Most discussions of decoherence in the literature consider the
high-temperature regime but it is also known that, in the presence of
dissipation, decoherence can occur even at zero temperature. Whereas most
previous investigations all assumed initial decoupling of the quantum system
and bath, we consider that the system and environment are entangled at all
times. Here, we discuss decoherence for a free particle in an initial
Schr\"{o}dinger cat state. Memory effects are incorporated by use of the single
relaxation time model (since the oft-used Ohmic model does not give physically
correct results)

Ford G W

G W Ford

Giulini D

R F O Connell

Walls D F

Zurek W H

Journal of Optics B Quantum and Semiclassical Optics

English

arXiv

Most discussions of decoherence in the literature consider the high temperature regime but it is also known that, in the presence of dissipation, decoherence can occur even at zero temperature. Whereas most previous investigations all assumed initial decoupling of the quantum system and bath, we consider that the system and environment are entangled at all times. Here, we discuss decoherence for a free particle in an initial Schrödinger cat state. Memory effects are incorporated by use of the single relaxation time model

Ford, G. W.

O\u27Connell, R. F.

LSU Scholarly Repository (Louisiana State Univ.)

Louisiana State University LSU Scholarly Repository Faculty Publications Department of Physics & Astronomy 12-1-2003 Decoherence at zero temperature G. W. Ford Dublin Institute for Advanced Studies R. F. O'Connell Dublin Institute for Advanced Studies Follow this and additional works at: https://repository.lsu.edu/physics_astronomy_pubs Recommended Citation Ford, G., & O'Connell, R. (2003). Decoherence at zero temperature. Journal of Optics B: Quantum and Semiclassical Optics, 5 (6) https://doi.org/10.1088/1464-4266/5/6/010 This Conference Proceeding is brought to you for free and open access by the Department of Physics & Astronomy at LSU Scholarly Repository. It has been accepted for inclusion in Faculty Publications by an authorized administrator of LSU Scholarly Repository. For more information, please contact ir@lsu.edu. INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICSJ. Opt. B: Quantum Semiclass. Opt. 5 (2003) S609–S612 PII: S1464-4266(03)60538-0Decoherence at zero temperatureG W Ford1 and R F O’Connell2School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road,Dublin 4, Republic of IrelandReceived 28 February 2003, accepted for publication 14 March 2003Published 16 October 2003Online at stacks.iop.org/JOptB/5/S609AbstractMost discussions of decoherence in the literature consider the hightemperature regime but it is also known that, in the presence of dissipation,decoherence can occur even at zero temperature. Whereas most previousinvestigations all assumed initial decoupling of the quantum system andbath, we consider that the system and environment are entangled at all times.Here, we discuss decoherence for a free particle in an initial Schrödinger catstate. Memory effects are incorporated by use of the single relaxation timemodel.Keywords: Decoherence, dissipation, zero-temperature heat bathMost discussions of decoherence obtain results for decoher-ence decay times which are proportional to the inverse of Tor T 1/2, where T is the temperature, and which apply only tothe high temperature regime, kT  h̄γ , where γ is a typicaldissipative decay rate [1–5]. In fact, whereas most papers inthe early literature give decay rates proportional to γ −1, weshowed that decoherence can occur at high temperature evenfor vanishingly small dissipation [4, 5]. For a pedagogical dis-cussion of the high T regime, we refer the reader to [6]. Here,we look at the opposite end of the temperature spectrum andpresent explicit results at temperature zero. In contrast to thecase for the high temperature regime, we find that a dissipativeenvironment is necessary to achieve decoherence.The zero temperature case was also considered by others,most notably by Walls and Milburn [2]. They considered a‘Schrödinger cat’ state consisting of a pair of coherent statesof an