Chattaraj P.K. (Ed.) Quantum Trajectories
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194 Quantum Trajectories
12.7 CONCLUDING REMARKS
For many-electron systems such as atoms, molecules and solids under the action of
TD external fields, which may include interaction with radiation and thus correspond
to both scalar and vector potentials, QFD is a versatile approach for the description of
equilibrium as well as dynamical properties. In this short review, a brief overview of
the underlying principles of the quantum fluid dynamical approach has been presented.
The emphasis has been on the formal aspects of QFD in 3-D space in the presence
of external TD electric and magnetic fields with arbitrary time-dependence. This
formalism is suitable for the treatment of interactions of radiation with atomic and
molecular systems. The Kohn–Sham like TD equations of TDDFT have been used
to derive the QFD equations and the basic picture that emerges is that of a multi-
component fluid mixture moving in common effective electric and magnetic fields
and component-specific quantum potentials.
The QFD-based approach within the TDDFT framework as presented here is a
versatile tool for the investigation of dynamical properties of many-electron systems.
The limitation on practical applications of TDDFT and hence QFD based on this the-
ory due to lack of exact and accurate XC functionals of density and current density
is gradually being overcome through ongoing research in the area of energy density
functional development. The formalism presented here has neglected the spin polar-
ization, although it can be incorporated easily. The latter is particularly important in
view of the many recent developments in the areas of magnetism and spintronics.Also,
only selected aspects of QFD have been covered here to provide a glimpse of the basic
formalism in 3-D space through a single-particle framework like TDDFT, although
recent years have witnessed many new developments in this exciting area of research.
ACKNOWLEDGMENTS
I am indebted to Professor B.M. Deb for introducing me to the subject of quan-
tum fluid dynamics and density functional theory. It is a pleasure to thank Professor
P.K. Chattaraj for inviting and encouraging me to write this chapter and also for his
patience.
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13
An Account of Quantum
Interference from
a Hydrodynamical
Perspective
Ángel S. Sanz and Salvador Miret-Artés
CONTENTS
13.1 Introduction.................................................................................................197
13.2 A First Look into Quantum Interference.....................................................199
13.2.1 Some Basic Theoretical Background............................................199
13.2.2 The Two-Slit Experiment..............................................................201
13.2.3 Many Slits: The Talbot Effect and Multimode Cavities ...............204
13.2.4 Interference in Two-Dimensions: Vortical Dynamics...................205
13.3 Quantum Flux Analysis ..............................................................................206
13.4 Effective Interference Potentials.................................................................209
13.5 Quantum Interference in the Complex Plane..............................................214
13.6 Concluding Remarks...................................................................................217
Acknowledgments..................................................................................................219
Bibliography ..........................................................................................................219
13.1 INTRODUCTION
Quantum interference is ubiquitous to many (if not all) quantum processes and phe-
nomena, for it is a direct manifestation of their inherent coherence. Experiments
based on matter wave interferometry [1] constitute very nice empirical evidences of
this property. Though the first diffraction experiments were carried out with elec-
trons [2], nowadays the (matter) particles utilized range from large atoms and single
molecules [3–6] to very complex, large molecules [7] and Bose–Einstein conden-
sates (BECs) [8]. This kind of experiments, apart from confirming the wave nature
of matter, has also given rise to important practical applications. For example, sens-
ing, metrology or quantum information processing are potential applications which
have motivated in the last few years the remarkable development of BEC interfer-
ometry [9–11]. This can be observed in the increasing amount of (theoretical and
197
198 Quantum Trajectories
FIGURE 13.1 Basic scheme illustrating the essentials of BEC interferometry (see text for
details).
experimental) work appearing in the literature [12–18] since the first experimen-
tal evidence of interference between two freely expanding BECs [19]. The basic
scheme upon which this type of interferometry is based is depicted in Figure 13.1.
In the first stage (left), the atomic cloud is cooled down in a magnetic trap until
condensation takes place. Then (right), the BEC is split up coherently by means of
radio-frequency [16] or microwave fields [17]. Finally, the double-well-like trapping
potential [20] is switched off, releasing the two resulting BECs, which will interfere by
free expansion (e.g., see Figure 6 of Ref. [18] as an illustration of a real experimental
outcome).
