Chattaraj P.K. (Ed.) Quantum Trajectories

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164 Quantum Trajectories

Probably the most well-known (although not exclusive) quantum trajectory approach

in physics and chemistry is based on the quantum hydrodynamic, or Bohmian mechan-

ical, approach [1–7]. The literature is rich with discussions of the interpretation of

the hydrodynamic approach to quantum mechanics [5, 6, 8] and over the last decade

there has been a surge of interest in developing numerical quantum trajectory tech-

niques [9–18]. Most quantum trajectory based approaches have focused on pure states,

i.e., wavefunctions. However, the central theme of this chapter is the quantum hydro-

dynamics of mixed states.

In the quantum dynamical treatment of mixed states the equations of motion are

generally formulated for the density matrix ρ or, by a suitable transformation, for a

phase-space density ρ

(q, p). Both these representations require 2N dimensions to

describe an N-dimensional system. The hydrodynamic description of mixed states

is an alternative approach that involves a reduction of the 2N-dimensional space to

N dimensions. The hydrodynamics approach may be derived from either the phase-

space (q, p) representation or density matrix in the positional (x, x



) representation.

The hydrodynamic formulation of mixed quantum states dates back to the work of

Takabayasi [19], Moyal [20] and Zwanzig [21] in the late 1940s and early 1950s

and has subsequently been investigated in a number of studies [22–25]. Although

the mixed state hydrodynamical theory has been developed since the 1940s, rela-

tively little connection to the corresponding pure state Bohmian mechanics was made

[19,26–28]. In the last two decades, work by Muga and collaborators [27] as well as

our work [26,28–31] contributed to establishing this connection explicitly.

Recently, we also formulated a novel hybrid hydrodynamic-Liouvillian approach

that is ideally suited for developing a mixed quantum-classical scheme [32–34]. The

method is known as the quantum-classical moment (QCM) approach and involves a

partition of the global system into a hydrodynamic part that remains quantum mechan-

ical and a Liouville phase-space part that involves making a classical approximation

to the Liouville sector. The classical approximation involves ignoring all  terms in

the phase-space version of the quantum Liouville equation and the resulting mixed

quantum-classical description is then equivalent to the quantum-classical Liouville

equation [35–40]. However, if the potentials for the Liouville sector are no more than

quadratic polynomials, i.e., harmonic, then the equations of motion for the Liouville

sector contain no  terms and the QCM approach is then quantum mechanically exact

in such cases. For the Hamiltonians described in this chapter the potentials for the

Liouville sector are harmonic and so no classical approximations are made to the

equations of motion.

The hybrid hydrodynamic-Liouville equations are deﬁned in terms of a partic-

ular type of moments, obtained by integrating over the momentum p of the quan-

tum part of the Wigner phase-space distribution ρ

(q, p;Q, P ) of the composite

system spanned by q, p, Q, P , i.e., P

ρ

qQP



dp p

(q, p;Q, P ), see Sec-

tion 11.2.6. We will refer to these moment quantities as partial moments. Exact equa-

tions of motion for the moments are then derived before transforming the equations of

motion to a Lagrangian trajectory framework. The resulting trajectory equations for

the dynamical q, Q, P variables involve a q, Q, P dependent generalized quantum

force. The approach was demonstrated for a completely harmonic composite sys-

tem by Burghardt and Parlant [32], and later for a double well potential system by

A Hybrid Hydrodynamic–Liouvillian Approach 165

Hughes et al. [34]. In this study we extend the approach to simulate non-Markovian

system-bath dynamics of open quantum systems.

In the following sections a non-Markovian approach to dissipative dynamics is

described which is based on the construction of a hierarchy of effective environ-

mental modes that is terminated by coupling the ﬁnal member of the hierarchy to a

Markovian bath. The effective modes in question can be understood as collective coor-

dinates, or generalized Brownian oscillator modes [41–43] by which the dynamics is

successively unraveled as a function of time. The deﬁnition of these modes emerges

naturally from spin-boson type models and linear vibronic coupling models for non-

adiabatic dynamics [44]. It has been shown [45–49] that the short-time dynamics of

the system is entirely accounted for by the ﬁrst member of the effective-mode hierar-

chy. Inclusion of higher order members of the chain captures the dynamics for longer

times.

