We show that the unitary evolution of a harmonic oscillator coupled to a
two-level system can be undone by a suitable manipulation of the two-level
system -- more specifically: by a quasi-instantaneous phase change. This
enables us to isolate the dissipative evolution to which the oscillator may be
exposed in addition. With this method we study the decoherence of a photon mode
in cavity QED, and that of the quantized harmonic motion of trapped ions. We
comment on the relation to spin echoes and multi-path interferometry.Comment: 4 pages, 2 figures; submitted to Physical Review A as a Rapid
  Communicatio

American Physical Society

Englert, Berthold-Georg

Morigi, Giovanna

Solano, Enrique

Walther, Herbert

English

arXiv

We show that the unitary evolution of a harmonic oscillator coupled to a two-level system can be undone by a suitable manipulation of the two-level system-more specifically, by a quasi-instantaneous phase change. This enables us to isolate the dissipative evolution to which the oscillator may be exposed in addition. With this method we study the decoherence time of a photon mode in cavity QED, and that of the quantized harmonic motion of trapped ions. We comment on the relation to spin echoes and multipath interferometry

Morigi, G.

Solano, E.

Englert, B.

Walther, H.

MPG.PuRe

Measuring irreversible dynamics of a quantum harmonic oscillator

Diposit Digital de Documents de la UAB

RAPID COMMUNICATIONSPHYSICAL REVIEW A, VOLUME 65, 040102~R!Measuring irreversible dynamics of a quantum harmonic oscillatorGiovanna Morigi,1 Enrique Solano,1,2 Berthold-Georg Englert,1 and Herbert Walther11Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany2Seccio´n Fı´sica, Departamento de Ciencias, Pontificia Universidad Cato´lica del Peru´, Apartado 1761, Lima, Peru~Received 20 August 2001; published 10 April 2002!We show that the unitary evolution of a harmonic oscillator coupled to a two-level system can be undone bya suitable manipulation of the two-level system—more specifically, by a quasi-instantaneous phase change.This enables us to isolate the dissipative evolution to which the oscillator may be exposed in addition. With thismethod we study the decoherence time of a photon mode in cavity QED, and that of the quantized harmonicmotion of trapped ions. We comment on the relation to spin echoes and multipath interferometry.DOI: 10.1103/PhysRevA.65.040102 PACS number~s!: 03.65.Yz, 32.80.Qk, 42.50.CtDecoherence is currently under investigation theoreticallyand experimentally @1#, thereby searching for the boundarybetween the quantum and classical realms @2,3#. From thepoint of view of technical applications, scalability of coher-ent control—like that needed in a quantum computer—is be-ing pursued intensively and is limited mainly by the presenceof decoherence @1,4#. With the tools of quantum optics, it hasbecome possible to create quantum states of high purity @4#and to monitor their decay @5,6#. The measurement of aquantum state, however, can introduce additional coherencedestruction, caused by the nonunitarity of the state reduction.Methods like quantum nondemolition measurements @7#, par-tially solving this problem, have been implemented forstudying nonclassical states of a harmonic oscillator andtheir decoherence @5#. Nevertheless, the decoherence of ar-bitrary initial states of the quantum oscillator deserves fur-ther studies.In this contribution, we propose a method for measuringthe decoherence time of a harmonic oscillator prepared in anarbitrary initial state by suitably coupling it to a stable two-level system ~TLS!, whose final level statistics carry the in-formation of the oscillator decay. The composite systemevolves following a Jaynes-Cummings ~JC! interaction @8#for a total duration T. At an intermediate instant, after time thas elapsed, the JC evolution is interrupted by a phase kick,that is, a fast unitary operation applied to the TLS that intro-duces a relative phase between the levels. We will show that,as a consequence of the phase kick, the coherent dynamics iseffectively reversed, mimicking a time-reversal operation@9–11#. When t5T/2, the final state of the composite systemmatches the initial one, provided that the