Chattaraj P.K. (Ed.) Quantum Trajectories
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274 Quantum Trajectories
As compared with the regular multisurface QTM Equations 17.4 and 17.9, the
decoupled representation QTM equations 17.19 exhibit extra terms that depend on
the slip velocity [10,30]
p
j
− P
j
/m (that links the GaussianALE frame with the true
Lagrangian frame) as well as extra quantum potentials Q
cpl
jj
=−
2
/(2m)
∇
2
α
j
/α
j
and Q
cross
jj
=−
2
/m
∇a
j
/a
j
∇α
j
/α
j
.
The “total” equations of motion Equation 17.19, together with the single-state
Equations 17.17, are formally exact and, in principle, may be propagated as such.
However, in the example given below—two electronic states coupled by an ultrashort
laser pulse—the radiative coupling is so big that the source/sink terms λ
jk
and the
off-diagonal quantum potentials Q
jk
dominate the behavior of Equation 17.19. In
this case, the Bohmian equations of motion for the “coupled” function χ
j
(x, t ) can
be simplified [35] to give
dω
j
/dt ≈−λ
jk
/f
j
, (17.20a)
dσ
j
/dt ≈−Q
jk
. (17.20b)
We now apply our decoupled-representation QTM approach described above to a
model problem [48,49] of two adiabatic potential energy curves coupled through a
laser pulse with a single frequency Ω.Assuming the rotating wave approximation [50],
the laser shot on state 1 results in (i) a shift Ω bringing states 1 and 2 energetically
closer to each other, and (ii) a time-dependent radiative dipole coupling between
states 1 and 2. Following Grossmann [48], our matrix TDSE Equation 17.2 translates
for the present model system into the dimensionless TDSEs:
i
∂
∂τ
ψ
1
(
ξ, τ
)
=
−
∂
2
∂ξ
2
+ C
2
ξ
2
ψ
1
(
ξ, τ
)
+ D/ cosh
[
(
τ − τ
0
)
/T
c
]
ψ
2
(
ξ, τ
)
,
(17.21a)
i
∂
∂τ
ψ
2
(
ξ, τ
)
=
−
∂
2
∂ξ
2
− Aξ + B
ψ
2
(
ξ, τ
)
+ D/ cosh
[
(
τ − τ
0
)
/T
c
]
ψ
1
(
ξ, τ
)
,
(17.21b)
where the potential surfaces for states 1 and 2 are a harmonic oscillator and a straight
line, respectively, and where the off-diagonal coupling is independent of the dimen-
sionless coordinate ξ. Since both potential curves V
jj
(
ξ
)
are quadratic functions,
the “decoupled” wave functions φ
j
can be chosen as GWPs [51], whose Bohmian
equivalent is easily propagated by means of Equations 17.17.
We study Grossmann’s [48] model I which corresponds to the following choice
of parameters: A = 1, B =−12, C = 1/
√
2, D = 100, τ
0
= 0.25, and T
c
=
0.025. This model is characterized by large radiative coupling values that allow us, as
already mentioned above, to derive the simplified equations of motion Equation 17.20.
Moreover, considering the very short interaction time, roughly 2.5/100 of the vibration
period of state 1, we will assume that the “decoupled” GWPs φ
j
(
ξ, τ
)
do not move or
disperse during the laser pulse.As a consequence, in this particular case the associated
ALE frames are reduced to fixed grids, which can be chosen as identical, so that grid
interpolation from one state onto the other is not needed.
Nonadiabatic Dynamics with Quantum Trajectories 275
Initially the system is in the ground vibrational state of the harmonic oscilla-
tor state 1, φ
1
(
ξ, τ = 0
)
=
(
2γ/π
)
(
1/4
)
exp
−γξ
2
, with γ = 1/2
√
2, and with an
identical GWP for state 2. In practice, a very small non-zero weight is assigned to
state 2 in order to avoid divergence of the off-diagonal quantum potential in Equa-
tion 17.8a: ω
2
(
ξ, τ = 0
)
= 1×10
−12
and ω
1
(
ξ, τ = 0
)
= 1−1×10
−12
. The coupled
differential Equations 17.17 and 17.20, where j = 1, 2 and k = 1, 2 (j = k), are
propagated in time by means of a Runge–Kutta algorithm in the same conditions as in
Section 17.3.1.
