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Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004, pp. 303∼307
A Full-Quantum Three-Dimensional Analysis of the Dynamics of a Single
Atom in an All-Optical Trap
Young-Tak Chough∗
School of Computers, Electronics & Communications Engineering, Gwangju University, Gwangju 503-703
Kyungwon An
School of Physics, Seoul National University, Seoul 151-742
(Received 12 November 2003)
We analyze two recent “state-of-the-art” experiments on the dynamics of a single atom in an
all-optical trap by using computer simulations based on the theory of quantum trajectories. We
compare the so-called “quasi-classical” simulations of the experiments with our full-quantum calculations and discuss how far the two models agree. We point out that the atomic wave packet
spreading in quantum calculations is mimicked by an interesting behavior of atom skipping around
from a potential minimum to another in the quasi-classical model.
PACS numbers: 42.50, 32.80.-t, 42.62.Fi
Keywords: All-optical-trap, Quantum trajectory, Quasi-classical, Tracking atomic trajectories
I. INTRODUCTION
momentum uncertainty principle, and this is indeed the
issue that the present article addresses. That is, in quantum mechanics, an atom is after all to be described as a
wave packet, and it is well known that the wave packet
undergoes rapid changes in morphology in the course of
the atom-field exchanges of energy-momentum quanta.
Then, without a specific mechanism which continuously
localizes the atomic wave packet in the cavity, it may
be difficult to think about such a classical trajectory of
an atom inside the cavity. Nevertheless, when an atom
gets out of the interaction region and hits a scintillation screen, everything about the wave packet changes
all of a sudden, and we do see a single atom as a shining
spot on the screen. Thus, it is indeed tempting to think
of such a corpuscular atom flying along a unique trajectory, on the other hand. Figure 1 contrasts these two
different concepts schematically. In (a), the transmission
signal at a given instant of time is the outcome of the
collective effect of every “fraction” of the atom within
the wave packet whereas in (b), the signal depends on
the particular location of the “whole” atom. No doubt,
the quantum mechanical picture is (a), so if one readily admits that (a) is ultimately the correct view, the
immediate question would be to what extent the (much
simpler) classical description, (b), can be valid. It would
be also very meaningful to ask whether these two different views might give different outcomes that would
be experimentally discernible. We wish to present some
interesting clues in answering these questions in this article. We will perform a fully quantum-mechanical, threedimensional treatment of the system, and for this pur-
Recently two major workgroups in cavity-quantum
electrodynamics(cavity-QED) announced successful capture of a single atom by the radiation pressure force of
a single photon in the so-called “all-optical trap” [1, 2]
within the setting of the cavity-QED, demonstrating an
advance in the ability to observe and manipulate the dynamical processes of individual quantum systems. They
even claimed the capability of tracking the trajectories
of the atoms inside the trap cavity from the experimentally observable variations of the intracavity field. The
idea is that the strong atom-field coupling will cause the
presence of an atom to significantly modify the intracavity field, thereby providing a means to track the atomic
motion by way of the light emerging from the cavity.
By introducing what they called “quasi-classical” model,
Doherty et al. reported atomic trajectories reconstructed
from the cavity transmission signals [3]. They solved the
Ito-type stochastic equations derived from the FokkerPlanck equation for the atomic center-of-mass motion,
approximating the dynamics of the Wigner operator of
the system by adiabatically separating the atomic motion from the photonic dynamics. Thus, the solution was
essentially a classical stochastic trajectory of an atom
moving in the trap.
One can, however, raise a legitimate question about
this idea of “determining the trajectory of an individual atom” when one rigorously adheres to the position∗ E-mail:
[email protected]
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Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004
Fig. 1. Two different viewpoints: (a) wave packet point of
view and (b) particle point of view.
γ) enters a standing-wave cavity (resonance frequency
ωC , photon loss rate 2κ) driven by an external field (amplitude E, frequency ωL ), along a direction transverse to
the cavity axis and encounters a spatially varying interaction energy ~g(~r) = ~g0 exp −ρ2 /w2 cos(kx) ≡
~g0 ψ(ρ) eikx + e−ikx , where k = 2π/λ and ρ2 =
y 2 + z 2 . As the atom approaches the interaction region, the cavity transmission will change since the eigenfrequencies of the atom-cavity system depend on the
atomic position ~r via the coupling g(~r), thereby signaling
the arrival of the atom. Then, one raises the potential
as illustrated in Fig. 3 by increasing the drive field amplitude E.
