Phase-intensity uncertainty relation from ... - Science Direct

I June 1997

OPTICS COMMUNICATIONS ELSJZVIER

Optics Communications

138 (1997) 311-316

Phase-intensity uncertainty relation from quasiprobability distributions Arkadiusz Orlowski a, Harry Paul b, Bernhard Bijhmer b a Inst_vtut Fiqki, Polska Akademia Nauk, Aieja L.atnik&v32 / 46. 02-668 Warsaw, Poland h Arbeitsgruppe “Nichtklassische Strahlung” der Max-Plan&Gesellschaji an der Humboldt-Unirersitiit ;a Berlin, Rudower Chaussee 5, 12489 Berlin, Germany Received 5 November

1996: revised 6 February

1997; accepted 7 February

1997

Abstract

Extendingthe conceptof Wehrl’s entropy to s-parametrized quasiprobability distributions, as they are observed in optical homodyne detection with realistic detectors, we show that the marginal entropies with respect to phase and amplitude obey a rigourous inequality relation. Relating those entropies to phase and intensity uncertainties, respectively, we arrive at an uncertainty relation for operationally defined phase and intensity that takes explicit account of the noise introduced by non-ideal detectors.

1. Introduction It is well known that the quantum mechanical phase concept suffers not only from the formal difficulty to define a proper phase operator (actually, a strictly Hermitian phase operator could be defined for an artificial finite-dimensional Hilbert space only [1,2]) but, still more seriously from the physical point of view, from the absence of an adequate realistic measuring prescription. On the other hand, several practical schemes have been devised for measuring a quantity that deserves to be named phase (for an overview see Refs. [3,4]). Common to them is the feature that additional noise enters the apparatus. As a result, the measured phase distribution will be broader one to be evaluated as a POM than the “true” (projection-operator valued measure) relying on London’s phase states [5]. In practice, the non-unit detection efficiency will give rise to further broadening. In fact, it has recently been shown [6] for both the eight-port homodyne detection scheme [7,8] with strong local oscillators and the amplification scheme [9-l 11 that the effect of detector efficiency is to smooth the measured distribution functions. Hence, what is actually measured in those circumstances is certain s-parametrized quasiprobability distributions, instead of the Q function. It appears natural to quantify the 0030-4018/97/$17.00 Copyright PI/ SOO30-4018(97)00086-2

uncertainty inherent to those distributions by deriving a Heisenberg-like uncertainty relation for two orthogonal field quadratures x and p (the analogs of position and momentum) that are actually measured in homodyne detection. However, the right-hand side of this new uncertainty relation proves now to be greater than Heisenberg’s value l/2 characteristic of the intrinsic quantum noise. and its actual value can be taken as a measure of the total noise being present which includes extra noise due to optical elements such as beam splitters and inefficient detectors. We have recently shown [12] that the uncertainty relation in question is readily calculated for any s-parametrized quasiprobability function with s < 0, i.e. any distribution obtained by smoothing (convoluting with a Gaussian) from the Wigner function corresponding to s = 0. In particular, our result applies to the distribution functions that can be observed in the eight-port homodyne detection scheme with strong local oscillators. Then, for ideal detectors s equals - 1, and for realistic, i.e. non-unit efficiency detectors, s will be smaller than - 1. In the present paper, it is our goal to study quantitatively the effect the noise in question has on the accuracy of phase measurement. To this end, we will establish an uncertainty relation for phase and intensity that is based on an entropic inequality rela-

0 1997 Elsevier Science B.V. All rights reserved.

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A. Orlowski et al./Optics Communications 138 (1997) 311-316

tion following rigourously from the concept of Wehrl’s entropy when applied to the measured s-distribution. The phase is simply the polar angle cp in the phase space, while the conjugate quantity is the squared radius, r’, as an analog of the photon number. The paper is organized as follows. In Section 2 we will derive an inequality relation for the marginal entropies with respect to radius and phase. In Section 3 we establish connections between those marginals and the uncertainties of the phase cp and the intensity I defined as I= r2. This gives us the desired phase-intensity uncertainty relation. For comparison, we present numerical results for the phase-intensity uncertainty product for relevant physical states, thereby taking account of various definitions of the phase uncertainty used in the literature. Finally, a short discussion is given in Section 4.

2. Entropic inequalities

We start from the well known Klein inequality x(lnx-lny)>>*-y

(x,y>O)

which is equivalent lnrlt-1

to the elementary

(7)

With the identifications x=Q(r,cp>,

.v=w(cp)w(r).

Q(r, cP>lnQ(r9 ‘P> - Q(r? v)ldw(v>w(r)l 2 Q(r, v) - w(cp)w(r).

