Continuous measurements: partial selection

Transcription

1 0 May 00 Pysics Letters A wwwelseviercom/locate/pla Continuous measurements: partial selection YuA Rembovsky Department of Pysics, MV Lomonosov State University, Moscow , Russia Receive 7 February 00; receive in revise form 5 Marc 00; accepte 7 Marc 00 Communicate by VM Agranovic Abstract Tis Letter is evote to te consieration of te continuous inirect measurement in te case of approximate measurement of te QRS observable Te equation of obect matrix evolution is obtaine Tis equation is compare wit te equation obtaine by restricte pat integral RPI meto Te general solution of equation tat as been obtaine is foun 00 Elsevier Science BV All rigts reserve Keywors: Inirect measurement; Continuous measurement; Restricte pat integral 1 Introuction For te past few ecaes, te teory of continuous measurements as receive a great eal of attention A number of various approaces to te escription of continuous measurements ave been put forwar 7 In tis Letter, two of tese approaces are examine Te first approac is base on restricte pat integrals RPI It as been evelope in etail by Mensky,3 In te secon approac te continuous measurement is propose to be simulate as a sequence of single instantaneous measurements see 6,8 In consiering te problems relate to measurements, it is usually assume tat te initial state of quantum rea-out system QRS is pure an te measurement of QRS observable tat is use to obtain estimation of obect observable is precise But in real- aress: yar76@mailru YuA Rembovsky ity te QRS state migt be mixe an te measurement of QRS observable is always approximate Te possibility of QRS to be in a mixe state was taken into account in 9 Te possibility of QRS observable measurement to be approximate was taken into account in, Capter 3 see also te case of ortogonal measurements in 10 In bot of tese cases, te pure initial obect state becomes mixe as a result of continuous monitoring of te obect observable Accoring to te RPI meto, te pure obect state remains pure in te course of a continuous measurement process So one can conclue tat RPI meto escribes ieal continuous measurements tat correspon to pure QRS state an precise measurement of QRS observable In te present Letter we are consiering te case of te approximate QRS observable measurement wit continuous monitoring of obect measuring Our goal is to obtain a ifferential equation tat woul escribe te evolution of obect uner measurement an, ultimately, to solve tis equation /0/$ see front matter 00 Elsevier Science BV All rigts reserve PII: S

2 YuA Rembovsky / Pysics Letters A Equation tat escribes evolution of obect uner continuous inirect measurement Te peculiarity of te approac tat is being use ere consists in te fact tat te continuous measurement is consiere as a sequence of single instantaneous measurements separate by small time interval uring wic te obect free evaluation takes place Te state after single ieal measurement is etermine by te relation 9 see also te case of ortogonal measurements in 10 ˆρ Ỹ = ˆRỸˆρ 0 ˆR + Ỹ 1 Tr ˆR + Ỹ ˆRỸρ 0, were Ỹ is te result of precise measurement of QRS observable, ˆRỸ is te reuction operator, an ˆρ 0 is te initial obect state Te state after measurement tat is etermine by 1 is normalize However, it is more convenient to use unnormalize ensity matrices wen ealing wit questions concerne wit continuous measurements 1 Te norm of suc a matrix is equal to probability ensity of obtaining te measurement result or te sequence of measurement results So, in orer to obtain te expression for unnormalize ensity matrix, one as to multiply te rigt sie of expression 1 by te probability ensity of te measurement result Ỹ tat is equal to Tr ˆR + Ỹ ˆRỸρ 0 Ten one woul ave ˆρ Ỹ = ˆR Ỹ ˆρ 0 ˆR + Ỹ Te matrices use in calculations below are te unnormalize ensity matrices Relation can be generalize to take into account te case of approximate QRS observable measurement Wen te measurement is approximate, te connection between te obtaine value Ỹ an actual value Y is caracterize by te conitional probability ensity WỸ Y Te state after suc a measurement is etermine by relation ˆρ Ỹ = W Ỹ Y ˆRY ˆρ 0 ˆR + Y Y 3 Wen te measurement is precise, te conitional probability ensity is δ-function an relation 3 canges to relation 1 Let us assume tat a sequence of single measurements takes place: te measurement is at te time t an between moments t 1 an t te obect state is being cange as a result of an action of te free evolution operator Û Ten te obect state after measurement is etermine by te following relation: ˆρ Ỹ = W Y Y 1 = ˆR + Y Û + Y, Û ˆRY ˆρ 0 4 were Ỹ is te sequence of measurement results Accoring to te stanar measurement sceme, te reuction operator can be expresse in te following form: ˆRY = Ψ Y α 0 Â, were ΨY is some function tat satisfies conition ΨY Y = 1, an α 0 is coefficient of interaction In orer to obtain te continuous measurement out of a sequence of single measurements, te time intervals between consequent measurements of sequence must be ecrease In orer to fix te inaccuracy of continuous measurement over a fixe time interval, value α 0 must be in inverse proportion to square root of measurement number over unit of time ν: α 0 = α ν ow, in orer to obtain te ifferential equation for obect ensity matrix, let us fin a ratio of ensity matrix increment over te small time interval δt to value of tat interval Te increment of obect ensity matrix as to be linearize over te interval δt During te process of linearization, te reuction operators ˆRY of expression 4 an te free evolution operators Û appear as a power series wit te small parameter τ/ Members of te zero orer in expansion of te reuction operator, as well as of te free evolution operator, are numbers Terefore, in a linear approximation, te multiplication of operators from expression 4 will become a sum In case of a particle, suc a result makes it possible to separate te terms tat relate to te free evolution an te measurement process So, te calculation of te cange in obect state can only be mae wit consieration of te influence of measurement process Te influence

