Generalized minimum-uncertainty squeezed states

E. Shchukin,* Th. Kiesel, and W. VogelArbeitsgruppe Quantenoptik, Institut für Physik, Universität Rostock, D-18051 Rostock, Germany

�Received 21 December 2007; published 29 April 2009�

Generalized squeezed states, related to a broad class of observables, are analyzed. Methods for characteriz-ing the properties of such states are developed, which are based on numerical solutions of ordinary differentialequations. As typical examples we deal with nonlinear generalizations of quadrature squeezed states anddeformed nonlinear squeezed states, which may be useful for optimized measurements at a reduced level ofquantum noise. The realization of such states is studied for the quantized center-of-mass motion of a trappedion.

DOI: 10.1103/PhysRevA.79.043831 PACS number�s�: 42.50.Ct, 03.65.Wj, 42.50.Ar, 42.50.Dv

I. INTRODUCTION

It is a well-known fact that the ground-state and thevacuum noise levels of a harmonic oscillator and a mode ofthe radiation field, respectively, are required to fulfill theHeisenberg uncertainty relation. In fact, this noise level justdefines the minimum-uncertainty level of the quantum noisewhich is required to obey the uncertainty principle. It is im-portant to note that the uncertainty principle sets a limit forthe product of the variances of two observables, such as po-sition and momentum of a harmonic oscillator and two or-thogonal field quadratures for a radiation mode. Therefore itis a natural conclusion that one may find quantum states forwhich the noise in one of the chosen pair of observables isreduced below the vacuum �or ground-state� noise level, atthe expense of increased noise in the other observable �1,2�.Nowadays such states are usually called squeezed states. Inthe early days of their study also the notion of two-photoncoherent states was used �3,4�, since the structure of the uni-tary operator leading to such states is formally a two-photongeneralization of the coherent displacement operator.

More than two decades ago squeezed radiation could berealized experimentally �5�. There has been some interest inpossible applications of the noise reduction in a given ob-servable, for example, in the context of interferometric de-tection of gravitational waves �6,7�. It could be shown thatthe squeezing effect indeed improves interferometric �8,9�and spectroscopic measurements �10�. Very recently squeez-ing could be realized with a reduction of the noise power by10 dB �11�, which makes the squeezed states indeed usefulfor gravitational wave astronomy.

Based on this encouraging progress in the generation andapplication of the so-called quadrature squeezed states, it isof some interest to raise the question of whether one mayconsider useful generalizations of the concept of squeezedstates. For example, a generalization of squeezing has beenproposed which is based on the uncertainty relation of twogeneral noncommuting observables �12�. The further inves-tigation of squeezing in such a general sense is closely re-lated to the uncertainty relation of the chosen Hermitian op-

erators, say F and G, which reads as

�F�G �12 ���F,G��� , �1�

with �A= ���A�2�1/2 being the dispersion of A. States forwhich the uncertainty relation reduces to an equality havebeen called intelligent states �13�. These states also representa type of generalized squeezed state. Generalized squeezingof this kind has been considered for special choices of thebasic operators, such as amplitude-squared squeezing �14�and its higher-order generalizations �15,16�; for a review seealso �17�. Other generalizations of quadrature squeezingwere based on the consideration of higher-order moments�18�, and on higher powers of the annihilation or creationoperators in the squeeze operator �19�.

In the present paper we will consider intelligent states fortwo general noncommuting observables. Pure quantum stateswhich fulfill this requirement will be constructed as the so-lutions of an eigenvalue problem, which can be analyticallysolved in special cases, such as quadrature squeezing andamplitude-squared squeezing. For more general choices ofthe two Hermitian operators, we develop a systematic ap-proach to finding the intelligent states numerically in theFock-Bargmann representation. This method is applied toother types of states, such as generalized quadraturesqueezed states and deformed nonlinear squeezed states. Itallows one to obtain the properties of intelligent states for thechosen observables. This may provide a powerful tool foroptimizing a given measurement principle with respect to therelevant level of quantum noise in a chosen observable. Therealization of different examples of intelligent states is stud-ied for the quantized center-of-mass motion of a laser-driventrapped ion.

The paper is organized as follows. In Sec. II we considerintelligent states and characterize their properties by solu-tions of ordinary differential equations. As examples of suchstates, generalized squeezed states are studied in Sec. III. InSec. IV we deal with the experimental realization of intelli-gent states in the quantized center-of-mass motion of a laser-driven trapped ion. A summary and some conclusions aregiven in Sec. V.