oscillator. They solved the master equation to obtain anexpression for the coherence attenuation coefficient, definedin terms of the off-diagonal elements of the density matrix.However, this approach is subject to serious assumptions:(a) initial decoupling of the system and environment; (b) weakcoupling, which implies a restriction to a finite oscillatorpotential.The assumption (a) used by Walls and Milburn and indeedmany other investigators leads to results which are seriouslyat variance with a model in which the quantum particle andthe heat bath are entangled at all times. In fact, at T = 0,we are simply dealing with the zero point oscillations of1 Permanent address: Department of Physics, University of Michigan, AnnArbor, MI 48109-1120, USA.2 Permanent address: Department of Physics and Astronomy, Louisiana StateUniversity, Baton Rouge, LA 70803-4001, USA.the heat bath which are necessarily always entangled withwhatever quantum particle is of interest. In fact, the resultsthat we obtain here are applicable to the whole low temperatureregime, defined by the inequality kT  h̄γ . General resultsfor all regimes, corresponding to arbitrary T and γ , wereobtained in our initial paper on this subject [4], where ourstarting point was the density matrix for the system of theparticle coupled to the bath at all times. However, thereexplicit results were presented only for the Ohmic model inthe high temperature regime (kT  h̄γ ). Later, we showedthat the same results could be obtained simply by startingwith the wavefunction describing a pure Schrödinger cat statemoving with an initial velocity v [5]. This was then used tocalculate the wavefunction and then the coordinate probabilitydistribution at time t . Finally, this probability distribution wasthen averaged over a thermal distribution of initial velocitiesto obtain the probability distribution corresponding to a finitehigh temperature T . This led immediately to a quantitativemeasure of decoherence (the destruction of the interferencepattern). In other words, the calculation of the decoherencedecay time at high temperature could be carried out simplywithin the framework of elementary quantum mechanics andequilibrium statistical mechanics, with a result independent ofγ [5, 6].In contrast to the high temperature case, results at lowtemperature depend essentially on γ and must be obtainedwithin the framework of nonequilibrium statistical mechanics.Thus, we will now give a synopsis of our general modeldescribing a system and its environment as being entangledat all times [4].First, we emphasize that, not only are our resultsin general applicable to arbitrary dissipation and arbitrary1464-4266/03/060609+04$30.00 © 2003 IOP Publishing Ltd Printed in the UK S609G W Ford and R F O’Connelltemperature, but also they are not subject to the weak couplingapproximation and so apply in the case of a free particle.Arbitrary dissipation refers to reservoir models which give riseto memory terms in the quantum Langevin equation (see (6)below and the subsequent discussion) which describes thequantum system in a dissipative environment. By contrast,the Ohmic model (the model considered by most authors) isthe simplest dissipative model and corresponds to the lack ofmemory terms in the Langevin equation. As we shall seebelow, the Ohmic model has the difficulty that the mean squarevelocity at zero temperature is divergent and a cut-off must besupplied to make it finite. For this purpose we use the singlerelaxation time model (which reduces to the Ohmic model inthe limit of zero relaxation time).As in our previous discussions [4–6], we considerdecoherence in terms of the simple problem of a freeparticle moving in one dimension that is placed in an initialsuperposition state (‘Schrödinger cat’ state) corresponding toa pair of Gaussian wavepackets, each with variance σ 2 andseparated by a distance d  σ . For such a state the probabilitydistribution at time t can be shown to be of the formP(x, t) = 12(1 + e−d2/8σ 2){P0(x − d2, t)+ P0(x +d2, t)+ 2e−d2/8w2(t)a(t)P0(x, t) cos[x(0), x(t)]xd4iσ 2w2(t)}, (1)where P0 is the probability distribution for a single wavepacket,given byP0(x, t) = 