Phase space master equations for quantum Brownian motion in a periodic potential: comparison of various kinetic models

Transcription

1 Phase space master equations for quantum Brownian motion in a periodic potential: comparison of various kinetic models L. Cleary, W. T. Coffey, W. J. Dowling, Yu. P. Kalmykov and S. V. Titov

2 Quantum Mechanics W. Heisenberg ( ) Wave and Matrix Mechanics Approaches E. Schrödinger ( ) Path Integral Approach R. Feynman ( )

3 Wigner (193) formulation of quantum mechanics as quasi-probability distributions on phase space (x,p) / (, ) ˆ ipy W x p = ρ x+ y, x y e dy π ˆ ρ( x, x ) = x ρ x - density matrix 1 E. P. Wigner, Phys. Rev. 40, 749 (193) E. P. Wigner ( ) Eigenvalue E equation 1 i p+ + V x+ W = EW m i x p Evolution equation for W ˆ ˆ p W 1 i i W + MWW = 0 MWW = V x+ V x W m x i p p W is a joint quasi-distribution. Marginal distributions which are true pdfs, yield correct QM dfs for displacements and momenta.

4 Phase space representation The time evolution of the joint qpdf W(x,p,t) for a closed system is where M W is the closed evolution operator t W + Mˆ W = 0 quantum analogue of the Liouville equation W ( i /) ( ) r r+ 1 r+ 1 p W V W V W W = r+ 1 r+ 1 r= 1 + 1! Mˆ W m x x p r x p

5 Phase Space Master Equations Generalise to open systems (reduced Wigner function) t closed system evolution operator W + Mˆ W = Mˆ W W D collision operator Interactions with the surrounding heat bath

6 Kinetic model (CKT) Semiclassical Approach applied to open systems i) Use a Kramers-Moyal-like expansion for the collision kernel Mˆ D W, truncated at the second term (just as in the classical Brownian motion) ˆ W W W M D W = D p pw + D pp + D xp + D xx p p x x x where are parameters to be determined D, D, D, D p pp xp xx W + M W = Mˆ W ii) Ansatz: Impose the Wigner stationary distribution of the closed system as the stationary distribution of the open system. ˆ W D

7 Kinetic Model (CKT) This semiclassical approach results in the QME (cf. Einstein 1905) W ˆ m β W + MWW = γ pw + 1+ V t p β 1 m p i) Closed system evolution ii) Quantum term in the collision kernel Clearly m β xp = xx = pp = + β D D 0, D 1 V ( x) 1m It is truncated at terms o(ħ ) so the result is valid at relatively high temperatures. However, the solution can be calculated to any order in ħ Coffey et al. Phys. Chem. Chem. Phys. 9, 3361 (007).

8 Caldeira-Leggett Model The commonly used quantum mechanical approach to Brownian motion is the Caldeira-Leggett master equation (Boson bath) ρ i iγ mγ + [ HS, ρ] = x, { p, ρ} x, { x, ρ} t β or in Wigner s phase space W ˆ M m W + W W = γ pw + t p β p i) Closed system evolution operator ii) Classical collision kernel A. O. Caldeira and A. J. Leggett, Physica A 11, 587 (1983).

9 Average relaxation time vs. damping Coffey et al. Phys. Rev. E 75, (007). Comparison with C-L V( x) = gcosx CL QME (symbols) CKT QME (open circles) QMRRT (Kramers) Mel nikov (Georgievskii- Pollak) (solid lines) C-L fails to reproduce expected decrease of (Wigner Quantum TST 193) τ

10 Melnikov, Rips-Pollak, etc. (Isolated well for simplicity) ϒ Γ =ΓIHDϒ is the depopulation factor. Classically 1 ln{1 exp[ δ ( x + 1/ 4)]} ϒ= exp dx π x + 1/4 0 Γ IHD ( ) δ βγ I E C where = EC = IHD Kramers rate. is the separatrix energy Quantum rate Γ IHD essentially = Wigner QTST Classical IHD This is not a bad approximation (Wolynes, generalized Kramers to QM) λsin λ ln[1 Px ( i/ ] ϒ QM = exp dx π cosh( xλ) cos λ Γ IHD

