ll signals.nberg limit

Physics Letters A 329 (2004) 8–13

www.elsevier.com/locate/pla

Nonlinear transformations and the Heisenberg limit

Alfredo Luis

Departamento de Óptica, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain

Received 7 March 2004; received in revised form 27 June 2004; accepted 28 June 2004

Available online 6 July 2004

Communicated by P.R. Holland

Abstract

We show that nonlinear transformations provide better resolution than linear transformations for the detection of smaWe show that the uncertainty decreases for increasing photon number faster than the rate established by the Heisefor phase shifts and displacements. 2004 Elsevier B.V. All rights reserved.

PACS: 42.50.Dv; 03.65.-w

Keywords: Quantum limits; Nonclassical states; Precision measurements

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The quantum nature of physical processes limitsprecision of measurements. Some relevant examin very diverse areas are the detection of gravitatiowaves, spectroscopy, metrology, or optical imagingexample[1–15]. The quantum limit acquires differenforms and expressions in different contexts and careferred to as shot noise, standard quantum limit, pjection noise, or Heisenberg limit[1,2,11,12,16]. Allof them refer to the maximum accuracy allowedthe quantum fluctuations of the measured observaGiven the continual improvement of experimentechniques, the thorough examination of quantum l

E-mail address: alluis@fis.ucm.es(A. Luis).

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.06.080

its, their actual range of application, and the possibiof overcoming them, becomes a matter of practicaterest[5–9]. For example, it has been recently demostrated that some quantum limits previously takengranted are actually not universal and they arepassed by suitable detection schemes[3–9,14,15].

On the other hand, from a theoretical point of viethis topic involves basic concepts of the quantum tory such as complementarity, measurement, and estmation theory for example[17–22]. Furthermore, thisincludes the study and practical generation of nononclassical states devised to reach the quantumits, such as quadrature and SU(2) squeezed statesrelated number states, and Schrödinger cats, for exple [10,23–46].

.

A. Luis / Physics Letters A 329 (2004) 8–13 9

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In this work we focus on the Heisenberg limit thstates that the minimum detectable phase shift ithe order of the inverse of the number of photons,equivalently, proportional to the inverse of the eneresources employed. We show that this statemenplies only when we restrict ourselves to linear phshifts, i.e., when the generator of the transformatencoding the signal in the output field state is pportional to the photon number operator. On the othand, measuring strategies that encode the signanonlinear transformations have also their own qutum limits. The point to be developed in this wois that for the nonlinear case the dependence onnumber of photons is significantly different from thlinear case. More specifically, for nonlinear transfmations the uncertainty decreases with photon nummore favorably than the rate established by the Heisberg limit, even when using semiclassical input staTherefore, nonlinear transformations can improve thcapabilities of current technology via a more efficienuse of the energy resources.

We denote byG the generator of an input–outptransformation parametrized by an adimensional varableφ that is to be determined by the measuremena suitable observableM performed on the output sta|ψ〉

(1)|ψ〉 = eiφG|ψ〉,where|ψ〉 is the input state.

The problem we are addressing is an exampleparameter estimation[10,17–22,47–50]. In quantummeasurement a proper data analysis is crucialdemonstrated by previously failed attempts to bquantum limits [51]. Fortunately, in most casesvery simple treatment that highlights the fundamtal points without involved numerics is possible. Tanalysis simplifies further since the parameter to bdetected is small enoughφ � 1, which is the case ointerest in precision measurements, so that a linear aproximation is adequate

(2)e−iφGMeiφG � M + iφ[M,G] + · · · .This linearization leads to a simple but meaningfulpression for the uncertainty�φ of the inferred valueof φ. In order to be detected the signal should be larthan the noise|〈M(φ)〉− 〈M(0)〉| > �M, and this oc-

curs provided thatφ > �φ, being[26,50]

(3)�φ = �M

|d〈M〉/dφ| = �M

|〈[M,G]〉| � 1

2�G,

where�M, �G are the uncertainties ofM and G,respectively, and we have used the uncertaintylation �M�G � |〈[M,G]〉|/2. Note thatφ is notan operator so that(3) actually defines�φ via asimple linearized propagation of uncertainties. Tsimple criterion has been used extensively incontext of precision measurements in diverse a[1–4,6,12,16,18,19,23,25,26,28,29,31,34,37,40,4152–54]. Other related uncertainty relations involvinparameters and operators simultaneously are sotimes referred to as parameter-based uncertaintylations (see references in[10]).

We can see that for every transformation (linearnonlinear) there is a quantum limit established by�G.The point addressed in this work is that the scalof �φ with photon number is different for linear annonlinear transformations.