1√2πw2(t)exp{− x22w2(t)}. (2)Here and in (1) w2(t) is the variance of a single wavepacket,which in general is given byw2(t) = σ 2 − [x(0), x(t)]24σ 2+ s(t), (3)where σ 2 is the initial variance, [x(0), x(t)] is the commutator,ands(t) = 〈{x(t) − x(0)}2〉, (4)is the mean square displacement. In (1) the first two termswithin the braces correspond to the two wavepackets, centredat ±d/2, expanding independently, while the third term is theinterference term. Decoherence refers to the destruction ofinterference, a measure of which is given by the attenuationcoefficient a(t) which can be defined as the ratio of thefactor multiplying the cosine in the interference term to twicethe geometric mean of the first two terms [4, 5]. Using amethod of successive measurements introduced by Ford andLewis [7], in which a particle in equilibrium and entangledwith the environment is placed in the initial state by a firstmeasurement and then, after a time interval t , is probed by asecond measurement, we obtained the following exact generalformula for the attenuation coefficient [4, 7]:a(t) = exp{− s(t)d28σ 2w2(t)}. (5)The quantities appearing in (3) and (4) are evaluatedby use of the quantum Langevin equation [8], which isa Heisenberg equation of motion for x(t), the dynamicalvariable corresponding to the coordinate of a Brownian particleinteracting with a linear passive heat bath. For the case of afree particle, this equation for the stationary process has thewell known formmẍ +∫ t−∞dt ′ µ(t − t ′)ẋ(t ′) = F(t), (6)where µ(t) is the memory function and F(t) is a fluctuatingoperator force with mean zero. The solution of the quantumLangevin equation (6) can be written asx(t) =∫ t−∞dt ′ G(t − t ′)F(t ′), (7)where G(t), the Green function, can in turn be written asG(t) = 12π∫ ∞−∞dω α(ω + i0+)e−iωt , (8)in which α(z) (the Fourier transform of the Green function)is the response function. For the free particle the responsefunction has the general formα(z) = 1−mz2 − izµ̃(z) , (9)in which µ̃(z) is the Fourier transform of the memory function,µ̃(z) =∫ ∞0dt µ(t)eizt , Im{z} > 0. (10)Using these results, we find that [8, 9] the mean squaredisplacement is given by the formulas(t) = 2h̄π∫ ∞0dω Im{α(ω + i0+)} coth h̄ω2kT(1 − cos ωt),(11)while the commutator, which is temperature independent, isgiven by the formula[x(t1), x(t1 + t)] = 2ih̄π∫ ∞0dω Im{α(ω + i0+)} sin ωt. (12)These expressions are valid for arbitrary temperature andarbitrary dissipation. (Indeed, with the appropriate expressionfor the response function, they are valid in the presence ofan external oscillator potential.) In our earlier discussions,for brevity, we confined our attention to the case of hightemperature. Here we consider the case of zero temperature.Most discussions in the literature are confined to the so-called Ohmic case, where µ̃(z) = ζ , the Newtonian frictionconstant but we analyse the more general case of the singlerelaxation time model [10]. This model corresponds to amemory function of the formµ(t) = ζτe−t/τ θ(t), (13)where τ is the relaxation time of the bath and where θ is theHeaviside function. Note that in the limit τ → 0 this becomesthe Ohmic memory function µ(t) = 2ζδ(t)θ(t). With thisform of the memory function,µ̃(z) = ζ1 − izτ . (14)S610Decoherence at zero temperatureNext, with this form in the expression (9) for the responsefunction, we find with T = 0 that the expression (11) for theresponse can be expressed in the forms(t) = 2h̄πζ2V (γ t) − γ 2V (t)2 − γ 2 , (15)while the expression (12) for the commutator becomes[x(0), x(t)] = ih̄ζ2(1 − e−γ t) − γ 2(1 − e−t)2 − γ 2 , (16)where we have introduced the quantities = 1 +√1 − 4ζτ/m2τ, γ = 1 −√1 − 4ζτ/m2τ,(17)and where [11]V (x) =∫ ∞0dyx2y(y2 + x2)(1 − cos y)= log x + γE − 12 [e−x Ēi(x) + ex Ei(−x)]= −(log x + γE)(cosh x − 1)− 12[e−x∞∑n=1xnn!n+ ex∞∑n=1(−x)nn!n]. (18)Here γE = 0.577 215 665 is Euler’s constant. Note that thesums in (18) are absolutely convergent. We should point outthat the quantities  and γ were referred to as γ+ and γ− in [10].We feel that the present choice is more suggestive since, forτ → 0, we have  → 1/τ and γ → ζ/m.Note the expansions [12], for small x ,V (x) ∼= − 