11 P = FT over energy (transform variable x) normalized by β 1, of QM Green function. Classically (got by reducing FPE in barrier region to an energy-action diffusion Eq.) Px = + ( ) exp δ ( x 1/4) β Ω, γ ω γ c λ = Ω= + 4 = frequency of unstable barrier crossing mode In quantum situation Hˆ particles tunnel 0 = pˆ / m+ V( q) + qˆ η( t) through the barrier and classical GF must be replaced by density matrix calculated semiclassically viz. PEE (, ) = AEE (, T

12 (, ) ˆ exp i A EE = ET ˆ η() tqtdt ˆ() E is amplitude of a quantum transition from energy state to state in one cycle of the periodic motion in the well under the influence of the noise ˆ( η t), E and E are the unperturbed wave function, T ˆ = time ordering operator. We assume that the spectral density of the noise operator is governed by the Planck distribution. γ QM E E modified for periodic potential by Georgiveskii and Pollak. In QM unlike classical the FT P must be calculated for each particular case from semi-classical matrix elements (cf Fermi Golden Rule) (e.g.coffey et al. JCP 17, 07450, 007)

13 P = QM Green function. Classically P = exp δ ( x + 1/ 4) Ω γ γ λ = β, Ω= + ωc 4 = frequency of unstable barrier crossing mode Modified for periodic potential by Georgievskii and Pollak. In QM unlike classical, P must be calculated for each particular case from semi classical matrix elements. (e.g. Coffey et al JCP )

14 L. Diósi (1993) Three More Models Generalised Caldeira-Leggett master equation (under the restriction of small damping) to Lindblad form Relaxation dynamics of a quantum Brownian particle in an ideal gas - B. Vacchini and K. Hornberger (007) Quantum linear Boltzmann equation Lorentz kinetic model - Gross & Lebowitz (1956). Simple equilibrium difference collision operator

15 Diósi generalization of C-L QME to Lindblad form i.e. if the density operator is represented in a Hilbert space basis set, and brought into diagonal form, its eigenvalues must be positive. In other words, when the time dependent operator is transformed to the position representation it must represent the correct quantum mechanical probability for the displacements. W ˆ m W γ β M W γω β + W WW = γ pw t p β p 1m x 6π p x ii) Classical collision operator iii) Additional terms L. Diósi, Physica A 199, 517 (1993).

16 Four models for collision term: Diósi Mˆ W D m W γ β W γω β W = γ pw p β p 1m x 6π p x Vacchini Mˆ W D = γ pw + + p β p 1m x m W γ β W Lorentz ˆ D ( ) M W = γ W W 0 Coffey Kalmykov & Titov ˆ m β W MDW = γ pw + 1+ V p β 1 m p

17 In normalized variables W W 1 V W γ W = p + + pw + + DW t x x p p p Generic Form DW V W W W =Λ + γ + 4γ Ω x p x p x Diósi DW V W 3γ W =Λ x p x Vacchini DW V W V W =Λ + γ x p x p Coffey et al.

18 Density Operator Forms i i m ρ = [ H, ρ] γ x, { p, ρ} γ x, [ x, ρ] β β Ωβ γ p, [ p, ρ] γ x, [ p, ρ] 6m 3π Diósi i i m β ρ = [ H, ρ] γ x, { p, ρ} γ x, [ x, ρ] γ p, [ p, ρ] β 8m Vacchini Compare Coffey et al. i i 1 im i β dv ρ = [ H, ρ] γ x, { p, ρ} x, [ x, ρ] x,, ρ β 6 dx

19 To compare models consider a quantum Brownian particle moving along the x-axis in V( x) = g cosx To obtain the intermediate scattering function ik[ x( t) x(0)] Skt (, ) = e and greatest relaxation time τ we need the process whereby the particle traverses the periodic potential V and so we must obtain via Floquet s theorem, the nonperiodic solution W (Risken, The FPE, 1989, Springer Verlag) with the wave vector k restricted to the first Brillouin zone viz. ( is the periodic solution) w 1/ ikx W( x, p, t) = w( k, x, p, t) e dk 1/

