Physical properties of a laser beam and the intracavity quantum state

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1 Physical properties of a laser beam ad the itracavity quatum state uthor Pegg, David Published 2012 Joural Title Physics Letters DOI Copyright Statemet 2012 Elsevier B.V.. This is the author-mauscript versio of this paper. Reproduced i accordace with the copyright policy of the publisher. Please refer to the joural's website for access to the defiitive, published versio. Dowloaded from Griffith Research Olie

2 1 Physical properties of a laser beam ad the itracavity quatum state David T. Pegg Cetre for Quatum Dyamics, School of Biomolecular ad Physical Scieces, Griffith Uiversity, Natha, Brisbae Q 4111, ustralia bstract We use the impossibility of sedig iformatio faster tha light to show that it is impossible to distiguish betwee various types of laser cavity field state, such as a Fock state of ucertai eergy ad ay superpositio of Fock states of ukow free evolutio time, by meas of the physically measurable properties of the laser beam. Superpositios ca iclude coheret states, squeezed states ad cat states, ideed ay pure itracavity state will produce the characteristic properties of the beam such as itrisic phase coherece. Keywords: Laser beam, coheret states, phase coherece. Correspodig author: Prof. D. T. Pegg, Cetre for Quatum Dyamics, School of Biomolecular ad Physical Scieces, Griffith Uiversity, Natha, Brisbae Q 4111, ustralia Phoe: Fax: address: David.Thomas.Pegg@gmail.com

3 2 1. Itroductio The state of a beam of light emitted by a cavity cotaiig a field i a pure photo umber, or Fock, state will be very differet from that of a beam emitted if the cavity field is i a coheret state. The itrisic phase coherece of a laboratory laser beam is well explaied if the beam is cosidered to be i a coheret state, which implies that the itracavity field must also be i a coheret state. If a beam i a coheret state is divided ito separate successive packets the each packet will be i a coheret state ad all the packets will have a well-defied phase relatioship with each other for times shorter tha the phase diffusio time. ll experimetal evidece so far supports the widely held belief that the laser beam is i a coheret state resultig from a itracavity field i a coheret state, icludig a early experimet by Pfleegor ad Madel [1] showig iterferece betwee two laser beams. coheret state beam has attractive properties i additio to phase coherece, icludig ot beig etagled with the itracavity field ad beig able to be treated as a classical field i may circumstaces. It came, therefore, as somewhat of a surprise whe Mølmer [2,3] challeged the covetioal wisdom cocerig the state of a laser beam by usig a simple model of light leakig from a cavity. He showed that light beams from cavities cotaiig pure Fock states give similar balaced homodye iterferece sigals to those from cavities cotaiig coheret states. Examiatio of the way i which the itracavity state is geerated i a typical laboratory laser led him to the coclusio that the use of coheret states to describe laser light is merely a coveiet fictio. Mølmer [3] stated that, for large photo umbers, the Poisso distributio of a coheret state is so arrow that the average over differet photo umber state compoets equals the result obtaied with just oe characteristic choice of

4 3 a pure umber state equal to the mea of the distributio; so eve sigle pure Fock states must yield the same iterferece pheomea as pure coheret states. Not surprisigly, Mølmer s work stimulated a cotroversial debate o the subject [4-11] ad the issue is ot yet settled. The preferred esemble fallacy, that is, that there ca be o privileged partitioig of a desity operator [12], has bee ivoked i support of Mølmer s view [5]. This implies that the itracavity desity operator ca represet either a mixture of coheret states or a mixture of photo umber states ad ay choice is a matter of coveiece oly. The fallacy is based o the otio that withi the framework of stadard of quatum mechaics the desity operator is the complete descriptio of the quatum state. By cosiderig pheomea other tha iterferece, however, defeders of the reality of coheret states as a descriptio of laser light have maitaied that there are good physical reasos, as opposed to mathematical coveiece, for usig a coheret state descriptio. Va Ek ad Fuchs [6] poited out that the argumets ivolvig the preferred esemble fallacy are based o a applicatio of the stadard descriptio of the laser field iside the cavity whereas the relevat properties are those of the light outside the cavity. They give a formulatio, ivolvig the quatum de Fietti theorem, showig why the coheret state plays a privileged ad uique role i the descriptio of propagatig laser fields. Noh ad Carmichael [11] proposed that a importat property for establishig the laser state as a coheret state is disetaglemet of the light from its target qubit ad i this respect coheret states have some uique properties. If the exteral light ca be determied by some experimetal meas to be i a coheret state, the the itracavity field must also be i a coheret state ad the questio