Let us consider the state populations, defined as Π
j
(
τ
)
=
+∞
−∞
ρ
j
(
ξ, τ
)
dξ. The
population of the harmonic oscillator, plotted as a function of time in Figure 17.4,
exhibits about two and a half oscillations (Rabi flops [50]) during the short laser
pulse. Despite the very strong radiative interaction, responsible for nearly complete
population swaps between states 1 and 2, one can observe an excellent agreement
between exact quantum calculations and our quantum trajectory approach using only
20 trajectories. It is remarkable that such a good result is obtained with such a small
number of trajectories. Moreover, the total population Π
1
(
τ
)
+ Π
2
(
τ
)
(not shown)
is conserved over time within an error range smaller than 10
−6
. Unitarity is in fact
expected from Equations 17.4 and 17.5 of the exact QTM formulation which ensure
1
0.8
0.6
0.4
0.2
0
0.1 0.2 0.3 0.4
Time τ
State 1
Population Π
1
0.5
FIGURE 17.4 Population Π
1
(
τ
)
as a function of dimensionless time τ for Grossmann model
I of two potential surfaces coupled by ultrashort laser pulses [48]; solid line: full quantum calcu-
lation; open circles: quantum trajectory result obtained with 20 trajectories; sum of probabilities
Π
1
(
τ
)
+Π
2
(
τ
)
(not shown) conserved to better than 10
−6
. The laser pulse profile is indicated
by the gray area.
276 Quantum Trajectories
the combined flux continuity. It appears to be also satisfied for the present model
system by the approximate Equation 17.20a.
In conclusion, our decoupled-representation QTM was used to compute the dynam-
ics of an ultrashort laser pulse excitation model system [48, 49], and proved to be
very fast and accurate for this task. Despite very large interstate coupling values, only
20 quantum trajectories were needed to obtain accurate populations, a number that
can be contrasted with the ∼10
4
trajectories used in Ref. [48] to obtain a reasonable
accuracy on the same system with a semiclassical method. Moreover, our approach
ensures the conservation of total probability to an excellent level. Similar results
have been obtained [35] on Grossmann’s [48] model II, which involves even larger
radiative coupling values.
17.3.3 B
IPOLAR WAVE PACKETS
In this section, we briefly introduce the bipolar decomposition theory developed by
Poirier et al. [18–24] to circumvent the node problem. The bipolar QTM equations
of motion are formally equivalent to those of a two electronic state problem given
in Section 17.2.1. As we mentioned in the Introduction, the scattering of a wave
packet by a potential barrier that generates interferences between the incident wave
and the reflected wave is a difficult problem for QTM because of the huge quantum
forces occurring about the nodes or quasi-nodes of the wave function. In the bipolar
decomposition, the wave function is expressed as the sum of two counter-propagating
waves,
ψ
(
x, t
)
= ψ
+
(
x, t
)
+ ψ
−
(
x, t
)
, (17.22)
where ψ
+
and ψ
−
represent the incident/transmitted wave and the reflected wave,
respectively. The main advantage of this approach is that the probability densities
ρ
+
=|ψ
+
| and ρ
−
=|ψ
−
| are much less oscillatory than the total probability
density, and consequently the quantum trajectories associated with ψ
+
and ψ
−
will
be much more regular than those associated with the total wave function ψ. Besides
this—very important—practical aspect, the bipolar decomposition is actually related
to a more theoretical aspect of quantum trajectories, i.e., the fact that they do not
satisfy classical correspondence [7,21].
Qualitatively, when the left-incident wave packet ψ
+
(x, t ) encounters the poten-
tial barrier, it starts to transfer probability to the ψ
−
(x, t ) component (initially
ψ
−
(x, t = 0) = 0), because ψ
+
(x, t ) and ψ
−
(x, t ) are coupled together through
the interaction potential (Equation 17.23). The ψ
−
(x, t ) wave first grows in place
until it reaches an appropriate size and leaves the interaction region, heading to the
left, while the ψ
+
(x, t ) wave moves straight through the interaction region.A detailed
description of this process is given in Ref. [19].