Now, despite all the experimental and theoretical work grounded on the concept of
quantum interference, very little attention beyond the implications of the superposition
principle has been devoted to better understanding this phenomenon at a more funda-
mental level. In this regard, Bohmian mechanics [21,22] constitutes a very convenient
tool to explore and understand the quantum system dynamics as well as the associ-
ated processes and phenomena, as pointed out by John S. Bell in his already famous
book Speakable and Unspeakable in Quantum Mechanics [23]. Although Bohmian
mechanics is at the same predictive level as standard quantum mechanics regarding
statistical properties (observables) [24], the former provides a more “intuitive” insight
into the problem under study since it allows us to monitor the probability “flow” with
(quantum) trajectories—this is exactly the same as when tiny charcoal particles are
released on the surface of a fluid in order to determine the flow of the fluid stream.
As an analytical approach, Bohmian mechanics has been applied, for example,
to the study of systems, processes and phenomena with interest in surface physics
[25–30] or quantum chemistry [31, 32]. As a synthetic approach, the quantum tra-
jectory method [33] has been developed as a computational implementation to the
hydrodynamic formulation of quantum mechanics [34] to generate the wave function
by evolving ensembles of quantum trajectories [35]. Furthermore, based on the com-
plex version of Bohmian mechanics [36], the problems of stationary states [37–39]
or wave-packet scattering dynamics [40–43] have also been treated. Here, we present
an analytical study of quantum interference from the viewpoint of Bohmian mechan-
ics (or, in terms of standard quantum mechanics, of quantum probability fluxes),
which will render some interesting properties that go beyond what standard quantum
mechanics textbooks teach us about this phenomenon [44]. Accordingly, we have
Quantum Interference from a Hydrodynamical Perspective 199
organized the chapter as follows. In Section 13.2 we introduce the basic theoretical
ingredients of quantum interference, Bohmian mechanics and quantum fluxes, as well
as several illustrative examples. In Section 13.3 we provide a formal discussion of
interference in terms of fluxes, showing how the Bohmian non-crossing property
already appears in standard quantum mechanics. As a consequence of this property,
any interference process can be understood as the action of some effective potential
on one of the interfering beams, as explained in Section 13.4. To be self-contained, in
Section 13.5 we provide a discussion of quantum interference when it is considered
within the complex plane, which requires a complexification of Bohmian mechanics
(a further development can be found in Chapter 18). Finally, the main conclusions
derived from this chapter are briefly summarized in Section 13.6.
13.2 A FIRST LOOK INTO QUANTUM INTERFERENCE
13.2.1 SOME BASIC THEORETICAL BACKGROUND
Due to the linearity of the Schrödinger equation, any solution will satisfy the super-
position principle. That is, if ψ
1
and ψ
2
are solutions of this equation,
Ψ(r, t ) = c
1
ψ
1
(r, t ) + c
2
ψ
2
(r, t ) = c
1
ψ
1
(r, t ) +
√
αψ
2
(r, t )
(13.1)
will also be a solution, with α = (c
2
/c
1
)
2
(there should also be an additional phase
factor e
iδ
multiplying ψ
2
, but we will consider δ = 0, for simplicity). The particular
interest in solutions like Equation 13.1 relies on the fact that they express pretty well
the essence of interference, more apparent through the associated probability density,
ρ =|Ψ|
2
= c
2
1
ρ
1
+ αρ
2
+ 2
√
α
√
ρ
1
ρ
2
cos ϕ
, (13.2)
where both ψ
1
and ψ
2
are expressed in polar form (ψ
i
= ρ
1/2
i
e
iS
i
/
, i = 1, 2) and
ϕ = (S
2
− S
1
)/. The presence of the oscillatory term in Equation 13.2 constitutes
observable evidence of interference, possible because ψ
1
and ψ
2
form a coherent
superposition. As can be noticed, ρ does not satisfy the superposition principle, as
also happens with the associated probability density current,
J =
1
m
Re
Ψ
∗
ˆ
pΨ
=−
i
2m
Ψ
∗
∇Ψ − Ψ∇Ψ
∗
, (13.3)
where
ˆ
p =−i∇ is the momentum (vector) operator. This vector field provides
information about the flow of ρ in configuration space, according to the continuity
equation,
∂ρ
∂t
+∇J = 0. (13.4)
Substituting Equation 13.1 into Equation 13.3, with ψ
1
and ψ
2
in polar form, yields
J =
c
2
1
m
ρ
1
∇S
1
+ αρ
2
∇S
2
+
√
α
√
ρ
1
ρ
2
∇(S
1
+ S
2
) cos ϕ
+
√
α(ρ
1/2
1
∇ρ
1/2
2
− ρ
1/2
2
∇ρ
1/2
1
) sin ϕ
, (13.5)
which, effectively, does not satisfy the superposition principle.