20 Using Risken s method (as for the classical FPE) the periodic w(k, x, p, t) can then be expanded in an orthonormal basis of trignometric and Hermite functions viz. p + ( g/)cosx e cnq, ( k, t) (,,, ) ( ) iqx wk x pt = H 3/ n p e π n n= 0 q= n! Thus for each QME by reducing it to a linear combination of the (linearly independent) basis vectors we have a differential recurrence relation for the Fourier coefficients c n,q (k,t)

21 For example, the eleven term differential recurrence relation in n the index of the H n and q of the trignometric functions is for the Coffey et al. model d c + nc Λ g n n c + c dt ( ) nq, γ nq, ( 1) n, q+ 1 n, q 1 = i n/ q+ k cn 1, q + g cn 1, q+ 1 cn 1, q 1 /4 ( ) ( ) + i ( n+ 1)/ q+ k cn+ 1, q g cn+ 1, q+ 1 cn+ 1, q 1 /4 ( ) ( ) ( ) +Λ i g n( n 1)( n )/8 c c. n 3, q+ 1 n 3, q 1 n = 0,1,, q=, 1, 0,1,,

22 While for Diósi ( ) and Vacchini ( B = Ω= ) B = 3/, 0 d dt g c n B q k c 8 nq, + γ + ( + ) Λ nq, n g = i ( q k) 1 8 c 1 8 c c + + Ω Λ + Ω Λ 4 ( γ ) n 1, q ( γ )( n 1, q+ 1 n 1, q 1 ) n+ 1 g n( n 1)( n ) + i ( q k) c c c i g c c + + Λ 4 8 ( ) ( ) n+ 1, q n+ 1, q+ 1 n+ 1, q 1 n 3, q+ 1 n 3, q 1 B 1 1 g +Λ γ g q+ k+ cnq, + 1 q+ k c c + c 8 ( ) nq, 1 nq, + nq,

23 For the Lorentz master equation (Gross-Lebowitz, Phys. Rev , 1956) the Fourier coefficients satisfy the differential recurrence relation d c 1 n, q = τ ( c n, q c n, q(0) ) + i n / ( q + k ) c n 1, q + g ( c n 1, q+ 1 c n 1, q 1) / 4 dt ( ) n+ 1, q ( n+ 1, q+ 1 n+ 1, q 1 ) + i ( n+ 1)/ q+ k c g c c /4 ( n 3, q+ 1 n 3, q 1 ) +Λ i g n( n 1)( n )/8 c c.

24 Solution by continued fractions in the Laplace domain, i.e. c, ( k, s) Trick is to reduce to matrix 3 term DRR because then a formal solution always exists. Diósi B = 0, Ω= 0, Coffey, Lorentz etc. c = c ( k, t) +Λc ( k, t) 0 1 nq, nq, nq, nq C c ( k, t) () t = c ( k, t) ( k, t) 0 n 1, n n 1,0 0 cn 1,1 C c ( k, t) () t = c ( k, t) ( k, t) 1 n 1, n n 1,0 1 cn 1,1 C C C 0 1 n() t = n() t +Λ n() t

25 d dt C () t = Q C () t γ ( n 1) C () t + Q C () t n n n 1 n n n+ 1 d dt C () t = Q C () t γ ( n 1) C () t + Q C () t + R () t n n n 1 n n n+ 1 n (PRE , 007) Q ± n k 1 g/4 0 n 1 1 = i ± g/4 k g/4 4 0 ± g/4 k+ 1 R () t = s C () t + p C () t + r C () t Diósi n n n 1 n n n n 3 R () t = q C () t + r C () t Coffey et al n n n n n 3 s n = 8Ωγ 1 1/ nq + n

26 Stationary Distributions (Coffey, Diósi models) { ( ) } W x p = Z e +Λ V V + p V (, ) 1 p V 1 3, 0 π 0 V( x) ( ) [ ] Z = π 1 +Λ V ( x) Λ V ( x) e dx = Zcl 1 ΛgI1( g)/ I0( g) Wigner { ( ) γ γ γ } W x p = Z e +Λ V V + p V + + Ω V p V 1 p V D(, ) D ( ) 4, π { ( ) } 1 V( x) Z D = π 1 V ( x) V ( x) 4 γ( γ ) V( x) +Λ + + Ω e dx 0 1 cl { 1 [ 1 4 γ( γ )] 1( )/ 0( )} = Z +Λ + + Ω gi g I g 1 Diósi subject to [ γ γ ] 1 0 Λ 1+ 4 ( + Ω ) gi ( g)/ I ( g) << 1

