Squashed states of light: theory and applications to quantum spectroscopy

This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 130.102.42.98

This content was downloaded on 07/06/2014 at 08:22

Please note that terms and conditions apply.

Squashed states of light: theory and applications to quantum spectroscopy

View the table of contents for this issue, or go to the journal homepage for more

1999 J. Opt. B: Quantum Semiclass. Opt. 1 459

(http://iopscience.iop.org/1464-4266/1/4/317)

Home Search Collections Journals About Contact us My IOPscience

J. Opt. B: Quantum Semiclass. Opt.1 (1999) 459–463. Printed in the UK PII: S1464-4266(99)00874-5

Squashed states of light: theory andapplications to quantumspectroscopy

H M Wiseman

Centre for Laser Science, The Department of Physics, The University of Queensland,Queensland 4072, Australia

E-mail: wiseman@physics.uq.edu.au

Received 12 January 1999

Abstract. Using a feedback loop it is possible to reduce the fluctuations in one quadrature ofan in-loop fieldwithout increasing the fluctuations in the other. This effect has been knownfor a long time, and has recently been called ‘squashing’ (Buchleret al 1999Opt. Lett.24259), as opposed to the ‘squeezing’ of a free field in which the conjugate fluctuations areincreased. In this paper I present a general theory of squashing, including simultaneoussquashing of both quadratures and simultaneous squeezing and squashing. I show that atwo-level atom coupled to the in-loop light feels the effect of the fluctuations as calculated bythe theory. In the ideal limit of light squeezed in one quadrature and squashed in the other, theatomic decay can be completely suppressed.

Keywords: Squeezing, squashing, quantum feedback, quantum spectroscopy, quantumnoise, quantum measurement, line narrowing

1. Introduction

Squeezed states of light are nonclassical [1]. The foremostconsequence of this is that they can produce a homodynephotocurrent having a noise level below the shot-noise limit.The shot-noise limit is what is predicted by a theory in whichthe light is classical, with no noise, but the process of photo-electron emission is treated quantum mechanically.

There is, however, a simple way to produce a sub-shot-noise photocurrent without squeezed light: modulating thelight incident on the photodetector by a current originatingfrom that very detector. This was first observed [2, 3]around the same time as the first incontestable observation ofsqueezing [4].

The sub-shot-noise spectrum of an in-loop photocurrentis not regarded as evidence for squeezing for a number ofreasons. First, the two-time commutation relations for anin-loop field are not those of a free field [5]. This meansthat it is possible to reduce the fluctuations in the measured(amplitude) quadrature without increasing those in the other(phase) quadrature. Second, attempts to remove some of thesupposedly low-noise light by a beam splitter yields onlyabove shot-noise light, as verified experimentally [2,6].

Because of these differences, the presence of a sub-shot-noise photocurrent spectrum for in-loop light has byand large been omitted from discussions of squeezing [7–9].Nevertheless, it has been argued [10] that the in-loop field canjustifiably be called ‘sub-Poissonian’ (even if not squeezed),because a perfect quantum-nondemolition (QND) intensitymeter for the in-loop light would register the same sub-shot-

noise statistics as the (perfect) in-loop detector. Furthermore,it was proposed in [10] that the apparent in-loop noisereduction could be used to improve the signal to noise ratiofor a measurement of a modulation in the coupling coefficientof such a QND intensity meter.

Following on from [10], it has been shown that in-loopoptical noise suppression may have other, more practical,applications. Buchleret al showed [11] that such in-looplight can suppress radiation pressure noise in a gravitationalwave detector by a factor of two. Even more strikingly, Irecently showed [12] that a two-level atom coupled to the in-loop field can exhibit linewidth narrowing exactly analogousto that produced by squeezed light [13].

In this paper I will discuss the properties of ‘squashed’states of light, as the in-loop analogues of squeezed states oflight are called in [11]. Section 3 covers the general theory ofsquashed states of light, including states which are squashedin both quadratures and states which are simultaneouslysquashed and squeezed. In section 4, I generalize the analysisof [12] by considering the effects of these more general stateof light on a two-level atom. But to begin, I review theproperties of squeezed states of light in the following section.