3 30 YuA Rembovsky / Pysics Letters A of free evolution must be taken into account at te final stage If te interval of time δt is very small, every measurement result from sequence Ỹ tat was obtaine uring δt as te same istribution In particular, tey ave te same average, wic epens on te value of te measure obect observable Â, an tey ave te same finite ispersion Dispersion of measurement result as a non-zero value, so te sequence of measurement results cannot be treate as values of some smoot function, because suc a function woul ave infinite large fluctuations wen ν tens to Soin te recoring part of evice tere must be a unit tat woul average measurement results over an interval of time τ To avoi aitional influence of averaging process, te interval τ must be muc sorter tan te typical perio of obect evolution see, Capter 5 or 11 for etails For example, averaging interval τ migt be equal to δt In aition, te frequency of measurements as to be muc greater tan te value 1/τ From matematical point of view, it is equal to te ouble limit: 1 Ỹt= lim lim τ 0 ν τν Ỹ lim τ 0 Ỹ τ t Dispersion of ranom quantity Ỹ τ t tens to zero in limit ν wen τ is infinitely small but non-zero Let us assume tat te averaging interval is equal to small interval of time δt Function value Ỹt Ỹ τ t can be realize by ifferent sets of measurement results Final ensity matrix ˆρỸt,t + δt is a mixture of ensity matrix ˆρỸ,t+δt in calculations below te time is roppe to make relations sorter, eac of tese correspons to te single sequence of measurements results tat satisfies te conition ˆρ Ỹ = ˆρ Ỹ Ỹ 5 Ỹ Ỹ=0 Since our goal is to watc te obect observable Â, let us recalculate function Ỹt to function ãt tat correspons to an estimate of obect observable Â: ν ãt = α Ỹt Density matrix ˆρã iffers from ˆρỸ by normalization factor α/ ν After rewriting relation 5, taking into account relation 4 an te rearrangement of multipliers, we will obtain ˆρã = α ν 1 = Ỹ Ỹ=0 ˆRY ˆρ 0 W Ỹ Y Ỹ ˆR + Y Y 6 If estimate of QRS observable is unbiase, ten te average value of Ỹ is equal to Y Let us also assume tat conitional ispersion of Ỹ oes not epen on Y an is equal to D Te terms in te roun brackets at te last expression represent noting else but te probability ensity of ranom value 1/ Ỹ Y Accoring to te limit teorem 1 te istribution of suc a value tens to normal istribution wit ispersion equal to D an an average equal to zero wen As a result, we obtain te following: ˆρã = α exp Ỹ Y ν πd D 1 ˆRY ˆρ 0 ˆR + Y Y, 7 = were Y is equal to 1/ Y After a matematical transformation see Appenix A one can ave an expression for ensity matrix ˆρãt,t + δt at te limit : α ˆρã = δt πσ Y + D exp α δt 4σ Y + D i σ Â, YP ã { Â ã }, 1 D σ P /4 + σ Y σ P σyp /4 /4 Â Â, ˆρ 0 8