II. INTELLIGENT STATES

Let us start with a brief discussion of the Fock-Bargmannrepresentation of pure states, which we need below. Any pure*evgeny.shchukin@gmail.com

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quantum state ���=�n=0+� cn�n� can be written as the action of

the operator ��a†�, which is a function of the creation opera-tor only, on the vacuum state ���=��a†��0�, where the func-tion ��z� is defined via ��z�=�n=0

+� cnzn /n!. This series con-verges, producing an entire analytical function. Thisrepresentation of quantum states by means of entire analyti-cal functions is referred to as Fock-Bargmann representation�20,21�. The normalization of a state ��� is given by thecondition �� ���=���z��2e−�z�2d2z=1. If an entire analyticalfunction ��z� satisfies a weaker condition, N−2

=���z��2e−�z�2d2z� +�, then the state ��� is not normalized.Its normalization is the number N defined by this integral.

The general uncertainty relation for two Hermitian opera-

tors F and G is given in Eq. �1�. In this paper we study thosestates for which the uncertainty relation reduces to the equal-ity

�F�G = 12 ���F,G��� . �2�

Such states are called intelligent states �13�. Here we con-sider pure states only. Any solution ��� of the eigenvalueproblem

�F + i�G���� = ���� , �3�

where � is a positive real number and � is arbitrary complex,is an intelligent state; cf. �22�.

In general, it is impossible to solve Eq. �3� analytically,but one can rewrite it as an ordinary differential equation,whereby making it possible to solve it numerically. To trans-form Eq. �3� to an ordinary differential equation, we use thefollowing relations for a coherent state ���:

���a† = �����, ���a = ��

2+

�

������� . �4�

We assume that the operators F and G are written in thenormally ordered form. From Eq. �3� we get the followingdifferential equation for the scalar product �� ���:

F���,�

2+

�

���� + i�G���,�

2+

�

���������� = ������ .

�5�

We can look for the solution �� ��� in the form �� ���=�����e−���2/2, whereby Eq. �5� can be simplified to an ordi-nary differential equation of a complex variable

F���,d

d��� + i�G���,d

d��������� = ������ . �6�

To get an ordinary differential equation of a real variable, letus represent �� in polar coordinates as ��=re−i. The un-known function ����� can be considered as a function of theradius r for a fixed phase according to �����=��re−i�=��r�. The derivative ���r� �with respect to r� can be cal-culated with the help of the standard chain rule

���r� =d�����

d��

d��

dr=

d�����d��

e−i. �7�

Thereby the derivative with respect to complex argument ��

is related to the derivative with respect to the radius �for thephase fixed� via d /d��=eid /dr. Now Eq. �5� can be writtenas an ordinary differential equation

F�re−i,ei d

dr� + i�G�re−i,ei d

dr����r� = ���r� ,

�8�

or, more precisely, as a family of equations parametrizedwith the phase � �0,2�. Solving this equation for allphases and combining the solutions ��r�, we get a solution����� of Eq. �6�. The initial conditions to Eq. �8� depend onthe phase , but the final solution ����� must be an analyti-cal function of ��. Thereby one cannot take arbitrary func-tions of as initial conditions to Eq. �8�. For ����� to beanalytical the initial conditions must be chosen as follows:�

�k��0�=cke−ik and k=0,1 , . . ., with ck being arbitrary com-

plex numbers. The deficiency of this method is that, in gen-eral, for arbitrarily chosen initial conditions, the solution����� we get in this way is not normalized, so one mustnumerically calculate the normalization constant.

III. GENERALIZED SQUEEZING

The relation between the dispersions �F and �G of astate satisfying Eq. �3� reads as �F=��G, so the parameter� plays the role of a degree of generalized squeezing. Unless�F=�G, exactly one of the following inequalities is valid:

��F�2 �12 ���F,G��� or ��G�2 �

12 ���F,G��� . �9�

A state that satisfies any of these inequalities was called gen-eralized squeezed state �12,23�. For 0���1 the first of in-equalities �9� is satisfied, and for ��1 the second one is. Thesolutions of Eq. �3� for �=1 are unsqueezed in the general-ized sense under consideration. Nevertheless, the resultingstates can be nonclassical ones.