2. Squeezed states

2.1. Single-mode squeezing

The noise in the quadratures of a single-mode light field ofannihilation operatora,

x = a + a†, y = −ia + ia†, (1)

1464-4266/99/040459+05$30.00 © 1999 IOP Publishing Ltd 459

H M Wiseman

is limited by the Heisenberg uncertainty relation

VxVy > | 12[x, y]|2 = 1, (2)

whereV denotes variance. Squeezed states of light are statessuch that one of the quadrature variances, sayVx , is lessthan one [1, 14, 15]. Clearly the other quadrature variance,sayVy , must be greater than one. If the equality is attainedin the uncertainty relation then these are called minimumuncertainty squeezed states. The only other sort of minimumuncertainty state is the coherent state withVx = Vy = 1.

2.2. Continuum squeezing

Single-mode squeezing was generalized early to multimodesqueezing [16]. In this work I wish to consider the limit of aninfinite number of modes: the electromagnetic continuum.Considering polarized light propagating in one direction,only a single real-valued index is needed for the modes, andwe take that to be the mode frequencyω = k (using unitssuch that the speed of light is one). Then the continuum fieldoperatorsb(k) obey

[b(k), b†(k′)] = δ(k − k′). (3)

For light restricted in frequency to a relatively narrowbandwidthB around a carrier frequency� it is possible toconvert from the frequency domain to the time or distancedomain by defining

b(t) =∫ �+B

�−Bdk b(k)e−i(k−�)t . (4)

These obey[b(t), b†(t ′)] = δ(t − t ′), (5)

whereb†(t) ≡ [b(t)]†, and theδ function is actually a narrowfunction with width of orderB−1. The operatorb†(t)b(t) canbe interpreted as the photon flux operator.

Defining continuum quadrature operators

X(t) = b(t) + b†(t), Y (t) = −ib(t) + ib†(t), (6)

one obtains

[X(t), Y (t ′)] = 2iδ(t − t ′). (7)

The singularity in the associated uncertainty relations can beavoided by quantifying the uncertainty by thespectrum[8]

SX(ω) = 〈X(ω)X(0)〉ss− 〈X(ω)〉ss〈X(0)〉ss. (8)

Then one can derive the finite uncertainty relations [5]

SX(ω)SY (ω) > 1. (9)

For very high frequencies the spectra always go tounity. This represents the shot-noise or vacuum-noise level.However for finite frequencies it is possible to have forexampleSX(ω) < 1. This indicates squeezing of theXquadrature. The uncertainty in the conjugate quadraturewould of course be increased.

The quadrature operatorsX(t), Y (t) can be directlymeasured (one at a time) using homodyne detection [17].

For detection of efficiencyε 6 1 the photocurrentIXhom hasthe same statistics as (and therefore can be represented by)the operator

IXhom(t) =√εX(t) +

√1− εξX(t), (10)

where ξX(t) is a Gaussian white noise term. Here thenormalization has been chosen so that the photocurrentspectrum

SXhom(ω) = εSX(ω) + (1− ε), (11)

remains equal to unity (the shot-noise limit) at highfrequencies, whereSX(ω) = 1.

2.3. An atom in a squeezed bath

Now consider the situation where a two-level atom isimmersed in a beam of squeezed light with annihilationoperatorb0(t). If the degree of mode matching of thesqueezed light to the atom’s dipole radiation mode isη,then the dipole coupling Hamiltonian in the rotating-waveapproximation is

H0(t) = −i[√ηb0(t) +

√1− ην(t)

]σ †(t) + h.c. (12)

Here σ is the atomic lowering operator and the atomiclinewidth has been set to unity. The operatorb0(t) representsthe squeezed field with spectraSX0 (ω) and SY0 (ω), andν(t) represents the vacuum field interacting with the atom,satisfying〈ν(t)ν†(t ′)〉 = δ(t − t ′).

Now if the quadrature spectra of the squeezed light aremuch broader than the atomic linewidth then one can makethe white noise approximation that they are constant. Forminimum-uncertainty squeezing, one has

SX0 (ω) = L = 1/SY0 (ω), (13)

and the atom will obey the master equation [13]

ρ = (1− η)D[σ ]ρ +η

4LD[(L + 1)σ − (L− 1)σ †]ρ, (14)

whereD[A]B ≡ A†BA − 12{A†A,B} as usual. This leads

to the following atomic dynamics:

Tr[ρσx ] = −γx Tr[ρσx ], (15)

Tr[ρσy ] = −γy Tr[ρσy ], (16)