For the hybrid hydrodynamic-Liouville approach described in this chapter, the

effective modes are harmonic and are described in Liouville phase-space. The system

part is described hydrodynamically. Due to the harmonicity of the effective-mode

environment the equations of motion in the Liouville phase-space contain no  terms

and so the equations are equivalent to the corresponding classical Liouville phase-

space description.

11.2 THEORY

We begin the following theory section by describing the Hamiltonian associated with

the non-Markovian system-bath dynamics (Section 11.2.1). It is then shown how the

Hamiltonian may be transformed to a chain like representation where each member

of the chain is bilinearly coupled to its nearest-neighbor (Sections 11.2.2–11.2.4). To

ensure irreversible dissipative behavior the ﬁnal member of the chain is coupled to

a Markovian bath. However, the dynamics of the reduced system is strongly non-

Markovian.

In the following sections the equations of motion associated with the compos-

ite chain Hamiltonian are initially formulated in Liouville phase-space, in terms of

the Wigner function (Section 11.2.5). Subsequently, a transformation to the hybrid

hydrodynamic-Liouville representation is made (Section 11.2.6). Mass-weighted

coordinates are used throughout this chapter.

11.2.1 S

YSTEM-BATH HAMILTONIAN

The starting point is the familiar partition of the Hamiltonian into a system H

, bath

, and a system-bath coupling part H

H = H

+ H

. (11.1)

Here, the system is taken to be one-dimensional, with the following Hamiltonian in

mass-weighted coordinates,

+ V (x

) (11.2)

166 Quantum Trajectories

where p

=−i∂/∂x

and V (x

) is the system potential energy. The bath consists

of a number N of harmonic oscillators, with N →∞to represent truly irreversible

dynamics,



n=1

+ ω

) (11.3)

where ω

is the frequency of the nth-harmonic oscillator. The bath is assumed to be

bilinearly coupled to the system coordinate x

= x



n=1

(11.4)

where c

are the system-bath coupling coefﬁcients.

The combination of H

of Equation 11.3 and H

of Equation 11.4 deﬁnes the

spectral density function J (ω) which reﬂects the spectral distribution of the bath,

weighted by the system-bath couplings,

J (ω) =

N→∞



n=1

δ(ω − ω

). (11.5)

11.2.2 E

FFECTIVE-MODE REPRESENTATION

The strategy as described in Refs. [48, 49] is to generate a series of approximate

spectral densities from reduced-dimensional analogs of the system-bath Hamiltonian

equation 11.1. These reduced-dimensional models are obtained from a transformed

version of the Hamiltonian equation 11.1,

H = H

+ H

eff

+ H

eff,B



+ H



(11.6)

where the transformation H

→ H

eff

eff,B



is based upon a

transformed set of bath modes comprising an effective-mode X

and a set of residual

modes {X

, ...,X

In the transformed representation H

is represented in terms of a single collective

mode X

= Dx

(11.7)

with the effective-mode Hamiltonian H

eff

given by

eff

+ Ω

). (11.8)

The effective mode is coupled to the residual bath modes by

eff,B



= X



n=2

1,n

. (11.9)

A Hybrid Hydrodynamic–Liouvillian Approach 167

The residual bath part H



is given by





n=2

+ Ω

) +



n=2



m>n

. (11.10)

The total transformed Hamiltonian H has now been partitioned into the following

form

H = H

+ H

eff

+ H

eff,B



+ H



. (11.11)

