The heating of trapped ions due to the interaction with a {\it quantized
environment} is studied {\it without performing the Born-Markov approximation}.
A generalized master equation local in time is derived and a novel theoretical
approach to solve it analytically is proposed. Our master equation is in the
Lindblad form with time dependent coefficients, thus allowing the simulation of
the dynamics by means of the Monte Carlo Wave Function (MCWF) method.Comment: 4 pages, 3 figure

Intravaia, F.

Maniscalco, S.

Messina, A.

Piilo, J.

English

arXiv

The heating of trapped ions due to the interaction with a quantized environment is studied without performing the Born–Markov approximation. A generalized master equation local in time is derived and a novel theoretical approach to solve it analytically is proposed. Our master equation is in the Lindblad form with time dependent coefficients, thus allowing the simulation of the dynamics by means of the Monte Carlo Wave Function (MCWF) method

Intravaia, F

Maniscalco, Sabrina

Piilo, Jyrki

Messina, A

Heriot Watt Pure

Physics Letters A 308 (2003) 6–10www.elsevier.com/locate/plaQuantum theory of heating of a single trapped ionF. Intravaia a, S. Maniscalco a,∗, J. Piilo b, A. Messina aa INFM and MIUR, Dipartimento di Scienze Fisiche ed Astronomiche dell’Università di Palermo, via Archirafi 36, 90123 Palermo, Italyb Helsinki Institute of Physics, PL 64, FIN-00014 Helsingin yliopisto, FinlandReceived 10 December 2002; accepted 18 December 2002Communicated by P.R. HollandAbstractThe heating of trapped ions due to the interaction with a quantized environment is studied without performing the Born–Markov approximation. A generalized master equation local in time is derived and a novel theoretical approach to solve itanalytically is proposed. Our master equation is in the Lindblad form with time dependent coefficients, thus allowing thesimulation of the dynamics by means of the Monte Carlo Wave Function (MCWF) method. 2003 Elsevier Science B.V. All rights reserved.Single ions confined in miniaturized radio fre-quency traps can be cooled down by means of lasercooling techniques very efficiently [1]. Experimentshave shown that the center of mass of a single cooledtrapped ion undergoes a quantized harmonic oscilla-tory motion [2]. During the last decade a great deal ofattention has been devoted to such systems since theyturn out to be very weakly coupled to external envi-ronment, thus allowing the manipulation and observa-tion of the coherent dynamics of a single quantum sys-tem [3]. Indeed, by irradiating the ion with properlyconfigured laser beams, it is possible to manipulatethe vibronic state of the ion. The easiness to engineerat will the interaction between the motional and inter-nal degrees of freedom of the ion makes it possible toperform experimental tests of fundamental features ofquantum mechanics. For example, many non-classical* Corresponding author.E-mail address: sabrina@fisica.unipa.it (S. Maniscalco).states of the oscillatory motion of the ion have beenprepared and measured [4]. By using multiple ions in alinear Paul trap, experiments on quantum non-localityhave been performed [5] and many-particle entangledstates have been realized [6]. Furthermore cold trappedions have been recently proposed as a physical imple-mentation for quantum computation [7].Both for fundamental studies and for technologi-cal applications it is of great interest to understand andstudy those factors limiting the fidelity of the opera-tions performed to manipulate coherently the quantumstate of the ion. Heating of the center of mass oscilla-tory motion of one or more trapped ions seems to beone of the major practical sources of decoherence [8].The process of heating of a single trapped ion is dueto the electromagnetic coupling between noisy elec-tric fields and the ion. Such fields give rise to fluctu-ating forces acting on the ion which can cause an in-crease in its motional energy. The physical origin ofthe noisy electric fields in the center of the trap has notyet been unambiguously identified. In fact, measure-0375-9601/03/$ – see front matter  2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0375-9601(02)01778-4F. Intravaia et al. / Physics Letters A 308 (2003) 6–10 7ments of the heating rate are very difficult to performsince they require high sensitivity and they may de-pend on parameters related to the specific trap geome-try used [8]. For this reason a general theory of heatingwould help in identifying and possibly reduce the upto now unknown sources of noise.Most of the theoretical studies on the heating oftrapped ions deal with the interaction between a sin-gle harmonic oscillator and a reservoir which is ei-ther described as a classical stochastic field [9–12] oras an infinite