Tr[ρσz] = −γz Tr[ρσz] − C, (17)

whereγx = 1

2[(1− η) + ηL], (18)

γy = 12[(1− η) + ηL−1], (19)

γz = γx + γy, C = 1. (20)

Note that forL < 1 the decay rate of thex componentof the atomic dipole is reduced below the vacuum level of1

2,while the decay rate of the other component is increased.The reduction or increase in the decay rates are directlyattributable to the reduction or increase in the fluctuations ofthe respective quadrature of the input continuum field. Theprediction of this effect by Gardiner [13] began the study of

460

Quantum spectroscopy with squashed light

0

I (t)Xhom

b (t)1b (t)

Detector

Electro-OpticModulator Homodyne

Figure 1. Simplified diagram of the production of squashed light.The circle in the path of the in-loop beam is where an in-loop atomcould be positioned.

quantum spectroscopy (that is, the interaction of nonclassicallight with matter) [18].

For sufficiently largeη this effect would be easilydetectable experimentally in the fluorescence powerspectrum of the atom (into the vacuum modes):

P(ω) = 1− η2π〈σ †(−ω)σ(0)〉ss (21)

= (1− η)(γz − C)8πγz

[γx

γ 2x + ω2

+γy

γ 2y + ω2

]. (22)

For L < 1 the spectrum consists of two Lorentzians, onewith a sub-natural linewidth and one with a super-naturallinewidth. The overall linewidth (defined as the full-width athalf-maximum height) is reduced.

This line-narrowing is only noticeable if the degreeof mode matchingη of the squeezed light to the atom issignificant, which is hard to do with a squeezed beam. Oneway around this is to make the atom strongly coupled to amicrocavity, which can be driven by a squeezed beam. Themicrocavity enhances the atomic decay rate, but a squeezedinput should suppress this enhancement in one quadrature.Unfortunately, experiments to date have failed to see thissuppression, due to imperfections of various kinds [19].

3. Theory of squashed states

3.1. Generation of squashed states

Consider the feedback loop shown in figure 1. The fieldentering the modulator isb0(t) = 1

2[X0(t) + iY0(t)].The quadrature operators are assumed to have independentstatistics defined by the spectraSX0 (ω), SY0 (ω). Themodulator simply adds a coherent amplitude to this field.There are various ways of achieving this, one of which isdiscussed in [12]. The field exiting is in any case given by

b1(t) = 12[X0(t) + iY0(t) + χ(t) + iυ(t)], (23)

whereχ(t), υ(t) are real functions of time. This field nowenters a homodyne detection device, set up so as to measuretheX quadrature. If the efficiency of the measurement isεXthen the photocurrent is given by

IXhom(t) =√εX[X0(t) + χ(t)] +

√1− εXξX(t). (24)

Now, through the feedback loop, this current maydetermine the classical field amplitudesχ, υ. Obviously inthe case of measuring theX quadrature, the only interestingresults will come from controllingχ(t), and we set

χ(t) =∫ ∞

0gXh(s)I

Xhom(t − τ − s)/

√εX ds. (25)

Here τ is the minimum delay in the feedback loop,h(s) is the feedback-loop response function normalized to∫∞

0 h(s) ds = 1 andgX is the round-loop gain [20]. Takingthe fourier transform of this expression and substituting intoequations (23) and (24) yields

b1(ω)

= 1

2

[X0(ω) +

√θXgXeiωτ h(ω)ξX(ω)

1− gXeiωτ h(ω)+ iY0(ω) + iυ(t)

].

(26)

whereθX ≡ ε−1

X − 1. (27)

TheX quadrature spectrum of this light is

SX1 (ω) =SX0 (ω) + θXg2

X|h(ω)|2|1− gXeiωτ h(ω)|2 . (28)

Say we are only interested in frequencies well inside thebandwidth ofh(ω), much less thanτ−1, and much less thanthe bandwidth ofSX0 (ω). Then we can replace eiωτ h(ω) byunity andSX0 (ω) by a constantL. This gives

SX1 =L + g2

XθX

(1− gX)2 >L

1 +L/θX, (29)

where the minimum is achieved for negative feedbackgX =−L/θX.