200 Quantum Trajectories
Within the hydrodynamical picture of quantum mechanics provided by ρ and J
one can further proceed and determine probability streamlines (lines along which the
probability flows), as in classical hydrodynamics. Consider the total wave function
in polar form, Ψ = ρ
1/2
e
iS/
. After its substitution into the Schrödinger equation,
we find the continuity equation 13.4, with J = ρ∇S/m, and the so-called quantum
Hamilton–Jacobi equation,
∂S
∂t
+
(∇S)
2
2m
+ V +Q = 0, (13.6)
where
Q =−
2
2m
∇
2
ρ
1/2
ρ
1/2
(13.7)
is the quantum potential. According to Equation 13.6, quantum mechanics can be
understood in terms of the eventual, well-defined (in space and time) trajectories
that a quantum system can follow (as in classical mechanics). These trajectories are
obtained by integrating the equation of motion
˙
r =
J
ρ
=
∇S
m
=−
i
2m
Ψ
∗
∇Ψ − Ψ∇Ψ
∗
Ψ
∗
Ψ
=
1
m
ρ
1
∇S
1
+ αρ
2
∇S
2
+
√
α
√
ρ
1
ρ
2
∇(S
1
+ S
2
) cos ϕ
ρ
1
+ αρ
2
+ 2
√
α
√
ρ
1
ρ
2
cos ϕ
+
√
α
m
(ρ
1/2
1
∇ρ
1/2
2
− ρ
1/2
2
∇ρ
1/2
1
) sin ϕ
ρ
1
+ αρ
2
+ 2
√
α
√
ρ
1
ρ
2
cos ϕ
. (13.8)
In order to reproduce the same results as in standard quantum mechanics it is only
necessary to consider a large ensemble of initial positions distributed according to
the initial probability density, ρ
0
(r).
From Equation 13.8, one can notice that Bohmian trajectories never cross the same
(configuration) space point at the same time—unlike classical trajectories, for which
this restriction takes place in phase space. This is the so-called non-crossing property
of Bohmian mechanics. Suppose that two velocities are assigned to the same point r,
v
1
(r) and v
2
(r), with v
1
= v
2
. According to Equation 13.8, these velocities will be
related to some wave functions ψ
1
(r) and ψ
2
(r), respectively. If both wave functions
represent the same problem, they can only differ in a constant or time-dependent
phase at r, i.e.,
S
2
(r, t ) = S
1
(r, t ) + φ(t). (13.9)
Now, since the velocities are the gradients of these functions at r, v
2
and v
1
must be
equal. This means that, given a superposition, there will be as many domains (regions
which cannot be crossed) as interfering waves; trajectories started within one of the
domains will remain confined within it at any time. Next, we illustrate this property
as well as some interesting effects derived from it by means of some examples (for
specific details, the reader can go to the corresponding references).
Quantum Interference from a Hydrodynamical Perspective 201
13.2.2 THE TWO-SLIT EXPERIMENT
Consider a simple two-slit diffraction experiment described in terms of a superposi-
tion, like Equation 13.1, which represents the interference of the outgoing diffracted
beams. In principle, the position of the interference maxima is independent of the
diffraction process (they only depend on the zeros of the cosine, according to Equa-
tion 13.2). However, the envelope of the interference pattern in the far-field or Fraun-
hofer region [25, 26] does depend on the transmission function associated with the
slits and, therefore, on the kind of diffraction process that takes place. For example, let
us assume two identical slits with total transmission along the whole distance covered
by each slit, in one case, and Gaussian transmission in the other. In the Fraunhofer
region, the envelope is given by the Fourier transform of the transmission function,
which in our case will correspond to a sinc-function and a Gaussian, respectively, as
shown in Figure 13.2a and b.
