Continuous Measurements of Quantum Phase

Transcription

1 Continuous Measurements of Quantum Phase V.P.Belavkin Mathematics Department University of Nottingham, UK C.Bendjaballah Ecole Superieure d'electricite, Gif sur Yvette, France 3 May 199 Published in: Quantum Opt. 6, (1994) 169{186 Abstract The problem of phase measurement for a quantum oscillator using the quantum nondemolition principle (NDP) which makes the selfadjoint operator and wave packet reduction postulate unnecessary. It is shown that this approach is an appropriate one, not only in order to introduce in a proper way the unsharp measurements of the quantum phase, but also to describe the relevant quantum jumps under such measurements. Our analysis suggests that in contrary to other quantum measurement problems (e.g. the position and the momentum of a particle), the operator to represent the phase of a quantum eld cannot be separated from the quantum measurement process. Moreover, it allows us to describe the time continuous process of quantum phase jumps and also to study the stochastic dynamics of spontaneous localization for the oscillator under phase measurement. Introduction The problem of a phase operator, obeying the laws of quantum mechanics, has motivated several physicists since Dirac's unsatisfactory assumption of the existence of a Hermitian phase operator conjugate to the number operator. Numerous studies have been devoted to this subject and recent analysis 1

2 in connection with the study of squeezed states of light, has renewed the interest in the quantum phase operator measurements. The rst point we may wonder what the quantum phase operator, discussed in [?], is about, specically, how is it related to the quantum phase observable introduced in [1]. We may also consider to nd which quantum phase distribution among those considered [?] corresponds to the observation of phase of a quantized light eld. Which scheme of quantum phase measurements is optimal with respect to a given criterion say for optical communication? Another question of interest in relation with the quantum phase jumps, is to describe the state of the quantum oscilator after its phase has been measured. Among the dynamical problems not yet solved for quantum phase there are following questions: Is there an interaction quantum mechanical model to reproduce the quantum phase jumps and the relevant phase probability distributions? How to describe the sequential and time continuous quantum phase measurements? What is happenning with the quantum oscillator in the process of the phase measurements? Is there a general physical principle or reduction equation which plays the role of the projection postulate for the timed measurements of such unusual observable as the quantum phase? All these questions are essential from the theoretical point of view and deserve to be studied from a practical point of view. The aim of this paper is to show how they can be solved in the developed framework of the unsharp nondemolition measurement. Although it seems that there is no Hermitian operator that directly corresponds to the phase of quantum electromagnetic eld, it is known that measurements of the phase (actually phase dierence or phase jump) in quantum optics do not involve such fundamental problems. There are several experiments in which the quantum phase can be measured indirectly. Furthermore, in order to apply techniques such as heterodyning, to use squeezed light or the phase shift keying modulation in optical space communications, it is important to compare the dierent models of indirect phase measurements and to determine which one is the most sharp and most adapted to quantum optical communications, for instance. The diculty of dening an appropriate quantum phase operator is not the reection of the impossibility of quantum phase measurements but is simply related with the asence of a dynamically conjugated operator to the photon number which would generate the shifts of this quantum number.

3 Indeed, the elementary shift n 7! n + 1 of n = ; 1; ; : : : cannot be inverted on + = f; 1; ; : : :g and hence it does not generate a unitary group but only the semigroup jni 7! jn + si, s + which does not posses a selfadjoint generator in the Hilbert space H of photon states. This situation would occur for the momentum operator of a quantum particle if its coordinate operator bx had its domain of variation restricted, say to the positive half{line R +. The translation generator bp = ih d dx of one parameter shift semigroup of the isometries e i h sbp, s R + would not be a selfadjoint operator in the Hilbert space H = L (R + ), but it does not mean the absence of a possibility to measure the momentum of a quantum particle in R +. This means that a proper representation of quantum phase must be made in terms of the quantum measurement involving a nonunitary scattering matrix for the isometric shifts of quantum states due to object{meter interaction rather than in terms of selfadjoint generators. This can be realized within the quantum theory of counting nondemolition measurements [3] dening both a probability distribution of the measurement and collapsed states after the measurement by generalized reduction transformations. Notice that due to the continuity of the phase spectrum [; [, the collapsed states couldnot be found by application of von Neumann projection postulate even if the phase operator had existed in the usual sense as a momentum operator bp with such spectrum for a quantum particle in the discrete space. Before showing how to solve this problem for the quantum phase, let us illustrate the nondemolition measurement scheme for a simple case of Hermitian operator bp generating the cyclic unitary shift e ibp ji = jsi; e ibp jni = jn 1i; n = 1; : : : ; s in the s + 1{dimensional space H = f P s n= c n jnig where jni are the eigenvectors for the values n = ; 1; : : : ; s of the number operator. The operator bp considered in [?] as a discrete model of oscillator phase has the normalized eigenvectors # = (s + 1) 1= P sn= e in# jni corresponding to the eigenvalues # = l, l = ; : : : ; s. s+1 The nondemolition measurement of bp is described by the pointer observable bq that is the coordinate operator of a meter acting in a copy E = f P s k n= k a njnig of H exactly as the operator bp acts in H: e ibq j ki = js ki; e ibq jni = jn 1i; n = 1 k; : : : ; s k: 3