Evidently this minimum is less thanL. This meansthat even starting with shot-noise limited light (L = 1)it is possible to produce sub-shot-noise light. However, itis important to note that this is not squeezed light in theordinary sense. For example, it is impossible to removeany of the squeezed light by putting a beam splitter in thepath of the in-loop beam. Under the above conditions theresulting out-of-loop beam actually has a noise level abovethe shot noise [2, 5, 6, 10]. Nevertheless, the fluctuations ofthe in-loop light do produce genuine physical effects in othercircumstances, as investigated in section 4.

3.2. Violation of the uncertainty relations

A curious point in the apparent squeezing of theX quadratureis that the feedback has no effect on theY quadrature ofb1. From equation (26),SY1 = L−1, assuming a minimumuncertainty inputb0 and υ = 0. Thus the uncertaintyrelation (9) is violated for this in-loop light. For this reason,the in-loop light exhibiting sub-shot-noise fluctuations hasbeen called ‘squashed light’ [11]. By feedback, the noise inone quadrature can be squashed (reduced), but there is no‘squeezing’ of the phase-space area into an increased noisein the other quadrature.

The reason that the uncertainty relation (9) is violatedis that the commutation relations (5) are no longer validfor time differences|t − t ′| > τ , the minimum feedback-loop delay. This is a direct consequence of the feedbackloop. It is important to realize that the parts of the in-loopfield separated in time by greater thanτ never actually existtogether. That is because the propagation time from themodulator to the detector is necessarily less thanτ . Thusthe fundamental commutation relations [7] between parts of

461

H M Wiseman

the field at different points inspaceat the same time are neverviolated.

For freely propagating fields there is no real distinctionbetween space and time separations, but for an in-loop fieldit is a crucial distinction. The temporal anticorrelations inthe in-loop squeezed light only exist for time separationsgreater thanτ , and hence greater than the time for whichany part of the in-loop light exists. There is never anyanticorrelation between parts of the in-loop field in existenceat any given time. In contrast, conventional squeezed lightcan propagate for an arbitrarily long time before detection,so the anticorrelations are between parts of the field whichcan exist simultaneously (even if they may not actually do soin a given experiment).

3.3. Simultaneous squashing in both quadratures

It is interesting now to consider feedback in both theX andYquadratures. Obviously one cannot simultaneously measureboth of these quadratures with unit efficiency, but one canmeasureY with efficiencyεY 6 1− εX. Carrying throughthe same sort of analysis as above shows that theY quadraturespectrum can be simultaneously reduced to

SY1 =L−1

1 +L−1/θY. (30)

For the special case ofL = 1 (a vacuum input), one finds

SY1 + SX1 = 2− εX − εY > 1. (31)

For feedback based on heterodyne detection (which isequivalent to homodyne detection on both quadratures withequal efficiency), one can haveSY1 = SX1 = 1− 1

2ε, whichgoes to one half in the limit of perfect detectors.

3.4. Simultaneous squeezing and squashing

In the general case ofL 6= 1, the sum of the quadraturespectra need not even be greater than one. Rather, for fixedL < 1 and fixedε = εX + εY one finds, forεX = 0 andεY = ε

SY1 + SX1 = L +ε−1− 1

L(ε−1− 1) + 1> 0, (32)

where the limit of zero noise in both quadratures isapproached forL → 0 andε → 1. For example, withexperimentally realizable parameters of 6 dB squeezing [19]and detection efficiencyε = 0.95 [21], one could obtainSX1 = 0.25 andSY1 = 0.05, givingSY1 +SX1 = 0.30, comparedwith the limit of 2 implied by the uncertainty relation (9).

4. Application to quantum spectroscopy

Since the quadrature spectra calculated above apply to an in-loop field, which cannot be extracted using a beam splitter, itmight seem that they have no physical significance. However,this is not the case. As I showed recently [12], placing a two-level atom in a squashed bath leads to linewidth narrowingof one atomic dipole quadrature, entirely analogous to thatproduced by a squeezed bath. The master equation is not

identical, however, because the nonsquashed quadrature isstill shot-noise limited, so there is no line broadening of theother atomic dipole quadrature.