-i2ndiFcrdrQ(r,

O(V)

cp)ln W(r)

2 1 -/n2ndpo(q)[rdrw(r), or, by virtue of Eqs. (l), (2) and the definitions

to the marginals,

(11)

S, + S, >_S 2 Scoh = 1 + In rr.

(3)

(4) rdr.

(4) and (5),

(2)

i.e., we introduce the marginal entropies

r)lnw(r)

(10)

It has been shown that Wehrl’s entropy becomes minimum for coherent states in which case it takes the value 1 + In 7~ [16]. We thus end up with the fundamental inequality

We have put CY= r exp(icp) ((Y complex amplitude) to ensure the correspondence between r2 and the photon number. We have assumed that the Q function is normalized to unity which implies that the same holds also for the marginals, Eqs. (1) and (2). We will first show that an entropic in-equality relation for phase and amplitude can be rigourously derived. To this end, we extend the concept of Wehrl’s entropy which is defined as [ 15,161 -~‘ndBlfixrdrQ(r,~)lnQ(r,ip)

(9)

Note that, because we will integrate this inequality over the whole phase space (r, q plane), we need not worry about discrete zeros of the functions involved, which eventually might occur. On performing the integration, we get

(1) and

S=

(8)

the Klein inequality, Eq. (6), reads

s, + s, 2 s.

w(r) = L2-Q(r, cp)dq.

inequality

(t>O).

-S-~21id~~~rdrQ(r,~)ln

We start from the Q function of a given field state, which, in fact, describes the distribution observed in an eight-port scheme [13,14], provided the local oscillator field is very intense and the detectors have unit detection efficiency. The corresponding phase and amplitude distributions are given by the marginals of the Q function written as a function of polar coordinates r, cp

(6)

(5)

(12)

Similar entropic uncertainty relations can also be derived for the position and momentum variables 117-191. It is also obvious that the procedure that led us to Eq. (12) can be applied to any non-negative distribution function. What is specific for the Q function is only the value 1 + In rr for the absolute lower bound which is actually given by Wehrl’s entropy for coherent states [16]. We will concentrate on so-called s-parametrized quasidistributions with s I - 1. These are smoothed Q functions (convolutions with a Gaussian). Their physical importance stems from the fact that they describe the distributions that are actually measured in eight-port homodyne detection [6] when intense local oscillator fields and realistic, i.e. non-unit efficiency detectors are used. The s-parameter has been shown to be related to the detector efficiency ~7 in a simple way, s = -(2 - 71)/n [6]. Let us note that there is a relationship between our measured quasiprobabilities and the so-called phase-space propensities [20]. We can immediately generalize our basic inequality, Eq. (12), to the s-parametrized distribution functions just

A. Orlowski et al. / Optics Communications 138 (1997) 311-316

313

mentioned. Since the generalized Wehrl entropy will take its minimum value for coherent states too, we may write in place of Eq. (12)

the simplest case that the distribution functions for phase and intensity, respectively, can both be approximated by Gaussians.

S’“‘+S~“2S’“‘2S2.~i,=l P

3. I. Phase uncertainty

+lnn+ln[(l

-s)/2]. (13)

We

have

thus arrived at an inequality that holds rigourously. Now, the question arises as to what physical interpretation it can be given. This problem will be addressed in the following section.

3. Phase-intensity

uncertainty

a phase distribution

function of the form

(15) where A9 assumed to be small compared to 2n-, denotes the mean square root deviation of cp from its average value (pa. Calculating now the corresponding entropy S,, Eq. (4). we have

relations

The main objective of the eight-port homodyne detection scheme is to provide an operational approach to the problem of quantum phase. In fact, it appears natural to identify the marginal of the measured distribution function (histogram) with respect to amplitude as a measured phase distribution which, of course, is smeared out compared to the “ideal” one that is based on the concept of London [5] or, equivalently, Pegg and Bamett [1,2]. This is easily understood as a result of additional noise introduced by the beam splitter for the signal and, in addition, by inefficient detectors. In the measuring scheme in question the squared amplitude, r’, corresponding to the photon number plays the role of the observable that is conjugate to the phase. Note that we want to investigate the original field solely in a scheme that allows to determine experimentally a (positive definite) quasiprobability distribution. Hence we will characterize the intensity by the measured values of r’ = i(x’ +p’) which have to be distinguished from the intensity (photon number) in the original field. In particular, the “intensity” r’ is a continuous variable, in contrast to the photon number. Similar to the phase, the intensity r2 will have an uncertainty that is enhanced in comparison with that of the photon number. (In particular, it is finite for a Fock state.) Now, it is well known that an uncertainty relation for phase and photon number holds, at least to a good approximation. Actually, Holevo [21] succeeded in giving a rigourous derivation which, however, is based on an unfamiliar definition of phase uncertainty. What we are looking for is a generalization of this uncertainty relation to the situation encountered in homodyne detection. In the following, we will show that Eq. (13) provides a clue to achieve this goal. First, we pass from Eq. (13) to the equivalent relation e.Y;‘)e”!” > ,S”’ 2 eTr(1 - s)/2,