4 Here k is any integer which one can take as k > s in order to have have the positivity of the pointer momentum M : jni 7! njnjrangle ine, generating the shift e i#m = # (modulo ) of the eigenvectors = (s + 1) 1 sx k n= k e in jni; = ; "; : : : ; s" for the discrete coordinate bq with " =. The interaction of the quantum s+1 oscillator and the meter, initially independent, is described by the scattering operator S = e ibpm in the product space H E. The unitary operator S shifts the initial value of the pointer bq to the value = # + (modulo ) corresponding to the eigenvalue # of the phase bp by the Heisenberg transformation S y (1 bq) S = bp bq (modulo ). Hence, if the initial state of the meter is xed as the eigenvector = (s + 1) 1= P sn= jni corresponding to = and is the reading of the pointer bq after the interaction, then we will denitly have # =. This means that the quantum state of the oscillator after such measurement is given by the eigenvector of bp for any initial state vector Hof the oscilator. It can be obtained [?] by the normalization P =kp k of the orthogonal projection P = 1 s + 1 sx k n= k e i(bp )n = 1 s + 1 sx sx c m e im e in jni = h ji m= n= given by the partial matrix element P = h jsj i of the scattering operator for any = P s n= c n jni. Thus, the projection postulate appears as a conjecture of the possibility of choosing an eigenvector of bq as the initial state of the meter and of the ltering by after the interaction accordingly to the pointer reading. In this paper, most of the questions of fundamental interest will be considered. We will try to nd an answer which is as complete as possible. More specically, we show how the non{demolition principle of the quantum measurements theory [?], free of a particular type of reduction postulate, helps to solve these problems, exactly in the same way as they have been solved for the ordinary quantum observables like the position of a quantum particle. As a result we obtain a stochastic wave reduction equation for the state vector of the quantum oscillator under the time continuous phase observation. In contrast to the stochastic diusive models of continuous reduction and 4

5 spontaneous localizations [?, 4, 8], the continuous phase reduction equation is governed on the individual level by not the Wiener but the Poisson process which can even have singular intensity distribution on the phase space [; [. 1 The quantum phase operators Let us consider a quantum system having a purely discrete energy spectrum, a quantum harmonic oscillator for instance, with the countable family of eigenvectors jni (n = ; 1; : : :). We represent the state vectors of such a system in the Hilbert space H of the linear combinations P 1 n= c n jni, where fc n g are complex coecients such that kk = P 1 n= jc n j < 1, by the {analytic functions (z ) = P 1 n= c n z n in the unit disc jzj < 1, similarly to [?,?]. Each function is completely dened by the boundary values h(#) = lim r"1 (re i# ) and the scalar product hji = R (e i# ) (e i# ) d# is considered as the correlation (hjf) = hh fi of h(#) with another such a function f(#) = lim r"1 (re i# ) respectively to the shift invariant (mod. ) probability measure d#. The z {transformation fc n g 7! (z) constitutes a phase representation via the natural embedding (z) 7! (e i# ) = P 1 n= c n e in# of space H into space E of all periodic functions f(#) modulo with hjfj i = 1 R jf(#)j d# < 1. A periodic wave function f(#) = P 1 n= 1 a n e in# represents a vector H only if it is analytic in the complex domain # C, =# < such that (z ) = f(i ln z), 8 jzj 1 is its z {transformation. In this enlarged space E, there exists a self{adjoint phase operator b' dened by the multiplication [ b'f](#) = '(#)f(#) for '(#) = # b # c with the generalized eigenvectors j#i of the spectral decomposition, b' = 1 R j#i # h#jd#, where bxc is the integer part of x R and h#j is the evaluating functional: f 7! f(#) dened for any periodic function f(#). This operator does not leave the subspace H invariant due to the non{ analyticity of '(#) at =# < but the unitary operator e ib' functions as an isometry [ y ](z) = z (z) in H because [e ib' h](#) = e i# h(#) is analytic and e ib' jni = jn + 1i for all n. Note that the operator y is not adjoint to the inverse operator ^z = e i ^' which maps H onto the bigger subspace H C E but to the non{ unitary operator = E^z given by the orthoprojector E in E onto H, and known as the exponential phase operator [](z) = 1 ((z) ()) [?]. The z latter operator is not normal because the commutator [ y ; ] = jihj is not 5

