CCollisional Quantum Brownian Motion
I. Kamleitner and J. Cresser
Centre for Quantum Computer Technology, Physics Department,Macquarie University, Sydney, New South Wales 2109, Australia
We derive a quantum master equation from first principles to describe friction in one dimensional,collisional Brownian motion. We are the first to avoid an ill-defined square of the Dirac delta functionby using localized wave packets rather than plane waves. Solving the Schr¨odinger equation for twocolliding particles, we discover that the previously found position diffusion is not a physical process,but an artifact of the approximation of a coarse grained time scale.
PACS numbers: 05.20.Dd,03.65.Yz,03.75.-b,47.45.Ab
More than a hundred years ago, Einstein and Smolu-chowski [1] derived a successful classical description ofBrownian motion, the erratic movement of a particle af-fected by ambient gas molecules. In the last few decades,the field of quantum Brownian motion (QBM) has be-come established [2], [3]. Not only is such a quantumdescription of friction and diffusion desirable, but QBMalso sheds light on the non-classical process of decoher-ence which is believed to be of importance in the quantumclassical transition.The field of QBM is still the source of open questions.Of particular concern is deriving a valid master equation(ME) for the reduced density operator of the Brownianparticle in the Markov limit, in which case a ME of Lind-blad form is to be expected. What should be noted hereis that all such MEs include position diffusion. This pro-cess is impossible in classical dynamics (although the po-sition does diffuse in the long term due to randomness ofthe momentum) as it derives from position jumps whichare at odds with continuous trajectories of classical par-ticles. Therefore it is currently believed that positiondiffusion arises as a quantum effect accompanying colli-sional quantum friction [4]. In this letter we show thecontrary: We find that position diffusion is not a realphysical process in collisional QBM, but results from us-ing a coarse grained time scale on which collisions ap-pear instantaneous. Unproblematic in classical dynam-ics where collision times of hard-core particles are indeedshort, the assumption of instantaneous collisions has tobe used with care in quantum dynamics, where collisiontimes depend on the widths of the colliding wave packets.There are several “non-collisional” models of QBM inthe literature (see e.g. [5]-[7]). Here we concentrateon collisional models which are the closest in spirit tothe original Einstein approach, and which can be testedin experimental setups [8]. Important contributions inthe literature include [9]-[12] and can be reviewed in [4].Mostly, the analysis is based on using scattering theoryto describe the effects of a single collision.To point out some problems which usually occur incollisional QBM when using scattering theory, we brieflyreview Hornberger’s contribution [12] as it seems to be the most complete one. Due to momentum conservationa two particle collision reduces to a one particle problem,where the transition operator T with momentum matrixelements related to the scattering amplitude (cid:104) p (cid:48) | T | p (cid:105) = δ ( p (cid:48) − p ) f ( p (cid:48) , p ) / ( π (cid:126) ) is of importance. In particularit appears in an expression of the form X = (2 π (cid:126) ) Ω (cid:104) p (cid:48) + q | T | p + q (cid:105)(cid:104) p (cid:48) − q | T † | p − q (cid:105) , (1)where Ω → ∞ is a box-normalization volume. For q = 0this term contains an ill-defined square of the Dirac deltafunction and a physically motivated replacement rule(2 π (cid:126) ) Ω |(cid:104) p (cid:48) | T | p (cid:105)| → δ (cid:18) p (cid:48) − p (cid:19) | f ( p (cid:48) , p ) | σ ( p ) | p | . (2)is used. However, for q (cid:54) = 0 Eq. (1) is well defined. Infact, using standard relations for Dirac delta functions, X = π (cid:126) Ω pq δ ( p − p (cid:48) ) δ (cid:0) ( p − p (cid:48) ) (cid:107) q (cid:1) × f ( p (cid:48) + q , p + q ) f ∗ ( p (cid:48) − q , p − q ) (3)is found, where q = | q | and p (cid:107) q is p projected on q . Nev-ertheless, this result Eq. (3) is not without its problems asthe off-diagonal matrix elements (cid:104) p (cid:48) | ρ | p (cid:105) of the Brown-ian particle’s density matrix would be found to disappearin a single collision due to Ω → ∞ , no matter how small p − p (cid:48) . This seemingly strange observation comes withthe fact that the collision time with a gas particle in anon-localized momentum eigenstate is infinite.To construct a Markovian (memory less) ME, Horn-berger assumed instantaneous collisions. But within thisassumption, the complete vanishing of coherences is notacceptable any more. To get around this problem, Horn-berger substituted the square root of the replacement ruleEq. (2) for (cid:104) p (cid:48) ± q | T | p ± q (cid:105) in Eq. (1) to effectively bringback the otherwise lost coherences of momentum states.The replacement rule (2) has to be used with cautionin the first place, but its application to a well definedexpression is even more questionable. Whether decoher-ence in momentum bases and the closely related positiondiffusion are correctly described by approaches of thiskind is surely not certain. a r X i v : . [ qu a n t - ph ] J u l To resolve this matter we study a single collision interms of localized gas states. Then a collision timecan be precisely defined, and the low-density and high-temperature (LDHT) limit will be quantified by ‘ colli-sion time ’ × ‘ collision rate ’ (cid:28)
1. To examine a collisionevent time resolved, and to investigate under which con-ditions a complete collision of two wave packets occurs,we analytically solve the two-particle Schr¨odinger equa-tion. This is in contrast to scattering calculations whichpostulate a complete collision.To simplify the problem, we consider one-dimensionalQBM and assume a hard core interaction potential be-tween gas and Brownian particles. Our notations are asfollows. Position and momentum operators are denotedwith a hat, whereas wave function variables are primed.The index g refers to gas particles.The two particle Hamiltonian H = ˆ p g m g + ˆ p m + aδ (ˆ x − ˆ x g ) , a → ∞ (4)forbids any tunneling of the gas particle through theBrownian one. The two orthogonal sets of energy eigen-states in position space are ψ a ˜ k ¯ k ( x (cid:48) g , x (cid:48) ) = e i ¯ k ( x (cid:48) + αx (cid:48) g ) sin(˜ k ( x (cid:48) g − x (cid:48) )) , (5) ψ s ˜ k ¯ k ( x (cid:48) g , x (cid:48) ) = e i ¯ k ( x (cid:48) + αx (cid:48) g ) sin(˜ k | x (cid:48) g − x (cid:48) | ) , (6)with ¯ k ∈ R , ˜ k ∈ R + , α = m g /m , and energy E (˜ k, ¯ k ) =(1 + α )(˜ k + α ¯ k ) (cid:126) / (2 m g ).The collisional analysis is based on noting that thedensity operator of a thermal ideal gas particle at tem-perature T can be written as convex decomposition of Gaussian states ρ g = (cid:90) d x g L (cid:90) d p g µ σ g ( p g ) | x g , p g (cid:105) σ g (cid:104) x g , p g | , (7)where L is a normalization length and where | x, p (cid:105) σ g de-notes a Gaussian wave packet with position variance σ g : (cid:104) x (cid:48) | x, p (cid:105) σ g = e − ixp/ (cid:126) (cid:112) √ πσ g e ix (cid:48) p/ (cid:126) e − ( x − x (cid:48) ) / σ g (8)Hornberger and Sipe [13] showed that Eq. (7) is validif µ σ g ( p g ) = √ πm g k B T σg e − p g / m g k B T σg and T σ g = T − (cid:126) m g k B σ g .We therefore undertake this analysis by considering aninitial state that approaches the product state | x g , p g (cid:105) ⊗ | x, p (cid:105) (9)in the limit of large separation, i.e. we assume vanishingoverlap of the two initial wave packets | x g − x | (cid:29) (cid:113) σ g + σ . (10) Thus we have to construct an initial two-particle state,expanded in terms of energy eigenstates Eq. (5) and (6),which satisfies the boundary condition ψ ( t = 0 , x g , x = x g ) = 0 required by the hard core interaction, and whichapproaches Eq. (9) in the limit Eq. (10). As the proce-dure is quite lengthy, we only present the