5 4 would be settled. The assigmet of a coheret state would the o loger be merely a coveiece as Mølmer suggests. The questio the is whether or ot there are physical properties of the exteral laser beam that deped o whether the itracavity state is a pure Fock state, for example, or a coheret state. The mathematical descriptio of coheret states are very differet from those of Fock states, ideed the former are sometimes described as classical states with the latter beig described as quatum states, so oe might expect that there should be some physical properties of a beam that ca be used to distiguish betwee a itracavity coheret state ad a itracavity Fock state. s poited out by va Ek ad Fuchs [6], a obvious property of a itracavity coheret state is the productio of a beam with itrisic phase coherece, that is, with differet packets havig a well-defied phase relatioship with each other for a time less tha the phase diffusio time. Further, o this time scale each packet is i the same coheret state. I priciple at least, oe should be able to recostruct the itracavity coheret state from measuremets doe o the idividual packets. t first sight it may seem ulikely that a itracavity Fock state, which has a uiform phase distributio, ca produce a beam with precisely the same phase coherece. This was the situatio ivestigated i Ref. [13] by explicit calculatio of the itrisic phase coherece of a beam obtaied from a strog itracavity Fock state by meas of a partially reflectig mirror. lthough the beam had o particular phase it was foud to have the same itrisic phase coherece as a coheret state. Eve the observed phase diffusio effect [13,14] was recovered. lthough strog itracavity Fock states seem to produce exteral beams with some properties similar to those of beams produced by strog itracavity coheret states, which is cosistet with Mølmer s assertio above, perhaps it still may be possible to

6 5 distiguish betwee coheret ad photo umber itracavity states if they are weak eough. It should be possible to adopt the approach of Ref. [13] for light leakig from a cavity without makig the large umber state approximatio to examie this questio, but it is more useful to adopt a more geeral approach that ca be applied to ay measurable property of the beam. We do so i this paper, where we examie the questio as to how much the characteristic properties of a beam leakig from a cavity actually deped o the type of itracavity state. 2. Depedece of beam properties o the cavity state To determie whether differet beams have differet physical properties that ca be used to distiguish betwee types of itracavity states, we cosider the case where lice ad Bob are spatially separated ad share a etagled state which at a particular time t 0 is give by f = c B where is the photo umber ad the subscripts refer to fields held by lice ad Bob i their ow separate perfect cavities. Both lice ad Bob kow the coefficiets c. With the prior agreemet of lice, Bob makes either a photo umber or a phase measuremet at this time but does ot tell lice which measuremet is made or the outcome. lice the replaces oe of the perfectly reflectig mirrors of her cavity with a partially trasmittig mirror ad examies the beam of light slowly leakig out. We assume that the separatio betwee lice ad Bob is sufficiet for lice to complete her experimets o her beam before a sigal from Bob set at t 0 ca reach her.

7 6 If Bob makes a photo umber measuremet, with outcome m photos, the state iside lice s cavity becomes the projectio of the etagled state f oto the umber state m, that is, it becomes the umber state m B. phase measuremet ca be represeted by a positive operator valued measure [15,16] with elemets where the phase state 2 ϕ = (2π ) 1/ exp( i ϕ). B B ϕ B B ϕ dϕ Thus, if Bob makes a phase measuremet with outcome θ istead of a photo umber measuremet, the state iside lice s cavity becomes the projectio of f oto θ, B that is, the superpositio state α ( θ ) = c exp( iθ ) = exp( i N ˆ θ ) c where Nˆ is the photo umber operator actig o the field i lice s cavity. If there were ay measuremet of ay physical property that lice could make o either her itracavity field or o the beam of light emitted from the cavity that would distiguish the photo umber state m from the superpositio state α (θ ), or that would eve partially favour oe of these, the Bob could sed her iformatio at a speed greater tha that of light. These states must therefore be completely idistiguishable by lice by meas of ay physical measuremet. s the coefficiets c are kow, i the absece of a kowledge of θ, α (θ ) ca be described as a kow superpositio state with a ukow phase shift θ or

8 7 equivaletly, by writig θ = ωt where ω is the frequecy of the light, as a kow superpositio state of ukow free evolutio time t. I the special case where the superpositio state is a coheret state, we see that the exteral beam must have the same measurable physical properties whether the itracavity field is i a coheret state of appropriate phase or a photo umber state of appropriate itesity. For example if we divide the exteral beam ito packets, we must fid the itrisic phase coherece discussed above. This applies for all itracavity states α (θ ), whether they are coheret states or quatum states such as squeezed states or Schrödiger cat states. It is impossible to distiguish betwee a itracavity Fock state of ukow photo umber ad ay superpositio state of ukow evolutio time by ay experimets performed o the cavity field itself, o the exteral beam or o packets of the exteral beam. s such experimets iclude iterferece experimets, we see that our result justifies ad stregthes Mølmer s statemet above. It goes further tha this, however, as the idistiguishability is ot restricted to strog Fock ad coheret states but applies to all Fock ad superpositio states icludig weak states. itracavity field i ay pure state, eve if it does ot have a well-defied phase, will produce a beam with observable properties, such as itrisic phase coherece, that are characteristic of a laser beam. Our result is also cosistet with the preferred esemble fallacy. Ideed it is possible to use our result to derive the fallacy for this case. I the absece of ay iformatio from Bob, lice would describe her state by ρˆ, the trace of the desity operator f f over Bob s cavity states. It is ot difficult to show that this ca be expressed as