6 zero, but it has the overcomplete family fjz)jjzj < 1g of non{orthogonal left eigenvectors in H 1X jz) = z n jni : jz) = jz)z (1.1) n= describing the unsharp phase vectors Ej#i as the limits je i# ) = lim r"1 jre i# ). The corresponding positive operator{valued measure = f g, dened for any Borel subset [; [ as = 1 je i# )(e i# jd# : 7! 1 je i# )h(#)d# with (e i# j = h(#) = h#jh for any H, is normalized to the identity operator I = [;[ in H, and describes the probabilities Pr(# ) = hj ji of an unsharp measurement of the oscillator phase in an initial state H. This measurement gives the best results among all the unsharp measurements of the phase parameter of a quantum coherent state under the maximum likelihood criterion, as was rst shown in [1] and also in [9,?,?]. However, the nonorthogonal vectors je i# ) = Ej#i are unnormalizable because (e i# je i# ) = lim r"1 (1 r ) 1! 1. Hence they are not in the Hilbert space and cannot be regarded as the vectors of a posteriori states, i.e. the states after the phase measurements, or even proportional to such vectors. A family f # : # [; [g of normalized vectors # H; k # k = 1 is called a posteriori state vectors if there exists a probability amplitude c(#), R jc(#)j d#= = 1 such that the vector{function (#) = c(#) # reproduces the state-vector of the combined system "ocilator plus phase meter" in the pointer representation after the interaction. According to the vector superposition principle, there must be a complete family fg(#) : # [; [; G(#) y G(#)d# = Ig of linear transformations G(#) in H giving the vector{function (#) = G(#). We say that the reduction transformations G(#)) satisfy the compatibility condition with respect to the phase vectors if they commute with these vectors in the sense that each (zj intertwines them with c-numbers g z (#): (zjg(#) = g z (#)(zj; 8 # [; [; jzj < 1 (1.) 6

7 Now we can prove that the ideal phase measurement demolish the quantum system because there is no way to get the posterior state vectors using the projection or any other reduction postulate G(#) : 7! (#), which would be compatible with such measurement. Indeed, the normalized posterior state vectors # = (#)=c(#)are uniquely dened up to Arg c(#) by the square root jc(#)j of the probability density jc(#)j = 1 jg z (#) j j (z) j dz (1.3) i jzj=1 for the quantum phase measurement respectively to the measure d#. As follows from (1.), the reduction transformations G(#) must be diagonal [G(#)](z) = (zjg(#) = g z (#)(z) ; jg z (#)j d# = in the phase representation, corresponding to jzj " 1. However, as one can easily see from (1.3), this condition is in contradiction with the reproducibility condition jc(#)j = jh(#)j of the probability densities jh(#)j corresponding to the best unsharp phase measurement (e i# j = h(#) = (e i# ). This is because there is no square integrable complex function # 7! g z (#) on [; [, with jg z (#)j = 1 (# Arg z) for jzj = 1 and hence (1.3) does not hold for all H and jc(#)j = j(e i# )j. This diculty which makes all attempts to study a sequential process of such demolition measurements meaningless, is not exceptional for the phase. It is inseparable from any quantum sharp measurement of usual observables like a position with a continuous spectrum. In order to obtain a reduction transformation for the sequential or even time continuous phase measurements satisfying condition (1.), it is necessary to consider a scheme with indirect phase observation or a discretization of the phase spectrum as was proposed in [?]. According to the NDP theory [?], we must consider another quantum system to describe an apparatus with a given pointer observable whose spectral values [; [ are rst assumed as discrete ones f; "; : : : ; s"g s, where s k and " =, k is any integer. The minimal (s + 1){ s+1 dimensional Hilbert space E s for this apparatus consists of the set of all discrete complex functions f : s! C which can be considered as the restrictions on s [; [ of the trigonometric functions f() = P s k n= k a ne in. 7