In this work I generalize the results of [12] to includelight which is simultaneously squashed in both quadratures,or simultaneously squeezed and squashed. If the atom iscoupled to the in-loop beamb1(t) with mode-matchingηthen the atomic Hamiltonian is

H(t) = −i[√ηb1(t) +

√1− ην(t)

]σ †(t) + h.c. (33)

Following the methods of [12], the expression forb1 ismodified from (26) by the addition of the atom’s radiatedfield in the direction of the beam,

√ησx(ω), to the input

operatorX0(ω). Thus the total Hamiltonian can be written

H(t) = Hfb(t) +H0(t), (34)

whereH0(t) is as given in equation (12) and the Hamiltoniandue to the feedback is

Hfb(t) = λX 12σy(t)

{σx(t

−) +[X0(t

−) +√θXξX(t

−)]√

η}

+λY 12σx(t)

{σy(t

−) +[Y0(t

−) +√θY ξY (t

−)]√

η}.

(35)

The feedback parameters are defined as

λX = gXη

1− gX ; λY = gY η

1− gY . (36)

For the feedback of the homodyne currentIYhom are definedanalogously to those from the feedback ofIXhom.

In equation (35) the limit of broad-band feedback hasbeen taken, withh(ω)eiωτ in equation (26) set to unity. ThisMarkov approximation is justified provided the bandwidth ofthe feedback is very large compared with the characteristicrates of response of the system [22]. In the present contextthe rate of atomic decay is unity so we require, for instance,τ � 1. For a typical electro-optic feedback loop witha bandwidth in the MHz range, the atom would have tohave be metastable to satisfy this inequality. The preciserequirements for the validity of the Markov approximationwill be investigated in a future publication. Of course evenin the broad-band limit the feedback from the measurementof a particular part of the field must act after that part of thefield has interacted with the atom. This is the reason for theuse of the time argumentt− rather thant in equation (35).

Now to describe the evolution generated by the totalHamiltonian (33), the theory of homodyne detection andfeedback in the presence of white noise is required. This wasfirst detailed in [23], generalizing the earlier work in [24]and [22]. The basic equation is

dρ (t) = 〈exp[−iHfb(t) dt ] exp[−iH0(t) dt ]ρ(t)

× exp[iH0(t) dt ] exp[iHfb(t) dt ]〉 − ρ(t). (37)

Here the ordering of the unitary operators has been chosensuch that the time delay of the feedback has been taken intoaccount and one can replacet− in equation (35) byt .

In equation (37) the expectation value indicates anaverage over the bath operatorsb0(t) and ν(t), and the

462

Quantum spectroscopy with squashed light

detector noise termsξX(t), ξY (t). This is effected by makingreplacements such as

[X0(t) dt ]2→ L dt; ν(t) dt ν†(t) dt → dt. (38)

The result is

ρ = (1− η)D[σ ]ρ +η

4LD[(L + 1)σ − (L− 1)σ †]ρ

−iλX[ 12σy,

12{(L + 1)σ − (L− 1)σ †}ρ + ρ 1

2{(L + 1)σ †

−(L− 1)σ }] + iλY [ 12σx,−i 1

2{(L−1 + 1)σ

+(L−1− 1)σ †}ρ + iρ 12{(L−1 + 1)σ †− (L−1− 1)σ }]

+λ2X(L + θX)

ηD[σy

2

]ρ +

λ2Y (L

−1 + θY )

ηD[σx

2

]ρ. (39)

This again produces the atomic dynamics of equations (15)–(17), but with

γx = 12[(1−η)+ηL(1+λX/η)

2+λ2XθX/η] = 1

2[(1−η)+ηSX],(40)

γy= 12[(1−η)+ηL−1(1+λY /η)

2+λ2Y θY /η]= 1

2[(1−η)+ηSY ],(41)

γz = γx + γy, C = 1 +λX + λY . (42)

In the above equations, the introduction of the spectrumSX is based on equation (29), with the identificationλX =ηgX/(1−gX), and similarly forY . Note that the expressionsfor γx and γy depend upon the quadrature spectra (in theabsence of the atom) in precisely the same way as those forpure squeezed (not squashed) light in equations (18) and (19).This suggests that the natural explanation for the change inthe decay rates is again the reduced fluctuations of the inputlight. It seems that squashed fluctuations are much the sameas squeezed fluctuations as far as the atom is concerned. Theone difference is that the constantC in the equation of motionfor 〈σz〉 is also altered by the feedback.