The transformation from Equation 11.1 to Equation 11.11 inevitably transforms the

diagonal frequency matrix ω to a non-diagonal symmetric form ω

→ T

†

T = W

that contains the following elements,

Ω



m=1

, d



k=1

(11.12)

where t

are the elements of the transformation matrix T. In Equation 11.11 the

effective-mode couples to the system and to all the residual bath modes via the d

terms. The couplings {d

}within the residual bath part H



can be chosen in various

ways, since there is some freedom in deﬁning the transformation T. In particular,

the couplings between the residual modes may be completely eliminated while all

modes couple to the effective-mode X

via H

eff,B



of Equation 11.9. Alternatively,

the coupling matrix d may be cast into a chain-type, band-diagonal form [48,50–53]

with nearest-neighbor couplings within the residual bath space, along with a nearest-

neighbor (X

-X

) coupling in H

eff,B



eff,B



= d





n=2

+ Ω

2

) +



n=2

nn+1

n+1

. (11.13)

This representation has been referred to as the hierarchical electron–phonon (HEP)

model [50,51].

11.2.3 E

FFECTIVE-MODE CHAIN INCLUDING MARKOVIAN CLOSURE

In the following non-Markovian model, a band-diagonal (tri-diagonal) form of the

residual bath couplings is adopted [50, 52, 53] in conjunction with a Markovian clo-

sure acting only on the Mth ﬁnal member of the Mori-type effective-mode chain.

In the following discussion the Caldeira–Leggett model [54] is used to describe the

inﬂuence of the bath on X

. To connect to the Caldeira–Leggett model, a potential

renormalization counter-term for X

= X



n=M+1

M,n

2Ω

(11.14)

168 Quantum Trajectories

is included in the Hamiltonian to give

H = H

+ Dx



n=1

+ Ω

)



n=2

n−1,n

n−1

+ X



n=M+1

M,n

+ X



n=M+1

M,n

2Ω

. (11.15)

In Equation 11.15 the second summation represents the coupling between succes-

sive modes of the chain. The third summation represents the coupling between the

remaining bath modes and X

is the ﬁnal member of the chain.

11.2.4 S

PECTRAL DENSITIES

In the Caldeira–Leggett model the coupling between the bath modes and X

is deﬁned

by an Ohmic spectral density J (Ω). In mass-weighted coordinates J (Ω) is given by

J (Ω) =



M,n

Ω

δ(Ω − Ω

) = 2γΩ exp(−Ω/Λ) (11.16)

where γ is a frequency independent friction coefﬁcient and Λ is a cut-off frequency

for the bath modes.Although the dynamics of the simple Caldeira–Leggett model of a

system coupled to a bath is Markovian (within the high-temperature approximation),

for the hierarchical representation presented here, where the coupling between the

system and the Mth mode is ohmic, the overall system-bath dynamics is strongly

non-Markovian. For the case where the chain is truncated at the ﬁrst order M = 1

effective-mode level it was demonstrated by Garg et al. [43] that the transformation

from the Hamiltonian of Equations 11.1 through 11.15 has a spectral density given

(in mass-weighted coordinates) by

(1)

eff

(ω) =

2γωD

(Ω

− ω

)

+ 4γ

. (11.17)

Here, D is the transformed coupling coefﬁcient of Equation 11.7, Ω

is the effective-

mode frequency Ω



m=1

of Equation 11.8, and γ is the friction coefﬁcient

that deﬁnes the Ohmic spectral density of Equation 11.16 for the residual bath modes.

It is shown in Refs. [48, 49] that truncation at a higher order for the chain enables

the construction of more complicated spectral densities that could provide suitable

models for reproducing various experimental spectral densities.

If the residual bath conforms to an Ohmic distribution, so that the Hamiltonians of

Equations 11.7 through 11.10 and Equations 11.3 through 11.4 are related by a unitary

transformation, Equation 11.17 is obtained under the condition that the cut-off Λ of

Equation 11.16 is large, see Refs. [43,48,49]. (Another caveat again concerns the fact

that this is strictly valid only in the high-temperature limit, where the residual bath is

memory-free.)