–1.0
5.0
4.0
3.0
2.0
z (m)
ρ
(z = 5 m)
(x)
ρ
(z = 5 m)
(x)
1.0
0.0
5.0
4.0
3.0
2.0
z (m)
1.0
0.0
–0.5
(a) (b)
(b) (c)
0.0
x (mm)
0.5 1.0 –1.0 –0.5 0.0
x (mm)
0.5 1.0
–400 –200 0
x (µm)
200 400 –400 –200 0
x (µm)
200 400
FIGURE 13.2 Top: Two-slit diffraction patterns observed along x (at a distance z = 5m
from the two slits), arising when the slit transmission function is: (a) a “hat”-function and (b) a
Gaussian. Bottom: Quantum trajectories illustrating the probability flow from the slits to the
detection screen of the cases displayed in the corresponding upper panels. The results shown
here describe a typical cold neutron diffraction experiment [26].
202 Quantum Trajectories
The associated Bohmian trajectories will show the flow of the probability density
from the slits to the detection screen, placed in the Fraunhofer region. The transmis-
sion function will also influence the topology of the trajectories, which spread out
faster (see Figure 13.2c) or slower (see Figure 13.2d) within the near-field or Fres-
nel region [25, 26] in order to satisfy the requirement that, in the detection region,
the swarm of trajectories has to cover the whole range covered by ρ
0
(x). Regarding
interference, we note that the trajectories start to group in bundles as they evolve from
the Fresnel to the Fraunhofer region, each bundle contributing to a particular interfer-
ence intensity maximum. However, the most interesting feature is that, as mentioned
above, trajectories arising from one of the slits, i.e., starting with the domain of one
of the interfering waves, will remain within the same domain all the time, without
crossing the borderline that separates them; because of symmetry, here both domains
are equal and the trajectories present mirror symmetry.
The previous description of the two-slit experiment is a quite simplified one which
makes possible the derivation of analytical results. However, a more realistic sim-
ulation of the experiment requires us to consider an appropriate potential model to
describe the interaction between the incident particles and the atoms constituting the
material where the slits are “carved” [45]. These models are usually obtained from
ab initio calculations or from reasonable fitted functions, whose validity is tested by
running some type of dynamical calculation (either molecular dynamics for many
degree-of-freedom systems or wave packets for simpler systems), and then compar-
ing with the corresponding experimental data. In Figure 13.3 we have considered a
purely repulsive potential energy surface, which describes the scattering/diffraction of
an incident electron beam (here, simulated by a quasi-plane wave) when it interacts
with two slits [26, 29]. The backward-scattered and transmitted (diffracted) beams
are represented in Figure 13.3a in terms of the probability density, while the angu-
lar intensity distribution is displayed in Figure 13.3b. In Figure 13.4 we show the
600(a) (b)
300
–300
y (a.u.)
–600
0
600300–300
x (a.u.)
θ
f
(deg)
–600
–100
ρ (deg)
100–50 500
0
FIGURE 13.3 (a) Probability density after an electron beam interacts with two slits simulated
by a soft, repulsive potential energy surface [26, 29]. (b) Angular intensity distribution after
diffraction.
Quantum Interference from a Hydrodynamical Perspective 203
500(a) (b)
400
y (a.u.)
x (a.u.)
300
200
100
0
–600 –300 0 300 600
500
400
y (a.u.)
x (a.u.)
300
200
100
0
–600 –300 0 300 600
(c)
500
400
y (a.u.)
x (a.u.)
300
200
100
0
–600 –300 0 300 600
FIGURE 13.4 Quantum trajectories illustrating the probability flow associated with the scat-
tering process represented in Figure 13.3. In each frame, the trajectories have been launched
from the same x
0
initial position (varying y
0
), which has been taken at the rearmost (a), central
(b) and foremost (c) positions on the initial wave function. For the sake of clarity, we have only
plotted the trajectories in the region y>0 and, in cases (a) and (b), only their scattered parts.
trajectory picture of this process, where only the region y>0 has been plotted due
to symmetry with respect to the borderline at y = 0. The sets of trajectories dis-
played are all associated with the same initial wave function, but selected according
to their x
0
initial condition; trajectories within the same frame have all the same x
0
value, which has been chosen at the rearmost (a), central (b) and foremost (c) part
of the incident wave function. Apart from the non-crossing rule, we also observe
that, depending on the particular initial conditions, trajectories display very different
dynamics. The trajectories associated with the rearmost part of the wave function (see
Figure 13.4a) “bounce” backwards quite far from the physical potential, on a sort of
effective potential energy surface [26–28], while those in the foremost part (see Fig-
ure 13.4c) are the only ones (at the energy considered) which can pass through the
two slits.