12 x2(log x + γE − 32 ). (19)Also, asymptotically, for large x ,V (x) ∼ log x + γE − 1x2− 3!x4− 5!x6. (20)For very short times (t  τ), one can show in general thats(t) ∼= 〈v2〉t2, (21)where 〈v2〉 is the mean square velocity, given for the singlerelaxation time model by〈v2〉 = h̄γ πm( − γ ) logγ∼= − h̄ζπm2logζτm. (22)We note that, for τ → 0 (corresponding to the Ohmic model),the mean square velocity, given in (22), is logarithmicallydivergent. The conclusion is that, in order to obtain finiteresults for 〈v2〉 at short time, one must use a cut-off model,such as the single relaxation time model used here.For intermediate times (τ  t  (ζ/m)−1), we obtains(t) ∼= − h̄ζπm2t2{logζ tm+ γE − 32}, (23)which is the Ohmic model result. We can compare these resultswith the weak coupling result which for a free particle leadsto s(t) being identically zero. (For a free particle the meansquare velocity vanishes at zero temperature.)Thus, to investigate the decay of coherence (which isgenerally a short time phenomenon) in both time ranges, weconsider the behaviour for times ζ t/m  1. In this case, thecommutator becomes that of a non-interacting particle:[x(0), x(t)] ∼= i h̄tm. (24)Hence, it is clear that for very short times the wavepacket widthis the initial width:w2(t) ∼= σ 2. (25)Thus, the attenuation coefficient (5) for short times takes theforma(t) ∼= exp{− s(t)d28σ 4}, (26)where s(t) is given by (21) and (23). Hence, in particular, wecan writea(t) = exp{(tτ0)2logζτm}, t  τ, (27)anda(t) = exp{(tτ0)2[logζ tm+ γE − 32]},τ  t  (ζ/m)−1, (28)is the Ohmic result, withτ0 ≡ mσ2d√8πh̄ζ. (29)The characteristic time for decoherence to occur, τd, is defined,as usual [4–6], as the time at which a(t) = exp(−1). Hence,for example it follows from (27) that, for t  τ ,τd = τ0| log(ζ τ/m)|−1/2 < τ0, (30)where the inequality sign follows from the fact that ζ t/m  1.It is also clear that τd < τ0 in the case t  τ .Next, we consider some numerical results. As mentionedabove, here ζ/m ≈ γ and, as a point of reference, it is usefulto note thatkTh̄γ= T (K )γ (1011 s−1). (31)Now, one of the few experimental investigations ofdecoherence is the work of Myatt et al [13] on the decoherenceof trapped 9Be+ ions for which γ values of ≈6×103 are quoted.It is clear from (31) that for such low values of γ we are inthe low temperature regime even for T values which wouldnormally be considered very high. Finally, as an illustration,if we consider a 9Be ion with γ ≈ 6 × 103 and if we takeσ = 1 Å and d = 1 cm then, from (29), we obtain a τ0value of ≈6 × 10−16 s−1 and, as mentioned above, τd willbe even smaller; i.e. decoherence occurs in a very short timeeven at zero temperature and indeed throughout the whole lowtemperature regime.AcknowledgmentThe authors wish to thank the School of Theoretical Physics,Dublin Institute for Advanced Studies, for their hospitality andthe Director, Dr Tony Douglas, for an enlightening remarkconcerning the evaluation of the integral in equation (18).S611G W Ford and R F O’ConnellReferences[1] Giulini D, Joos E, Kiefer C, Kupisch J, Stamatescu I-O andZeh H D 1996 Decoherence and the Appearance of aClassical World in Quantum Theory (New York: Springer)[2] Walls D F and Milburn G J 1985 Phys. Rev. A 31 2403[3] Zurek W H 1991 Phys. Today 44 36[4] Ford G W, Lewis J T and O’Connell R F 2001 Phys. Rev. A 64032101[5] Ford G W and O’Connell R F 2001 Phys. Lett. A 286 87[6] Ford G W and O’Connell R F 2002 Am. J. Phys. 70 319[7] Ford G W and Lewis J T 1986 Probability, StatisticalMechanics, and Number Theory (Advances in MathematicsSupplemental Studies vol 9) (Orlando, FL: Academic)[8] Ford G W, Lewis J T and O’Connell R F 1988 Phys. Rev. A 374419[9] Ford G W and O’Connell R F 1989 J. Stat. Phys. 57 803[10] Ford G W and O’Connell R F 2001 Phys. Rev. D 64105020 especially the appendix[11] Bateman Manuscript Project 1954 Tables of IntegralTransforms vol 1, ed A Erdélyi (New York: McGraw-Hill)[12] Bateman Manuscript Project 1954 Higher TranscendentalFunctions vol 2, ed A Erdélyi (New York: McGraw-Hill)[13] Myatt C J, King B E, Turchette Q A, Sackett C A,Kielpinski D, Itano W M, Monroe C and Wineland D J2000 Nature 403 269S612