chain of quantum harmonic oscillators.In the first case an analytic solution for the dynam-ics of the system can be found without performing theBorn–Markov approximation [9,11]. In other words,it is possible to study the short time evolution of theheating function when the field noise spectrum is notflat, as it is indeed in any real experimental situation.However, the analytical treatments, in this case, cannotdescribe the thermalization process, i.e., the long timebehavior, since the stochastic field continuously feedsenergy in the system, leading to an infinite growth ofthe heating function. On the other hand, modelling athermal reservoir as an infinite chain of harmonic os-cillators at temperature T allows, under certain ap-proximations, to describe the dynamics of the systemin terms of a master equation of the form [13]∂ρˆ∂t=−Γ (N + 1)[aˆ†aˆρˆ − 2aˆρˆaˆ† + ρˆaˆ†aˆ](1)− ΓN[aˆaˆ†ρˆ − 2aˆ†ρˆaˆ + ρˆaˆaˆ†],where aˆ† and aˆ are the creation and annihilation oper-ators of vibrational quanta, Γ is the heating constantand the environment is assumed to be at thermal equi-librium, N being the mean number of reservoir exci-tations at temperature T . In Eq. (1) the terms propor-tional to Γ (N + 1) account for spontaneous and stim-ulated emission, that is a transfer of energy from thesystem to the reservoir, while the terms proportional toΓ N account for absorption, that is a transfer of energyfrom the reservoir to the system. The presence of thespontaneous emission term, arising from a quantizeddescription of the reservoir, ensures the thermalizationprocess. Indeed, from Eq. (1), it is possible to derivethe following expression describing the time evolutionof the heating function:(2)〈nˆ(t)〉= n(ω0)(1− e−Γ t),with n(ω0)= (eβh¯ω0 − 1)−1, β = (KT )−1 and ω0 fre-quency of the trap. In order to describe the dynam-ics of the system by means of the master equation (1),however, it is necessary to assume the validity of theBorn–Markov approximation. As a consequence sucha treatment is valid only for times t much bigger thanthe correlation time of the reservoir and thus does notgive the correct short time behavior.The aim of our Letter is to study the heating of asingle trapped ion interacting with a quantized reser-voir without performing the Born–Markov approxima-tion. One of the main results of the Letter is the deriva-tion of an analytical expression for the heating func-tion describing both the non-Markovian short time be-havior and the asymptotic thermalization process.The Hamiltonian describing the interaction be-tween the ion motion and a quantized reservoir mod-eled as an infinite chain of harmonic oscillators, can bewritten as follows Ĥ = h¯√α∑∞j=0 cj (aˆ + aˆ†)(bˆj +bˆ†j ), where bˆj is the annihilation operator of reser-voir excitations and α is the coupling strength. By us-ing the time-convolutionless projection operator tech-nique [14] and performing the rotating wave approxi-mation, we have derived the following master equationdescribing the non-Markovian dynamics of the systemin the interaction picture:∂ρˆ∂t=− ∆¯(t)+ γ (t)2[aˆ†aˆρˆ − 2aˆρˆaˆ† + ρˆaˆ†aˆ](3)− ∆¯(t)− γ (t)2[aˆaˆ†ρˆ − 2aˆ†ρˆaˆ + ρˆaˆaˆ†].The functions ∆¯(t) and γ (t) are defined in termsof the two reservoir characteristic functions [15]:the correlation functionκ(τ )= α∞∫0ω|g(ω)|2(2n(ω)+ 1) cos(ωτ) dωand the susceptibilityµ(τ)= α∞∫0ω|g(ω)|2 sin(ωτ) dω,where n(ω) = (eβh¯ω − 1)−1. Moreover, assuming aLorentzian spectral density |g(ω)|2 = ω2c/[π(ω2 +ω2c )] (Ohmic environment [16]) with ωc reservoirfrequency cut, in the high temperature regime defined8 F. Intravaia et al. / Physics Letters A 308 (2003) 6–10by the condition (h¯β)−1  ωc and in the weakcoupling limit, the real functions ∆¯(t) and γ (t) canbe written in a simple analytic form:∆¯(t)= αh¯βω0rr2 + 1(4)× {e−ωct sin(ωct/r)+ r[1− e−ωct cos(ωct/r)]},γ (t)= α2r2r2 + 1(5)× [1− e−ωct cos(ωct/r)− re−ωct sin(ωct/r)],where r = ωc/ω0. Remembering that in the hightemperature regime n(ω0)  (h¯βω0)−1, it is easyto check that in the Markovian limit, that is whent  τR with τR = 1/ωc correlation time of thereservoir, Eq. (3) reduces to Eq. (1). The analyticexpression of ∆¯(t) and γ (t) for a generic temperatureT is, in general, more complicated. Nonetheless thetemperature dependence of these coefficients of themaster equation brings to light interesting features inthe dynamics of the system and, for this reason, it iscarefully discussed in a follow up paper [17].Our non-Markovian master equation (3) has thegeneral form of the time-convolutionless master equa-tions [18,19]. However, it is worth noting that it is notonly local in time but also, as long as the coefficients∆¯(t)+ γ (t) and ∆¯(t)− γ (t) are positive, in the Lind-blad form. We have verified that, for the value of para-meters currently used in the experiments