4 YuA Rembovsky / Pysics Letters A Te norm of te ensity matrix ˆρã is equal to te probability ensity of obtaining te single value ã uring te time interval δt Our goal is for te norm to be equal to te probability of obtaining te curve ãt uring te time of monitoring t It can be sown tat te measure on te space of curves of tis kin must be equal to α δt t/δt t/δt 1 ã πσ Y + D see 13, Capter 9 an, Appenix 3 So, to write relation between ˆρãt,t + δt an ˆρãt,t, were ãt means obtaine curve, one must reect te multiplier α δt/πσ Y + D in relation 8 Density matrix ˆρãt,t + δt can be presente wit an accuracy up to te first power of δt by relation ˆρ ãt,t + δt = 1 + α δt 4σ Y + D { Â ã }, 1 i σ Â, YP ã D σ P /4 + σ Y σ P σyp /4 /4 Â Â, ˆρ ãt,t 9 In orer for te influence of free evolution to be taken into te account, we ave to a te term i/ Hˆ ob, ˆρδt to te rigt sie of 9 Subtracting ensity matrix ρãt,t from bot sies of te obtaine relation, at te time moment t, an iviing tem by δt an ten tening δt to zero, one will arrive at a ifferential equation t ˆρ = ī H ˆob, ˆρ + ī α σ YP Â, ã ˆρ σ Y + D α {Â ãt, ˆρ} 4σ Y + D α D + σ Y σ P σ YP /4 /4σ P /4 Â, Â, ˆρ 10 4σ Y + D Te basic ifference between Eq 10 an equation wit complex Hamiltonian iscusse in,3 is te term in rigt part of 10, wic is proportional to Â, Â, ˆρ Because of tis term, te state of obect, tat was initially pure, becomes mixe uring te measurement process One can see tat tis term appears because of inaccuracy of QRS observable measurement conitional ispersion of estimation D is non-zero an because of averaging unit in recoring part of te evice as well Te only case wen averaging unit oes not lea to mixing of obect state is wen QRS state is in a squeeze base state, so tat σ Y σ P σ YP = /4 Te presence of correlation between measure QRS observable an an observable conugate wit it, leas to an aitional action aime at te obect real a-on to Hamiltonian α σ YP /σ Y + DÂ ã For example, if obect is a mecanical oscillator an measure observable is a coorinate, te correlation leas to te appearance of an extra rigiity negative or positive epening on te sign of te correlation Ŷ an Pˆ Y Te questions concerne wit te reverse action on an obect as a result of evice observables correlation are consiere explicitly in 9 for te case of single measurements Eviently, te influence of correlation can be eliminate by coosing te optimal combination of observable Ŷ an Pˆ Y as te QRS observable to be measure 3 Solution of te equation One can verify irectly tat relation ˆρ ãt,t D + σ Y σ P σ = α YP /4 t /4σ P /4 πσ Y + D ˆRξ,t ˆρ 0 ˆR + ξ, t ξ, 11 were ˆRξ,t is reuction operator of continuous inirect measurement, efine by relation ˆRξ,t = exp ī t 0 Hˆ ob τ α t 0 Â ãτ τ 4σ Y + D α σ YPÂ ãτ τ σ Y + D

5 304 YuA Rembovsky / Pysics Letters A D α + σ Y σ P σ YP /4 /4σ P /4 Â ξ 1 4σ Y + D t, is a solution of te ifferential equation 10 Eq 10 is linear ifferential equation of te first orer So its solution 11 is unique Relation 11 escribes state of obect uner continuous measurement in te case wen QRS state is pure Tere are taken into account te inaccuracy of QRS observable measurement caracterize by conition ispersion D, te correlation of QRS observables correlation coefficient σ YP an non-fulfillment of te conition σ Y σ P σyp = /4 te conition takes place if QRS state is a squeeze base state In general case, obect state escribe by relation 11 is mixe It is cause not only by te inaccuracy of QRS observable measurement but by te fact tat QRS state is not a squeeze base state as well 4 Comparison of te results wit te ones obtaine by restricte pat integral RPI meto RPI meto leas to te equations below, wic escribe te evolution of obect wose observable is uner continuous monitoring : t ˆρ = ī H ˆob, ˆρ k { Â ãt }, ˆρ in te selective approac an t ˆρ = ī H ˆob, ˆρ k Â, Â, ˆρ in te non-selective approac Differential equation 10 obtaine in tis Letter correspons to te intermeiate case at te absence of correlation in QRS In orer to emonstrate tis, let us re-write it in te following form: t ˆρ = ī H ˆob, ˆρ k1 γ { Â ãt }, ˆρ k γ Â, Â, ˆρ, were k = α 4σ Y σ P 4σ Y, γ = D σ Y + D 4σ Y σ P 15 Quantity γ varies from 0 to 1 Wen γ = 0, Eq 15 turns to 13 tat correspons to selective approac, an wen γ = 1, Eq 15 turns to 14 tat correspons to a non-selective approac Quantity γ as sense of measurement non-selectivity egree 5 Conclusion In tis Letter we ave obtaine te equation escribing te evolution of an obect state, wen its observable is uner continuous inirect measurement for te case of te approximate measurement of QRS observable It as been emonstrate tat non-selectivity wic always causes initially pure obect state to become mixe arises not only as a result of te fact tat measurement of te QRS observable is approximate but as a result of te averaging of te measurement results in te laboratory evice as well Te only case wen te process of averaging oes not cause non-selectivity is wen QRS state is a squeeze base state If te measurement of QRS observable is precise, te QRS state is a squeeze base state an tere is no correlation between measure QRS observable an observable conugate wit it, te equation obtaine ere coincies wit te equation obtaine by RPI meto in te selective approac If QRS observable measurement is approximate D, te equation obtaine ere coincies wit te equation obtaine by RPI meto in te non-selective approac inepenently of QRS state It as also been emonstrate tat, wen tere is correlation between measure QRS observable an observable conugate wit it, an extra real a-on to obect Hamiltonian arises Te general solution for te equation as been foun Acknowlegement I woul like to express my gratitue to YuI Vorontsov wo encourage tis work an contribute some critical an elpful remarks