Consider now the case as in section 3.4 whereL < 1,εX = 0 andεY = ε. Then choosingλY = −η/(1 +Lθ), soas to minimize the in-loopY quadrature spectrum, gives

γx = 12[(1− η) + ηL], (43)

γy = 12[(1− η) + ηθ/(1 +Lθ)], (44)

γz = γx + γy, C = 1− η/(1 + θL), (45)

where θ = ε−1 − 1 as usual. Choosingε = 0.95 andL ≈ 0.25, as before, givesγz ≈ 1 − 0.85η. That is, inthe limit ofη→ 1, the rate of decay of the atomic populationwould be slowed by 85%. In the ideal limit ofL → 0, andε, η → 1, the atom would be frozen in its initial state andwould not decay at all.

5. Conclusion

In this work I have presented for the first time the theoryfor a new class of in-loop light, namely light whichmay be both squashed (in either or both quadratures) andsqueezed. Squeezing here refers to conventional quantumnoise reduction, whereas squashing refers to the noisereduction produced by the feedback loop. Even withouta squeezed input it is possible to reduce the noise inboth

quadratures of the in-loop field below the shot-noise limit.With a squeezed input it is possible, in principle, to reducethe noise in both quadratures to zero (by squeezing one andsquashing the other).

I next derived the effect of this arbitrarily squeezed andsquashed light on an in-loop atom. The calculated in-loopspectra precisely reflect the noise to which the atom responds,provided the bandwidth of the squeezing and the bandwidthof the feedback are much greater than the atomic linewidth.As the quantum fluctuations seen by the atom are reduced,the decay rates for the quadratures of the atom’s dipole arereduced. In the limit that the atom is coupled only to in-loop light which is perfectly squeezed in one quadrature andperfectly squashed in the other, the atomic decay rates vanishand the atom’s dynamics are frozen.

As discussed above, the main experimental difficultywith seeing squeezing-induced line-narrowing is related toefficiently coupling the squeezed light onto the atom. Usingsquashed light rather than squeezed light would not overcomethis difficulty. However, highly squashed light shouldbe easier to generate than highly squeezed light, becauseit is limited only by the homodyne detector efficiency.Also, it can be produced at any frequency for which acoherent source and the appropriate electro-optic equipmentis available. These factors, plus the intriguing possibility ofobserving simultaneous linewidth narrowing on both atomicquadratures, suggest an important role for squashed light inexperimental quantum spectroscopy.

References

[1] Walls D F 1986Nature234210[2] Walker J G and Jakeman E 1995Proc. Soc. Photo-Opt.

Instrum. Eng.492274[3] Machida S and Yamamoto Y 1986Opt. Commun.57290[4] Slusher R Eet al 1985Phys. Rev. Lett.552409[5] Shapiro J Met al 1987J. Opt. Soc. Am.B 4 1604[6] Yamamoto Y, Imoto N and Machida S 1986Phys. Rev.A 33

3243[7] Gardiner C W 1991Quantum Noise(Berlin: Springer)[8] Walls D F and Milburn G J 1994Quantum Optics(Berlin:

Springer)[9] Bachor H-AA Guide to Experiments in Quantum Optics

1998 (Berlin: Wiley)[10] Taubman M S, Wiseman H M, McClelland D E and Bachor

H-A 1995J. Opt. Soc. Am.B 121792[11] Buchler B Cet al 1999Opt. Lett.24259[12] Wiseman H M 1998Phys. Rev. Lett.813840[13] Gardiner C W 1986Phys. Rev. Lett.561917[14] Caves C M 1981Phys. Rev.D 231693[15] Yuen H P 1976Phys. Rev.A 132226[16] Caves C M and Schumaker B L 1985Phys. Rev.A 313068[17] Yuen H P and Shapiro J H 1980IEEE Trans. IT2678[18] Ficek Z, Drummond P D 1997Phys. Today5034[19] Turchette Q Aet al 1998Phys. Rev.A 584056[20] Stefano J J, Subberud A R and Williams I J 1990Theory and

Problems of Feedback and Control Systems 2e(New York:McGraw-Hill)

[21] Schiller Set al 1996Phys. Rev. Lett.772933[22] Wiseman H M 1994Phys. Rev.A 492133

Wiseman H M 1994Phys. Rev.A 495159 (erratum)Wiseman H M 1994Phys. Rev.A 504428

[23] Wiseman H M and Milburn G J 1994Phys. Rev.A 494110[24] Wiseman H M and Milburn G J 1993Phys. Rev. Lett.70548

463