Decoherence at zero temperature

Most discussions of decoherence in the literature consider the high temperature regime but it is also known that, in the presence of dissipation, decoherence can occur even at zero temperature. Whereas most previous investigations all assumed initial decoupling of the quantum system and bath, we consider that the system and environment are entangled at all times. Here, we discuss decoherence for a free particle in an initial Schrödinger cat state. Memory effects are incorporated by use of the single relaxation time model

R F O’connell

CiteSeerX

Most discussions of decoherence in the literature consider the high-temperature regime but it is also known that, in the presence of dissipation, decoherence can occur even at zero temperature. Whereas most previous investigations all assumed initial decoupling of the quantum system and bath, we consider that the system and environment are entangled at all times. Here, we discuss decoherence for a free particle in an initial Schrödinger cat state. Memory effects are incorporated by use of the single relaxation time model (since the oft-used Ohmic model does not give physically correct results)

O'Connell, R. F.

DIAS Access to Institutional Repository

arXiv:quant-ph/0311019v1  5 Nov 2003Decoherence at zero temperatureG. W. Ford† and R. F. O’Connell‡School of Theoretical Physics, Dublin Institute forAdvanced Studies,10 Burlington Road, Dublin 4, Ireland(January 6, 2018)AbstractMost discussions of decoherence in the literature consider the high-temperature regime but it is also known that, in the presence of dissipation,decoherence can occur even at zero temperature. Whereas most previous in-vestigations all assumed initial decoupling of the quantum system and bath,we consider that the system and environment are entangled at all times. Here,we discuss decoherence for a free particle in an initial Schro¨dinger cat state.Memory effects are incorporated by use of the single relaxation time model(since the oft-used Ohmic model does not give physically correct results).Typeset using REVTEX1Most discussions of decoherence obtain results for decoherence decay times which areproportional to the inverse of T or T 1/2, where T is the temperature, and which apply onlyto the high temperature regime, kT >> h¯γ, where γ is a typical dissipative decay rate[1–5]. In fact, whereas most papers in the early literature give decay rates proportional toγ−1, we showed that decoherence can occur at high temperature even for vanishingly smalldissipation [4,5]. For a pedagogical discussion of the high T regime, we refer to [6]. Here,we look at the opposite end of the temperature spectrum and present explicit results attemperature zero. In contrast to the high temperature regime, we find that a dissipativeenvironment is necessary to achieve decoherence.The zero temperature case was also considered by others, most notably by Walls andMilburn [2]. They considered a ”Schro¨dinger cat” state consisting of a pair of coherent statesof an oscillator. They solved the master equation to obtain an expression for the coherenceattenuation coefficient, defined in terms of the off-diagonal elements of the density matrix.However, this approach is subject to serious assumptions: (a) initial decoupling of thesystem and environment; (b) weak coupling, which implies a restriction to a finite oscillatorpotential.The assumption (a) used by Walls and Milburn and indeed many other investigators,leads to results which are seriously at variance with a model in which the quantum particleand the heat bath are entangled at all times.. In fact, at T = 0, we are simply dealingwith the zero-point oscillations of the heat bath which are necessarily always entangled withwhatever quantum particle is of interest. In fact, the results we obtain here are applicableto the whole low temperature regime, defined by the inequality kT << h¯γ. General resultsfor all regimes, corresponding to arbitrary T and γ, were obtained in our initial