Any pair of Hermitian operators F and G can be repre-

sented in the form F= f + f† and G=−i� f − f†�, respectively,

for a properly chosen operator f . This is easily verified by

just setting f = �F+ iG� /2. In all the examples considered be-low we use this representation with different choices of the

operator f .

A. Deformed nonlinear squeezing

Here we consider the case of the operator f of the form

f =g�n�a, where we assume g�z� to be real. Now eigenvalueproblem �3� reads as

��1 + ��g�n�a + �1 − ��a†g�n����� = ���� . �10�

To formulate the corresponding differential equation �8�, weneed to normally order the functions of the photon numberoperator. After straightforward algebra we get

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g�n� = �k=0

+���kg��0�

k!:nk

ª :�en�g��0�: , �11�

where � is the difference operator defined via ��g��n�=g�n+1�−g�n�. The powers of this operator, calculated at the ar-gument zero, read as ��kg��0�=�i=0

k �−1�k−i� ki �g�i�. Now we

can write differential equation �8� explicitly as

�k=0

+�

Ck�r�dk��r�

drk = ���r� , �12�

where the functions Ck�r��Ck�� ,r ,�, with k�1, are de-fined via C0�r�= �1−��g�0�re−i,

Ck�r� = �1 + ����k−1g��0�

�k − 1�!rk−1ei + �1 − ��

��kg��0�k!

rk+1e−i.

�13�

The initial conditions are given as before, but in this case thecoefficients of the equation are not constants and thus theinitial conditions cannot be chosen arbitrarily. It is shown inthe Appendix that there are the following relations for theinitial conditions to Eq. �12�:

k�1 − ��g�k − 1�e−i��k−1��0� − ��

�k��0�

+ �1 + ��g�k�ei��k+1��0� = 0. �14�

Usually �but not always, as we will see shortly� this meansthat the solution of Eq. �12� is unique. In the very specialcase of �=1 the solution of the Eq. �10� reads as �24,25�

��� = N��0� + �n=1

+�1

n!

��/2�n

g�0� ¯ g�n − 1��n�� , �15�

provided that all the numbers g�n�, with n=0,1 , . . ., are notequal to zero.

As an example let us take g�n�= n. In this case Eq. �10�reads as

�1 + ��reid2��r�dr2 + �1 − ��r2e−id��r�

dr= ���r� .

�16�

Constraints �14� on the initial conditions in this case simplyread as ���0�=0. Thereby if ��0 then ��0�=0 and if �=0 then both the derivatives ��0� and ���0� can be arbi-trary. Since it is a second-order equation and if one of itsinitial conditions is fixed, then its �normalized� solution isunique. The Q function of the solution is shown in Fig. 1. Inthe case of �=1 it is possible to solve Eq. �16� analytically,but we cannot directly use the expression given in Eq. �15�since in this case we have g�0�=0. If ��0 then the solutionfor �=1 reads as ���= 0F2�;1 ,2 ; ���2 /4�−1/2

� 0F1�;2 ;�a† /2��1�, where 0F1�;b ;z� is the regularized con-

fluent hypergeometric function defined via 0F1�;b ;z�= 0F1�;b ;z� / �b�, where 0F1�;b ;z�=�k=0

+� zk / ��b�kk!�. If �=0then the solution is just a linear combination of the first twoFock states ���=c0�0�+c1�1�; hence the solution is notunique.

B. Nonlinear quadrature squeezing

In the following we present a nonlinear generalization ofquadrature squeezed states and we prove their nonclassical-

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(a) (b) (c)

(d) (e) (f)

FIG. 1. �Color online� The Q functions corresponding to the solutions of Eq. �16�, with initial conditions ��0�=0 and ���0�=e−i, for�=1 and different values of �. �a� �=0.2 �b� �=0.5 �c� �=1 �d� �=2 �e� �=5 and �f� �=10. Dark color corresponds to small values of Qfunction and bright color corresponds to large values.