Formally, ψ
+
(x, t ) and ψ
−
(x, t ) are solution to the following TDSE (see Ref. [19])
i
∂ψ
±
∂t
=
ˆ
H ψ
±
±
V
2
Ψ
∆
, (17.23)
Nonadiabatic Dynamics with Quantum Trajectories 277
where
ˆ
H =−
2
2m
∂
2
∂x
2
+ V
(
x
)
, (17.24)
is the Hamiltonian operator, and where the unusual off-diagonal coupling term
involves the wave function integral
Ψ
∆
(x) =
x
−∞
ψ
+
x
− ψ
−
x
dx
, (17.25)
as well as the space derivative of the potential V
. Here and in the following primes
denote spatial differentiation, except in Equation 17.25.
From Equations 17.23, together with ψ
±
= R
±
exp
[
iS
±
/
]
and v
±
= S
±
/m,itis
straightforward to derive Lagrangian QTM equations of motion [18] similar to those
obtained for a two electronic state problem (Section 17.2.1):
dR
±
dt
=−
1
2
R
±
v
±
±
V
2
Im
[
Ψ
∆
exp(−iS
±
/)
]
, (17.26)
dS
±
dt
=
1
2
mv
2
±
− V −Q
±
∓ Q
∆±
, (17.27)
dv
±
dt
=−
1
m
(
V +Q
±
± Q
∆±
)
, (17.28)
where
Q
±
(x, t ) =−
2
2m
R
±
(x, t )
R
±
(x, t )
, (17.29)
and
Q
∆±
(x, t ) =
V
(x)
2R
±
(x, t )
Re
[
Ψ
∆
(x, t ) exp(−iS
±
(x, t )/)
]
. (17.30)
The coupling term in Equation 17.26, referred to as ±λ
±
(x, t ), describes the rate of
transfer of probability amplitude between the (±) components and is comparable to
±λ
jk
(x, t ) in Equation 17.4 which transfers probability density between two different
electronic states.
∗
From these equations, Poirier et al. [18] have been able to perform numerically
exact synthetic QTM calculations for a scattering system that exhibits realistic reflec-
tion and interferences.
In order to illustrate the bipolar decomposition approach, analytic bipolar quantum
trajectories x
i
±
(
t
)
from a wave packet scattered by an Eckart potential barrier are
presented in Figure 17.5, where the local probability density is represented by a
color, continuously varying along each trajectory i. Panels a and b correspond to
trajectories x
i
+
and x
i
−
, respectively, with a superposition of both given in panel c.
∗
Notice, however, that λ
+
is not equal to λ
−
, implying that the combined flux continuity relation is not
satisfied here [19], contrary to the case of two different electronic states.
278 Quantum Trajectories
10
5
0
–5
–10
10
5
0
–5
–10
0 4000
X
+
(a.u.)X
–
(a.u.)
8000 12000
0 4000
(b)
10
5
0
–5
–10
X
±
(a.u.)
(c)
(a)
8000 12000
0 4000 8000
Time (a.u.)
12000
–4
–3
–2
–4
–3
–2
–4
–3
–2
–1
FIGURE 17.5 Color representation of analytical bipolar quantum trajectories (a) x
+
and
(b) x
−
for an Eckart barrier centered at x = 0. In panel (c), the x
+
and x
−
trajectories are
superimposed. The color palette shown on the right represents the base-10 logarithm of the
probability density value; values smaller than 10
−4
are invisible. On grayscale reproductions,
dark regions represent high-probability density. (From Park, K., Poirier, B., and Parlant, G.,
J. Chem. Phys., 129, 194112–16, 2008. With permission.)
Nonadiabatic Dynamics with Quantum Trajectories 279
The x
i
+
trajectories divide into two groups. The first group is made of very regular
trajectories which continue to propagate past the interaction region (centered at x = 0)
into the product asymptote, thus forming the transmitted wave. The probability along
each trajectory is conserved (except for a slight decrease due to dispersion), which
translates in the figure by the fact that color does not change along a given trajectory.
In the second group, trajectories tend to merge along the x = 0 axis. At the same
time they change color and become gradually invisible. Thus, they do not conserve
probability but rather, lose all of their probability which is transferred to the ψ
−
component.
As for the x
i
−
trajectories, they arise, as expected, in the interaction region. Initially,
they stay in place for a while as they grow in probability, but eventually start heading
toward negative x values to form the reflected ψ
−
wave. A peculiar “fanout” effect
is observed by which the x
i
−
trajectories tend to spread out exponentially quickly
from each other in the interaction region. Once out of the interaction region, the x
i
−
trajectories exhibit a very regular shape, like the x
i
+
trajectories of the first group.