and for r > 1the time dependent sum and difference coefficients arepositive at all times t . Note that the condition r > 1simply means that the spectral density of the reservoiroverlaps with the frequency of oscillation of the ion,which is a reasonable and commonly done assump-tion. In other words, we demonstrate for the first timethat the non-Markovian dynamics of the reduced den-sity matrix of the system here considered, is describedby a Lindblad-type master equation. Thus, it is pos-sible to use the standard MCWF method to unravelEq. (3) [20]. We note that up to now, only quantumstate diffusion unravelings have been known and usedfor simulating the temporal behavior of a harmonicoscillator interacting with a non-Markovian quantizedenvironment [21]. We would like to emphasize that theadvantage of working with a master equation of Lind-blad type whose analytical solution may be success-fully found, as in our case, is that the positivity of thedensity matrix is ensured. In the following we will usethe quantum jump MCWF method to obtain the tem-poral behavior of the heating function and compare itwith the analytic solution.We now briefly sketch the novel analytical treat-ment we have developed to solve the master equa-tion (3) and derive a closed expression for the heatingfunction describing both its short and its long time be-havior. A detailed description of the method is givenelsewhere [22]. The first step of our approach consistsin expanding the density matrix of the system as fol-lows [23]:(6)ρˆ(t)= 1π∫χt(λ,λ∗)exp(λ∗aˆ − λaˆ†)d2λ.In Eq. (6), λ ∈ C and χt(λ,λ∗) = Tr{exp(λaˆ† −λ∗aˆ)ρˆ(t)} is the quantum characteristic function. In-serting Eq. (6) into Eq. (3) and by using some usefulalgebric properties of the chronologically ordered timeevolution superoperators Tc(t) of the system, definedby ρˆ(t)= Tc(t)ρˆ(0), we obtain the following expres-sion for the quantum characteristic function:χt(λ,λ∗)= exp(−∆(t)|λ|2)(7)× χ0(T 1/2γ (t)λ;T 1/2γ (t)λ∗).In this equation χ0 is the quantum characteristicfunction at the initial time instant t = 0,(8)Tγ (t)= exp(−2t∫0γ (t ′) dt ′),and(9)∆(t)=t∫0Tγ (t)T−1γ (t′)∆¯(t ′).The functions ∆¯(t) and γ (t) are defined by Eqs. (4)and (5). For example, assuming as initial state avibrational Fock state |k〉, Eq. (7) becomesχt(λ,λ∗)= exp[−(∆(t)+ Tγ (t)2)|λ|2](10)×Lk(Tγ (t)|λ|2),where Lk(y) is the Laguerre polynomial of order kin y . Eq. (7) together with Eq. (6) gives the analyticF. Intravaia et al. / Physics Letters A 308 (2003) 6–10 9Fig. 1. Short time behavior of the heating function for couplingconstant α = 0.1 Hz, trap frequency ω0 = 107 Hz, r = 10 andT = 300 K. We compare the analytic solution and the MCWFsimulation with 107 histories.Fig. 2. Long time behavior of the heating function showing thethermalization process, for coupling constant α = 0.1 Hz, trapfrequency ω0 = 107 Hz, r = 10 and T = 10 mK. We compare theanalytic solution and the MCWF simulation with 104 histories. Wenote that, even for such a small value of the reservoir temperature,the high temperature approximation holds since (h¯β)−1  109 Hzwhich is one order of magnitude bigger than ωc.expression of the density matrix of the system in anoperatorial form at any time instant t . From Eq. (7)it is easy to derive the following expression for themean vibrational quantum number, i.e., for the heatingfunction:(11)〈nˆ(t)〉=∆(t)+ 12(Tγ (t)− 1)+ kTγ (t).In Figs. 1 and 2 we show the short and long time be-havior of the heating function of a trapped ion ini-Fig. 3. Comparison between the short time Markovian andnon-Markovian time evolutions of the heating function for cou-pling constant α = 0.1 Hz, trap frequency ω0 = 107 Hz, r = 10and T = 300 K.tially prepared in its vibrational ground state. We com-pare the analytical solution and the MCWF simula-tion performed starting from the master equation (3).The figures show a very good agreement betweenthe analytical and the numerical approaches. We notethat our quantum theory of heating predicts the ini-tial quadratic behavior typical of non-Markovian dy-namics. In Fig. 3 we compare the non-Markovian andMarkovian time evolution for times t smaller than thereservoir correlation time τR = 1/ωc. A similar resultwas deduced by James and by Budini in the case of aclassical environment described in terms of a stochas-tic noisy electric field [9,11]. Our approach, however,allows also to describe the asymptotic thermalizationprocess as shown in Fig. 2. Indeed, for times t  τRand for a flat reservoir spectrum r  1, the heatingfunction given by Eq. (11) reduces to the well-knownMarkovian equation given by Eq. (2), as one can eas-ily derive substituting the asymptotic expressions for∆¯(t →∞) and γ (t →∞) into Eqs. (8), (9) and (11),in the limit r  1.In conclusion we have proposed a new approach forstudying