We consider

(14)

which has already the form of an uncertainty relation. The main question, however, is whether it is actually justified to consider the exponentials on the left-hand side as measures of uncertainties. To this end we calculate them for

-4v)ln

o(cp) =

.

(16) The integration

over cp yields

-/~(~)lnw(cp)dcp=fln2~+lnA~+f,

(17)

and we arrive at the result es* = (2n-e)“’

Ap,

(18)

which indicates that exp S,, apart from a normalization factor, equals the phase uncertainty. 3.2. Intensity uncertainty Similarly we find the quantity exp S, to be related to the uncertainty of the intensity I = rz. Since r dr = d1/2, we may consider, as a convenient example, a Gaussian intensity distribution o(r)

=2W(Z),

(19)

where

Here, I is the mean value and AI the uncertainty of the intensity. For the sake of mathematical simplicity we will assume that I, x=- Al. This allows us to formally extend the integration over I from - 00 (instead of 0) to +m. Then our former calculation for the phase distribution applies equally well to the intensity distribution, we have only to take properly into account the factor 2 in Eq. (19) which leads to es’. = i(2rre)“2

hf.

(21)

A. Orlowski et al/Optics

Communications 138 (1997) 311-316

0.5 0.4 -

‘?7

I 4

I

-5

I

-4

I

-3

-2 s

-1 CY

Fig. 1. Different measures of phase uncertainty versus parameter s (characterizing the underlying quasiprobability distribution) for a

Fig. 2. Different measures of phase uncertainty for a Glauber state and s = - 1 (Q function) as a function of cz.

Glauber state with (Y= 2.

and Holevo’s measure [21] 3.3. Uncertainty relations (Acp);=Jexp(icp)J-‘The results (18) and (21) suggest to adopt those relations as new definitions of phase and intensity uncertainties. Note that they apply, in the sense of on operational approach to phase and intensity measurements, only to special measurement schemes, such as eight-port optical homodyne detection with strong local oscillators, heterodyne detection or amplification (cf. Refs. [3,4]), that allow for an experimental determination of a phase-space distribution. From the latter the uncertainties in question can be uniquely evaluated, first forming the marginals of the distribution function and then utilising Eqs. (4) and (5). Clearly, the advantage of this approach is that the uncertainties A+Dand AI thus defined obey a rigourous uncertainty relation. In fact, it follows immediately from the exact inequality (14) that A~Alz~e”“‘>(l ae

-s)/2.

(22)

This is our main result. It is interesting to note that, in contrast to familiar uncertainty relations such as Heisenberg’s, the inequality (22) contains, in addition to an absolute lower bound (depending on the parameter s and, hence, on the detection efficiency), a second lower bound (ne>- ’ exp S’“’ that is specific of the quantum state under study. Of special interest is also the fact that the right-hand side in Eq. (22) is precisely the same as in the Heisenberglike uncertainty relation for two orthogonal quadratures 1121. Hence non-unit efficiency detectors affect the two uncertainty relations in just the same way, which, in fact, is no matter of surprise. In the following, we will illustrate the uncertainty relation (22) by some examples studied numerically. First, we compare our new measure of phase uncertainty with familiar measures, namely the variance

1,

(24)

where the bar indicates averaging with respect to the phase distribution. Holevo’s measure is distinguished by the fact that, when applied to the initial field, it obeys the uncertainty relation

(Aq)” An

2 f

(25)

strictly, as was proved by Holevo [21]. Accordingly, this measure becomes infinite for a random phase distribution, in particular for a Fock state (An = 01, and this is a feature not shared by any other measure (including ours!. Similarly, we compare our new measure of intensity uncertainty with the intensity variance. In Fig. 1 we have plotted the different measures of phase uncertainty for a Glauber state with (Y= 2 versus the parameter s that is related to the detection efficiency 7 in the simple form [6] s = -(2 - 77)/v. The general tendency exhibited by this figure, of the phase uncertainty to increase with decreasing s is, of course, what one expects from the fact that the phase-space distribution becomes more and more extended, as a result of smoothing (convo-

7

I

I

I

I

I

entropic

I l-7

I

I

I

I

I

-6

-5

-4

-3

-2

I

-1

s

(23)

Fig. 3. Different measures of intensity uncertainty versus parameter s for a Glauber

state with (Y= 2.