8 This constitutes the isometric embedding sx E s E; jf()j d = jf("l)j " l= into the Hilbert space E = L () of all square{integrable functions on = [; [ given by the interpolation formula f() = P s l= l " ()f("l)", where l " () stands for 1 P s k n= k ein("l ). We assume that at the initial time, the oscillator and the apparatus are independent and the state of the apparatus is xed g K, kgk = 1, in the (k + 1){dimensional subspace K = E k of the trigonometric polynomials g() = P k m= b m e im. The compound system at this stage is described by the product state vectors g, in H K so that their joint wave function (z; ) in the (z; ){representation (jzj 1; ), is initially written as (z; ) = (z)g(). At the instant of the measurement, the oscillator and the apparatus are coupled by the scattering operator S = exp( i b' M) where M is the angular momentum operator i d in E, having the simple nite spectrum f; 1; : : : ; kg on the invariant subspace K E, because d Mjni = njni, 8 n. Thus dened in E E, the operator S is also unitary in the invariant Hilbert subspace E K and it is isometric on H K E K as follows from the phase{momentum compatibility property (e i# j hnjs = (e i# je ib'n hnj = e in# (e i# j hnj (1.4) for n = ; 1; : : : ; k as a result of the positivity of the spectrum m = n of M = P n jnimhnj in K and the property (e i# je ib' = e i# (e i# j (1.1). Note that although the unitary scattering S t = 1 t S, where 1 t = if t and 1 t = 1 if t >, admits the formal Hamiltonian h b' (t)m in the extended space E K, the correspondent isometry ym = Sj(H K) has no Hamiltonian as being adjoint to the partial isometry M = P n jni n hnj. Let us denote by fji : s g, ji = P s k n= k ein jni, s, the orthogonal family of eigenvectors for the diagonal operator [^qf]() = f() in E s of the discrete angle s, considered in [?] as an approximate nite dimensional phase operator. The partial matrix elements G () = hjsg = g ( ^') ; s (1.5) for a given g K, dene the linear reduction transformations H! H, [G () ](z) = g ( i ln z ) (z) (1.6) 8

9 satisfying the nondemolition condition (1.) with the trigonometric polynomial function g() = g( #) for all z = e i#, s. If the initial state vector g of the meter was localized at the point =, then the posterior state vectors = G() in the phase representation x c() (#) = lim r"1 (re i# ) are given by the localization x (#) = g( #)h(#) c() (1.7) of the initial wave function h(#) = lim r"1 (re i# ) around the result of the measurement s with c() = kg()k. This result also holds for the phase measurement with the continuous spectrum = [; [, because the reduced description of the non{demolition measurement in terms of the transformations (1.5), only depends on the initial state-vector g of the k +1 dimensional space K and does not depend on the dimensionality s + 1 of the space E s so that the continuous limit s! 1 appears straightforward. Note that the periodic function g() = P k m= b m e im, representing the initial state vector g K on the measurement scale s, can be sharply localized at a point, say = if and only if s = k and k < 1. The unsharply localized g() denes the (mod ) additive noise s to the quantum phase # under the nondemolition measurement = # + in the sense that the probability density (1.3) is given by the convolution jc()j = 1 jg ( #) j jh (#) j d# (1.8) of the periodic distributions jg () j and jh (#) j. As an example, let us consider an initial state vector g E s, (k = s), specied by the following limiting cases: a): s < 1, r! 1 and b): s! 1, r < 1 of the geometric sequence b m = q 1 r r m, m = ; : : : ; s. The 1 r (s+1) reduction transformations (1.5) in the phase representation, are dened by v u g( #) = t 1 r 1 r# s+1 ; r 1 r (s+1) # = re i# 1 r # The conditional probability density jg( #)j is given by jg()j = 1 r 1 r cos + r 1 r s+1 cos(s + 1) + r (s+1) 1 r (s+1) (1.9) 9

10 which is localized on s at the point = if r < 1, with P s jg()j = s+1. The characteristic sequence c # s (m) = 1 P s+1 s jg( #)j e im, ( m s) of this distribution, has the form of the convolution (b # b # ) m = mod (s+1) X k+n=m of b # n = b n e i#n and (b # n) = b # n, so that s m c # s (m) = e i#m X = = n= 1 r mod(s+1) X 1 r (s+1) 1 r(s+1 m) (b # n) b # k m 1 b nb n+m + e i#(s+1 m) X k+n=m r n # rk # r m 1 r (s+1) # + 1 rm 1 r b n+s+1 n= (s+1) rs+1 m mb n # (1.1) Since c # s (m) = e i#m for r = 1 and # s, the conditional distribution jg( #)j is sharply localized on s at # = only in the limit case a) and it is almost sharply localized in case b) if " = 1 r 1. This is because c # (m) = lim s!1 c # s (m) = r# m ' (1 m")e i#m for all m = ; 1; : : :. The sequential phase measurements Now let us x a countable set = ft 1 ; t ; : : :g of time instants ft 1 < t < : : :g for the nondemolition phase measurements which are described as previously in terms of the independent meters at the dierent t. We shall suppose that t = \ [; t[ is a nite subsequence for each t < 1, i.e. j t j < 1, where jj is the cardinality n = ; 1; : : : of = ft 1 ; ; : : : ; t n g with n = corresponding to the empty sequence = ;. Let us denote by (): 7! () a partition = X s () = [ s (); () \ ( ) = ;; 8 6= into the subsets () = ft n j n = g, corresponding to the xed values s in a sequence ( 1 ; ; : : :) of the measurement data n s at the 1