6 YuA Rembovsky / Pysics Letters A Appenix A Te expression for ensity matrix at te limit Function ΨYcan be expresse troug its Fourier image ψk accoring to te relation 1 ΨY= 1 ψkexp iky k π After substituting it in 7 one can obtain ˆρã = α exp Ỹ Y ν πd D 1 π ψ k ψ k exp i exp i Y k k k Y α ν  ˆρ 0 Y α  ν k A1 Let us cange te variables Y = 1,tovariables Y = 1, 1, Y After integration by variables Y = 1, 1antenbyvariables k = 1, 1, relation A1 is rearrange to ˆρã = α δt π πd exp Ỹ Y D ψ k ψ k + k k exp i α k ν + k k  ˆρ 0 exp i α ν exp iy k k Y k k A Let us integrate by variable Y an make a cange of variable accoring to relations k = k κ/, k  k = k + κ/ As a result, A is rearrange to ˆρã = α δt exp κ D π iα δtãκ ψ k + κ ψ k κ exp i α k  + i ακ ν ν  ˆρ 0 exp i α k  + i ακ ν ν  κ k = α δt exp κ D π iα δtãκ δω exp i ακ ν  ˆρ 0 exp i ακ ν  κ, A3 were δω ˆρ is linear operator tat acts in linear space of finite Hermitian matrix It is efine by relation δω ˆρ = ψ k + κ ψ k κ exp i α k  ˆρ exp i α k  ν ν A4 ow let us cange ν to /δt, make an expansion terms uner integral in 1/ power series wit Peano remainer up to secon orer of vanising After integrating te relation obtaine, taking into account relations ψk k = 1, 1 ψ ψ k ψψ k k = iȳ, ψk i kk= P Y, ψ k k = 1 ψ ψ k ψψ k k = σ Y + Ȳ, i ψ ψ k ψψ k kk= σ PY + P Y Ȳ,

7 306 YuA Rembovsky / Pysics Letters A ψk k k = σ P Y + P Y A5, one can ave expansion of δω ˆρ in 1/ power series wit Peano remainer up to secon orer of vanising: δω ˆρ = 1 + i κȳ + α 1 δtp Y 1 D σ Y σ P σyp /4 σ P /4 /4 Â Â, ˆρ 0 References A7 κ σ Y α δtκ σ YP + α δt σ P Â, Â, Â, 1 + o ˆρ A6 Te only case wen tere is limit of δω is wen te vanising term of first orer in expansion A6 tat is proportional to 1/ is equal to zero It is possible wen Ȳ = 0anP Y = 0 So, relation A3 at te limit can be rearrange to α ˆρã = δt πσ Y + D α δt i exp 4σ Y + D σ Â, YP ã { Â ã }, 1 MB Mensky, Te Groups of Pats: Measurements, Fiels, Particles, auka, Moscow, 1983; MB Mensky, Continuous Quantum Measurements an Pat Integrals, IOP Publising, Bristol, 1993 MB Mensky, Quantum Measurements an Decoerence, Kluwer Acaemic, MB Mensky, Pys Lett A DF Walls, MJ Collet, GJ Milburn, Pys Rev D G Linbla, Comm Mat Pys FYa Kalily, Vestnik Moskov Univ Ser AA Kulaga, Pys Lett A YuI Vorontsov, in: VB Braginsky E, Teory an Metos of Macroscopic Measurements, auka, Moscow, VB Braginsky, FYa Kalily, in: KS Torne E, Quantum Measurement, Cambrige University Press, CW Helstrom, Quantum Detection an Estimation Teory, Acaemic Press, ew York, MB Mensky, Pys Lett A GA Korn, TM Korn, Matematical Hanbook, McGraw- Hill, C Itzykson, J-B Zuber, Quantum Fiel Teory, Vol, McGraw-Hill, 1980