result and referto our more detailed paper [14]. It may be checked that ψ (0 , x (cid:48) g , x (cid:48) ) = i √ π (cid:90) ∞∞ d¯ k (cid:90) ∞ d˜ k (cid:101) ψ (0 , ˜ k, ¯ k ) × (cid:0) ψ a ˜ k ¯ k ( x (cid:48) g , x (cid:48) ) − ψ s ˜ k ¯ k ( x (cid:48) g , x (cid:48) ) (cid:1) (11)with (cid:101) ψ (0 , ˜ k, ¯ k ) = (1 + α ) σ (cid:112) π √ α exp (cid:20) i xp (1 + α )2 α (cid:126) − ¯ k α ) σ (cid:21) × (cid:26) e i ˜ kx (1+ α ) /α exp (cid:20) − (cid:16) ˜ k + p (cid:126) (cid:17) α α σ (cid:21) − e − i ˜ kx (1+ α ) /α exp (cid:20) − (cid:16) ˜ k − p (cid:126) (cid:17) α α σ (cid:21)(cid:27) has indeed the required properties. Without loss of gen-erality we used the reference frame of centre of mass(i.e. p = − p g and x = − αx g ) as well as x g < x and p g > p . Furthermore, we related the widths ασ g = σ of the Gaussian wave packets to the relative mass of thecolliding particles. This turns out to be the crucial re-quirement for the two particles not being entangled afterthe collision (see also [15]), and | x, p (cid:105) σ can be seen asthe pointer bases of a measurement performed by a gasparticle in the state | x g , p g (cid:105) σ [14].The time evolution is now easily achieved by multi-plying the integrand of Eq. (11) with e − iE (˜ k, ¯ k ) t/ (cid:126) . Afterstraight forward integration we find ψ ( t, x (cid:48) g , x (cid:48) ) = σ (cid:112) √ α √ π (cid:0) σ + i (cid:126) tm (cid:1) exp (cid:20) i x g p g + xp (cid:126) (cid:21) × exp (cid:34) − (cid:0) αα (cid:1) σ (cid:0) x + ptm (cid:1) + (cid:0) x (cid:48) + αx (cid:48) g (cid:1)(cid:0) σ − i (cid:126) tm (cid:1) (cid:0) σ + (cid:126) t m (cid:1) (cid:35) × exp − ( x (cid:48) g − x (cid:48) ) σ (cid:0) x + ptm (cid:1) + i (cid:16) pσ (cid:126) − x (cid:126) tm (cid:17)(cid:0) σ + (cid:126) t m (cid:1) − exp ( x (cid:48) g − x (cid:48) ) σ (cid:0) x + ptm (cid:1) + i (cid:16) pσ (cid:126) − x (cid:126) tm (cid:17)(cid:0) σ + (cid:126) t m (cid:1) (12)for x (cid:48) g < x (cid:48) and ψ ( t, x (cid:48) g , x (cid:48) ) = 0 otherwise. At time t = 0the first term in the curly brackets is the dominant one. Ifrelation (10) holds, then we can neglect the second termand Eq. (12) becomes the position representation of thedesired initial state (9). If the particles’ relative velocityis large compared to their velocity uncertainty | p g | (cid:29) (cid:114)
11 + α (cid:126) σ g , (13)then we can neglect the first term sufficiently long afterthe collision. In this case Eq. (12) is the position repre-sentation of the remarkable simple product state U g ( t ) |− x g , − p g (cid:105) ⊗ U ( t ) |− x, − p (cid:105) , (14)where U ( t ) is the free particle evolution operator. UsingEq. (12) we can now define the collision time t c = (cid:114)
81 + α σ g m g | p g | (15)as the time it takes from the first term in the curly brack-ets to dominate, to the second term to dominate.To examine the behavior of the Brownian particleduring a collision in detail we trace out the gas parti-cle from Eq. (12). Its position probability distribution p ( t, x (cid:48) ) = (cid:82) d x (cid:48) g (cid:12)(cid:12) ψ ( t, x (cid:48) g , x (cid:48) ) (cid:12)(cid:12) can be obtained analyti-cally in terms of the error function and does not showany position jumps. An example is plotted in Fig. 1(a),and reminds of a classical collision. As Gaussian statesbuild an over complete set of states, this result showsthat no position jumps, and hence no position diffusioncan occur in collisional QBM. Using the Fourier trans- FIG. 1: (color online) Position (a) and momentum (b) prob-ability distributions for the Brownian particle during a col-lision. While the position probability distribution “flows”,the momentum distribution “jumps” from the initial value tothe final one. This disproves position diffusion in collisionalQBM. Parameters are: x = 10, p = − m = 1, α = 0 . (cid:126) = 1, and σ = 4. formation of Eq. (12) with respect to x (cid:48) , we find themomentum distribution of the Brownian particle whichshows the expected momentum jump (see