11 given time instants t n and by n t () = j t ()j the corresponding cardinalities of t () = () \ [; t[, dening a number distribution n t () : 7! n t (); X s n t () = n t on the nite subset s [; [ for each t < 1. The partitions () describe the graphs = fy 1 ; y ; : : :g s ; y n = (t n ; n ) of the maps y : t n 7! n s, giving a measurement result y(t) = at a time instant t R + if and only if d t n() = 1, where d t n() = n t+dt () n t (), < dt < t n+1 t n for each t [t n ; t n+1 [. In these terms one can dene the stochastic propagator T (t; ) : H! H, corresponding to a random sequence t = (y 1 ; y ; : : : ; y n ) of the events y t s for the sequential phase measurements up to a time instant t [t n ; t n+1 [ as a chronological product T (t; ) = U(t t n )G( t ); G( t ) = G(y n ) : : : G(y 1 ) (.1) Here U(t) : H! H is the unitary time{evolution operator of the quantum harmonic oscillator 1X U(t) = jnie i!nt hnj = e i!nt (.) n= where N is the photon number operator. (1.5) are taken in the Heisenberg picture The reduction transformations G(t; ) = U(t) y G()U(t) = g I i ln (t) y (.3) where we have (t) = U(t) y U(t) = e i!t as follows from property (1.1) and also U(t)jz) = jze i!t ). Such oscillating operators G(t; ) = G( +!t) commute for the dierent values of (t; ) R + s due to the commutativity of the operators G() = g I i ln y at dierent s. The reduced state vector (t; t ) = T (t; ) is not normalized for a xed sequence of measurement events y = (t; ), but it is normalized in the mean square sense hk (t) k i o = kg ( t ) k P o (d t ) = 1; t 8 : kk = 1 (.4) 11

12 with respect to P (d t ) = (d t ) (s + 1) nt which is the probability measure of d t = d t f 1 ; : : : ; n g. The measure P is dened on t = P n t(n) n s by any probability measure (d) on the space t = S n t (n), the union of the spaces t (n) of all nite = f t 1 < : : : < t n < tg; n = ; 1; : : : with = ; for n =, and = t for n = 1. Indeed, the integral over t can be written, by denition, as an integral over t kg ( t ) k P o (d t ) = t 1 X kg (t t (s + 1) nt nt ; nt ) : : : G (t 1 ; 1 ) k o (d t ) (.5) s and as X s kg (t n ; n ) : : : G (t 1 ; 1 ) k = (s + 1) n kk kg (t n ; n ) : : : G (t 1 ; 1 ) k = 1 g ( n # n ) : : : g ( 1 # 1 ) h(#) d# ; (.6) where # n = #!t n. One can easily show [5] that the map (t) : 7! (t; ) as a H{valued stochastic processes, satises the ltering wave equation d(t) + i!n(t)dt = X h i G () I (t) dt n () (.7) s in the Ito sense: d(t) = (t + dt) (t), with the independent increments d t n() f; 1g, according to the following multiplication table: d t n()d t n( ) = d t n() ; = ( 1 ; = ; 6= (.8) Thus, the stochastic dierential equation (.7) for an initial state vector () = denes uniquely the statistics of the sequential phase measurements by the probability density k (t; ) k of the output measure P (d t ) = kg ( t ) k P o (d t ) ; t t (.9) 1

13 restricted to any nite t R + with respect to a given input probability measure P o of d t. It also denes the a posteriori state vector (t) = (t;) if the probability density of the observation of the sequence is not k(t;)k equal to. To study a specic example, let us x the stationary Poisson probability measure o (d) given on the set = t [t of all sequences = t ; [t with nite t [; t[ and innite t [t; 1[ for any t < 1, by the marginal distributions o (d) = nt exp( t)d t ; d t = n t Y n=1 dt n (.1) with an intensity > and d t = 1 if n t =. The Lebesque measure d on corresponds to the exponential integral taking the value f()d = t 1X : : : n= t 1 ::: <t n<1 f(t 1 ; : : : ; t n )dt 1 : : : dt n 1 f()d = exp f(t)dt if f() = Q t f(t) for all. The output probability measure (.9) on the space = 1 of all innite sequences = ( 1 ; ; : : :) with nite t = (y n ) nnt t is completely dened by the limit t! 1 of the characteristic functional ' s (t ; l) = t exp t i X l(t; )d t n () P(d) s as the expectation value of f() = Q y e il(y), given by Therefore ' s (t ; l) = ' # s (t ; i ln f) = t y f()p (d) = 1 ' # s (t ; l) jh(#)j d# Y f(y)jg# (y)j P o (d) 13