the heating of a single trapped ion due to theinteraction with a quantized reservoir. Our approachdoes not rely on the Born–Markov approximation andthus allows to describe the initial quadratic behaviorof the heating function. Moreover, due to the quantumdescription of the environment, our solution describescorrectly the thermalization process. Stated anotherway our quantum theory of heating bridges a gap10 F. Intravaia et al. / Physics Letters A 308 (2003) 6–10existing in the literature. Until now, indeed, onlynon-Markovian analytical solutions, in the case ofinteraction with a classical environment, or Markoviananalytical solutions, for interactions with a quantumenvironment, were known.It is important to emphasize that measurements ofthe initial quadratic behavior of the heating functionare difficult to perform for two reasons. Firstly be-cause this would mean to make many measurementsin an interval of time of the order of τR  10−8 s, forthe value of parameters used in the Letter. The secondand more important reason is related to the difficultyin revealing so small variations in 〈nˆ(t)〉 as the onesshown in Fig. 1. Very recently, however, it has beenexperimentally demonstrated the possibility of engi-neering both the type of reservoir interacting with asingle trapped ion and the coupling between the sys-tem and the environment [24]. The analytical solutionwe have presented in this Letter makes it possible tolook for ranges of the relevant parameters of both thesystem and the reservoir in correspondence of whichthe non-Markovian quadratic behavior becomes exper-imentally observable.The experimental ability in engineering reservoirsuggests also another application. As we have alreadymentioned in the Letter, we have analyzed the depen-dence of the coefficients of our master equation (3) onreservoir parameters such as its temperature T and fre-quency cut ωc. We have found that by changing suchparameters the master equation passes from Lindbladtype to non-Lindblad type. Such a modification doesreflect a deep change in the dynamics of the system.This result will be discussed in more detail in a fol-low up paper since we believe that the possibility ofstudying analytically such changes may give more in-sight in understanding the fundamental properties ofthe heating process of single trapped ions.AcknowledgementsWe acknowledge K.-A. Suominen for helpful com-ments and the Finnish Center for Scientific Computing(CSC) for computing resources. S.M. acknowledgesthe Helsinki Institute of Physics, where part of thework was done, for the hospitality. J.P. acknowledgesfinancial support from the National Graduate Schoolon Modern Optics and Photonics.References[1] F. Diedrich, J.C. Bergquist, W.M. Itano, D.J. Wineland, Phys.Rev. Lett. 62 (1989) 403.[2] D.M. Meekhof, et al., Phys. Rev. Lett. 76 (1996) 1796;Ch. Roos, et al., Phys. Rev. Lett. 83 (1999) 4713.[3] W. Vogel, R.L. de Matos Filho, Phys. Rev. A 52 (1995) 4214.[4] D. Leibfried, et al., J. Mod. Opt. 44 (1997) 2485;C. Monroe, et al., Science 272 (1996) 1131;R. Blatt, Phys. World 9 (1996) 25.[5] M.A. Rowe, et al., Nature 409 (2001) 791.[6] C.A. Sackett, et al., Nature 404 (2000) 256.[7] J.I. Cirac, P. Zoller, Phys. Rev. Lett. 74 (1995) 4091;C. Monroe, et al., Phys. Rev. Lett. 75 (1995) 4714.[8] D.J. Wineland, et al., J. Res. Natl. Inst. Stand. Technol. 103(1998) 259;Q.A. Turchette, et al., Phys. Rev. A 61 (2000) 063418.[9] D.F.V. James, Phys. Rev. Lett. 81 (1998) 317.[10] S.K. Lamoreaux, Phys. Rev. A 56 (1997) 4970.[11] A.A. Budini, Phys. Rev. A 64 (2001) 052110.[12] A.A. Budini, R.L. de Matos Filho, N. Zagury, Phys. Rev. A 65(2002) 041402.[13] C.W. Gardiner, Quantum Noise, Springer-Verlag, Berlin, 1991;H.J. Carmichael, An Open System Approach to QuantumOptics, Springer-Verlag, Berlin, 1993.[14] S. Chaturvedi, J. Shibata, Z. Phys. B 35 (1979) 297;N.H.F. Sgibata, Y. Takahashi, J. Stat. Phys. 17 (1977) 171.[15] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-Photon Interaction, Wiley, New York, 1992.[16] U. Weiss, Quantum Dissipative Systems, 2nd Edition, WorldScientific, Singapore, 1999.[17] F. Intravaia, S. Maniscalco, J. Piilo, F. Petruccione, A. Messina,in preparation.[18] H.-P. Breuer, B. Kappler, F. Petruccione, Phys. Rev. A 59(1999) 1633.[19] H.P. Breuer, F. Petruccione, The Theory of Open QuantumSystems, Oxford Univ. Press, 2002.[20] J. Dalibard, Y. Castin, K. Mølmer, Phys. Rev. Lett. 68 (1992)580;M.B. Plenio, P.L. Knight, Rev. Mod. Phys. 70 (1998) 101.[21] W.T. Strunz, L. Diósi, N. Gisin, T. Yu, Phys. Rev. Lett. 83(1999) 4909.[22] F. Intravaia, S. Maniscalco, A. Messina, quant-ph/0211196.[23] W. Vogel, D.-G. Welsch, S. Wallentowitz, Quantum Optics,Wiley-VCH, Berlin, 2001.[24] C.J. Myatt, et al., Nature 403 (2000) 269;Q.A. Turchette, et al., Phys. Rev. A 62 (2000) 053807.