Fig. 1(b)).In a general reference frame the collision event withGaussian states can be written as | x g , p g (cid:105)⊗| x, p (cid:105) scattering −−−−−−→ U g ( t ) | ¯ x g , ¯ p g (cid:105)⊗ U ( t ) | ¯ x, ¯ p (cid:105) (16)where positions and momenta after the collision relatewith the initial values in the classical way [17]:¯ x g = 2 x − (1 − α ) x g α , ¯ x = 2 αx g + (1 − α ) x α , (17)¯ p g = 2 αp − (1 − α ) p g α , ¯ p = 2 p g + (1 − α ) p α . (18) The transformation of a general Brownian particle’s den-sity matrix due to a collision with a gas particle in thestate | x g , p g (cid:105) can be written in terms of Kraus operators ρ ( t ) = (cid:90) (cid:90) d˜ x d˜ p U ( t ) K x g ,p g (˜ x, ˜ p ) ρ (0) K † x g ,p g (˜ x, ˜ p ) U † ( t ) . (19)In [14] we will present a derivation of the Kraus operators K x g p g (˜ x, ˜ p ) = (cid:98) D (cid:18) α α ( x g − ˜ x ) , α ( p g − α ˜ p ) (cid:19)(cid:112) ˆ π (˜ x, ˜ p )where (cid:98) D ( x, p ) = e i ( p ˆ x − x ˆ p ) / (cid:126) is the Glauber displacementoperator andˆ π (˜ x, ˜ p ) = (cid:90) (cid:90) d x d p π (cid:126) w ( x, p ) | ˜ x + x, ˜ p + p (cid:105)(cid:104) ˜ x + x, ˜ p + p | w ( x, p ) = 2 απ (cid:126) (1 − α ) exp (cid:20) − α (1 − α ) (cid:18) x σ + σ p (cid:126) (cid:19)(cid:21) are effect operators corresponding to a measurementwhich is indirectly performed by the gas particle [14].The transformation (19) may be confirmed by applica-tion to Gaussian states and comparison to Eq. (16).Next we determine whether during a time interval δ the Brownian particle actually collides with a gas particlewhich is in the state | x g , p g (cid:105) . For this purpose we assume δ (cid:29) t c (20)so that we can neglect partial or uncompleted collisions.Then we find that the two particles collide if ( x, p ) iswithin the phase space region S ( x g , p g ) = (cid:8) ( x, p ) (cid:12)(cid:12) < x − x g p g /m g − p/m < δ (cid:9) . Therefore we have to project the Brow-nian particle’s density operator onto S ( x g , p g ) before ap-plying the transformation Eq. (19). Of course, phasespace projections do not exist in quantum mechanics, butas long as (20) is satisfied we can use the approximateprojection operatorΓ δ ( x g , p g ) = (cid:90) (cid:90) S ( x g ,p g ) d x d p π (cid:126) | x, p (cid:105)(cid:104) x, p | . (21)By multiplying this operator with the probability offinding a gas particle in the state | x g , p g (cid:105) , we define thecollision probability operator P δ ( x g , p g ) = n g µ ( p g )Γ δ ( x g , p g ) , (22)where n g is the gas number density. In fact, the expecta-tion value Tr[ ρP δ ( x g , p g )] gives the probability of a colli-sion of the Brownian particle and a gas particle describedby | x g , p g (cid:105) . We also define P δ = (cid:82)(cid:82) d x g d p g P δ ( x g , p g ) cor-responding to the total collision probability.To find the density operator for the Brownian particleat time δ , quantum trajectory theory [16] suggests tomultiply each Kraus operator by the square root of theprobability operator. We use the low density assumptionTr( ρP δ ) (cid:28) , (23)to neglect two or more collisions during δ and find ρ ( δ ) = U ( δ ) (cid:26)(cid:90)(cid:90)(cid:90)(cid:90) d x g d p g d˜ x d˜ p (cid:104) K x g ,p g (˜ x, ˜ p ) × (cid:113) P δ ( x g , p g ) ρ (0) (cid:113) P δ ( x g , p g ) K † x g ,p g (˜ x, ˜ p ) (cid:105) + (cid:112) − P δ ρ (0) (cid:112) − P δ (cid:27) U † ( δ ) . (24)The usual approach to a ME is to expand all operatorsin δ and to take the limit δ →
0. However, this limit isnot allowed in Γ δ ( x g , p g ) as then uncompleted collisionswere not negligible. A finite δ will ultimately lead toerrors which we minimize by performing the transitionto continuity in the interaction picture and to secondorder in δρ int ( δ ) − ρ int (0) δ ≈ ρ int (cid:0) δ + d t (cid:1) − ρ int (cid:0) δ (cid:1) d t . (25)Transforming back to the Schr¨odinger picture we findd ρ ( t )d t = − i (cid:126) [ H, ρ ( t )] −