27 Stationary Distributions Stationary solutions i.e. W = 0 (with zero current) for C-L and Vacchini models cannot be presented in the analytic form of a series of powers of Λ. However, can be calculated numerically using matrix continued fractions by setting C 0 in DRR. For Ω = 0 only difference between the Diósi and Vacchini models lies in the coefficients and 3/. Thus we may consider the Diósi model when Ω=0 as a simplified version of the Vacchini model. n =

28 1/W st (x, p) : p=0 : p=0.7 1 Classical Wigner Diosi Ω = 0 Λ =0.0 g = γ =1 0 1/W st (x, p) : γ =0.1 : γ =1 3: γ = 4: γ = Λ =0.0 g = p = π x Diósi (and by inference Vacchini) agrees with Wigner only for small damping. For large γ deviation is significant. γ < 0.1 π

29 Inverse stationary distributions of the Diósi (dash-dotted lines) kinetic model (with Ω=0) vs. the coordinate x for various values of the momentum p = 0, p = 0.7, and γ = 1 and various values of the damping parameter γ = 0.1, γ = 1, γ =, and γ = 3 and p = 0.7 with g = and Λ = 0.0. The inverse of the Wigner stationary distribution 1/ W0 ( x, p) (solid lines) and of the classical Maxwell-Boltzmann distribution (dashed lines, Λ = 0) are also shown.

30 Time dependent solutions Need nonperiodic solution in order to describe wandering and escape because cosine potential has only one well in one period. Skt (, ) = e ik[ x( t) x(0)] = 0 ik ( x x0 ) e W( x, p, x, p, t) dxdx dpdp 1/ = 1/ ik ( x x0) k1( x x0) e e w( k, x, p, x, p t) dk dxdx dpdp 1 0 0, Here x(0) = x, x( t) = x. The joint nonperiodic qpdf 0 1/ 0 0 = satisfies the various 1/ 0 0 ik ( x x0 ) W( x, p, x, p, t) e w( k, x, p, x, p, t) dk master equations with the initial condition W( x, p, x, p,0) = W ( x, p ), 0 0 st 0 0

31 Now for periodic ( ) and f x 1/ kk, 1 1/ π ik ( k1 ) x e f( x) dx= δ ( k k ) f( x) dx Thus with the orthogonality of the π π 0 0 π H n S( k, t) = w( k, x, p, x, p, t) dxdx dpdp = wk (, x, ptdxdp, ) = ac ( kt, ), q= π 1 iqx U ( x) / aq = ( π ) e dx 0 π wk (, x, pt, ) wk (, x, px,, p, tdx ) dp. 0 = q 0, q Thus the dynamic structure factor Skω becomes (, ) Sk iωt (, ω) = ac q 0, q( k, ω), q = c 0, q( k, ω) = c0, q( k, t) e dt. 0

32 Re[S(k,ω) / S(k,0)] Im[S(k,ω) / S(k,0)] g = 4 k = 0.4 (b) (a) Classical (Λ=0) Coffey et al. (Λ=0.0) 3 3 Diosi (Λ=0.0, Ω = 0) Lorentz (Λ=0) Lorentz (Λ=0.0) 1: γ = 0.5 : γ = 5 3: γ = ηω 1 1

33 At long times Greatest average relaxation time Skt hke τ t/ ( k) (, ) = ( ). Because Skω (, ) may be represented as a linear combination of the c k ω, it may be exactly calculated using the MCF method. 0, q(, ) Then one may estimate Γ because τ ( k), with k restricted to the first Brillouin zone, represents a characteristic time associated with the long time behavior of Skt (, ). In the frequency domain as ω 0 this approximation (on eliminating hk ( )) yields τ ( k) = lim ω 0 Sk (,0) Sk (, ω) iωs ( k, ω)