Quantum theory of heating of a single trapped ion

    Heriot-Watt University Research Gateway                                                          Heriot-Watt UniversityQuantum theory of heating of a single trapped ionIntravaia, F; Maniscalco, Sabrina; Piilo, Jyrki; Messina, APublished in:Physics Letters ADOI:10.1016/S0375-9601(02)01778-4Publication date:2003Link to publication in Heriot-Watt Research GatewayCitation for published version (APA):Intravaia, F., Maniscalco, S., Piilo, J., & Messina, A. (2003). Quantum theory of heating of a single trapped ion.Physics Letters A, 308(1), 6-10. 10.1016/S0375-9601(02)01778-4General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Physics Letters A 308 (2003) 6–10www.elsevier.com/locate/plaQuantum theory of heating of a single trapped ionF. Intravaia a, S. Maniscalco a,∗, J. Piilo b, A. Messina aa INFM and MIUR, Dipartimento di Scienze Fisiche ed Astronomiche dell’Università di Palermo, via Archiraﬁ 36, 90123 Palermo, Italyb Helsinki Institute of Physics, PL 64, FIN-00014 Helsingin yliopisto, FinlandReceived 10 December 2002; accepted 18 December 2002Communicated by P.R. HollandAbstractThe heating of trapped ions due to the interaction with a quantized environment is studied without performing the Born–Markov approximation. A generalized master equation local in time is derived and a novel theoretical approach to solve itanalytically is proposed. Our master equation is in the Lindblad form with time dependent coefﬁcients, thus allowing thesimulation of the dynamics by means of the Monte Carlo Wave Function (MCWF) method. 2003 Elsevier Science B.V. All rights reserved.Single ions conﬁned in miniaturized radio fre-quency traps can be cooled down by means of lasercooling techniques very efﬁciently [1]. Experimentshave shown that the center of mass of a single cooledtrapped ion undergoes a quantized harmonic oscilla-tory motion [2]. During the last decade a great deal ofattention has been devoted to such systems since theyturn out to be very weakly coupled to external envi-ronment, thus allowing the manipulation and observa-tion of the coherentdynamicsof a single quantumsys-tem [3]. Indeed, by irradiating the ion with properlyconﬁgured laser beams, it is possible to manipulatethe vibronic state of the ion. The easiness to engineerat will the interaction between the motional and inter-nal degrees of freedom of the ion makes it possible toperform experimental tests of fundamental features ofquantum mechanics. For example, many non-classical* Corresponding author.E-mail address: sabrina@ﬁsica.unipa.it (S. Maniscalco).states of the oscillatory motion of the ion have beenpreparedandmeasured[4].By usingmultipleions inalinear Paul trap, experiments on quantum non-localityhave been performed [5] and many-particle entangledstateshavebeenrealized[6].Furthermorecoldtrappedions have been recently proposed as a physical imple-mentation for quantum computation [7].Both for fundamental studies and for technologi-cal applicationsit is of greatinterest to understandandstudy those factors limiting the ﬁdelity of the opera-tions performedto manipulatecoherentlythe quantumstate of the ion. Heating of the center of mass oscilla-tory motion of one or more trapped ions seems to beone of the major practical sources of decoherence [8].The process of heating of a single trapped ion is dueto the electromagnetic coupling between noisy elec-tric ﬁelds and the ion. Such ﬁelds give rise to ﬂuctu-ating forces acting on the ion which can cause an in-crease in its motional energy. The physical origin ofthe noisy electric ﬁelds in the center of the trap has notyet been unambiguously identiﬁed. In fact, measure-0375-9601/03/$ – see front matter  2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0375-9601(02)01778-4F. Intravaia et al. / Physics Letters A 308 (2003) 6–10 7ments of the heating rate are very difﬁcult to performsince they require high sensitivity and they may de-pend on parameters related to the speciﬁc trap geome-try used [8].For thisreasona generaltheoryofheatingwould help in identifying and possibly reduce the upto now unknown sources of noise.Most of the theoretical studies on the heating oftrapped ions deal with the interaction between a sin-gle harmonic oscillator and a reservoir which is ei-ther described as a classical stochastic ﬁeld [9–12] oras an inﬁnite chain of quantum harmonic oscillators.In the ﬁrst case an analytic solution for the dynam-ics of the system can be found without performing theBorn–Markov approximation [9,11]. In other words,it is possible to study the short time evolution of theheating function when the ﬁeld noise spectrum is notﬂat, as it is indeed in any