A Hybrid Hydrodynamic–Liouvillian Approach 169

The generalization of the construction of the spectral density to higher orders has

been described in detail in Ref. [48]. Here, we summarize the key aspects of the deriva-

tion. For a bilinear x

coupling and a zeroth-order system Hamiltonian of the form

= p

/2 + V (x

), the spectral function may be derived from the Fourier–Laplace

transform of the equations of motion for the dynamical variables {x

, X

...X

}.In

the transformed form the equation of motion for x

(z) is given by [43, 48]

(M)

(z) ˆx

(z) =−V



) (11.18)

where

ˆx

(z) =



∞

(t) exp(−izt)dt,Imz<0 (11.19)

for z = ω −i. In Equation 11.18, V



) is the Fourier–Laplace transform of V



the spatial derivative of the system potential.

It is shown in Ref. [48] that

(M)

(z) is given by the continued fraction

(M)

(z) ˆx

(z)

=−z

ˆx

(z) −

Ω

− z

−

1,2

Ω

− z

−···

M−2,M−1

Ω

M−1

− z

−

M−1,M

Ω

− z

+ i2γz

ˆx

(z)

(11.20)

where the {Ω

}and {d

}are the diagonal and off-diagonal components of the trans-

formed frequency matrix, see Equation 11.13. The spectral density J

(M)

eff

(ω) associ-

ated with the Mth-order model Equations 11.3 through 11.4 is related to

(M)

(z)by

Refs [43,55]

(M)

eff

(ω) = lim

→0

(M)

(ω − i). (11.21)

11.2.5 R

EDUCED-DIMENSIONAL EFFECTIVE-MODE CHAIN INCLUDING

MARKOVIAN CLOSURE

In line with the above form of the Mth order spectral density, the explicit represen-

tation of the bath modes (M + 1, ...,N ) in Equation (11.15) can be replaced by a

formulation of the dynamics in terms of the following master equation:

∂ρ

(M)

(t)

∂t

=−



(M)

, ρ

(M)

]+L

(M)

diss

(M)

. (11.22)

Here, ρ

(M)

is a reduced-density matrix comprising all modes of the effective-mode

hierarchy that are included explicitly in the dynamics, i.e., ρ

(M)

= Tr

n>M

{ρ} where

ρ is the density matrix of the full-dimensional system comprising N bath modes.

170 Quantum Trajectories

The dissipative Liouvillian L

(M)

diss

acting at the Mth order of the hierarchy chosen

as a Caldeira–Leggett [54] form is given by

(M)

diss

(M)

=−

iγ



, [P

, ρ

(M)

]

]−

2γE



, [X

, ρ

(M)

]] (11.23)

where γ is the friction coefﬁcient and E

= Ω

/2 coth[Ω

/(2k

T )] is the mean

equilibrium energy of the Mth harmonic mode; k

is Boltzmann’s constant and T

is the temperature. Here, the Caldeira–Leggett equation has been formally extended

from the high-temperature limit (where the equation is rigorously valid) to low temper-

atures and zero temperature. This extension is consistent with a quantum ﬂuctuation-

dissipation theorem [58,59], but does not correctly account for non-exponential effects

that are typical of the low-temperature regime. Still, remarkably good agreement is

often observed with explicit quantum-dynamical calculations at zero temperature,

and with predictions of zero-temperature decoherence rates [60, 61]. This trend is

also conﬁrmed by the results of Refs. [48,49].