GENERALIZED MINIMUM-UNCERTAINTY SQUEEZED STATES PHYSICAL REVIEW A 79, 043831 �2009�

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ity. Consider the case of the operator f being a function of

the annihilation operator only, f = f�a�. In order to not over-load the notation and without loss of generality, we assumein the following the function f�z� to be real �i.e., for realargument z its value f�z� is also real�, which implies that therelation f�a�†= f�a†� is valid. Let us write the generalizedsqueezing conditions given by inequalities �9� explicitly. Thevariance ��F�2 reads as

��F�2 = ��� f†�2 + �� f�2 + � f†� f + � f� f†� , �17�

and for the commutator of F and G we have the equality

��F , G��=2i�� f , f†��. In our case the relation �f , f†�=�k=1

+� f �k�†f �k� /k!, guarantees that the latter commutator is al-

ways non-negative: �� f , f†��= ��� f ,� f†���0. Now the first

inequality �9� can be written as ��� f†�2�+ ��� f�2�+2�� f†� f��0. Since f = f�a�, the left-hand side of this in-

equality is just the normally ordered mean value �:��F�2:�,so the first condition �9� reads as �:��F�2:��0. From thesecond inequality in Eq. �9� we can conclude in the same

way that �:��G�2:��0. We see that generalized quadraturesqueezing always implies nonclassical behavior.

Eigenvalue problem �3� reads as

��1 + ��f�a� + �1 − ��f�a†����� = ���� , �18�

and the corresponding differential equation �8� is of the form

f�ei d

dr���r� = �� − 1

� + 1f�re−i� +

�

� + 1���r� . �19�

In general, for an arbitrary function f�z�, it is impossible tosolve this equation analytically. But in a special case of �=1 it is possible to find the general solution, provided thatwe know how to find the roots of entire functions. In the case

of �=1, Eq. �18� simply reads as f�a����=����, where �=� /2. Each root � of the equation f���=� gives a partialsolution of the form Pk�−1�a†−�����, where k� is the multi-plicity of the root � and Pk�−1�z� is an arbitrary polynomialof the degree k�−1. In particular, any simple root �of multi-plicity 1� gives the solution which is proportional to the co-herent state ���. The general solution ��� of Eq. �18� for �=1 is a linear combination of all the partial solutions ���=� f���=�Pk�−1�a†−������, where the sum here is taken overall the roots of the equation f���=�. If all the roots of thisequation are simple, then the general solution is just a linearcombination of coherent states. Below we consider onlypolynomial functions f�z�.

It is clear to see that Eq. �19� with a first-order polynomialf�z� corresponds to the definition of quadrature squeezing,and it is easy to prove that a second-order polynomial leadsto the definition of amplitude-squared squeezing. That is whyit makes sense to consider polynomials of third and higherorders. As an example, let us consider the polynomial f�z�=z3+z. Differential equation �19� with this polynomial readsas

e3i���r� + ei���r� = �� − 1

� + 1�r3e−3i + re−i� +

�

� + 1���r� .

�20�

The contour plot of the Q functions obtained from solutionsof this equation are shown in Fig. 2, for special values of thesqueezing parameter �.

IV. REALIZATION

In this section we demonstrate how the states discussed inSecs. II and III can be realized experimentally as the quan-tum states of the center-of-mass motion of a laser-driven

�3 �2 �1 0 1 2 3�3

�2

�1

0

1

2

3

Re�Α�

Im�Α�

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2

3

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Im�Α�

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1

2

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Im�Α�

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�2

�1

0

1

2

3

Re�Α�

Im�Α�

(a) (c)

(d) (e) (f)

FIG. 2. �Color online� The Q functions corresponding to the solutions of Eq. �20� with the initial conditions ��0�=1, ���0�=0, and���0�=e−2i for �=5 and different values of �. �a� �=0.2 �b� �=0.5 �c� �=1 �d� �=2 �e� �=5 and �f� �=10. Colors are used as in Fig. 1.

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trapped ion. Our approach is based on the techniques pro-posed for preparing nonlinear coherent states �25� and evenand odd coherent states �26� in the form of motional darkstates. The basis ��i ,�i�� of the Hilbert space of quantumstates of the ion is described by two electronic states i=1,2and the motional energy levels �i�N. By exciting a set ofvibronic transitions by lasers and choosing appropriateLamb-Dicke parameters and Rabi frequencies, we obtaineigenstates of the desired kind. Each laser, tuned to the elec-tronic transition on the nth blue �red� motional sideband,creates �annihilates� n vibration quanta during electronic ex-citation of the ion. For the blue-detuned excitation the inter-action part of the Hamiltonian reads as �27�

Hint,n =�

2A21�e−�2/2�

k=0

+��i��2k+n

k ! �k + n�!a†k+nak + H.c. �21�

For red detuning we have to replace A21↔ A12. Here Aij isthe flip operator for the transition between the electronicstates �i�→ �j�, � is the Rabi frequency, and � is the Lamb-Dicke parameter. The operators a , a† represent the annihila-tion and creation operators of the vibrational levels.