The entire system is, therefore, described by the following master equation:
1
[H, %] + κ(2a%a† − a† a% − %a† a)
i~
γ
+ (2σ− %σ+ − σ+ σ− % − %σ+ σ− ),
2
%̇ =
Fig. 2. Scheme of the actual experiments.
(1)
where % is the density operator of the atom-cavity system, a† (a) the photon creation (annihilation) operator, σ+ (σ− ) the atomic excitation(de-excitation) operator such that [σ+ , σ− ] = σz . H is the Hamiltonian of
the system in the interaction picture and is given by
p2y
p2x
p2
+
+ z
2M
2M
2M
ikx
+ i~g0 ψ(ρ) e + e−ikx eiδt σ− a† − e−iδt σ+ a
+ i~E ae−i∆t − a† ei∆t ,
(2)
H =
Fig. 3. How the trap works.
pose, we will make direct comparisons with the aforementioned “quasi-classical” description.
where δ = ωC − ωA and ∆ = ωC − ωL [4]. We will work
in the framework of the quantum trajectory theory [5]
by expanding the state vector of the system as
∞ X Z
X
|Ψi =
d3 p Ca (n, px , py , pz )|n, px , py , pz , ai.
n=0a=e,g
II. 3-D FULL-QUANTUM MODEL
Figure 2 shows the schemes of the experiments in Refs.
1 and 2. The essential mechanism of the all-optical trap
is as simple as depicted in Fig. 3. When an atom comes
into the initial optical potential, the potential depth is
raised so that the atom has no other choice but to be
trapped inside. In the actual setup, a two-state atom
(mass M , transition frequency ωA , radiative decay rate
(3)
However, the radial and the axial motions can be easily
be shown to be separable in the fashion
Ca (n, px , py , pz ) = Ca (n, px )R(py , pz ), (a = e, g), (4)
which is one of the useful findings in this work. Thus,
the axial motion in the coherent evolution turns out to
be simply given by
√
√
√
d
De (n, qx ) = E n + 1ei∆τ De (n + 1, qx ) − E ne−i∆τ De (n − 1, qx )ψρ n + 1e−iδτ e−i(2qx +1)µτ
dτ
√
γ
× Dg (n + 1, qx + 1)ψρ n + 1e−iδτ ei(2qx −1)µτ Dg (n + 1, qx − 1) − nκ +
De (n, qx )
2
√
√
√
d
Dg (n, qx ) = E n + 1ei∆τ Dg (n + 1, qx ) − E ne−i∆τ Dg (n − 1, qx )ψρ neiδτ e−i(2qx +1)µτ
dτ
(5)
A Full-Quantum Three-Dimensional Analysis of the Dynamics· · · – Young-Tak Chough and Kyungwon An
√
× De (n − 1, qx + 1)ψρ neiδτ ei(2qx −1)µτ De (n − 1, qx − 1) − nκDg (n, qx ),
where τ = g0 t, px = qx ~k, and µ = ~k 2 /2M g0 , if transformations such that
Ca (n, qx ) = exp −iµqx2 τ Da (n, qx ),
(7)
d
i
hqξ i = −
dτ
2π
∂ψ iδt
†
−iδt
e σ− a − e
σ+ a ,
∂ξ
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(6)
(9)
are used, whereas the radial motion is given by
d
µ
hξi = hqξ i
dτ
π
(8)
where ξ = y, z. Note here that
Z
Z
√ ∗
2hξi X
0
0
0
ψ
dq
ρ
x dqx nDg (n, qx )De (n−1, qx )hqx | cos kx|qx i
w2
n
Z
√
hξi X
dqx nDg∗ (n, qx ) [De (n−1, qx −1) + De (n−1, qx +1)]
= − 2 ψρ
w
n
∂ψ
σ− a†
∂ξ
=−
(10)
since
1
[δ(qx , qx0 + 1) + δ(qx , qx0 − 1)] . (11)
2
The three-dimensional motion has thus been essentially
reduced to a one-dimensional problem.