paper onthis subject [4], where our starting-point was the density matrix for the system of particlecoupled to the bath at all times. However, there explicit results were presented only forthe Ohmic model in the high temperature regime (kT >> h¯γ). Later, we showed that thesame results could be obtained simply by starting with the wave function describing a pureSchro¨dinger cat state moving with an initial velocity v [5]. This was then used to calculate2the wave function and then the coordinate probability distribution at time t. Finally, thisprobability distribution was then averaged over a thermal distribution of initial velocities toobtain the probability distribution corresponding to a finite high temperature T . This ledimmediately to a quantitative measure of decoherence (the destruction of the interferencepattern). In other words, the calculation of the decoherence decay time at high temperaturecould be carried out simply within the framework of elementary quantum mechanics andequilibrium statistical mechanics, with a result independent of γ [5,6] .In contrast to the high temperature case, results at low temperature depend essentially onγ and must be obtained within the framework of nonequilibrium statistical mechanics. Thus,we will now give a synopsis of our general model describing a system and its environmentas being entangled at all times [4].First, we emphasize that, not only are our results in general applicable to arbitrarydissipation and arbitrary temperature, but they are not subject to the weak coupling ap-proximation and so apply in the case of a free particle. Arbitrary dissipation refers toreservoir models which give rise to memory terms in the quantum Langevin equation (see(6) below and the subsequent discussion) which describes the equation of the quantum sys-tem in a dissipative environment. By contrast, the Ohmic model (the model considered bymost authors) is the simplest dissipative model and corresponds to the lack of memory termsin the Langevin equation. As we shall see below, the Ohmic model has the difficulty thatthe mean square velocity at zero temperature is divergent and a cutoff must be supplied tomake it finite. For this purpose we use the single relaxation time model (which reduces tothe Ohmic model in the limit of zero relaxation time).As in our previous discussions [4–6], we consider decoherence in terms of the simpleproblem of a free particle moving in one dimension that is placed in an initial superpositionstate (“Schro¨dinger cat” state) corresponding to a pair of Gaussian wave packets, each withvariance σ2 and separated by a distance d≫ σ. For such a state the probability distributionat time t can be shown to be of the form:3P (x, t) =12(1 + e−d2/8σ2){P0(x−d2, t) + P0(x+d2, t)+2e−d2/8w2(t)a(t)P0(x, t) cos[x(0), x(t)]xd4iσ2w2(t)}, (1)where P0 is the probability distribution for a single wave packet, given byP0(x, t) =1√2piw2(t)exp{−x22w2(t)}. (2)Here and in (1) w2(t) is the variance of a single wave packet, which in general is given byw2(t) = σ2 −[x(0), x(t)]24σ2+ s(t), (3)where σ2 is the initial variance, [x(0), x(t)] is the commutator, ands(t) = 〈{x(t)− x(0)}2〉, (4)is the mean square displacement. In (1) the first two terms within the braces correspondto the two wave packets, centered at ±d/2, expanding independently, while the third termis the interference term. Decoherence refers to the destruction of interference, a measure ofwhich is given by the attenuation coefficient a(t) which can be defined as the ratio of thefactor multiplying the cosine in the interference term to twice the geometric mean of thefirst two terms [4,5]. Using a method of successive measurements introduced by Ford andLewis [7], in which a particle in equilibrium and entangled with the environment is placedin the initial state by a first measurement and then, after a time interval t , is probed bya second measurement, we obtained the following exact general formula for the attenuationcoefficient [4,7].a(t) = exp{−s(t)d28σ2w2(t)}. (5)The