evolution is unitary.Any mismatch represents a measurement of changes in theoscillator state due to external actions and an estimation ofthe time in which decoherence occurs.This method allows us to separate the irreversible decayof the oscillator state from the unitary JC dynamics. Thescheme can be easily implemented in two representative sce-narios in quantum optics, ion traps and cavity QED, forstudying the irreversible decay of the quantized motion orthe electromagnetic field, respectively. In cavity QED, theharmonic oscillator is a mode of the quantized radiationfield, coupled to a resonant electronic transition of atomssent through the resonator and undergoing JC dynamics @12#.In ion traps, the center-of-mass motion of the trapped ion is1050-2947/2002/65~4!/040102~4!/$20.00 65 0401harmonic, and it couples to an internal atomic transitionwhen the ion is irradiated by a laser @13#. We discuss belowhow the decay of these oscillators can be studied by meansof the proposed scheme.Let us consider the resonant interaction between a TLS,with lower state ug& and upper state ue& , and a harmonicoscillator, with creation and annihilation operators a† and a,respectively. The eigenstates of the number operator a†a aredenoted by un& with n50,1,2, . . . . The transition energy ofthe TLS is \v , equal to the energy of the oscillator quanta.The coherent dynamics is described by the HamiltonianH~ t !5HJC1Hkick5H01H int1Hkick . ~1!Here, HJC describes the JC dynamics, composed of the freeHamiltonianH05\v~s†s1a†a !, ~2!with s5ug&^eu, s†5ue&^gu, and of the interaction termH int5\g~s†a1sa†!, ~3!where g is the coupling constant. Hkick generates the instan-taneous phase kick at t5t and is defined asHkick5\pss†d~ t2t!. ~4!The effect of the kick alone is thus given byUkick5expS 2 i\Et20t10 dt H(t)5s†s2ss†5sz . ~5!After the duration T, the evolution operator for the wholeprocess is thenUt~T !5e2(i/\)HJC(T2t)Ukicke2(i/\)HJCt5szexpF i\S H012t2TT H intDTG , ~6!where the second line exploits @H0 ,H int#50 and $H int ,sz%50. So the net evolution amounts to propagation governedby a JC coupling of the effective strength©2002 The American Physical Society02-1RAPID COMMUNICATIONSMORIGI, SOLANO, ENGLERT, AND WALTHER PHYSICAL REVIEW A 65 040102~R!geff52t2TT g ~7!for time T, followed by the phase kick. Since the probabilityof finding the TLS in ug& or ue& is measured eventually, thisfinal phase kick can in fact be ignored. For t5T/2, the ef-fective interaction term in Eq. ~6! vanishes and the final stateis just the freely evolved initial state ~except for the irrel-evant phase kick!. So, if the initial state were ug& ^ ucosc&, forexample, the TLS would always be measured in the groundstate after the interaction. The effect of the kick consists thusin symmetrizing the JC evolution @10#, which is somewhatreminiscent of spin-echo techniques @14#, where the evolu-tion of spin-12 particles interacting with a classical field isundone. We remark that, with this scheme, the external ac-tion could come from any other coherent or even incoherentprocess, but not from the probing TLS. Nevertheless, it is itsfinal statistics that carry the information about the effect ofthe environment on the oscillator.We now turn to the application of this scheme to the par-ticular scenario of cavity QED. The quantized electromag-netic field in a cavity is probed by atoms that fly through theresonator and whose dipole transition couples resonantly to aprivileged field mode, undergoing the JC interaction de-scribed by HJC in Eq. ~1!. The phase kick could be imple-mented by a laser that quasiresonantly couples the groundstate of the atom to a third atomic level, thereby realizing afast 2p pulse @11#. We consider the proposed scheme formeasuring the decoherence time of the cavity field, coupledFIG. 1. ~a! Numerical values @solid line, Eq. ~8!# and approxi-mate analytical values @dashed line, Eq. ~10!