In Section 11.2.6 the hybrid hydrodynamic-Liouville approach is described for

the non-Markovian situation described in the previous sections. For the hybrid

hydrodynamic-Liouville approach the equations are more easily derived from a

Liouville phase-space representation for the system and effective modes. For the

phase-space distribution function ρ

(q, p, Q, P ), where q, p are the position and

momentum coordinates for the subsystem parts and Q, P are the M-dimensional posi-

tion and M-dimensional momenta for the effective modes of the chain, the equation of

motion that corresponds to Equation 11.22 with the dissipative form of Equation 11.23

is given in Liouville phase-space by

∂ρ

∂t

=−p

∂ρ

∂q

∞



k=1

odd







k−1

∂

∂q

∂

∂p

+ DQ

∂ρ

∂p

+ Dq

∂ρ

∂P



m=1



Ω

∂ρ

∂P

− P

∂ρ

∂Q





m=2

m−1,m



∂ρ

∂P

m−1

+ Q

m−1

∂ρ

∂P



+ γ

∂P

+ γE

∂

∂P

. (11.24)

This equation of motion for the Wigner function is exact for the bath model that we

are considering here. If the bath was anharmonic, the quantum-classical Liouville

equation would be employed [35–40] which disregards the  correction terms in the

classical sector.

11.2.6 P

ARTIAL MOMENTS

Central to the formulation of the hybrid hydrodynamic-Liouville approach is the

construction of partial hydrodynamic moments which may be derived from the den-

sity operator in the coordinate representation ρ(x

, x



, X, X



)(X =X

...X

)orby

A Hybrid Hydrodynamic–Liouvillian Approach 171

an integration over the momentum part of the quantum subsystem for the complete

phase-space,

P

ρ

qQP



dp p

(q, p;Q, P ) (11.25)

where q = 1/2(x



) is associated with the hydrodynamic space. In the coordinate

representation the moments may also be deﬁned by differentiation of ρ(x

, x



, X, X



)

with respect to the difference coordinate r = x

− x



P

ρ

qQP

2π



dR exp(−iPR/)







∂

∂r

ρ(q, r;Q, R)

r=0

. (11.26)

The coordinate-space density plays the role of a moment-generating function where

the hydrodynamic moments correspond to the coefﬁcients of the Taylor expansion of

the coordinate-space density with respect to the coordinate r,

ρ(q, r;Q, P ) =



P

ρ

qQP







. (11.27)

The equations of motion for the partial moments may be obtained from the coordi-

nate representation of the density matrix or from the phase-space distribution function

(q, p;Q, P ). The phase-space representation presents the easiest option for deriv-

ing the equations of motion, and these are obtained by applying Equation 11.25 to

Equation 11.24.The resulting equations of motion for the partial moments are given by

∂P

ρ

qQP

∂t

=−

∂

∂q

P

n+1

ρ

qQP

− n



∂V(q)

∂q

+ DQ



P

n−1

ρ

qQP

−



odd









−1

∂

V (q)

∂q

P

n−l

ρ

qQP



m=1



Ω

∂

∂P

− P

∂

∂Q



P

ρ

qQP

+ Dq

∂

∂P

P

ρ

qQP



m=2

m−1,m



∂

∂P

m−1

+ Q

m−1

∂

∂P



P

ρ

qQP

+ γ

∂P

P

ρ

qQP

∂P

+ γE

∂

P

ρ

qQP

∂P

. (11.28)

The equations of motion display coupling to both higher and lower moments—the

up-coupling result from the kinetic energy-term and the down-coupling results from

the terms involving the potential energy. The implication of the coupling is that an

inﬁnite hierarchy of equations of motion is obtained. For the general case of mixed

states the hierarchy is non-convergent and moment closure schemes [62–65] need

172 Quantum Trajectories

to be implemented to terminate the hierarchy. However, for Gaussian densities the

moment hierarchy can be terminated analytically by representing the third moment

in terms of the lower moments [66–68]

P

ρ

qQP

=¯p

(q)ρ

qQP

+ 3 ¯p

qQP

P

ρ

qQP

−¯p

qQP

ρ

qQP

(11.29)

where ¯p

qQP

is deﬁned in the next section. In this chapter we focus on Gaussian

densities in harmonic systems where it is well-known that Gaussian densities maintain

their Gaussian form throughout a propagation.