As to the implementation of the discontinuous evolution of the quantum trajectory, let us just briefly summarize it. The atomic decay process is implemented by
axial jumps such that
hqx | cos kx|qx0 i =
2
Dg (n, qx ) −→ e−iµτ (2qx ηx +ηx ) Dg (n, qx + ηx ),
De (n, qx ) −→ 0,
(12)
(13)
and by radial jumps such that
R(qy , qz ) −→ R(qy + ηy , qz + ηz ) ,
(14)
where ηi (i = x, y, z) are the direction cosines of the random spontaneous emissions. The cavity decay is simply
described by the jumps
√
Da (n, qx ) −→ n + 1Da (n + 1, qx ), (a = e, g). (15)
III. RESULT
We did separate computations for the experimental
situations of Refs. 1 and 2. The important experimental parameters in Ref. 1 are (δ, ∆) = 2π(−47, 78)MHz
and (g0 , γ, κ) = 2π(110, 2.6, 14.2)MHz with a mean
empty-cavity photon number n̄ ∼ 0.3 for 55 Cs atoms
whereas in Ref. 2, (δ, ∆) = 2π(−35, 5)MHz, (g0 , γ, κ) =
2π(16, 3, 1.4)MHz and n̄ ∼ 0.9 for 85 Rb atoms. As seen
in Figs. 4 (a)–(d), the two experiments gave characteristically different results, primarily due to the different
Fig. 4. Radial orbits in the yz-plane. (a) and (b) are for
the case of Hood et al. [1] (λ = 852 nm), and (c) and (d) are
for case of Pinkse et al. [2] (λ = 780 nm). Axes are in units
of λ.
potential depths in comparison to the heating strength.
From Eqs. (8) – (10), one can easily guess that the radial
motion is essentially subject to a central force except for
the diffusion introduced by quantum jumps. Just like in
the quasi-classical calculation, mostly the radial motion
has rose-shaped orbits with quite long periods in the case
of Ref. 1 as shown in Figs. 4(a) and (b). However, in the
case of Ref. 2, the orbits are much more diffusive, as
shown in Figs. 4(c) and (d).
The axial dynamics in quantum and quasi-classical
models, however, are apparently quite different. First,
Figs. 5(a)–(c) show the typical time evolutions of the
atomic wave packet in the coordinate space for the experimental situation of Ref. 1, starting from the initial
width ∆x = λ/4π. In (a), the initial mean atomic position is hxi = 0, i.e., at an antinode, in (b) hxi = λ/8,
halfway between the antinode and the node, and in (c)
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Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004
Fig. 5. Axial motions for the case of Hood et al.: Quantum description (a)–(c) and quasi-classical description (i)–
(iii). The horizontal axes are in units of λ. The time span
in the top figures is 0–80 µsec, and in the bottom, it is 0–
200 µsec.
hxi = λ/4, at a node. These figures are compared with
the quasi-classical counterparts, Figs. 5(i)–(iii). Note
that in Figs. 5(iii), the initial mean position of the atom
is not exactly at the node, but is hxi = 0.2499λ since
if hxi = λ/4, there will be no dynamics whatsoever. It
is seen that the quantum wave packet tends to migrate
into the nearby potential minima to settle down there,
ending up being split mostly into two pieces, quite reminiscent of a viscous liquid drop fallen into a double-well
potential. In the quasi-classical case, the atom described
by a mass point behaves similarly; i.e., instead of the
probability peaks, it shows a fast oscillation around a
potential minimum. However, one of the differences in
principle is that in the quantum model, the cavity transmission at a given time is determined by the atom-cavity
coupling strength averaged over the extent of the wave
packet whereas in the quasi-classical model, it is determined by the definite atomic position. Also, one can
easily expect the fluctuations in the cavity transmission
signal to be greater, at least in principle, in the quasiclassical model than in the quantum counterpart due to
the fast oscillations in the axial direction.
One may, however, argue that a meaningful quantumclassical comparison should be made between the quantum wave packet and the distribution of the classical trajectories obtained by evolving an ensemble of initial positions corresponding to the initial wave packet. In this
viewpoint, it is obvious that the quantum and the quasiclassical predictions are by no means in contradiction.