# of Pg as a function ofgT for different ratios k/g and the field initially in the number stateu3&. ~b! Numerical values of Pg for k50.05g and initial cavitystates (u2&1iu3&)/A2 ~solid line!, u2& ~dashed line!, coherent stateua& with a5exp(ip/4)/A2 ~dotted line!.04010to a Markovian bath at zero temperature @12#. We assumethat the atomic transition is not subject to dissipation byitself on the relevant time scale. The initial state r(0) of thejoint atom-field system is then evolved into the final state inaccordance withr~T !5eL(T2t)KeLtr~0 !, ~8!where Kr5UkickrUkick21 5szrsz is the effect of the kick attime t , and the Liouville operator L is given byLr52i\@HJC ,r#1kS ara†2 12 a†ar2 12 ra†a D , ~9!where k is the field decay rate. This particular choice of thenonunitary part of L is taken for simplicity, but the methodworks just as well for other cases. An explicit form of Pg ,the probability for measuring the atom at the exit in state ug& ,can be obtained from Eq. ~8! in the limits kT!1 and k/g!1. At first order in these parameters, for t5T/2 and r(0)5ug&^gu ^ (n ,mrnmun&^mu, it isPg512 (n52‘rn ,nH kT4 ~2n21 !1ksin~gTAn !4gAn2ksin~gTAn21 !4gAn212k4g FAn~4n23 !sin~gTAn !cos~gTAn21 !2An21~4n21 ! sin~gTAn21 !cos~gTAn !]J . ~10!The effect of the decay is evident, causing deviations of Pgfrom unity. These deviations are oscillatory functions of theduration T, resulting from the JC dynamics characterizing thesystem evolution. Note that in Eq. ~10! the summation startsat n52. In fact, for rnn5dn ,1 one has Pg51 in this pertur-bative limit, the deviations appearing at second order @15#.In Fig. 1, we plot Pg as a function of the duration T.Figure 1~a! compares the full numerical evaluation of Eq. ~8!with the approximation ~10! for different decay rates k . InFig. 1~b!, the curves are plotted for different initial states.Note that for kT@1 the cavity state tends to the vacuum, andthe probability of counting the atoms in the ground stateapproaches unity again. How big the deviation from Pg51should be for considering that the system has decohered is aquestion that will depend on each particular case or use.The decay of the field phase can be studied by adding tothe proposed scheme a Ramsey zone at the exit of the cavity,which performs the unitary transformation exp@2if(seiz1s†e2iz)#. In this way, the Ramsey zone and the detectorsconstitute a field-phase-sensitive device @16#. When only thecoherent atom-field interaction takes place, we have Pg5cos2f, independent of the initial state of the field, and thedecay of the field phase is measured through the deviationsfrom this value. In Fig. 2, we plot the phase sensitive prob-ability Pg , after the Ramsey zone with f5p/4, z50, as a2-2RAPID COMMUNICATIONSMEASURING IRREVERSIBLE DYNAMICS OF A . . . PHYSICAL REVIEW A 65 040102~R!function of the total interaction time for different initial fieldstates. Both signals contain a high frequency modulation atthe field frequency, chosen here for convenience as v510g .For reference, we plot in both figures the curve correspond-ing to an initial state characterized only by a diagonal densitymatrix ~dashed line!—a Fock state, say.We now study the application of the phase-kick scheme inion traps, where a JC dynamics between the internal transi-tion of the ions and its center-of-mass motion is implementedby a laser. Now, the Hamilton operator ~1! applies in thereference frame rotating at the laser frequency. The atomicphase kick can be realized here in a similar way as in thecavity QED case. Provided that the spontaneous emissionfrom the atom can be neglected and that the motion iscoupled to a Markovian bath at zero temperature, the evolu-tion of the density matrix of the composite system is againgiven by Eq. ~8!, and the results reported above for the atom-cavity case are recovered. In the ion-trap situation, however,the phase kick is more easily implemented than in cavityQED. For example, we can switch off the laser system, aftera fast initial coherent JC evolution and before the phase kick,and so interrupt the interaction during a convenient period oftime T free . By continuing with the phase kick and a fast finalcoherent JC evolution, the experiment is completed.This variation of our scheme has features in common withRamsey interferometry @17#. One could say that, during theinitial short interaction time, interferometric paths are cre-ated in the ion-motion Hilbert space and that, after suffering‘‘dephasing’’ during time T free , these paths are recombinedinto the initial geff50 