11.2.7 L

AGRANGIAN TRAJECTORIES

The equation of motion for the partial moments deﬁned in Equation 11.28 was for-

mulated for an Eulerian framework. To analyze trajectories for the partial moments

the equation of motion needs to be transformed to a Lagrangian framework. In a

Lagrangian framework the coordinates q, Q, P become dynamical variables and are

interpreted as ﬂuid particles that evolve along hydrodynamic-Liouville space trajec-

tories. For the non-dissipative case the trajectory equations of motion ˙q,

P can

be derived by interpreting the zeroth moment as a continuity equation in the hybrid

hydrodynamic-Liouville space,

∂ρ

qQP

∂t

=−∇

qQP

· j

qQP

(11.30)

with ∇

qQP

= (∂/∂q, ∂/∂Q, ∂/∂P) and the current

qQP

ρ

qQP





˙q









¯p

qQP

(∂H/∂P)

−(∂H/∂Q)





(11.31)

where the momentum ﬁeld ¯p

qQP

is given by

¯p

qQP

Pρ

qQP

ρ

qQP

. (11.32)

The quantity ¯p

qQP

represents the average momentum derived from the underlying

Wigner distribution for a given combination of independent variables (q, Q, P ). At

the Mth truncation of the chain the trajectories are explicitly expressed as

˙q =¯p

qQP

¯p

qQP

=−

∂

∂q

V (q) +DQ

+ F

hyd

= P

=−Ω

− Dq −d

...

= P

=−Ω

− d

M−1,M

M−1

(11.33)

A Hybrid Hydrodynamic–Liouvillian Approach 173

where the hydrodynamic force F

hyd

(q, Q, P ) is given by

hyd

(q, Q, P ) =

ρ

qQP

∂

∂q

qQP

ρ

qQP

∂

∂q



P

ρ

qQP

−¯p

qQP

ρ

qQP



. (11.34)

For the dissipative case described in this chapter a Langevin description is imple-

mented for the Mth order effective mode [69]

=−Ω

− d

M−1,M

M−1

− γP

+ R(t) (11.35)

where R(t) is a random force distributed according to the Gaussian [70]

Prob[R(t)]=

exp(0.5R

(t)/R

)



2πR



(11.36)

and satisfying R(t)=0, R(t)R(t



)=2γk

T δ(t −t



11.3 RESULTS AND DISCUSSION

In the following sections trajectories generated from Equations 11.33 and 11.35 are

illustrated for the reduced dimensional effective-mode chain model of Sections 11.2.5

and 11.2.6. For the examples demonstrated, the Markovian closure is implemented

at the M = 1 level of the hierarchy. The overall picture for this model is that of

a subsystem bilinearly coupled to a single effective mode that is itself Ohmically

coupled to a Markovian bath. Examples using higher order effective-mode hierar-

chies are described in Refs. [48, 49]. The system-bath spectral density J

(1)

(ω)atthe

M = 1 level is given by Equation 11.17 and corresponds to the form described by

Garg et al. [43]. In the following examples the inﬂuence of the residual bath on the

trajectories of the effective mode and the system are investigated in terms of the

friction strength γ and temperature.

The initial condition for the Gaussian density given by

ρ

qQP

√

πω

exp(−ω

(q − q

)

− Ω

(Q − Q

)

− (P − P

)

/Ω

) (11.37)

(at time t = 0, Pρ

qQP

= 0) are summarized in Table 11.1.Also shown in Table 11.1

are the frequency and coupling parameters used in the examples of this chapter.

Unless otherwise stated atomic units (a.u.) are used throughout. The initial condition

was a coherent state Gaussian displaced to q

= 47.14 along the subsystem coordinate

and Q

= P

= 0 for the effective mode.

For the parameters deﬁned in Table 11.1 the spectral density is depicted in Fig-

ure 11.1a for various strengths of γ. For the smallest value of γ = 5 ×10

−5

, J

(1)

eff

(ω)

is quite sharply peaked around Ω

. Increasing γ leads to a broadening of the peak,

and for a large friction term γ = 5 × 10

−3

the peak is shifted to lower frequencies