That is, in an ensemble in which atoms in both dressed
states are present, some atoms will go to the right and
some to the left, resulting in an overall distribution essentially the same as the quantum wave packet. It would
be very comforting if this were the case because, then,
Fig. 6. Axial motions for the case of Pinkse et al.: Quantum descriptions (a) and (b), and quasi-classical descriptions
(i) and (ii). The horizontal axes are in units of λ, and the
vertical axes are in units of µsec.
the formidable quantum mechanical calculation could be
replaced by a number of much simpler classical calculations without such an ambiguous wavy picture of atoms.
However, one can easily see that this type of argument
loses its logical ground when one considers the case of
an atom-cavity on resonance. As we have shown in our
previous publication [6], the wave packet splits even in
the case of an exact resonance, in general. The point is
that a single atom on resonance with the cavity is always in a 50-50 combination of the two dressed states;
and therefore, the net force that a single atom gets from
the optical potential is simply zero, so no center-of-mass
motion occurs, no matter where the “corpuscular” atom
is located in the potential. Indeed, this is precisely why
the optical dipole force vanishes when the atom-cavity
detuning is zero [7]. Thus, it appears that in this situation the wave packet never corresponds to the statistical
ensemble of anything in so far as there is no classical theory that gives a nonzero force to each atom in a cavity on
exact resonance. It is, however, true that in the present
setting with nonzero detunings, the quasi-classical treatment appears to be an excellent approximation to the
quantum description.
In Figs. 6, we contrast the results of the quantum and
quasi-classical calculations for the case of Ref. 2. The
quantum description says that, in this parameter region,
the wave packet rapidly spreads and tends to settle down
into every slot of the potential minima (at antinodes)
within its span. The quasi-classical description, on the
other hand, suggests quite an interesting atomic behav-
A Full-Quantum Three-Dimensional Analysis of the Dynamics· · · – Young-Tak Chough and Kyungwon An
Fig. 7. Shape of the wave packet at a given time.
ior. That is, the atom appears to be skipping around
from one potential minimum to another. Such skipping happens when an atom is heated up and kicked
out of a potential well, followed by cooling down in another potential minimum. The heated atom will move
around rather freely along the cavity axis with its acquired high energy, but when an appropriate sequence of
spontaneous emissions happens, it falls into the nearby
potential minimum. This type of behavior has also been
mentioned by Pinkse et al. [2] and Doherty et al. [3].
Let us, however, note that such skipping, in fact, happens only in rare occasions because, for it to happen, the
process of heating and cooling should occur in just the
right sequence and at just the right strengths. Thus, the
fact is that mostly the atoms are trapped in a single potential minimum until they leave the interaction region.
Nevertheless, it is interesting to see how the classical picture mimics the quantum description.
IV. DISCUSSION
At this stage, a meaningful question would be whether
one can experimentally probe the difference between
these two pictures. Again, if one compares the ensemble of the classical point atoms with the quantum wave
packet description, it does not seem easy at all to tell any
difference between the two. However, as we mentioned
before, the very first question of “whether the quantum
wave packet only and always represents a statistical ensemble, and whether there really is no way to describe a
single atom in the quantum-mechanical ideology” is still
there. Thus, the questions cast by these experiments
are, indeed, massive. If one could find a practical way
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and actually probe the existence, or absence, of the fast
axial oscillation and/or the skipping behavior of atoms
predicted in the quasi-classical description—for example, by measuring the multi-order correlation functions
as suggested in Ref. 2, it would mean a lot. The goal of
our next work is, therefore, none other than a theoretical
investigation of such a possibility.
Let us finally point out another piece of interesting
quantum-classical correspondence by using Fig. 7. This
is the shape of the wave packet shown in Fig. 6 (a) at
time t ≈ 200 µsec. It shows that every probability density peak has a dip at the center. This fact can be directly connected to the quasi-classical oscillations around
the potential centers; i.e., the probability density to find
a particle at the bottom of a potential well is lowest
because it moves fastest there. However, as the atom
moves, the depths of the dips change in time, but we
find that, when a peak has a dip, so do all the others,
implying the dynamics around one potential minimum
is correlated to the rest, quite reminiscent of the spatial
coherence of an optical wave front.
ACKNOWLEDGMENTS
This work was supported by a Korea Research Foundation grant (KRF-2002-070-C00044).
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