For sufficiently small Lamb-Dicke parameters, only a fewleading terms in this infinite sum must be considered. In thefollowing, we will concentrate on this regime. The time evo-lution of the system is determined by the master equation

d�

dt= −

i

��Hint, �� +

2�2A12�A21 − A22� − �A22� , �22�

where the last term accounts for spontaneous emission withan energy relaxation rate . We are interested in the station-ary state ��� of the vibrational motion of the laser-driven ionin the electronic ground state: �s= �1��������1�. When thesystem is in such a state, the ion cannot emit photons any-more. Hence a state of this type is called a motional darkstate. The remaining condition for the stationarity of the statereads as

Hint�1���� = 0, �23�

where the interaction Hamiltonian is usually composed oftwo or more different interactions of the type given by Eq.�21�. A typical excitation scheme, which is considered below,is shown in Fig. 3.

A. Deformed nonlinear squeezed states

For the realization of states obeying eigenvalue equation�10� with g�n�= n, we excite the pure electronic transitiontogether with the two first sidebands; see Fig. 3. However, inthe present case we need two lasers with different Lamb-Dicke parameters and Rabi frequencies for each sideband tocompensate for the leading �n-independent� term propor-tional to a and a†. This idea of using different lasers on thesame sideband was introduced for the engineering of the vi-bronic coupling Hamiltonian of a trapped ion �28�.

For small Lamb-Dicke parameters, this leads to the inter-action Hamiltonian

Hint ��

2A21 i�

j=1

2

� j−� je

−�j2/2�1 −

� j2

2a†a�a

+ i�j=1

2

� j+� je

−�j2/2a†�1 −

� j2

2a†a� + �0e−�0

2/2� + H.c.

�24�

Here � j� are the Rabi frequencies of the lasers exciting the

red �−� and blue �+� first sidebands; the index j=1,2 labelsthe values of the Lamb-Dicke parameters � j. In addition, onthe pure electronic transition the Rabi frequency and theLamb-Dicke parameter, �0 and �0, respectively, can be cho-sen independently.

The stationary state of the system fulfills the eigenvalueequation

��1−na + �1

+a†n���� = 2�0e��1

2−�02�/2

i�1��12 − �2

2���� , �25�

which is obtained from Eq. �23� together with Eq. �24�, andcompensation of the leading terms of the sideband contribu-tions by fixing the parameters such that the equality

�1��1e−�1

2/2=−�2��2e−�2

2/2 if fulfilled.This can be directly converted into Eq. �10� with g�n�

= n; the parameters � and � are given by

� = −4ie−��0

2−�12�/2

�1��12 − �2

2��0

�1− + �1

+ , � =�1

− − �1+

�1− + �1

+ . �26�

Thus we have also demonstrated the possibility of preparingthe nonlinear deformed squeezed states as motional darkstates of a trapped ion.

B. Limitations and generalizations

In this section we have shown how one can prepare theintelligent—or generalized squeezed—states discussed inSec. III A. To derive exactly these states, some limitations ofthe Hamiltonian of the laser-ion interactions have been used,such as operation close to the Lamb-Dicke regime and aproper engineering of the interaction Hamiltonian. In factthis limits the accessible parameter ranges of the states understudy to some extend. In this context a more detailed analysiswould be required in order to prepare such states in an ex-periment. We did not consider this problem here, since thiswould depend on details of the used experimental setup, for

FIG. 3. �Color online� Excitation scheme for the realizations ofdeformed nonlinear squeezed states.

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example, on the available range of Lamb-Dicke parameters.However, since the approximations in the Hamiltonian havebeen discussed in detail, the situation is well defined.

Today the experimental preparation and examination ofmotional states of trapped ions are well-established fields.One is able to generate well-defined states, e.g., the motionalground state �29� as well as Fock states and superpositions ofcoherent states �30,31�. Decoherence effects are often negli-gible or can be controlled by an engineered environment�32�. The reservoir engineering could also be demonstratedin experiments �33�. Furthermore, measurements on the statecan be performed with high efficiency �34�. The experimen-tal techniques in the above papers typically use trapped ionswhich are driven in the resolved sideband regime. This is thefirst important requirement for the state preparation dis-cussed in the present paper. In the experiments on sidebandRaman cooling �29� and reservoir engineering �33�, also thedissipation effects are realized, which are the second require-ment for preparing the motional dark states we are interestedin. Altogether, all the experimental techniques needed for thepreparation of the generalized squeezed states under studyhave been realized already in some other context. Last butnot least, the quantum state diagnostics can also be performin the resolved sideband regime �34�, in order to demonstratethe properties of the prepared states.