In conclusion, it is clear from Figure 17.5 that the bipolar quantum trajectories that
form the transmitted and reflected wave packets are very well-behaved—unlike their
unipolar counterparts (not displayed), which for this system exhibit very substantial
interference. The x
i
+
trajectories of the second group, which merge and eventually
“disappear,” as well the x
i
−
trajectories that “fanout” in the interaction region, are
discussed in Ref. [18].
17.4 SUMMARY AND CONCLUSION
In this chapter, the extension of the QTM to the dynamics of electronic nonadiabatic
transitions has been described. Wave packets representing nuclear motion on each of
the coupled PESs are discretized into Bohmian particles, and the exact equations of
motion for these particles reflect the classical and quantum forces on each single PES,
as well as transfer of probability density between states and interstate transfer forces.
On a simple two state model system, the method accurately predicts the oscillatory
structure of the wave functions.
For more complex systems, mixed representations used to formally separate inter-
state transitions from single-state nuclear motions have been introduced. An applica-
tion to ultrashort laser pulse dynamics shows that mixed representations can be very
fast and very accurate.
Interestingly, the bipolar decomposition of the wave function of a single electronic
state, developed by Poirier to circumvent the node problem, leads to a formalism
similar to the QTM formalism for two-electronic states. Bipolar quantum trajectories
have been described to illustrate the bipolar equations of motion, leading to very
regular trajectories as compared to the regular monopolar trajectories for the same
system.
So far, the multistate QTM has seen a relatively small development compared to
the single-state QTM [10]. We believe that considerable improvement of the method
can be achieved, in particular in the framework of mixed representations.
280 Quantum Trajectories
APPENDIX: DIABATIC AND ADIABATIC ELECTRONIC
REPRESENTATIONS
In Section 17.2.1, we set up the QTM equations in a diabatic—as opposed to the well-
known adiabatic—representation.Adiabatic electronic states are coupled through the
kinetic energy operator and may change drastically [52] over a short internuclear
distance range, for example in the vicinity of an avoided crossing, thus making the
propagation of wave packets—and quantum trajectories as well—very difficult. How-
ever, for a counterexample, see Ref. [53]. Diabatic electronic states [54–56], as far
as they are concerned, are built in such a way as to vary smoothly with internuclear
distances, and are thus decoupled (as much as is possible) from the nuclear motion,
but are coupled through the electronic Hamiltonian instead.
Although QTM formalism tends to be simpler in a diabatic basis, it has also
been described in the adiabatic representation (see Ref. [57] and Appendix A of
Ref. [4]).
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18
Recent Analytical Studies
of Complex Quantum
Trajectories
Chia-Chun Chou and Robert E. Wyatt
CONTENTS
18.1 Introduction.................................................................................................283
18.2 Quantum Trajectories in Complex Space ...................................................285
18.2.1 Time-Dependent Problems ...........................................................285
18.2.2 Time-Independent Problems.........................................................286
18.3 Analytical Studies of the Complex Quantum Trajectory Formalism..........287
18.3.1 Quantum Trajectories for One-Dimensional Stationary
Scattering Problems......................................................................287
18.3.2 Quantum Vortices and Streamlines within the Complex
Quantum Hamilton–Jacobi Formalism.........................................290
18.3.3 Quantum Caves and Wave-Packet Interference............................293
18.3.4 Complex-Extended Born Probability Density ..............................296
18.4 Concluding Remarks and Future Directions...............................................297
Acknowledgments..................................................................................................297
Bibliography ..........................................................................................................298
18.1 INTRODUCTION
Bohmian mechanics, developed by Bohm in 1952, provides an alternative interpre-
tation to nonrelativistic quantum mechanics [1–3]. In the hydrodynamic formulation
of quantum mechanics, the continuity equation and the quantum Hamilton–Jacobi
equation (QHJE) are obtained by substituting the wave function expressed in terms
of the real amplitude and the real action function into the time-dependent Schrödinger
equation. Bohm’s analytical approach has been used to compute and interpret real-
valued quantum trajectories from a precomputed wave function for a diverse range of
physical processes [3–8]. In the synthetic approach, the quantum trajectory method
has been developed as a computational tool to generate the wave function by evolving
ensembles of real-valued quantum trajectories through the integration of the hydro-
dynamic equations on the fly [9]. Remarkable progress has been made in the use of
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