows the nonbijective character of this transfor- mation and the particularities that this fact introduces. In Sec. III we apply these conclusions to the Pegg-Barnett formalism and in Sec. IV to theQ function. The conse- quences of the casting procedure are then discussed and compared. II. ANGLE SUM AND DIFFERENCE We begin our discussion with a brief description of the problem of the angle difference. For a system described by an angular momentum componentLz ~like a particle con- strained to move on a circle or a one-dimensional system obeying periodic conditions! the exponential of the angle E5eiw is given by the lowering operator@10# Eum&5um21&, ~2.1! whereum& are the eigenvectors ofLz , the integerm running from 2` to 1`. This unitary operatorE verifies the following commuta- tion relation withLz @11#: PHYSICAL REVIEW A JANUARY 1996VOLUME 53, NUMBER 1 531050-2947/96/53~1!/495~7!/$06.00 495 © 1996 The American Physical Society @E,Lz#5E, ~2.2! and its eigenvectors are uw&5 1 A2p ( m52` 1` eimwum&, ~2.3! with Euw&5eiwuw&. This operatorE, like Lz , is itself a com- plete set of commuting operators for the system. When we have two such systems, labeled 1 and 2, the exponentials of the angle sumE1 and angle differenceE2 are E15E1E2 , E25E1E2 † . ~2.4! These unitary operators verify the commutation relations FE2 , L1z1L2z 2 G50, FE1 , L1z1L2z 2 G5E1 , FE2 , L1z2L2z 2 G5E2 , FE1 , L1z2L2z 2 G50, ~2.5! showing that their canonically conjugate variables are the angular momentum sum and difference. The eigenvectors of E1 and E2 are of the form uw1 ,w2& with eigenvalues eiw15ei (w11w2) andeiw25ei (w12w2), respectively. Note that while (E1 ,E2), (L1z ,L2z), or (L1z1L2z ,L1z2L2z) are complete sets of commuting opera- tors, this is not true for (E1 ,E2), since uw1 ,w2& and uw11p,w21p& have the same angle sum and difference. Therefore another commuting operator must be consid- ered to describe the system. We propose to use even and odd combinations ofuw1 ,w2& and uw11p,w21p& to solve the degeneracy. So, we can take uw1 ,w2 ,p&5 e2 ipw1 2 @ uw1 ,w2&1~21!puw11p,w21p&], ~2.6! with p50,1; and w15 w11w2 2 , w25 w12w2 2 , ~2.7! and define an operatorP, Puw1 ,w2 ,p&5puw1 ,w2 ,p&. ~2.8! This operator, together withE1 andE2 , gives a complete set of commuting operators. The associated basis is~2.6!, allowing the resolution of the identity I5( p E E dw1dw2uw1 ,w2 ,p&^w1 ,w2 ,pu, ~2.9! wherew1 andw2 run over 2p intervals. To obtain the probability distribution functionP cast into a 2p range for the angle sum and difference associated with a system stater we must add the contributions from eachp value, P ~w1 ,w2!5 ( p50,1 ^w1 ,w2 ,puruw1 ,w2 ,p&. ~2.10! Taking into account ~2.6! and ~2.7! we can express P (w1 ,w2) in terms of the probability distribution for the individual anglesP(w1 ,w2)5^w1 ,w2uruw1 ,w2& in the form P ~w1 ,w2!5 1 2 FPS w11w2 2 , w12w2 2 D 1PS w11w2 2 1p, w12w2 2 1p D G . ~2.11! We see that the probability distribution for the angle sum and difference cannot be obtained from the one associated with the individual angles simply by the corresponding trans- formation of the variables~2.7!. This is because the same sum and difference can be obtained from two different values for the angles of each system such that the transformation becomes nonbijective. The true transformation is obtained only after adding these two contributions. In the Appendix we study this problem in the context of canonical transfor- mations. III. PEGG-BARNETT FORMALISM FOR THE PHASE DIFFERENCE We now turn to our problem of the phase difference be- tween two systems like two harmonic oscillators or a two- mode quantum field. As discussed in the Introduction, one possible way to de- scribe this variable is using a previous definition of the ab- solute phase for each system. Contrary toLz , now the spec- trum of the number operatorN is bounded from below. This precludes the unitarity of the operator exponential of the phaseE arising from the polar decomposition of the annihi- lation operator a5EAN, ~3.1! adding difficulties to its interpretation in describing the phase in quantum optics. Despite this fact, the eigenvectors ofE with unit-modulus eigenvalueseif, uf&5 