34 The escape rate or average decay rate Γ from the well characterised by the first Brillouin zone is 1/ 1 τ 0 Γ ( kdk ) One must use the average decay rate because in a multiwell potential the particle having escaped a particular well may again be trapped due to thermal fluctuations in another. Moreover, jumps of either a single lattice spacing or many are possible. Thus, Γ in a periodic potential is called the jump rate. The foregoing result pertains to the Caldeira-Leggett, Diósi, Vacchini, and Coffey et al. kinetic models while for the Lorentz kinetic model, can also be approximately presented as τ Skω (, ) (, ) hk ( ) S, L k ω 1 iω + τ where is now regarded as a phenomenological parameter determined by Mel nikov s estimate of the greatest relaxation time. τ M

35 Mel nikov s equation for the greatest relaxation time Mel nikov initially extended his solution to the classical Kramers turnover problem, to include quantum effects by simply inserting the quantum mechanical transmission factor for a parabolic barrier into the classical integral equation for the energy d.f. yielded by the Wiener- Hopf method in the Kramers turnover region, where the energy dissipated per cycle of the almost periodic motion of a particle on a noisy trajectory corresponding to the barrier energy is β 1. For Ohmic M 1 damping his formula Γ =τ (valid for all values of damping at M temperatures above the crossover temperature between tunneling and thermal activation) reads M Γ =Γ. IHD ϒ Here Γ IHD is the quantum escape rate from an isolated well in the IHD region ( γ 1) and ϒ is the quantum depopulation factor.

36 Later Rips and Pollak gave a consistent solution of the quantum Kramers turnover problem demonstrating how Mel nikov s result can be obtained without his ad hoc interpolation between the weak and strong damping regimes. Finally, Georgievskii and Pollak treated Γ in a periodic cosine poterntial, which is qualitatively different from that for a metastable well because the periodic potential is multistable, showing that ϒ= 1 4 sin ( ) ( ) 0 π kfkdk Fk ( ), which accounts for both the multistable V( x) and the Kramers turnover between VLD and IHD regimes, is R( x) asin a 1 e dx Fk ( ) = exp ln 1 R( x) R( x) π e e cos( πk ) + cosh( ax ) cos a Rx ( ) = πγ cosh( Λy) cos( Λxy) 3Λ ysinh( Λy)cosh [ π y/ ( 6 g)] dy

37 ( ) Here a = 3Λ γ + g γ and δ = 8γ g is the dimensionless classical action associated with the path of a particle librating in a well of the cosine potential with E = E C i.e. the critical energy trajectory on which escape may take place by dint of a thermal fluctuation. ( ) ( k ) ( ) In the classical limit, Rx ( ) δ x + 1/4 we have F ( ), 0, / M k = e σ δ σ δ, where 1 σ ( k, δ) = n erfc ( nδ / ) cos( πnk). n= 1 In calculating Γ in the IHD limit only, for the cosine potential it is sufficient to consider the escape rate from an isolated well.

38 The IHD rate Γ IHD is where Ξ= Ξ Γ = + πη n= 1 ( γ g γ) g IHD e, ω + ( πn/ β) + πnγ / β a c n ω + ( π / β) + πnγ / β is the quantum correction factor and c are the saddle and well angular frequencies. = = ω = V ( x )/ m ω V ( x ) / m If γβ << π and gλ << 1 and if only terms are retained we have the result (Wigner 193) c o( ) a a ( ) c a Ξ = β ω ω /4...

39 1 g = τ /η γ 1: Λ = 0 (classical) : Λ = 0.0

40 Fig. 3. (Color on line) Greatest relaxation time τ / η vs. dimensionless damping parameter γ for the barrier parameter g=5. Solid lines: the turnover Eq. for Λ=0 (classical case; curve 1) and Λ=0.0 (curve ). Dashed lines: IHD for Λ=0 (curve 1) and Λ=0.0 (curve ). Open circles: CKT master Eq. Filled circles: the Caldeira Leggett Eq. Asterisks: the Diósi master Eq. with Ω = 0 (fittings for other values of Ω 0 yield substantial deviations from the predictions of the escape rate theory).

41 Thank you References L. Cleary et al. in press (010) W. T. Coffey et al. Phys. Rev. E (007) B. Vacchini and K. Hornberger Eur. Phys. J. Special Topics (007). L. Diósi, Physica A (1993). E. P. Gross and J. L. Lebowitz, Phys. Rev (1956).