We simulate the center of mass motion of cold atoms in a standing, amplitude
modulated, laser field as an example of a system that has a classical mixed
phase-space. We show a simple model to explain the momentum distribution of the
atoms taken after any distinct number of modulation cycles. The peaks
corresponding to a classical resonance move towards smaller velocities in
comparison to the velocities of the classical resonances. We explain this by
showing that, for a wave packet on the classical resonances, we can replace the
complicated dynamics in the quantum Liouville equation in phase-space by the
classical dynamics in a modified potential. Therefore we can describe the
quantum mechanical motion of a wave packet on a classical resonance by a purely
classical motion

Hug, M.

Milburn, G. J.

English

arXiv

We simulate the center of mass motion of cold atoms in a standing, amplitude modulated, laser field as an example of a system that has a classical mixed phase-space. We show a simple model to explain the momentum distribution of the atoms taken after any distinct number of modulation cycles. The peaks corresponding to a classical resonance move towards smaller velocities in comparison to the velocities of the classical resonances. We explain this by showing that, for a wave packet on the classical resonances, we can replace the complicated dynamics in the quantum Liouville equation in phase-space by the classical dynamics in a modified potential. Therefore we can describe the quantum mechanical motion of a wave packet on a classical resonance by a purely classical motion

Milburn, Gerard J.