Beside the mentioned restrictions of the accessible param-eter range, the ion-trap experiments would also open a mani-fold of possibilities to study much more general types ofsqueezed states. This becomes clear by inspection of the gen-eral form of interaction Hamiltonian �21� of a laser which isresonant to the nth sideband. By Hamiltonian engineering,using more lasers with different Lamb-Dicke parameters, itis possible to generate rather general dependences of the cou-pling strength on the vibrational number operator n; for moredetails see Ref. �28�. In such cases of complex interactionsthe characterization of the states under study by solving theresulting differential equations would be a demanding task.The trapped-ion experiments open a huge variety of possi-bilities to prepare such kinds of generalized squeezed states,leading to a quantum noise reduction for a manifold of ob-servables.

V. SUMMARY AND CONCLUSIONS

In the present paper we have studied intelligent states orgeneralized squeezed states, which fulfill the uncertainty re-lation for two general Hermitian operators with the equalsign. The problem of finding the related generalizedsqueezed states has been reduced to solving ordinary differ-ential equations which are obtained in the Fock-Bargmannrepresentation. These equations can be solved analyticallyonly in some special cases, such as quadrature squeezing andamplitude-squared squeezing. For more general cases wehave developed techniques to solve the differential equationsnumerically.

To illustrate the power of our approach, we have studiedtwo examples of generalized squeezed states of types. Thefirst type is a nonlinear generalization of the quadraturesqueezed states, where in the definition of the quadrature

operators the annihilation operator is replaced by a functionof the latter. The second type is a deformed nonlinearsqueezed state, which is defined on the basis of the quadra-tures of a deformed algebra. For both types of generalizedsqueezed states, we illustrate their properties by calculatingthe phase-space distributions, i.e., the Q functions.

After illustrating the properties of the intelligent states,we have also studied the possibilities of their experimentalrealization. In particular we have shown how the consideredstates can be prepared as stationary states of the quantizedcenter-of-mass motion of a laser-driven trapped ion. Depend-ing on the chosen laser configuration driving the ion, a mani-fold of intelligent states can be prepared in this way.

In conclusion, we have studied several types of intelligentor generalized squeezed states. The reduction in the quantumnoise level of such states is related to different types of Her-mitian operators. In this sense these quantum states may beuseful if one wants to improve special measurementschemes. It may be of some interest to find the best squeezedstates in relation to the observable to be detected by a givendevice. When knowing the measurement scheme, ourmethod is useful in characterizing the squeezed states whichare adjusted to the observation scheme. This may be a steptoward the preparation of these states and their applicationsfor optimized measurements at a reduced level of quantumnoise.

APPENDIX: THE RELATIONS FOR THE INITIALCONDITIONS

In this appendix we prove relation �14� for the initial con-ditions for Eq. �12�. It is easy to see that for the derivativeswe have Ck

�k−1��0�= �1+����k−1g��0�ei, Ck�k+1��0�= �k+1��1

−����kg��0�e−i and all other derivatives are equal to zero.Differentiating Eq. �12� m times at r=0 we get the followingrelation:

�k=0

+�

�j=0

m �m

j�Ck

�j��0���k+m−j��0� = ��

�m��0� . �A1�

The left-hand side of this relation is

�j=0

m �m

j��Cj+1

�j� �0���m+1��0� + Cj−1

�j� �0���m−1��0�� . �A2�

The coefficient in front of the derivative ��m+1��0� can be

further simplified as follows:

�j=0

m �m

j�Cj+1

�j� �0� = �1 + ��ei�j=0

m �m

j��� jg��0�

= �1 + ��ei�Emg��0� = �1 + ��g�m�ei,

�A3�

where E=1+� is the step operator which acts as �Eg��n�=g�n+1�. In the same way one can show that the coefficientin front of �

�m−1��0� is � j=0m � m

j �Cj−1�j� �0�=m�1−��g�m−1�e−i.

This completes the proof of relation �14�.

SHCHUKIN, KIESEL, AND VOGEL PHYSICAL REVIEW A 79, 043831 �2009�

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