12 [ Rρ ( t ) + ρ ( t ) R ]+ (cid:90)(cid:90)(cid:90)(cid:90) d x g d p g d˜ x d˜ p (cid:104) U (cid:0) δ (cid:1) K x g ,p g (˜ x, ˜ p ) × (cid:113) R δ ( x g , p g ) U † (cid:0) δ (cid:1) ρ ( t ) U (cid:0) δ (cid:1)(cid:113) R δ ( x g , p g ) × K † x g ,p g (˜ x, ˜ p ) U † (cid:0) δ (cid:1) (cid:105) (26)where R = P δ /δ and R δ ( x g , p g ) = P δ ( x g , p g ) /δ are rateoperators corresponding to collision rates. This masterequation of Lindblad form is our main result. It describescollisional friction as well as decoherence for general massratios and velocities, and allows a clear interpretation interms of quantum trajectories.An interesting feature of Eq. (26) is the dependence on δ and σ g . These parameters persist in the ME becausethey are bound from above and below by the relations(10) and (13), and (20) and (23), respectively and there-fore the limits δ → σ g → ∞ are not allowed.However, we show in [14] that the effects of δ and σ g onthe dynamics vanish in the LDHT limit √ α ) n gas (cid:126) (cid:112) πm g k B T (cid:28) , (27)which is also the crucial condition to simultaneously sat-isfy relations (10), (13), (20), and (23) for most thermalgas momenta. Our ME is valid on a coarse grained timescale δ (cid:29) (1 + α ) (cid:126) / ( k B T ).We apply our ME to derive equations of motion forexpectation values of position and momentum. In thelimit of a slow Brownian particle | p/m | (cid:28) (cid:112) k B T /m g (the generalization to fast Brownian particles is found in[14] and includes non-linear friction) we findd (cid:104) ˆ x (cid:105) d t = (cid:104) ˆ p (cid:105) m , (28) d (cid:104) ˆ p (cid:105) d t = − f (cid:104) ˆ p (cid:105) , (29)d (cid:10) ˆ x (cid:11) d t = (cid:104){ ˆ x, ˆ p }(cid:105) m + n gas √ π (cid:18) k B Tm g (cid:19) / δ , (30)d (cid:104){ ˆ x, ˆ p }(cid:105) d t = 2 (cid:10) ˆ p (cid:11) m − f (cid:104){ ˆ x, ˆ p }(cid:105) . (31)d (cid:10) ˆ p (cid:11) d t = 2 f (cid:2) mk B T − (cid:10) ˆ p (cid:11)(cid:3) , (32)where f = 4 n g (cid:112) m g k B T / ( √ πm ) is the friction con-stant. The second term in Eq. (30) represents positiondiffusion. However, as shown in our single collision cal-culation this is not a true physical process. It ratherresults from neglecting partial or uncompleted collisions,which is necessary for any Markovian description of col-lisional QBM. This is confirmed by the observation thatthis term is small in the range of validity of our ME.Using only first principles we derived a quantum MEof Lindblad form for collisional QBM in one dimensionin the LDHT limit. We showed that the counter intu-itive position diffusion found in earlier work results fromapproximations necessary to derive Markovian dynam-ics, but is not a real physical process. The applicationto expectation values, where we recovered the expectedclassical equations served as a test of our ME. In a fu-ture paper we will use our ME to study the importantissue of collisional decoherence which is now in reach ofquantitative experiments [8]. [1] A. Einstein, Ann. d. Physik , 549 (1905); M. vonSmoluchowski, Ann. d. Physik , 756 (1906).[2] E. Joos and H. D. Zeh, Z. Phys. B: Condens. Matt. ,223 (1985).[3] A. O. Caldeira and A. J. Leggett, Physica A , 587(1983).[4] B. Vacchini and K. Hornberger, to appear in Phys. Rep.(2009). (doi:10.1016/j.physrep.2009.06.001)[5] B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D ,2843 (1992).[6] M. R. Gallis, Phys. Rev. A , 1028 (1993).[7] K. Jacobs, EPL , 40002 (2009).[8] K. Hornberger et. al., Phys. Rev. Lett. , 160401 (2003).[9] L. Di´osi, Europhys. Lett. , 63 (1995).[10] S. M. Barnett and J. D. Cresser, Phys. Rev. A , 022107(2005).[11] B. Vacchini, Phys. Rev. Lett. , 1374 (2000); B. Vac-chini, Phys. Rev. E , 066115 (2001).[12] K. Hornberger, Phys. Rev. Lett. , 060601 (2006).[13] K. Hornberger and J. E. Sipe, Phys. Rev. A , 012105(2003).[14] I. Kamleitner and J. D. Cresser, to be published.[15] F. Schm¨user and D. Janzing, Phys Rev A , 052313(2006).[16] C. W. Gardiner, A. S. Parkins, and P. Zoller, Phys. Rev.A , 4363 (1992).[17] Noting that U ( t ) shifts x to x + pt/mpt/m