14 1 Y 1 X f(t; )jg # (t n ; )j A o (d) t t s + 1 s 1 X 1 n 1 t X = e f(t; )jg # (t; )j o (dt) A n= n! s + 1 s 1 = t X [f(t; ) 1]jg # (t; )j dta s + 1 s where we considered that g # (t; ) = g( +!t #). These functions are normalized so that P s jg # (t; )j = s + 1, 8 # [; [. In particular, one obtains the characteristic sequence C s (t; m) = e imt() P(d) t of the second quantized phase t () = P s n t () at time instant t. It is given as the mean value of C # s (t ; m) = ' # s (t ; l) at l(t; ) = m, C s (t; m) = 1! t exp [c # s (t; m) 1]dt jh(#)j d# where c # s (t; m) = c #(t) s (m) is the characteristic sequence of the distribution on jg( +!t #)j, having form (1.1) in the case of the geometric sequence b m given by (1.9). 3 The spontaneous phase localization First of all, let us seek the general ltering wave equation for a sequential unsharp nondemolition measurements of the quantum phase. More precisely, we want to modify equation (.7) for the case of continuous observation of the stochastic process only t () = Xn t n n=1 $ 1 n t X n=1 n % = '! Xn t n n=1 (3.1) It is given by the random sums P n t n=1 n = P s n t () (mod.) of the angular positions n s of the independent pointers at the random instants t n R + with the Poisson distribution (.1). 14

15 This is a stochastic s {valued process having the spontaneous jumps n 6= at the time instants t n P (), where () = ft s n j n = g and the domain s = f"; "; : : : ; s"g, " = s+1 does not contain. When there is no jump of t (), t = P (), there is no reduction of the state s vector, and d t must be proportional to the innitesimal increment dt. The general ltering equation (.7) now takes the following linear form [6] d(t) + K(t)ddt = X s L()(t)d t n() (3.) where K and L() are the operators in H, such that K K y = i!n and L() =, due to the continuity of (t) at t (). The self{adjoint part of the operator K can be derived from the condition hdk(t)k i = of the mean square normalization (.4) by application of the Ito formula dkk = <(jd) + kdk = <(jk)dt + X s h <(jl()) + kl()k i d t n() From the conditions that the increments d t n() are independent and that the mean value is such that hd t n()i = ()dt, it is derived K + K y = X s h L() + L() y L() + L() yi () (3.3) where (), is a given intensity distribution = P s () of the Poisson process n t = P s n t (). This completely denes the modication of the Eq.(.7) if L() is taken as G() G() such that the summation in (3.) is important only over s. The stochastic propagator T (t; ) : 7! (t; ) for this equation is given as in (.1) by the nonunitary time evolution operator U(t) = e Kt and G(t; ) = e Kth L() + I i e Kt ; G(t; ) = I (3.4) It is dened only on the pairs v n with n 6= and normalized w.r.t. the Poisson measure P o (d t ) = exp (() )t Q h i nt n t() n=1 (n )dt n on the space of all sequences = fv 1 ; v ; : : :g; v n R + s with 6=. Let us now examine the limit at s! 1 corresponding to the continuous scale of the phase meter. First, we assume that the intensity () = s () of 15

16 the jumps per degree of freedom, is inversely proportional to their number s + 1, s () = _(). Therefore one can obtain the limit s! 1 s+1 h i d(t) + i!n(t)dt = G() I (t)dt n(d) (3.5) of the stochastic wave equation (.7) and in the same way, the continuous limit of the modied equation (3.). It is controlled by the stochastic Poisson measure n t () = j \ ([; t[j = R d n t, counting the number of events (t n ; n ) on the time interval [; t[ in a phase domain [; [ or ( ]; [) where d n t = n t;+d n t; n t (d), n t; = n t ([; [). The increments d t n() = n t+dt () n t () satisfy the multiplication table d t n()d t n( ) = d t n( \ ) (3.6) that is a generalization of the discrete table (.8). The mean value hn t ()i = t() of such a process is given by an intensity measure (), characterizing the operator K +K y in (3.) as the integral h K + K y = L() + L() y L() + L() yi (d) (3.7) respectively to (d) = _()d = d where d = +d, _() = d. d Note that due to () = _()! as s! 1, the operator{valued function s+1 L() cannot necessarily be equal to. The stochastic propagator (.1) resolving the equation (3.5) is then normalized as in (.4) respectively to the probability measure P o (d t ) on t = P n t(n) n s ; = [; [ having the density e t Q n t n=1 _( n ), = R _()d with respect to d t = Q n t n=1 (d n dt n ). Thus the continuous limit of the modied equation considered in (3.5) in the case L() = G() I and _() = as K + Ky = in this case. The operators G() = g(i ^') are dened in the continuous phase limit by a meter wave function g() = P 1 m= b m e im in the innite dimensional space K H. The output probability measure (.9) of the continuous phase measurement is now described by the characteristic functional # s (t ; l) = t exp i t 16 f(t; )d t n(d) P(d t ) (3.8)