real experimental situation.However,theanalyticaltreatments,inthiscase,cannotdescribe the thermalization process, i.e., the long timebehavior, since the stochastic ﬁeld continuously feedsenergy in the system, leading to an inﬁnite growth ofthe heating function. On the other hand, modelling athermal reservoir as an inﬁnite chain of harmonic os-cillators at temperature T allows, under certain ap-proximations, to describe the dynamics of the systemin terms of a master equation of the form [13]∂ ˆ ρ∂t=−Γ(N+1)ˆ a†ˆ a ˆ ρ −2ˆ a ˆ ρˆ a† +ˆ ρˆ a†ˆ a(1) −ΓNˆ aˆ a† ˆ ρ −2ˆ a† ˆ ρˆ a +ˆ ρˆ aˆ a†,where ˆ a† and ˆ a are the creation and annihilation oper-ators of vibrational quanta, Γ is the heating constantand the environmentis assumed to be at thermal equi-librium, N being the mean number of reservoir exci-tations at temperature T. In Eq. (1) the terms propor-tional to Γ(N+1) account for spontaneous and stim-ulated emission, that is a transfer of energy from thesystem to the reservoir,while the termsproportionaltoΓNaccountfor absorption,that is a transfer of energyfrom the reservoir to the system. The presence of thespontaneous emission term, arising from a quantizeddescription of the reservoir, ensures the thermalizationprocess. Indeed, from Eq. (1), it is possible to derivethe followingexpression describingthe time evolutionof the heating function:(2)ˆ n(t)= n(ω0)1−e−Γt,with n(ω0) = (eβ¯ hω0 −1)−1, β = (KT)−1 and ω0 fre-quency of the trap. In order to describe the dynam-ics of the system by means of the master equation (1),however, it is necessary to assume the validity of theBorn–Markov approximation. As a consequence sucha treatment is valid only for times t much bigger thanthe correlation time of the reservoir and thus does notgive the correct short time behavior.The aim of our Letter is to study the heating of asingle trapped ion interacting with a quantized reser-voir withoutperformingthe Born–Markovapproxima-tion. One of the mainresults of the Letter is the deriva-tion of an analytical expression for the heating func-tion describing both the non-Markovianshort time be-havior and the asymptotic thermalization process.The Hamiltonian describing the interaction be-tween the ion motion and a quantized reservoir mod-eledas aninﬁnitechainof harmonicoscillators, canbewritten as follows  H = ¯ h√α∞j=0cj(ˆ a +ˆ a†)(ˆ bj +ˆ b†j),w h e r eˆ bj is the annihilation operator of reser-voir excitations and α is the coupling strength. By us-ing the time-convolutionless projection operator tech-nique [14] and performing the rotating wave approxi-mation,wehavederivedthefollowingmasterequationdescribingthe non-Markoviandynamicsof the systemin the interaction picture:∂ ˆ ρ∂t=−¯ ∆(t) +γ(t)2ˆ a†ˆ a ˆ ρ −2ˆ a ˆ ρˆ a† +ˆ ρˆ a†ˆ a(3) −¯ ∆(t) −γ(t)2ˆ aˆ a† ˆ ρ −2ˆ a† ˆ ρˆ a +ˆ ρˆ aˆ a†.The functions ¯ ∆(t) and γ(t) are deﬁned in termsof the two reservoir characteristic functions [15]:the correlation functionκ(τ)= α∞ 	0ω|g(ω)|22n(ω)+1cos(ωτ)dωand the susceptibilityµ(τ)= α∞ 	0ω|g(ω)|2sin(ωτ)dω,where n(ω) = (eβ¯ hω − 1)−1. Moreover, assuming aLorentzian spectral density |g(ω)|2 = ω2c/[π(ω2 +ω2c)] (Ohmic environment [16]) with ωc reservoirfrequency cut, in the high temperature regime deﬁned8 F. Intravaia et al. / Physics Letters A 308 (2003) 6–10by the condition (¯ hβ)−1   ωc and in the weakcoupling limit, the real functions ¯ ∆(t) and γ(t) canbe written in a simple analytic form:¯ ∆(t) =α¯ hβω0rr2 +1(4)×e−ωct sin(ωct/r)+r1−e−ωct cos(ωct/r),γ(t)=α2r2r2 +1(5)×1−e−ωct cos(ωct/r)−re−ωct sin(ωct/r),where r = ωc/ω0. Remembering that in the hightemperature regime n(ω0)   (¯ hβω0)−1, it is easyto check that in the Markovian limit, that is whent   τR with τR = 1/ωc correlation time of thereservoir, Eq. (3) reduces to Eq. (1). The analyticexpression of ¯ ∆(t) and γ(t)for a generic temperatureT is, in general, more complicated. Nonetheless thetemperature dependence of these coefﬁcients of themaster equation brings to light interesting features inthe dynamics of the system and, for this reason, it iscarefully discussed in a follow up paper [17].Our non-Markovian master equation (3) has thegeneral form of the time-convolutionlessmaster equa-tions [18,19]. However, it is worth noting that it is notonly local in time but also, as long as the coefﬁcients¯ ∆(t)+γ(t)and ¯ ∆(t)−γ(t)are positive, in the Lind-blad form. We have veriﬁed that, for the value of para-meters currentlyused in the experimentsand for r>1the time dependentsum and differencecoefﬁcientsarepositive at all times t. Note that the condition r>1simply means that the spectral density of the reservoiroverlaps with the frequency of oscillation of the ion,which is a reasonable and commonly done assump-tion. In