1 A2p ( n50 ` einfun&, ~3.2! known as Susskind-Glogower phase states, are considered to have a well-defined phase, and are the starting point for some approaches to the problem@12,13#. One of them, the Pegg-Barnett approach@14#, is perhaps the most widely used in the last few years. It is equivalent to the prescription of a phase probability distribution for a sys- tem stater in terms of the Susskind-Glogower phase states ~3.2! P~f!5^furuf&. ~3.3! When we have a two-mode field the joint probability distri- bution is given by 496 53A. LUIS AND L. L. SÁNCHEZ-SOTO P~f1 ,f2!5^f1 ,f2uruf1 ,f2&, ~3.4! where uf1 ,f2& are two-mode Susskind-Glogower phase states. Since we are mainly interested in the phase difference, we may ask first for the corresponding probability distribu- tion P (f1 ,f2) associated with the phase-sum and phase- difference variables. Finally we integrate over the phase sum. At this point the reader is referred to the previous and careful analysis of Barnett and Pegg@9#. However, we think that the Barnett-Pegg procedure is perhaps obscure in the sense that the nonbijective character of the transformation is not explicit and must be supplied with a careful handling of the range variation of all the variables. Moreover, the general relation betweenP (f1 ,f2) and P(f1 ,f2) is not easily obtained. As is clear from the preceding section, the transformation to the phase sum and difference f15f11f2 , f25f12f2 , ~3.5! or equivalently f15 f11f2 2 , f25 f12f2 2 , ~3.6! is nonbijective because we are demandingf1 andf2 to be 2p periodic and not 4p periodic, as seems necessary from the previous equations. ThereforeP (f1 ,f2) cannot be ob- tained fromP(f1 ,f2) simply by means of the change of variables~3.6!. The proper form for the transformation is obtained by adding the probabilities corresponding to (f1 ,f2) and (f11p,f21p) which give the same phase sum and difference, getting@taking into account the Jacobean of ~3.6!# P ~f1 ,f2!5 1 2 FPS f11f2 2 , f12f2 2 D 1PS f11f2 2 1p, f12f2 2 1p D G , ~3.7! which is 2p periodic. Before going on, some remarks seem in order. First, P (f1 ,f2) contains less information thanP(f1 ,f2). For example, in generalP(f1) cannot be obtained from P (f1 ,f2). Contrary to the angle variable, here we do not have phase operators from which~3.7! could be directly de- rived as in~2.4!. However, the analysis of Sec. II supports this transformation law. In fact, one way to overcome the difficulties caused by the semiboundedness of the number operator is, precisely, enlarging the Hilbert space so as to include fictitious negative number states@15,16#. Physical results are recovered only when we are restricted to states having null projection over them. With this extension we are then formally in the same situation considered in Sec. II and therefore we arrive at~3.7! in the same way. If we were only interested in the calculation of mean val- ues ~such aseikf1eilf2, with k and l integers! all of this analysis would not be necessary, since they can be obtained simply as ^eikf1eilf2&5E E df1df2e ik~f11f2!eil ~f12f2!P~f1 ,f2!. ~3.8! Equation ~3.8! can also be used to obtainP (f1 ,f2) by noting that we must get the same mean values for any peri- odic function of the phase sum and difference whether we use the variables (f1 ,f2) or (f1 ,f2); i.e., E E df1df2e ikf1eilf2P ~f1 ,f2! 5E E df1df2e ik~f11f2!eil ~f12f2!P~f1 ,f2!. ~3.9! Since P (f1 ,f2) and P(f1 ,f2) are periodic functions, these equalities determineP (f1 ,f2) completely, as can be shown using Fourier analysis. The final result is the same as Eq. ~3.7!. From now on we shall deal exclusively with the phase difference whose probability distribution function is given by P ~f2!5E df1P ~f1 ,f2!, ~3.10! and, for simplicity, in what follows we shall omit the sub- script2 on it. A useful expression forP (f) is P ~f!5E du^u1f,uuruu1f,u&. ~3.11! Note that, as may be expected,P (f) is invariant under any phase-sum shift. Then the system stater and eif0~N11N2!re2 if0~N11N2! ~3.12! have the sameP (f), as is clearly seen from~3.11!. This property means~as it occurs in the angle case! that the phase difference and the total photon number are com- patible @17#. This compatibility can be expressed more ex- plicitly by noting that~3.10! and~3.11! are also equivalent to P ~f!5 ( n50 ` ^f~n!uruf~n!