quantities appearing in (3) and (4) are evaluated by use of the quantum Langevinequation [8], which is a Heisenberg equation of motion for x(t), the dynamical variablecorresponding to the coordinate of a Brownian particle interacting with a linear passive4heat bath. For the case of a free particle, this equation for the stationary process has thewell known form,mx¨+∫ t−∞dt′µ(t− t′)x˙(t′) = F (t), (6)where µ(t) is the memory function and F (t) is a fluctuating operator force with mean zero.The solution of the quantum Langevin equation (6 ) can be writtenx(t) =∫ t−∞dt′G(t− t′)F (t′), (7)where G(t), the Green function can in turn be writtenG(t) =12pi∫ ∞−∞dωα(ω + i0+)e−iωt, (8)in which α(z) (the Fourier transform of the Green function) is the response function. Forthe free particle the response function has the general formα(z) =1−mz2 − izµ˜(z), (9)in which µ˜(z) is the Fourier transform of the memory function,µ˜(z) =∫ ∞0dtµ(t)eizt, Im{z} > 0. (10)Using these results, we find that [8,9] the mean square displacement is given by theformulas(t) =2h¯pi∫ ∞0dωIm{α(ω + i0+)} cothh¯ω2kT(1− cosωt), (11)while the commutator, which is temperature independent, is given by the formula[x(t1), x(t1 + t)] =2ih¯pi∫ ∞0dωIm{α(ω + i0+)} sinωt. (12)These expressions are valid for arbitrary temperature and arbitrary dissipation. (Indeed,with the appropriate expression for the response function, they are valid in the presenceof an external oscillator potential.) In our earlier discussions, for brevity, we confined our5attention to the case of high temperature. Here we consider the case of zero temperature.Most discussions in the literature are confined to the so-called Ohmic case, where µ˜(z) = ζ ,the Newtonian friction constant but we analyze the more general case of the single relaxationtime model [10]. This model corresponds to a memory function of the formµ(t) =ζτe−t/τθ(t), (13)where τ is the relaxation time of the bath and where θ is the Heaviside function. Note thatin the limit τ → 0 this becomes the Ohmic memory function µ(t) = 2ζδ(t)θ(t). With thisform of the memory function,µ˜(z) =ζ1− izτ. (14)Next, with this form in the expression (9) for the response function, we find with T = 0 thatthe expression (11) for the response can be expressed in the form,s(t) =2h¯piζΩ2V (γt)− γ2V (Ωt)Ω2 − γ2, (15)while the expression (12) for the commutator becomes[x(0), x(t)] =ih¯ζΩ2(1− e−γt)− γ2(1− e−Ωt)Ω2 − γ2, (16)where we have introduced the quantities,Ω =1 +√1− 4ζτ/m2τ, γ =1−√1− 4ζτ/m2τ. (17)and where [11]V (x) =∫ ∞0dyx2y(y2 + x2)(1− cos y)= log x+ γE −12[e−xE¯i(x) + exEi(−x)]= −(log x+ γE)(cosh x− 1)−12[e−x∞∑n=1xnn!n+ ex∞∑n=1(−x)nn!n]. (18)Here γE = 0.577215665 is Euler’s constant. Note that the sums in ( 18) are absolutelyconvergent. We should point out that the quantities Ω and γ were referred to as γ+ and γ−6in [10]. We feel that the present choice is more suggestive since, for τ → 0, we have Ω→ 1τand γ → ζ/m.Note the expansions [12], for small x,V (x) ∼= −12x2(log x+ γE −32). (19)Also, asymptotically, for large x,V (x) ∼ log x+ γE −1x2−3!x4−5!x6. (20)For very short times (t << τ), one can show in general thats(t) ∼= 〈v2〉t2, (21)where 〈v2〉 is the mean square velocity, given for the single relaxation time model by〈v2〉 =h¯γΩpim(Ω− γ)logΩγ∼= −h¯ζpim2logζτm. (22)We note that, for τ → 0 (corresponding to the Ohmic model), the mean square velocity,given in (22), is logarithmically divergent. The conclusion is that, in order to obtain finiteresults at short time, one must use a cutoff model, such as the single relaxation time modelused here.For intermediate times (τ << t << (ζ/m)−1), we obtains(t) ∼= −h¯ζpim2t2{logζtm+ γE −32}. (23)We can compare these results with the weak coupling result which for a free particle leadsto s(t) being identically zero. (For a free particle the mean square velocity vanishes at zerotemperature.)Thus, to investigate the decay of coherence (which is generally a short-time phenom-ena) in both time ranges, we consider the behavior for times