superposition. With the aid of an arbi-trary phase kick, instead of the fixed 2p kick, we create amodulated output that is reminiscent of an interferometerpattern. This variation could also be implemented in theatom-cavity system, but the experimental setup would bemuch more demanding.FIG. 2. Phase sensitive Pg as a function of gT for k50.05g andv510g . The initial field states are ~a! (u2&1iu3&)/A2 and ~b! co-herent state ua& with a5exp(ip/4)/A2. In both plots, the constantPg50.5 of an initial Fock state is plotted for reference.04010In general, the scheme can be applied to studying deco-herence in the collective motion of a chain of N ions @18#,i.e., a linear crystal of ions, achieved in linear ion traps atsufficiently low temperature. In this limit, the chain motionalong its main axis is described by the collective excitationsof its N normal modes with frequencies n1 , . . . ,nN @19#. Adynamics of the JC type can be designed between the j th ioninternal state and the bth mode of the crystal, by laser irra-diation analogous to the one-ion case. Then, the total Hamil-tonian describing the interaction is given by HJC( j ,b)5H0( j ,b)1H int( j ,b) withH0( j ,b)5\nb~s j†s j1ab† ab! ~11!andH int( j ,b)5\g j ,b~s j†ab1s jab† !, ~12!where g j ,b is the Rabi frequency for ion j and mode b , ab ,and ab† are the annihilation and creation operators of a vibra-tional quantum \nb , while nb is the detuning of the laserfrom the atomic transition and equal to the frequency of theaddressed mode. The dynamics then follows the lines of thatdescribed for a single ion, except that here the sz kick is alaser pulse applied to ion j. In this way, we are able to studythe irreversible decay of a chosen collective mode. Similarly,it is possible to measure the irreversible loss of coherence ofan arbitrary motional state involving all modes. One wouldneed to pair the ions with the modes, so that the correspond-ing Hamiltonian has the form H5( ( j ,b)HJC( j ,b), a sum overthese pairs. One could then measure the decoherence time ofeach mode, and also of entangled states between differentnormal modes.Here are some remarks on the experimental feasibility ofthe proposed scheme in relation to the systems discussedbefore. First, we note that the phase kick is crucial for thewhole scheme, but it is not necessary to realize the ideal kickof Eq. ~5!. Indeed, the scheme is rather robust against experi-mental imperfections that would amount to multiplying Hkickof Eq. ~4! by 11e . In particular, even with an error as largeas ueu&7%, we would still cancel more than 99% of the JCdynamics.In the case of cavity QED, our scheme could be imple-mented in the microwave and optical regime. For a test ofthe decoherence time associated with the population and thephase of the intracavity field, convenient ratios k/g<1 canbe found in both regimes. In either case, atomic decay isirrelevant in a wide range of interaction times. Furthermore,the kick could be implemented by introducing a sheet oflight—transversal to the mode of interest and crossed by theatomic trajectory at the center of the cavity—by shining alaser through a hole drilled at the side of the cavity ~closedcavity!, or by focusing laser beams ~open cavity!. Note that,for measuring decoherence in atom-cavity systems, a qualityfactor that is not too large may be desirable. Further, the useof an external kick for testing this scheme is not necessary, ifthe field mode in question changes sign at the center of thecavity. In this case, the odd spatial symmetry of the field2-3RAPID COMMUNICATIONSMORIGI, SOLANO, ENGLERT, AND WALTHER PHYSICAL REVIEW A 65 040102~R!naturally undoes the JC dynamics, producing the same effectas the phase kick for the fundamental mode.Ion traps appear more fitting for testing and scaling thepresent scheme. For example, the atomic decay can be ne-glected by suitably coupling either a long-lived transition ortwo internal metastable states with a two-photon coherentprocess. Also, JC ~or anti-JC! dynamics like that discussedhere as well as pulses of well defined shape and duration arecurrently realized. Individual ions of a chain cannot be ad-dressed by a laser in all experiments, but this is not necessaryfor the present