9 8 ˆ ρ = c 2 2π 1 = (2π ) α( θ ) α( θ ) dθ. 0 This desity operator would describe both a mixture of Fock states ad a mixture of superpositio states α (θ ). The fact that she has o experimetal basis for distiguishig betwee the Fock states ad the superpositio states implies that she has o reaso, apart from mathematical coveiece, for preferrig oe descriptio over the other as far as predictig the results of experimets is cocered. 3. Discussio ad coclusio The above argumets show that a uique partitioig, or decompositio, of the desity operator for the itracavity field caot be determied from the results of measuremets o this field or o the beam. The desity operator itself is all that is eeded to predict the results of such measuremets ad it is a fallacy to assume that ay particular decompositio will predict physically measurable properties of the itracavity field or the beam that other decompositios will ot. We should be cautious, however, before usig the preferred esemble fallacy for purposes other tha predictig properties of the system described by the desity operator. For example we should ot coclude that the desity matrix is the best possible quatum descriptio of a mixture of pure states for all situatios, that is, to coclude that there is ever ay situatio i which a more accurate quatum descriptio is give by a particular decompositio. Cosider a situatio similar to the above but i which lice retais the field i her perfect cavity util she receives a message from Bob, who iforms her that he has performed a photo

10 9 umber measuremet but does ot iform her of the outcome. The desity operator that she would the use to describe her field would be ρˆ, just as for the previous situatio. Now, however, a better quatum descriptio of her field, i that it also cotais the iformatio received from Bob, would specify the set of states with the associated values of 2 c, which i this case are the probabilities that the states were prepared as a result of Bob s measuremet. It would ow be justifiable for lice to describe this field as beig i a Fock state of ukow umber ad ot i a superpositio state of ukow free evolutio time as the latter descriptio would ot be cosistet with the preparatio procedure that icludes Bob s measuremet. lthough the latter descriptio is just as good as the former i eablig lice to calculate the outcome probabilities for future measuremets o her field, the former descriptio also eables her to retrodict the outcome of Bob s measuremet by choosig to make a photo umber measuremet o her field. Her retrodictio could be verified by Bob. This is a situatio where a particular decompositio ca be used to determie a verifiable result that caot be foud from kowledge of the desity operator aloe ad which thus might be said, with some justificatio, to provide a better quatum descriptio tha the desity operator. The questio ow is whether or ot a similar argumet ca be applied to a laboratory laser i order to assig a preferred decompositio o grouds other tha coveiece. I a laboratory laser it is most likely that the itracavity light ad the atoms of the laser medium are i a etagled state [2,3], so a suitable measuremet o the atoms could i priciple prepare the light i a mixed state described by a particular decompositio of the desity operator. However such a measuremet is ormally ot doe, i which case there

11 10 is o justificatio i assigig a particular decompositio such as a coheret state o this basis. I coclusio, we have re-examied the questio of the state of a light emitted by a cavity from the poit view of the impossibility of sedig iformatio faster tha light. We have arrived at a stroger coclusio tha that of Mølmer [2,3] who stated that strog itracavity sigle pure Fock states must yield the same iterferece pheomea as pure coheret states. We fid that, without restrictio to strog states, all physical properties of the beam from a itracavity field i a Fock state of ucertai eergy but with a kow umber state probability distributio must be idistiguishable from those from a itracavity state i a kow superpositio state but of ukow free evolutio time. The latter icludes coheret states of ukow mea phase ad also quatum states such as squeezed or cat states. This approach offers additioal isight to oe based o the preferred esemble fallacy but leads to the same coclusio that there is o reaso, other tha coveiece, for preferrig a coheret state descriptio of a laboratory laser beam i order to determie its measurable properties or the outcome of experimets doe with it. Refereces [1] R. L. Pfleegor ad L. Madel, Phys. Lett. 24 (1967) 766. [2] K. Mølmer, Phys. Rev. 55 (1997) [3] K. Mølmer, J. Mod. Opt. 44 (1997) [4] J. Gea-Baacloche, Phys. Rev. 58 (1998) [5] T. Rudolph ad B. C. Saders, Phys. Rev. Lett. 87 (2001) [6] S. J. va Ek, ad C.. Fuchs, Phys. Rev. Lett., 88 (2002)

12 11 [7] B. C. Saders, S. D. Bartlett, T. Rudolph, ad P. L. Kight Phys. Rev. 68 (2003) [8] H. M. Wisema, J. Opt. B: Quatum Semiclass. Opt. 6 (2004) S849. [9] H. Cable, P. L. Kight, ad T. Rudolph, Phys. Rev. 71 (2005) [10] D. T. Pegg ad J. Jeffers: J. Mod. Opt. 52 (2005) [11] C. Noh ad H. J. Carmichael, Phys. Rev. Lett., 100 (2008) [12] P. Kok ad S. L. Braustei Phys. Rev. 61 (2000) [13] D. T. Pegg, Phys. Rev. 79 (2009) [14] S. M. Barett, S. Steholm ad D. T. Pegg, Opt. Commu. 73 (1989) 314. [15] C. W. Helstrom, Quatum Detectio ad Estimatio Theory, cademic Press, New York, [16] J..Vaccaro ad D. T. Pegg, Physica Scripta T48 (1993) 22. Correspodig author: David T. Pegg