University of Queensland eSpace

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Quantum slow motion
M. Hug and G. J. Milburn
Centre for Laser Science, Department of Physics, The University of Queensland, St. Lucia, Qld
4072, Australia
(December 24, 2002)
Abstract
We simulate the center of mass motion of cold atoms in a standing, amplitude
modulated, laser field as an example of a system that has a classical mixed
phase-space. We show a simple model to explain the momentum distribution
of the atoms taken after any distinct number of modulation cycles. The
peaks corresponding to a classical resonance move towards smaller velocities
in comparison to the velocities of the classical resonances. We explain this by
showing that, for a wave packet on the classical resonances, we can replace
the complicated dynamics in the quantum Liouville equation in phase-space
by the classical dynamics in a modified potential. Therefore we can describe
the quantum mechanical motion of a wave packet on a classical resonance by
a purely classical motion.
05.45,42.50.Vk,03.75.Be
Typeset using REVTEX
1
2To have an intuitive picture of the quantum mechanical dynamics of a wave packet we are
usually confined to the semi-classical regime, that is, to orbits with action large compared to
Planck’s constant [1,2], or to special systems like the harmonic oscillator, where the quantum
evolution equations in phase-space are identical to the classical ones [3]. In this letter we
propose a scheme which enables us to describe a wave packet, localized near a resonance
of a classical mixed phase-space, by classical dynamics in a modified potential. Hereby we
replace the potential in the high order quantum Liouville equation by an effective potential
in such a way that we obtain a classical Liouville equation. Then we describe the quantum
motion as classical motion in this modified potential. We are then able to characterize
the quantum effect by comparing the modified dynamics with the dynamics in the original
potential. This method is applicable well beyond the semi-classical regime for many different
potentials.
Usually quantum effects on wave packets express themselves in the revival and fractional
revival properties [4] or in the occurrence of tunneling phenomena [5]. Both take place on
a comparatively long time scale so that we intuitively don’t expect quantum effects to be
visible on a short time scale. We disprove this intuitive assumption in our model where we
use the center of mass motion of cold atoms in a standing amplitude modulated laser field.
Here we demonstrate that the momentum distribution after each cycle of the modulation is
peaked at smaller momenta than we would expect classically. This shows that the atoms
are traveling slower than we would expect from classical simulations and we can give a
very simple explanation of this “quantum slow motion” phenomenon. Since we here include
stimulated and spontaneous transitions we expect that this quantum mechanical effect is of
realistic order of magnitude to observe experimentally.
We investigate a cloud of two level atoms situated in a standing laser field, with a periodic
modulated amplitude. In this system the Hamiltonian of the center-of-mass motion in the
limit of large detuning is [6]
H(t) =
p2
2
− κ(1− 2 cos t) cos q, (1)
3where p and q denote scaled dimensionless momentum and position, t time, and κ and  are
the parameters defining the depth of the standing wave and the strength of the amplitude
modulation, respectively. Note that p and q fulfill the commutator relation [p, q] = ik,
where k is a scaled Planck’s constant that is in some sense a measure for the “quantum
mechanicality” of the problem since it defines the size of a minimum uncertainty wave
packet in relationship to the resonances [7].
In Fig. 1(left) we show as an example the classical stroboscopic phase-space portrait [8]
for  = 0.2 with κ = 1.2. This choice of parameters is capable to show classical stable period-
one resonances after each modulation period symmetrically situated along the momentum
axes. This specific phase-space structure allows a quantum mechanical wave packet, situated
initially near one of these resonances, to coherently tunnel to the other resonance. This takes
place on a long time scale in terms of cycles of the modulation. Note that this tunneling
cannot be understood in terms of the presence of a potential barrier as it is present in several
publications [5] regarding tunneling in mixed systems.
We simulate the tunneling dynamics by starting each realization with a minimum un-
certainty wave packet that may be squeezed [], centered on the classical resonance. We
then simulate the full quantum mechanical dynamics by applying a split operator algorithm
with adapted time step size [9] in the context of a standard quantum Monte Carlo integra-
tion scheme to include stimulated and spontaneous transitions [6]. We calculate the mean
momenta and the corresponding variance from the Poincare section of the momentum dis-
tribution taken after each cycle of the modulation at t = 2npi. In Fig. 2 (full line) we
show the result of this simulation for  = 0.2, κ = 1.2, and k = 0.25. Related to recent
experiments [10] we used the parameters for Rubidium to obtain a realistic scenario. We
plot the mean momentum after each cycle of the modulation of the standing wave against
the number of cycles. As expected and clearly indicated by the drop down of the variance,
we observe coherent tunneling of the mean momentum from the location of the resonance
at approximately p = 1 to the corresponding resonance at p = −1.
However there are additional oscillations that might lead to the conclusion that the wave
4packet is not sitting precisely on the classical period-one fixed point but is indeed circulating
around an alternative stable point in phase-space. It seems like the wave packet, centered
on the classical resonance is not appropriately centered on the ”true” resonance but sitting
beside it. Therefore the mean momentum at each kick strongly oscillates around its mean
motion.
This lead us to the conclusion that if we moved the initial wave packet onto this alter-
native stable point and started the simulation of the dynamics from there, we expect the
oscillations to vanish. This is exactly what we see in Fig. 2 (dashed line). The oscillations
are strongly compressed and we face essentially the situation of a well localized wave packet
which undergoes coherent tunneling on the longer time scale. For the dynamics of the wave
packet the classical resonance is obviously not important but a modified resonance, shifted
towards slower momentum. How can we explain this effect?
To give an explanation we first recall that a wave packet localized near a classical res-
onance has been shown [8] to remain localized without changing its shape, at least for a
long time. Therefore we may assume that a minimum uncertainty wave packet sitting near
a classical resonance will remain unchanged in shape for several cycles. This is the main
assumption we need to apply a theory of Henriksen et. al [11] where the effect of quantum
mechanics on a wave packet is described as classical motion, that is as motion following the
classical Liouville equations in phase-space, but in a modified potential.
The convenient quantum mechanical phase-space representation is the Wigner function
W (q, p, t),because it has the correct quantum mechanical marginal distributions. Since in
the experiments we are seeking to describe the momentum distribution and the position
distribution of the center of mass motion, this property of the Wigner function allows us to
compare the marginals directly with the measured distributions.
The phase-space dynamics of the Wigner function is given by [12,11]
∂W
∂t
= −p
∂W
∂q
+
i
k
(
∞∑
ν=0
1
ν!
(
k
2i
)ν ∂νV (q, t)