ζt/m << 1 In this case, thecommutator becomes that of a non-interacting particle,[x(0), x(t)] ∼= ih¯tm. (24)7Hence, it is clear that for very short times the wave packet width is the initial width,w2(t) ∼= σ2. (25)Thus, the attenuation coefficient (5) for short times takes the form:a(t) ∼= exp{−s(t)d28σ4}, (26)where s(t) is given by (21) and (23). Hence, in particular, we can writea(t) = exp{(tτ0)2logζτm}, t << τ (27)anda(t) = exp{(tτ0)2 [logζtm+ γE −32]]}, τ << t << (ζ/m)−1 (28)withτ0 ≡mσ2d√8pih¯ζ. (29)The characteristic time for decoherence to occur, τd, is defined as usual [4–6], as the time atwhich a(t) = exp(−1) . Hence, for example it follows from (27) that, for t << τ ,τd = τ0| log(ζτ/m)|−1/2 < τ0, (30)where the inequality sign follows from the fact that ζt/m << 1. It is also clear that τd < τ0in the case t >> τ .Next, we consider some numerical results. As mentioned above, here ζ/m ≈ γ and, as apoint of reference, it is useful to note thatkTh¯γ=T (K)γ(1011s−1). (31)Now, one of the few experimental investigations of decoherence is the work of Myatt et al.on the decoherence of trapped 9Be+ ions [13] for which γ values ≈ 6× 103 are quoted. It isclear from (31) that for such low values of γ that we are in the low-temperature regime even8for T values which would normally be considered very high. Finally, as an illustration, if weconsider a 9Be ion with γ ≈ 6 × 103 and if we take σ = 1A˚ and d = 1cm then, from (29)we obtain a τ0 value of ≈ 6× 10−16s−1 and, as mentioned above, τd will be even smaller i.e.decoherence occurs in a very short time even at zero temperature and indeed in the wholelow temperature regime.ACKNOWLEDGMENTSThe authors wish to thank the School of Theoretical Physics, Dublin Institute for Ad-vanced Studies, for their hospitality.9REFERENCES† Permanent address: Department of Physics, University of Michigan, Ann Arbor, MI48109-1120.‡ Permanent address: Department of Physics and Astronomy, Louisiana State University,Baton Rouge, LA 70803-4001.[1] D. Giulini, E. Joos, C. Kiefer, J. Kupisch, I.-O. Stamatescu and H. D. Zeh, Decoherenceand the appearance of a classical world in quantum theory (Springer, New York 1996).[2] D.F. Walls and G.J. Milburn, Phys. Rev. A 31, 2403 (1985).[3] W.H. Zurek, Physics Today 44(10), 36 (1991).[4] G.W. Ford, J.T. Lewis and R.F. O’Connell, Phys. Rev. A 64 , 032101 (2001).[5] G.W. Ford and R.F. O’Connell, Physics Letters A 286, 87 (2001).[6] G. W. Ford and R. F. O’Connell, Am. J. Phys. 70, 319 (2002).[7] G. W. Ford and J. T. Lewis, in Probability, Statistical Mechanics, and Number Theory,Advances in Mathematics Supplemental Studies, Volume 9 (Academic Press, Orlando,Florida 1986).[8] G. W. Ford, J. T. Lewis and R. F. O’Connell, Phys. Rev. A 37, 4419 (1988).[9] G. W. Ford and R. F. O’Connell, J. Stat. Phys. 57, 803 (1989).[10] G. W. Ford and R. F. O’Connell, Phys. Rev. D 64, 105020 (2001), especially theappendix.[11] Bateman Manuscript Project, Tables of integral transforms, A. Erde´lyi, Editor,(McGraw-Hill, New York 1954) Vol. 1.[12] Bateman Manuscript Project, Higher transcendental functions, A. Erde´lyi, Editor(McGraw-Hill, New York 1954) Vol. 2.10[13] C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano,C. Monroe and D. J. Wineland, Nature 403 , 269 (2000).11

Most discussions of decoherence in the literature consider the high temperature regime but  it is also known that, in the presence of dissipation, decoherence can occur even at zero  temperature. Whereas most previous investigations all assumed initial decoupling of the  quantum system and bath, we consider that the system and environment are entangled at  all times. Here, we discuss decoherence for a free particle in an initial Schrödinger cat  state. Memory effects are incorporated by use of the single relaxation time model.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48857/2/job3_6_010.pd

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Decoherence at zero temperature

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