application. It is always possible to illuminatethe chain of ions homogeneously and, by giving an equallyhomogeneous phase kick, the JC dynamics could be ‘‘timereversed’’ and tested by measuring the global atomic groundstate. In this case, we replace s by ( js j in Eq. ~12!, and thephase-kick operator sz by ) jszj.In summary, we have discussed a method for measuringthe decoherence time of a harmonic oscillator by coupling itto a two-level system and detecting its level statistics. The04010method is based on a phase kick applied to the two-levelsystem during its interaction with the oscillator, thereby un-doing the coherent interaction and isolating the effect of anyexternal disturbance affecting the oscillator. We have dis-cussed applications of the scheme for measuring decoher-ence times of the quantized field mode in cavity QED ex-periments and of the quantized motion of trapped atoms inharmonic traps. We think that the proposed scheme repre-sents a useful tool for measuring decoherence times in dif-ferent physical systems undergoing JC-like interactions. As ageneral feature, this method is sensitive to any deviationfrom the expected JC dynamics and could also be used, forinstance, for measuring corrections to the rotating wave ap-proximation, or for studying the anharmonic coupling amongthe normal modes of a chain of ions, or the influence ofspectator atoms and spectator modes.This work was partly supported by the European Com-mission ~TMR networks ERB-FMRX-CT96-0077 and ERB-FMRX-CT96-0087!.@1# See for example, Decoherence: Theoretical, Experimental andConceptual Problems, Vol. 538 of Lecture Notes in Physics,edited by P. Blanchard, D. Giulini, E. Joos, C. Kiefer, and I.-O.Stamatescu ~Springer, Berlin, 2000!.@2# P. Bertet, S. Osnaghi, A. Rauschenbeutel, G. Nogues, A.Auffeves, M. Brune, J.M. Raimond, and S. Haroche, Nature~London! 411, 166 ~2001!.@3# M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van derZouw, and A. Zeilinger, Nature ~London! 401, 680 ~1999!; M.Arndt and A. Zeilinger, Phys. Bl. 56, 69 ~2000!.@4# See, for example, The Physics of Quantum Information, editedby D. Bouwmeester, A. Ekert, and A. Zeilinger ~Springer, Ber-lin, 2000!.@5# M. Brune, E. Hagley, J. Dreyer, X. Maıˆtre, A. Maali, C.Wunderlich, J.M. Raimond, and S. Haroche, Phys. Rev. Lett.77, 4887 ~1996!.@6# C.J. Myatt, B.E. King, Q.A. Turchette, C.A. Sackett, D.Kielpinski, W.M. Itano, C. Monroe, and D.J. Wineland, Nature~London! 403, 269 ~2000!.@7# V.B. Braginsky and F.Y. Khalili, in Quantum Measurement,edited by K.S. Thorne ~Cambridge University Press, Cam-bridge, England, 1992!.@8# E.T. Jaynes and F.W. Cummings, Proc. IEEE 51, 89 ~1963!.@9# Phase kicks of this sort are at the basis of schemes for inhib-iting decoherence of a small system coupled to a bath. See, forexample, L. Viola and S. Lloyd, Phys. Rev. A 58, 2733 ~1998!;D. Vitali and P. Tombesi, ibid. 59, 4178 ~1999!; L. Viola, E.Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 ~1999!; L. Viola,S. Lloyd, and E. Knill, ibid. 83, 4888 ~1999!.@10# P. Zanardi, Phys. Lett. A 258, 77 ~1999!.@11# See, for example, G.S. Agarwal, M.O. Scully, and H. Walther,Phys. Rev. Lett. 86, 4271 ~2001!.@12# G. Raithel, C. Wagner, H. Walther, L.M. Narducci, and M.O.Scully, in Advances in Atomic, Molecular and Optical Physics,edited by P. Berman ~Academic, New York, 1994!, Suppl. 2.@13# D.J. Wineland, C. Monroe, W.M. Itano, D. Leibfried, B.E.King, and D.M. Meekhof, J. Res. Natl. Inst. Stand. Technol.103, 259 ~1998!.@14# See, for example, L. Allen, and J.H. Eberly, Optical Resonanceand Two-Level Atoms ~Wiley, London, 1975!.@15# For the initial state r(0)5ug,1&^g,1u, one can evaluate Pgexactly:Pg51264g2k2e2kT/2~16g22k2!2sin4S18A16g22k2T D ,which shows that the effect of the decay is O(k2/g2).@16# B.-G. Englert, T. Gantsog, A. Schenzle, C. Wagner, and H.Walther, Phys. Rev. A 53, 4386 ~1996!.@17# N.F. Ramsey, Molecular Beams ~Oxford University Press,New York, 1985!.@18# E. Solano, R.L. de Matos Filho, and N. Zagury, Phys. Rev.Lett. 87, 060402 ~2001!.@19# D.F.V. James, Appl. Phys. B: Lasers Opt. 66, 181 ~1998!.2-4

Measuring irreversible dynamics of a quantum harmonic oscillator

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