For linear phase shiftsG = N , whereN representsthe number of photons. Since the number operatopositive definite we have, roughly speaking, thatmaximum�N is of the order of〈N〉/2, so that fromEq.(3)

(4)�φ � �φHL � 1

〈N〉 ,

which is the Heisenberg limit. For semiclassical cherent states the uncertainty is�φ � 1/(2

√〈N〉), sothat in order to approach the Heisenberg limit via lear phase shifts it is necessary to use nonclassicalstates[10,23–42]

However, as we have mentioned above, the uslinear input–output transformations is not the optimuse of the energy resources. This is because in Eq(3)we can have�G > 〈N〉 for suitably chosen generatoand input states. Below we will present some examdemonstrating that the parameters of nonlinear traformations can be estimated with a higher precisthan the parameters of the linear ones.

We emphasize that nonlinear transformations halso their corresponding quantum limit expressed(3). The key point is that this limit can be substantialess restrictive for nonlinear transformations thanlinear ones since the dependence with photon numis significantly different.

10 A. Luis / Physics Letters A 329 (2004) 8–13

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It might be also argued thatφ is a different variablefor linear and nonlinear transformations. Howevfrom the estimation perspectiveφ is a parameter thamust be disclosed by a suitable measuring scheboth for linear and nonlinear transformations. Frompractical perspective,φ is a function of physical variablesA such as time, length, frequency, temperatustress, etc. For example, nonlinear transformationscur in the propagation of light in nonlinear mediathatφ depends on the length, density, and temperaof the medium, and the frequency, polarization, andtensity of the light, among other variables. From tperspective, the detection of the shiftφ serves to inferthe value of the physical variableA with uncertainty

(5)�A = ∂A

∂φ�φ.

This relation applies both to linear and nonlinear traformations with different factors of proportionalit∂A/∂φ. In this work we are focusing exclusively othe second factor�φ studying its scaling with thenumber of photons. If we focus exclusively on thpoint, nonlinear transformations would be the opmum choice for the encoding of signals, the motoring of physical variables, and the detection ofagents causing system variations. Nevertheless,more sensible comparison we should take into accosome other circumstances which are beyond the sof this work, such as the behavior of the functionA(φ)

and the feasibility of encoding the variable of interin a suitable nonlinear transformation.

The resolution depends simultaneously on thfactors: the input state, the operator that generateshift, and the observable measured. We can regargenerator as the most fundamental factor since it adetermines the maximum resolution achievable(3).We will say that the arrangement is optimal or not dpending on whether the example examined reaches thlimit �φ = 1/(2�G) or not. For the sake of clarity wpresent each example under asection heading specifying the trio{G,ψ,M}.

G photon number square, ψ coherent, M quadratureAs a very simple example let us consider a modethe electromagnetic field with complex amplitude oerator a prepared in a semiclassical coherent st|ψ〉 = |α〉, with a|α〉 = α|α〉. The phase-shift geneator isG = (a†a)2 while the measured observable

the field quadratureX

(6)a = 1√2(X + iY ).

Optimum results are obtained forα∗ = −α. In such acase for small enough signalsφ � 1 we have

〈ψ |M|ψ〉 � 2√

2|α|3φ + · · · ,(7)〈ψ |M2|ψ〉 = 1

2+ · · · ,

so that

(8)�φ = �M

|d〈M〉/dφ| = 1

4|α|3 = 1

4〈N〉3/2 ,

where 〈N〉 = 〈α|a†a|α〉 = |α|2 is the mean numbeof photons of the state undergoing the transformatThis result implies a much better resolution than lintransformations for the same energy resources. Itworth pointing out that this is achieved using semclassical coherent states, which are much moreily prepared and far more robust than the nonclassstates required to reach the Heisenberg limit via linphase shifts. It can be appreciated that this strategnot optimal since the maximum of�G scales roughlyas�G ∝ 〈N〉2 (note thatG is positive). An optimalscheme is examined next.

G Stokes operator square, ψ number, M Stokes opera-tor square It can be expected that the use of nonclsical input states will improve the resolution furthTo show this let us consider two field modes, wcomplex amplitude operatorsa1, a2, and a generatoof transformationsG = S2

x , whereSx is the Stokes operator

(9)Sx = a†1a2 + a

†2a1 = a

†+a+ − a

†−a−,

where the modesa± are

(10)a± = 1√2(a1 ± a2).

The last equality in Eq.(9) shows thatG is the squareof the difference of photon numbers in modesa±.