17 of the form s (t ; l) = 1 R # s (t ; l)jh(#)j, where # s (t ; i ln f) = exp t f(t; ) 1 jg# (t; )j ddt (3.9) with g # (t; ) = g( +!t #) normalized so that R jg # (t; )j d =. In particular, the characteristic sequence of the second-quantized continuous phase t () = R n t (d) at time t, is given by the mean value of t C # (t ; m) = exp [c # (t; m) 1]dt (3.1) respectively to the initial phase distribution jh(#)j d#=. Here, c # (t; m) = e im(!t #) c(m) is the characteristic sequence c(m) = 1 jg()j e im d; that is geometric c(m) = r jmj in the case jg()j = 1 r 1 r cos() + r ; 1 < r < 1 corresponding to b m = p 1 r r m m = ; 1; ; : : :. 4 The singular limit of quantum phase Now let us consider a singular measurement limit s! 1 for the spontaneous phase (3.1) with continuous jumps n [; [, when the intensity of the 1 P spontaneous jumps per degree of freedom is xed as s+1 s () = 1, i.e. = s + 1! 1. Suppose also that l s (m) = P s [e im 1]() has a nite limit l(m) at s! 1, 8 m = ; 1; ; : : : with the following properties l() = ; l( n) = l(n), X c n = ) X n n c nl(m n)c m ; 8 c n C: (4.1) 17

18 Therefore the normalized distribution s () = () s+1 on s in the limit that k! 1, s = k, is concentrated at =, () " (), where " () = 1 kx m= k e im = cos((k + 1)) cos(k) (cos() 1) = ( 1 " ; = ; s (4.) is the Dirac function () on [; [ at " =!. k+1 However, the probability of the jumps n 6= of t () does not tend 1 to because they have a non zero intensity function _() = lim k!1 (), " coinciding on ]; [ with the distribution _() = 1 1X m= 1 l(m)e im = d d (4.3) where = + +, with + = P 1 1 m=1 im l(m)eim# = is dened as the formal limit at s = k! 1 of ([; [) = P #< # s (#) (#) = 1 k + 1 kx m= k l s (m)e im# = (#) " (#) : (4.4) The function _() is the mean value h _v t ()i = t _() of the distribution{ valued process _v t () = 1 1X t (m)e im = d d v t; (4.5) m= 1 1 coinciding with _n t () = lim k!1 n " t() at 6=. It is given by the complex stochastic amplitudes t (n) = t ( n), t () = with the mean values h t (m)i = tl(m) and the Ito multiplication table: X c nd t (n)d t (m)c m = X c nd t (m n)c m (4.6) n;m n;m for the linear combinations P m t (m)c m with the complex coecients P m c m =. At the limit k!, the stochastic phase (3.1) can be formally dened in terms of the amplitudes t (m) as '(Y t ), where Y t = 1 _v t ()d = lim k!1 k kx m= k 1 im t(m) (4.7)

19 Moreover, since R _v t ()d =, one easily shows _n t ()d = _v t ()d = v t; d where v t; = v t; + v t; + and v+ t; = P 1 1 m=1 im t(m)e im = v t; is given by the formal limit s = k! 1 of v t ([; [) = P #< # s v t (#), where v t (#) = 1 k + 1 kx m= k s;t (m)e im# ; s;t (m) = X s (e im 1)n t () : (4.8) Hence the integral Y t = R v t; d is dened as the formal limit where v t; d: = lim Yt + Y + s!1 t Y t + = 1 k + 1 kx m=1 P km= k e im = X lim v t ([; [)" ; (4.9) s!1 s " e im" 1 s;t(m) = Y t since v t ([; [) = 1 k+1 e im" 1 s;t(m). Notice that s;t () = and the stochastic amplitudes s;t (m), m = 1; ; : : : ; k have the mean values h s;t (m)i = tl s (m) and satisfy the multiplication table (4.6) mod (k + 1) as their limits t (m) = lim k!1 s;t (m). The right hand side of the continuous reduction equation (3.) can be written in the phase representation x(t; #) = h#j(t) = (t; e i# ) as 1X h#jl()(t)d t _v()d = b m e im# x(t; #)d t (m) m= where d t (m) = t+dt (m) t (m), the sum being nite for the admissible wave functions g() = P 1 m= b m e im of the meters (b m = for m > k). In addition to the examples described in sections 1 and, let us consider the number distribution () given on s by the density (1.9) with the value of r given by r = 1, >, as () = s+1 jg()j. One can easily nd that l s (m) = (s + 1) (c s (m) 1)! l(m) = jmj; 19