other words, we demonstrate for the ﬁrst timethat the non-Markovian dynamics of the reduced den-sity matrix of the system here considered, is describedby a Lindblad-type master equation. Thus, it is pos-sible to use the standard MCWF method to unravelEq. (3) [20]. We note that up to now, only quantumstate diffusion unravelings have been known and usedfor simulating the temporal behavior of a harmonicoscillator interacting with a non-Markovian quantizedenvironment[21].We wouldlike to emphasizethattheadvantage of working with a master equation of Lind-blad type whose analytical solution may be success-fully found, as in our case, is that the positivity of thedensity matrix is ensured. In the following we will usethe quantum jump MCWF method to obtain the tem-poral behavior of the heating function and compare itwith the analytic solution.We now brieﬂy sketch the novel analytical treat-ment we have developed to solve the master equa-tion (3) and derive a closed expression for the heatingfunction describing both its short and its long time be-havior. A detailed description of the method is givenelsewhere [22]. The ﬁrst step of our approach consistsin expanding the density matrix of the system as fol-lows [23]:(6) ˆ ρ(t)=1π	χtλ,λ∗expλ∗ˆ a −λˆ a†d2λ.In Eq. (6), λ ∈ C and χt(λ,λ∗) = Tr{exp(λˆ a† −λ∗ˆ a)ˆ ρ(t)} is the quantum characteristic function. In-serting Eq. (6) into Eq. (3) and by using some usefulalgebricpropertiesofthechronologicallyorderedtimeevolution superoperators Tc(t) of the system, deﬁnedby ˆ ρ(t)= Tc(t) ˆ ρ(0), we obtain the following expres-sion for the quantum characteristic function:χtλ,λ∗= exp−∆(t)|λ|2(7) ×χ0T 1/2γ (t)λ;T 1/2γ (t)λ∗.In this equation χ0 is the quantum characteristicfunction at the initial time instant t = 0,(8) Tγ(t) = exp−2t 	0γ(t )dt ,and(9) ∆(t) =t 	0Tγ(t)T −1γ (t ) ¯ ∆(t ).The functions ¯ ∆(t) and γ(t) are deﬁned by Eqs. (4)and (5). For example, assuming as initial state avibrational Fock state |k , Eq. (7) becomesχtλ,λ∗= exp−∆(t) +Tγ(t)2|λ|2(10) ×LkTγ(t)|λ|2,where Lk(y) is the Laguerre polynomial of order kin y. Eq. (7) together with Eq. (6) gives the analyticF. Intravaia et al. / Physics Letters A 308 (2003) 6–10 9Fig. 1. Short time behavior of the heating function for couplingconstant α = 0.1 Hz, trap frequency ω0 = 107 Hz, r = 10 andT = 300 K. We compare the analytic solution and the MCWFsimulation with 107 histories.Fig. 2. Long time behavior of the heating function showing thethermalization process, for coupling constant α = 0.1 Hz, trapfrequency ω0 = 107 Hz, r = 10 and T = 10 mK. We compare theanalytic solution and the MCWF simulation with 104 histories. Wenote that, even for such a small value of the reservoir temperature,the high temperature approximation holds since (¯ hβ)−1   109 Hzwhich is one order of magnitude bigger than ωc.expression of the density matrix of the system in anoperatorial form at any time instant t. From Eq. (7)it is easy to derive the following expression for themean vibrationalquantumnumber,i.e., for the heatingfunction:(11)ˆ n(t)= ∆(t)+12Tγ(t) −1+kTγ(t).In Figs. 1 and 2 we show the short and long time be-havior of the heating function of a trapped ion ini-Fig. 3. Comparison between the short time Markovian andnon-Markovian time evolutions of the heating function for cou-pling constant α = 0.1 Hz, trap frequency ω0 = 107 Hz, r = 10and T =300 K.tially preparedin its vibrationalgroundstate. We com-pare the analytical solution and the MCWF simula-tion performed starting from the master equation (3).The ﬁgures show a very good agreement betweenthe analytical and the numerical approaches. We notethat our quantum theory of heating predicts the ini-tial quadratic behavior typical of non-Markovian dy-namics. In Fig. 3 we compare the non-Markovian andMarkovian time evolution for times t smaller than thereservoir correlation time τR = 1/ωc. A similar resultwas deduced by James and by Budini in the case of aclassical environment described in terms of a stochas-tic noisy electric ﬁeld [9,11]. Our approach, however,allows also to describe the asymptotic thermalizationprocess as shown in Fig. 2. Indeed, for times t   τRand for a ﬂat reservoir spectrum r   1, the heatingfunction given by Eq. (11) reduces to the well-knownMarkovian equation given by Eq. (2), as one can eas-ily derive substituting the asymptotic expressions for¯ ∆(t →∞ ) and γ(t→∞ ) into Eqs. (8), (9) and (11),in the limit r   1.Inconclusionwe haveproposeda newapproachforstudying the heating of a single trapped ion due to theinteraction with a quantized reservoir. Our approachdoes not rely on the Born–Markov approximation andthus allows to describe the initial quadratic behaviorof the heating