Chattaraj P.K. (Ed.) Quantum Trajectories

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174 Quantum Trajectories

TABLE 11.1

Parameters, Quoted in Atomic Units, Used for the Examples

Described in Section 11.3

Ω

D(×10

) q

0.0018 0.002 4.0 47.14 0 0

0 0.001 0.002 0.003 0.004

ω (a.u.)

2e–07

4e–07

eff

(ω)

0 500 1000 1500 2000 2500 3000

Time (fs)

500

1000

1500

<E>/cm

–1

γ = 5×10

–5

γ = 5×10

–4

γ = 5×10

–3

0 0.002 0.004

5e–08

(a)

(b)

FIGURE 11.1 (a) Spectral density J

(1)

eff

(ω) for three values of the friction γ = 5 × 10

−5

5 × 10

−4

and 5 × 10

−3

. Also depicted in the inset is a magniﬁcation of J

(1)

eff

(ω). (b) Energy

relaxation for the spectral densities depicted in (a) for a temperature T = 0 K and parameters

deﬁned in Table 11.1.

and the distribution becomes increasingly skewed (see the inset of Figure 11.1a). The

corresponding relaxation of the system mean energy E for temperature T = 0K

is depicted in Figure 11.1b. For γ = 5 × 10

−5

, E displays strong recurrences as a

result of energy transfer between q ↔ Q. The amplitude of the recurrences dimin-

ishes over time. The rate of energy transfer depends on the amplitude of the oscillation

dynamics along the Q coordinate. A large amplitude oscillation along Q facilitates

greater transfer of energy. This is clearly illustrated by the trajectories depicted in Fig-

ure 11.2b for the Q coordinate, where the trajectories oscillate with an increasingly

larger amplitude up until around 250–300 fs. The amplitude of oscillation then dimin-

ishes until around 550 fs where there is only a small amplitude along Q. For longer

timescales the oscillations along Q are damped by the coupling of the effective mode

A Hybrid Hydrodynamic–Liouvillian Approach 175

0 500 1000 1500 2000

–50

q (a.u.)

Time (fs)

–100

–50

Q (a.u.)

FIGURE 11.2 For temperature T = 0 K and friction γ = 5 ×10

−5

: (a) trajectories along the

subsystem q coordinate for initial condition Q = P = 0; (b) trajectories along the effective-

mode coordinate for initial condition q = P = 0.

to the residual bath. Throughout the propagation the hydrodynamic trajectories along

q maintain a coherent oscillatory pattern with a gradual damping of the oscillation.

For a largerfriction of γ = 5×10

−4

, Erelaxes with no observable back transfer of

energy from the effective mode to the system. Here, any oscillation along Q is rapidly

damped by the strong coupling of the effective mode to the residual bath. For the large

γ = 5 × 10

−3

, the Q mode becomes strongly overdamped and the relaxation rate of

Ereduces signiﬁcantly.The Q trajectories are depicted in Figure 11.3band due to the

increasing inﬂuence of the random force in the ﬂuctuation term in Equation 11.35, the

trajectories display stochastic motion. In Figure 11.3a the q trajectories still maintain

a coherent oscillatory pattern but due to the overdamping of the effective mode there

is hardly any noticeable damping of the amplitude of oscillation along q.

The stochastic nature of the trajectories for the effective-mode sector is more

prominent in the P trajectories. As depicted in Figure 11.4 ﬂuctuations in the momen-

tum trajectories are manifested in the smaller γ = 5 × 10

−5

case. The trajectories

oscillate with the frequency of the effective mode but the inﬂuence of the random

force is also discernable. For the larger γ values the P oscillation is damped so rapidly

that no large amplitude oscillation occurs, only stochastic motion around the mean

P =0. Due to the rapid damping of P , there is no discernable Q vibration which

is required to facilitate dissipation for the subsystem part.

176 Quantum Trajectories

0 500 1000 1500 2000

–50

q (a.u.)

500

1000

1500

2000

Time (fs)

–100

–50

Q (a.u.)

FIGURE 11.3 For temperature T = 0 K and friction γ = 5 ×10

−3

: (a) trajectories along the

subsystem q coordinate for initial condition Q = P = 0; (b) trajectories along the effective-

mode coordinate for initial condition q = P = 0.