&, ~3.13! where the vector uf~n!&5 1 A2p ( n150 n ein1fun1 ,n2n1& ~3.14! lies in the subspaceHn with total photon numbern. Equa- tion ~3.13! allows us to extract a joint probability distribution function for the total number and the phase difference P ~n,f!5^f~n!uruf~n!&, ~3.15! and in this way we have the compatibility expressed as P ~f!5 ( n50 ` P ~n,f!, P ~n!5E dfP ~n,f!, ~3.16! 53 497PROBABILITY DISTRIBUTIONS FOR THE PHASE DIFFERENCE whereP (n) is the probability of havingn photons in the system. Consequently, we can independently study the phase-difference properties of any field state in each finite- dimensional subspaceHn without losing information. EachP (n,f) is to some extent redundant. This can be shown by noting thatP (n,f) cannot be arbitrary. Its more general form is, according to~3.15! and ~3.14!, P ~n,f!5 ( k52n n cke ikf, ~3.17! with ck5c2k* . This has two interesting consequences. First, the mean value ofeimf with umu.n is 0 in the subspace Hn . Second,P (n,f) depends on 2n11 parameters and can be completely fixed by its values on 2n11 properly chosen points, such as, for example,f r (n)52pr /(2n11) with r an integer running from2n to n. We can then invert ~3.17! and obtain the coefficientsck as ck5 1 2n11 ( r52n n P ~n,f r ~n!!e2 i2pkr/~2n11!, ~3.18! which allows us to expressP (n,f) as P ~n,f!5 1 2n11 ( k,r52n n P ~n,f r ~n!!eik~f2fr ~n! !. ~3.19! Thus the knowledge of the distributionP (n,f) for 2n11 values of the phase difference is enough to characterize the behavior of the phase difference within theHn subspace. We can now express any mean value in terms of these P (n,f r (n)) as ^ f ~f!&5 ( n50 ` 2p 2n11 ( r52n n P ~n,f r ~n!! f ~n!~f r ~n!!, ~3.20! wheref (n)(f) is the function arising fromf (f) after remov- ing the Fourier frequencies higher thann. It is worth noting the close resemblance of these expressions with those ob- tained with a phase-difference operator arising from a polar decomposition@5#. Formally there are only two differences: the removing of Fourier frequencies and the number of phase-difference values, 2n11 instead ofn11. Otherwise, it turns up that they share more properties than expected at first glance. It seems that the phase difference has properties not evidently related to the absolute phase, as happens in the polar decomposition, here hidden in the casting procedure. A particular example is discussed in detail in Ref.@18#. IV. PHASE DIFFERENCE AND Q FUNCTION When one focuses attention on realistic schemes for phase measurements~such as amplification with a linear laser am- plifier @19#, heterodyning@20#, or beam splitting@21#! it be- comes clear that, in fact, all these techniques are suited to determine theQ function for the signal field@22#, defined as usual as Q~a!5 1 p ^aurua&, ~4.1! whereua& is a coherent state. In particular, a phase distribution is obtained fromQ as a marginal distribution after integrating over the radial vari- able. Moreover, theQ function was recently shown@23# to be a special case of the quantum propensity@24#, correspond- ing to the choice of the reference state as the vacuum state and of the phase-spacemotionas the Glauber displacement operator. Thus it is interesting to examine the corresponding casting procedure for this formalism that, to our best knowl- edge, has not been previously considered. The joint distribution for the phase of a two-mode field is P~f1 ,f2!5E 0 `E 0 ` dr1dr2r 1r 2Q~r 1e if1,r 2e if2!, ~4.2! wherea15r 1e if1, a25r 2e if2. Then we can express it in terms of the phase sum and difference by means of Eq.~3.7! and finally integrate over the phase sum to get the probability distribution for the phase difference. Instead of taking this direct way, we rather prefer to em- ploy a useful relation between the two-mode coherent states ua1 ,a2& and the SU~2! coherent statesun,u,f& @25#, ua1 ,a2&5 ( n50 ` r neinf2 An! e2r2/2un,u,f&, ~4.3! where un,u,f&5 ( n150 n S nn1D 1/2S cosu2D n1S sinu2D n2n1 3ein1fun1 ,n2n1&, ~4.4! and r5Ar 121r 2 2, f5f12f2 , tan u 2 5 r 2 r 1 , ~4.5! with 0