As the measuring observable we considerM = S2z

where

(11)Sz = a†1a1 − a

†2a2,

while the input state is

(12)|ψ〉 = |n〉1|n〉2,

A. Luis / Physics Letters A 329 (2004) 8–13 11

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where |n〉j , j = 1,2, represent number states in tcorresponding mode[38–42]. Among other possibil-ities, a simple and efficient way to prepare the st(12) is via spontaneous parametric down-conversin a nonlinear crystal[16,38–45]. Since we are alwayinterested in the casen � 1 it is possible to approximateSx in the form

(13)Sx � n(E+ + E−),

where the action ofE± in the number basis is

(14)E±|m〉1|m′〉2 = |m ± 1〉1|m′ ∓ 1〉2,

for m,m′ > 1. Moreover, since we will be always deaing with m,m′ � 1 we can treatE± as if they wereeffectively unitary,E+E− � E−E+ � 1. With all thisin mind, for small enough phase shiftsφ � 1 we get

〈ψ |M|ψ〉 � 25n4φ2 + · · · ,(15)〈ψ |M2|ψ〉 � 29n4φ2 + · · · ,

so that

(16)�φ = �M

|d〈M〉/dφ| = 1

2√

2n2=

√2

〈N〉2 ,

where〈N〉 = 2n is here also the total number of phtons involved in the measurement. This resolutionproves with photon number faster than both the seclassical result(8) and the Heisenberg limit(4). More-over this scheme is optimal, since the maximum fo�G scales again roughly as〈N〉2.

G quadrature square, ψ quadrature squeezed, M

quadrature We can carry out a similar analysis finput–output transformations generated by powerCartesian operators. For exampleM = Y , G = Xk ,wherek is a natural number, andX, Y are the quadrature operators(6) for a single field mode of complex amplitude operatora. It holds that[X,Y ] = i,�X�Y � 1/2, and

(17)e−iφGYeiφG = Y + kφXk−1.

It must be pointed out that there is a limitation to tpower k that can be actually used, as discussedRef. [55], although this limitation does not affect uhere since we focus on the lowest ordersk = 1,2. Inorder to determine the analog of the Heisenberg lifor displacements we begin by considering the lin

casek = 1 [52]

(18)�φ = �Y � 1

2�X.

We optimize this uncertainty looking for the maximuof �X for a fixed mean number of photons〈N〉 � 1

〈N〉 � 1

2

⟨X2 + Y 2⟩

(19)= 1

2

(〈X〉2 + 〈Y 〉2 + (�X)2 + (�Y)2).

The maximum of�X for fixed 〈N〉 is �X = √2〈N〉

and occurs when〈X〉 = 〈Y 〉 = 0 and�Y � 1. Thuswe get the analog of the Heisenberg limit for displaments

(20)�φ � �φHL = 1√8〈N〉 ,

which scales as the inverse of the square root ofnumber of photons.

On the other hand, for a nonlinear displacemwith k = 2 and considering the optimum case of mimum uncertainty states�X�Y = 1/2 we get

(21)�φ = �Y

2〈X〉 = 1

4〈X〉�X.

Using again Eq.(19), the minimum value for the righthand side of Eq.(21) is obtained for〈Y 〉 = 0 and〈X〉 = �X = √〈N〉, so that

(22)�φ = 1

4〈N〉 ,which decreases with〈N〉 faster than the Heisenbelimit (20). This scheme is optimal by construction.

The optimum states in this last example are nclassical squeezed states. For the sake of compness, let us examine the resolution when the sarrangement is illuminated with semiclassical cohent states for which�X = �Y = 1/

√2. If 〈Y 〉 = 0,

we have〈X〉 � √2〈N〉 and �φ = 1/(4

√〈N〉). Thismeans that the semiclassical coherent states usednonlinear arrangement reach the optimum resoluof linear displacements. A similar result has beenrived for detection schemes based on nonlinear atfield couplings[52–54].

In conclusion, we have demonstrated that nonear transformations provide detection schemes mucmore precise than linear transformations even ussemiclassical coherent states. It is worth noting t

12 A. Luis / Physics Letters A 329 (2004) 8–13

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there is an extensive literature focusing on the usenonlinear transformations in order to generate the omum states for the detection of linear phase shifts[10,16,23–46]. In this work we have changed the persptive by proposing that these nonlinear transformatishould be used to encode the signal to be detectethe output state. This new perspective would greimprove the precision of the measurement. Concing the feasibility of these approaches with currtechnology, the basic building block is the implemetation of nonlinear input–output transformations. Tcan be easily achieved by propagation in nonlincrystals. For example, the Kerr effect provides the cG = (a†a)2. To be more specific, we can mentiothe giant nonlinear effects arising even at low pholevels in the context of electromagnetic induced traparency[56–58]. The measurements considered in twork (intensity and quadrature) are standard, andimprovement of precision can be demonstrated simby using semiclassical coherent input states.

Acknowledgement

This work has been supported by project PR1/11630 of the University Complutense.

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