0 500 1000 1500 2000

Time (fs)

–2

P (a.u.)

γ = 5×10

–5

γ = 5×10

–4

γ = 5×10

–3

FIGURE 11.4 For temperature T = 0 K and the three γ terms shown the effective-mode

momentum trajectories are illustrated. For all three examples the initial condition q = Q =

P = 0 is used.

A Hybrid Hydrodynamic–Liouvillian Approach 177

0 500 1000 1500 2000

500

1000

<E>/cm

–1

T = 1000 K

T = 300 K

T = 0 K

Time (fs)

–50

q (a.u.)

FIGURE 11.5 (a) Energy relaxation at different temperatures computed using a ﬁxed value

of γ = 5 × 10

−5

. (b) Trajectories along the subsystem q coordinate for initial condition

Q = P = 0atT = 1000 K (black). Also shown for comparison are the corresponding

trajectories (grey) for the T = 0K.

The inﬂuence of temperature on the dynamics is depicted in Figure 11.5 for ﬁxed

γ = 5×10

−5

and temperatures of T = 0, 300 and 1000 K. In Figure 11.5a, the relax-

ation of E is depicted for the three temperatures. In all three cases the inﬂuence of

the effective mode on Eis manifested by the transfer of energy between the q ↔ Q

modes. Due to the coupling to the residual bath the subsystem eventually relaxes to

E=E

. The width of the relaxed equilibrium density ρ

(qQP) increases with

temperature and this is manifested in the trajectories. As depicted in Figure 11.5b the

trajectories for the q sector become more widely separated for the higher temperature

residual bath; thus indicating a wider distribution for ρ

(qQP).

As a ﬁnal example we consider the case where the subsystem frequency is resonant

with the effective-mode frequency. In Figure 11.6athe relaxation of Eis depicted for

the case where ω

is (i) resonant with Ω

and (ii) off-resonant with Ω

. The presence

of resonances increases the overall rate of energy dissipation to the bath. From the

pre-transformed direct system-bath perspective of Section 11.2.1, the spectral density

(1)

eff

(ω), which is a measure of the strength of the system-bath coupling, is sharply

peaked at Ω

, and so it is expected that the rate of energy dissipation is greatest

when ω

is resonant with the peak in J

(1)

eff

(ω). From a post-transformed effective-

mode interpretation (which has been the main focus of this chapter) the presence

of resonances enhances the energy transfer between the q ↔ Q modes. With more

178 Quantum Trajectories

500

1000

1500

2000

500

1000

1500

2000

500

1000

<E>/cm

–1

Ω = 0.0020

Ω = 0.0018

Time (fs)

–50

q (a.u.)

FIGURE 11.6 (a) Energy relaxation dynamics E. In one example (dashed curve) the system

harmonic frequency ω

= 0.0018 is off-resonant with the effective-mode frequency Ω

;

in the other example (solid curve) ω

is resonant with Ω

. (b) Trajectories along q for the

resonant case.

energy being transferred to the effective mode the overall rate of energy dissipation

increases as a result of the effective mode to the residual bath coupling.

The q trajectories for the resonant case are depicted in Figure 11.6b. The amplitude

of oscillation along q is strongly correlated to the rate of energy transfer. In Figure 11.6

the amplitude of trajectory oscillation along q varies synchronously with the rate

of energy transfer—the dips in E coincide with very small amplitude oscillations

along q.

11.4 CONCLUSION

In this chapter a hybrid hydrodynamic phase-space approach to non-Markovian

dynamics was described. The non-Markovian approach involves a hierarchical chain

of coupled effective modes that is terminated by coupling the ﬁnal Mth member of

the chain to a Markovian bath. This Mori-type approach gives rise to a hierarchy

of spectral densities that are deﬁned in Equation 11.21 as the imaginary part of a

continued fraction obtained from the classical equation of motion for the subsystem

coordinate variable x

(see Ref. [48] for a detailed derivation).