315

A. Orlowski et al. /Optics Communications 138 (IY97) 311-316

3.5 3-5 Fig. 4. Intensity-phase uncertainty products for squeezed states with squeezing parameter 3 versus o (displacement). (The axes of the uncertainty ellipses for the squeezed states are oriented in x and p direction, respectively.) The marks on the right ordinate indicate the respective absolute lower bounds of the uncertainty product, according to Eq. (22).

luting a Gaussian with a second Gaussian). Fig. 1 indicates that Holevo’s measure is always higher than the variance, whereas our entropic measure is lower. It turned out that this is, in fact, a general feature that is revealed, in particular, by Fig. 2. This figure shows that the differences between the three measures become negligible even for moderate mean photon numbers 1ai’. This is in accordance with the correspondence principle. Clearly, one has to require that in the limit of large mean photon numbers all measures should coincide. Further, our numerical analysis shows that the entropic measure of intensity uncertainty is smaller than the variance. An example is given by Fig. 3. In Fig. 4 the phase-intensity uncertainty product is plotted for a squeezed state. For comparison this product is presented also for the case that conventional measures for the uncertainties are adopted. It is interesting to note that the entropic uncertainty product comes very close to the absolute lower bound marked at the ordinate [see Eq. (2211, irrespective of the intensity, whereas familiar uncertainty

/I

I1

I

I

/

I

I

I

I

I

I

2

3

4

5

6

7

8

9

10

cr Fig. 5. Same as Fig. 4 for a displaced lowest curve represents the specific according to E!q. (22).

Fock state with n = 4. The lower bound exp S/err,

.._ A.5

-4

-3.5 I

-34

-2.5 I

-2I

-1.5 S

Fig. 6. Intensity-phase uncertainty product for a displaced state with n = 4, (Y = 10, versus parameter S.

-1 Fock

measures give rise to large deviations at low intensities. Fig. 5 indicates that for a displaced Fock state the entropic uncertainty product differs noticeably from those based on familiar uncertainty measures. This is mainly due to the fact that in the present case the entropic intensity measure is distinctly smaller than the variance. Obviously, the value of the uncertainty product is much greater than the absolute minimum (1 - s)/2 that now equals 1. It comes close, however, to the specific lower bound exp St”‘/rre which we call Wehrl limit [see Eq. (22)]. Finally, one observes from Fig. 6 that the entropic uncertainty product approaches the other ones for decreasing values of S, i.e.. when more and more noise is introduced by the detectors.

4. Summary Utilizing the concept of Wehrl’s entropy and extending it to more general distribution functions in phase space that can actually be observed in well known realistic experiments, we were able to rigourously prove an uncertainty relation for phase and intensity, Eq. (22). The corresponding distributions are defined operationally, i.e. as marginals of the measured phase-space distribution. The prize one has to pay for such a strict uncertainty relation is an unfamiliar definition of the uncertainties, both for phase and intensity. However, we could show that the new uncertainty measures based on Wehrl’s entropy, coincide with the familiar variances in the case of narrow Gaussian distributions for phase and intensity, respectively. Moreover, from numerical studies of Glauber and, more generally, squeezed states we observed that the entropic variances come close to the familiar ones. Hence, the same holds true also for the corresponding uncertainty products. In particular, our results take properly into account the effect of non-unit detection efficiency. Formally, this effect gives rise to s-parametrized quasiprobability distributions, with s < - 1, in place of the Q function. The smoothing process corresponding to this modification

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A. Orlowski et al./ Optics Communications

clearly leads to a broadening of both the phase and the intensity distribution, and hence enlarges the right-hand side of the phase-intensity uncertainty relation. It is interesting but, in fact, not surprising to note that the enhancement factor given simply by (1 - s)/2, is just the same as for the Heisenberg-like uncertainty relation for two orthogonal quadratures [ 121. References [l] D.T. Pegg, S. M Bamett, Europhys. Len. 6 (1988) 483. [2] D.T. Pegg, S.M. Barnett, Phys. Rev. A 39 (1989) 1665. [3] U. Leonhardt, H. Paul, Phys. Scripta T 48 (1993) 45. [4] U. Leonhardt, H. Paul, Progr. Quantum Electron. 19 (1995) 89. [5] F. London, Z. Phys. 40 (1927) 193. [6] U. Leonhardt, H. Paul, Phys. Rev. A 48 (1993) 4598. [7] N.G. Walker, J.El. Carroll, Opt. Quantum Electron. 18 (1986) 355.

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