function. Moreover, due to the quantumdescription of the environment, our solution describescorrectly the thermalization process. Stated anotherway our quantum theory of heating bridges a gap10 F. Intravaia et al. / Physics Letters A 308 (2003) 6–10existing in the literature. Until now, indeed, onlynon-Markovian analytical solutions, in the case ofinteractionwith a classical environment,orMarkoviananalytical solutions, for interactions with a quantumenvironment, were known.It is important to emphasize that measurements ofthe initial quadratic behavior of the heating functionare difﬁcult to perform for two reasons. Firstly be-cause this would mean to make many measurementsin an interval of time of the order of τR   10−8 s, forthe value of parameters used in the Letter. The secondand more important reason is related to the difﬁcultyin revealing so small variations in  ˆ n(t)  as the onesshown in Fig. 1. Very recently, however, it has beenexperimentally demonstrated the possibility of engi-neering both the type of reservoir interacting with asingle trapped ion and the coupling between the sys-tem and the environment [24]. The analytical solutionwe have presented in this Letter makes it possible tolook for ranges of the relevant parameters of both thesystem and the reservoir in correspondence of whichthenon-Markovianquadraticbehaviorbecomesexper-imentally observable.The experimental ability in engineering reservoirsuggests also another application. As we have alreadymentioned in the Letter, we have analyzed the depen-dence of the coefﬁcients of our master equation (3) onreservoirparameterssuch asits temperatureT andfre-quency cut ωc. We have found that by changing suchparameters the master equation passes from Lindbladtype to non-Lindblad type. Such a modiﬁcation doesreﬂect a deep change in the dynamics of the system.This result will be discussed in more detail in a fol-low up paper since we believe that the possibility ofstudying analytically such changes may give more in-sight in understanding the fundamental properties ofthe heating process of single trapped ions.AcknowledgementsWe acknowledge K.-A. Suominen for helpful com-mentsandtheFinnishCenter forScientiﬁc Computing(CSC) for computing resources. S.M. acknowledgesthe Helsinki Institute of Physics, where part of thework was done, for the hospitality. J.P. acknowledgesﬁnancial support from the National Graduate Schoolon Modern Optics and Photonics.References[1] F. Diedrich, J.C. Bergquist, W.M. Itano, D.J. Wineland, Phys.Rev. Lett. 62 (1989) 403.[2] D.M. Meekhof, et al., Phys. Rev. Lett. 76 (1996) 1796;Ch. Roos, et al., Phys. Rev. Lett. 83 (1999) 4713.[3] W. Vogel, R.L. de Matos Filho, Phys. Rev. A 52 (1995) 4214.[4] D. Leibfried, et al., J. Mod. Opt. 44 (1997) 2485;C. Monroe, et al., Science 272 (1996) 1131;R. Blatt, Phys. World 9 (1996) 25.[5] M.A. Rowe, et al., Nature 409 (2001) 791.[6] C.A. Sackett, et al., Nature 404 (2000) 256.[7] J.I. Cirac, P. Zoller, Phys. Rev. Lett. 74 (1995) 4091;C. Monroe, et al., Phys. Rev. Lett. 75 (1995) 4714.[8] D.J. Wineland, et al., J. Res. Natl. Inst. Stand. Technol. 103(1998) 259;Q.A. Turchette, et al., Phys. Rev. A 61 (2000) 063418.[9] D.F.V. James, Phys. Rev. Lett. 81 (1998) 317.[10] S.K. Lamoreaux, Phys. Rev. A 56 (1997) 4970.[11] A.A. Budini, Phys. Rev. A 64 (2001) 052110.[12] A.A. Budini, R.L. de Matos Filho, N. Zagury, Phys. Rev. A 65(2002) 041402.[13] C.W. Gardiner, Quantum Noise, Springer-Verlag, Berlin, 1991;H.J. Carmichael, An Open System Approach to QuantumOptics, Springer-Verlag, Berlin, 1993.[14] S. Chaturvedi, J. Shibata, Z. Phys. B 35 (1979) 297;N.H.F. Sgibata, Y. Takahashi, J. Stat. Phys. 17 (1977) 171.[15] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-Photon Interaction, Wiley, New York, 1992.[16] U. Weiss, Quantum Dissipative Systems, 2nd Edition, WorldScientiﬁc, Singapore, 1999.[17] F. Intravaia, S.Maniscalco, J. Piilo, F.Petruccione, A.Messina,in preparation.[18] H.-P. Breuer, B. Kappler, F. Petruccione, Phys. Rev. A 59(1999) 1633.[19] H.P. Breuer, F. Petruccione, The Theory of Open QuantumSystems, Oxford Univ. Press, 2002.[20] J. Dalibard, Y. Castin, K. Mølmer, Phys. Rev. Lett. 68 (1992)580;M.B. Plenio, P.L. Knight, Rev. Mod. Phys. 70 (1998) 101.[21] W.T. Strunz, L. Diósi, N. Gisin, T. Yu, Phys. Rev. Lett. 83(1999) 4909.[22] F. Intravaia, S. Maniscalco, A. Messina, quant-ph/0211196.[23] W. Vogel, D.-G. Welsch, S. Wallentowitz, Quantum Optics,Wiley-VCH, Berlin, 2001.[24] C.J. Myatt, et al., Nature 403 (2000) 269;Q.A. Turchette, et al., Phys. Rev. A 62 (2000) 053807.

Quantum theory of heating of a single trapped ion

Abstract

Similar works

Full text

Available Versions

Heriot Watt Pure

Heriot Watt Pure