In the reduced dimensional model obtained from the effective-mode construction,

the subsystem part was described in a hydrodynamic setting and the effective modes

A Hybrid Hydrodynamic–Liouvillian Approach 179

were represented in a Liouville phase-space setting. Starting from a 2M phase-space,

a reduction involving the partial moment construction of Equation 11.25 was car-

ried out, resulting in a hierarchy of equations of motion that are obtained for these

moment quantities. This hierarchy results from the up and down coupling between the

moments and is unrelated to the effective-mode hierarchy. The effective-mode hier-

archy determines the dimensionality of the reduced model while the hydrodynamic

hierarchy affects the number of coupled equations of motion for Equation 11.28.

For Gaussian densities of the completely harmonic reduced dimensional

model described here, the hydrodynamic hierarchy terminates at the second-order

level. However, for anharmonic systems the hierarchy can only be terminated by

implementing an approximate closure scheme, such as the Gauss–Hermite approach

recently developed by Hughes et al. [62]. A challenge in developing a closure scheme

for quantum hydrodynamics is ﬁnding a low-order closure to the hierarchy. A large

number of moments are generally required to account for quantum effects. For one-

dimensional problems a high-order closure to the hierarchy poses no computational

challenges. However, for large dimensional problems the number of moments grows

considerably and soon becomes computationally challenging.

The equations of motion could well have been formulated completely in a hydro-

dynamic setting; however, the advantage of a partial hydrodynamic representa-

tion becomes clear when considering anharmonic systems that require approximate

moment closure schemes. For a one-dimensional subsystem the number of moments

involved that includes the effective modes in a completely hydrodynamic setting is



n=0

(M +1)

where M is the number of effective modes in the reduced dimensional

model and N is the order of the highest moment in the closure. For the hybrid hydro-

dynamic phase-space approach described here only N moments would be required

for the closure.

For the examples described in this chapter the effective-mode hierarchy was trun-

cated at the M = 1 level. Higher order models have been discussed in Refs. [48,49]

where it was shown that reconstruction of the spectral density from the effective-mode

chain representation can account for quite complicated spectral densities.

The trajectories formulated in Equations 11.33 and 11.35 capture an intuitive pic-

ture of the non-Markovian behavior of the system-bath dynamics. From the trans-

formed effective-mode representation the trajectories clearly displayed the energy

transfer between the modes. The dissipative inﬂuence of the bath was buffered, or

delayed, by the coupling to the effective mode and this was also manifested in the

trajectories.

ACKNOWLEDGMENTS

We thank Klaus B. Møller for useful discussions.

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Quantum Fluid

Dynamics within

the Framework of Density

Functional Theory

Swapan K. Ghosh

CONTENTS

12.1 Introduction.................................................................................................183

12.2 Quantum Fluid Dynamical Transcription of Quantum Mechanics (QM)

of a Single Particle......................................................................................185

12.3 Quantum Fluid Dynamical Equations for Many-Particle Systems

in Conﬁguration Space ...............................................................................186

12.4 Density Functional Theory of Many-Electron Systems:ASingle-Particle

Framework for QFD in 3-D Space .............................................................188

12.5 Quantum Fluid Dynamical Equations within a Density Functional

Framework..................................................................................................191

12.6 Miscellaneous Aspects of the Quantum Fluid Dynamical Approach .........193

12.7 Concluding Remarks...................................................................................194

Acknowledgments..................................................................................................194

Bibliography ..........................................................................................................194

12.1 INTRODUCTION

The concepts of quantum mechanics (QM) are so radically different from those of

classical mechanics that soon after Schrödinger proposed the concept of quantum

mechanical wavefunction, there were attempts to provide a classical interpretation

of QM. In fact, the success of Madelung [1] in transforming the Schrödinger equa-

tion into a pair of hydrodynamical equations consisting of the continuity equation

and an Euler type equation of hydrodynamics, giving birth to so-called quantum

ﬂuid dynamics (QFD) [2–5], in 1926, itself is testimony to the strong urge to have

a classical interpretation of the strange world of QM. The concept was resurrected

with full vigor more than 25 years later through the pioneering works of Bohm [6]

and Takabayashi [7]. In the meantime, Wigner [8] introduced, in 1932, the concept

183