20 where c s (m) is given by (1.1) with # = and = e 1. The corresponding e +1 distribution (4.3) is dened as the formal series _() = 1X m sin(m) = i m=1 i d d [ () + ()] (4.1) where + () = 1 P 1m=1 e im = (). This distribution coisides with the positive one _() = =(1 cos ) on the trigonometric polynomials f() = g() g() = kx m= k c m e im = kx m= k b m (e im 1) given with c m = b m ; m 6= ; c = P m6= b m. This is derived from jf()j _()d = kx m;n= b njm njb m = 1 k jf()j d ; 1 cos where jm nj = jmj + jnj jm nj = 1 because 1 R cos m 1 d = jmj. 1 cos R (e in 1)(e im 1) d 1 cos Hence operator (3.6) denes the Hermitian form hj(k + K y )ji for the phase reduction equation (3.) as the nite sum hj L() + L() y L() + L() y ji _()d = jx(#)j kx m;n=1 kx (m ^ n) b mb n e i(m n)# m b m e im# + b me im# d# for any initial trigonometric wave function g() of the meter with admissible Fourier coecients (b m = if m < ). Note that the distribution m=1 ( #) = jg( #)j " ( #) denes at the limit s! 1 an innite dimensional Minkowski metrics in the space E = S s<1 E s of all trigonometric polynomials f(), giving an example of the Ito algebra [] which is particularly important for the phase measurements with the singular input Poisson noise, inducing the phase spontaneous localizations.

21 Conclusion It is demonstrated that the quantum phase operator of a discrete (or continuous) spectrum can be treated in the conventional Hilbert space as a conditional expectation (projection) of a proper observable of the phase meter without any limiting procedure. It is shown that under the NDP, the problems mentioned in [?] in the phase measurements, namely the convergence for s! 1 of the operators, commutators, states, eigenvalues,... dened in E to the usual ones in H, the normalization of the phase states, and the restriction to the "physical" states, are simply solved by the proper choice of the measurement apparatus and are not relevant to the quantum oscillator itself. It is shown, that by using a NDP approach, the time continuous process of the phase observations can be completely described. Moreover, this principle allows us to treat the sequential unsharp measurements of quantum phase in terms of non{projective reduction transformations. The time continuous measurements of the quantum oscillator phase are described as a stochastic process of a spontaneous localization of the phase pointer. The unsharp localization of the quantum phase under the observation at random instants is caused by the proper interaction of the oscillator with the meter. This quantum jump model shows once again that there is no universal individual-trajectory reduction dynamics but the stochastic reduction equation derived here for photon-phase countings has another type than in the diusive case [?,?]. Because of the boundness of the phase spectrum, the quantum phase jumps can not even be approximized by a diuion prrocess, but there is a singular Poisson limit in this case. But as in the diusive case the phase reduction equation depends on the quantum mechanical model of spontaneous interaction only via the chosen phase operator, the initial state{ vector of the meter, and the statistics of the input counting process, inducing the photon emission. These two only possible types of stochastic dynamics constitute the real limitation for a form of spontaneous reduction equations. References [1] Belavkin V. P. Hypothesis testing for quantum optical signals. Radio 1

22 Eng Physics, 1(1):95{14, [] Belavkin V. P. Chaotic states and stochastic integrations in quantum systems. Usp. Math Nauk (Russian Math Surveys), (1), 199. [3] Belavkin V.P. A continuous counting observation and posterior quantum dynamics. J Phys A Math Gen, ():L 119{L 1114, [4] Belavkin V.P. A new wave equation for a continuous non-demolition measurement. Phys Letters A, 14(78):355{358, [5] Belavkin V.P. A stochastic posterior Schrodinger equation for counting non-demolition measurement. Letters in Math Phys, ():85{89, 199. [6] Belavkin V.P. and A Barchielli. Measurements continuous in time and posteriori states in quantum mechanics. J. Phys. A. Math. Gen., 4:1495{ 1514, [7] Belavkin V.P. and Staszewski P. Nondemolition observation of a free quantum particle. Phys. Rev. A., 45(3):1347{1356, 199. [8] Ghirardi G.C., Pearle P., and Rimini A. Markov processes in Hilbert space and continuous spontaneous localization of systems of indentical particles. Phys. Rev. A., 4:78{89, 199. [9] Helstrom C.W. Quantum Detection and Estimation theory. Academic Press, New York, 1976.