∂qν
∂νW
∂pν
−
∞∑
ν=0
1
ν!
(
−
k
2i
)ν ∂νV (q, t)
∂qν
∂νW
∂pν
)
(2)
where V (q, t) = κ(1−  cos t) cos q denotes the potential. This, for the following convenient
5representation, corresponds to the well known one given by Wigner where only one sum over
odd derivatives occurs.
Due to the special spatial dependence of our cosine potential, where the odd derivatives
reproduce themselves, we can replace the infinite sum by defining an effective potential Veff
by
∂Veff
∂q
=
i
k
(
∞∑
ν=0
1
ν!
(
k
2i
)ν ∂νV (q, t)
∂qν
∂νW
∂pν
−
∞∑
ν=0
1
ν!
(
−
k
2i
)ν ∂νV (q, t)
∂qν
∂νW
∂pν
)
/
∂W
∂p
. (3)
Then Eq. 2 is replaced by the first order equation
∂W
∂t
= −p
∂W
∂q
+
∂Veff
∂q
∂W
∂p
(4)
which is identical to the classical Liouville equation describing the classical dynamics in the
modified potential Veff . In this sense the action of quantum mechanics can be described by
the classical motion in a modified potential.
Assuming a Gaussian squeezed minimum uncertainty wave packet with time dependent
squeeze parameter ξ(t), we take the Wigner function to be of the form
W (q, p, t) =
1
pik
exp
(
−
ξ
k
(q −〈q〉)2 −
1
kξ
(p−〈p〉)2
)
(5)
with the mean time dependent momentum and position, 〈p〉(t) and 〈q〉(t), respectively,
chosen in such a way, that the wave packet always stays centered on the resonance in order
the assumption of staying unchanged in shape to remain valid. It is not important to know
the explicit time dependence of these parameters. Then the effective potential is
Veff(q, t) = V (q, t) exp
(
−
k
4
)
sinh(p− 〈p〉)
p− 〈p〉
(6)
That means the motion of the wave packet is locally described by the original potential
compressed by a factor of exp(−k/4) since the sinh-factor can, for the sake of qualitative
discussion, locally be approximated by 1.
In Fig. 1 (middle and right) we show for k = 0.25 and k = 0.35 classical stroboscopic
phase-space portraits for the effective potential and compare them to the phase-space por-
trait of the original potential. Note, that our approximation is only valid in the vicinity
6of the period-one resonances. However, since we are interested in exactly these regions of
phase space this kind of representation gives an idea of what is going on, although the other
phase-space regions are not to be taken as a valid description of the dynamics there. The
main conclusion regarding the resonances is that the central resonance at (q, p) = (0, 0)
becomes smaller and the second order resonances we are interested in are pushed towards
smaller momenta p, which exactly corresponds to the observation made in Fig. 2, where we
could simulate the tunneling phenomenon best for initially situating the wave packet at the
shifted resonance.
Eq. 6 indicates that the effect scales with k which identifies it as a purely quantum
mechanical effect. We can clearly see this property by comparing Fig. 1 (middle) and
(right), where we can directly see the relocation of the classical resonance for two values
of k. In Fig. 3 we simulate wave packets for different values of k for only a few cycles.
We start each simulation with a minimum uncertainty wave packet in such a way that the
oscillations in the evolution are most suppressed and observe that in correspondence to
the modified potential the mean momenta and therefore the wave packets themselves are
relocated towards smaller velocities with increasing k. Note that the curve corresponding to
k = 0.25 corresponds to a situation where the conditions for tunneling are fulfilled, therefore
the mean momentum starts to decrease.
There is a second important consequence of this phenomenon in the scenario of present
experiments [10] of investigating the short time behavior of loading all the resonances from
a spatially uniform distributed cloud of atoms. In order to effectively load the resonances
we start with a phase shift of −pi/2, that is to say, we now investigate the Hamiltonian
H(t) =
p2
2
− κ(1− 2 sin t) cos q, (7)
and take the snapshots at t = pi/2 + 2npi. Then the resonances are initially aligned on
the q-axes and are therefore covered best by the cloud of atoms. A classical picture of
the dynamics suggests, that as time goes by only those atoms initially sitting close to a
resonance remain, whereas all the other atoms perform a nonlinear motion corresponding to
7the fact that they are sitting in a chaotic region [13] of phase-space. Therefore we expect
to observe after some time only the three peaks of loaded resonances. Since the assumption
of a durable wave packet is only valid for a wave packet initially situated on a resonance
and not for all the other wave packets this motivates us to believe that the local relocation
of the resonance described above only happens to those atoms trapped at the resonance.
This should change the overall momentum distribution in comparison to a pure classical
simulation.
In Fig. 4 (left) we compare the momentum distributions of snapshots at t = 9pi/2, that
is after 2.25 modulation cycles only, of three different simulations: a quantum mechanical
(top), a modified classical (middle), and a purely classical simulation(bottom). Note that
this is a very short time compared to tunneling and revival experiments. For the quantum
simulation (top) we start with a large number of wave packets of the width of the distribution
in momentum of the atom cloud. The width in position is chosen in order to have a minimum
uncertainty wave packet. We distribute them uniformly on the q-axis, apply the Monte
Carlo integration scheme and finally add the contribution of each wave packet up to get the
whole momentum distribution. In the purely classical simulation (bottom) we simply take
a cloud of point particles uniformly distributed in q−direction and Gaussian distributed in
p−direction. Then the individual motion of the atoms is treated classically by letting the
atoms evolve following the classical Liouville dynamics, but we still have included stimulated
and spontaneous transitions in a Monte-Carlo-Integration scheme.
Note that the quantum peaks are shifted towards smaller momenta. This shift becomes
larger with the scaled Planck’s constant k which is a further indication, that this effect
can be explained by the quantum mechanical effect described above. To show that the
occurrence of the effective potential may in principle be sufficient to explain this feature, we
simulate this by applying the classical simulation again (middle), where we now change the
trajectory according to the effective potential once we start on a resonance. This is a very
simple approach which is certainly only useful to show qualitatively that our explanation
is suitable to describe the quantum dynamics. But we note that this modified classical
8simulation indeed shows the essential features of the pure quantum simulations.
In Fig. 4(right) we show the same simulations but without any modulation. Here the
difference corresponding to the quantum mechanical effect vanish and now more or less
all three simulations show the same structure. This structure is due to classical transient
effects, which appear in the first few cycles and are closely related to the motion in the
standing wave, since they are independent of the modulation. This transient is always there
and interferes with the quantum mechanical effect investigated in this paper. However the
quantum mechanical effect is easily to identify since it vanishes for  = 0. Therefore this
effect is clearly related to the modulation and therefore shows a quantum feature of the
classical mixed phase space. Note that the effect is vanishing for  = 0 is consistent with
our theory since in this case we face classical integrable motion. A wave packet in such a
system is not stabilized but spreads and changes its shape and therefore the assumption for
applying the theory of Henriksen et al. is no longer valid.
To conclude, we have shown that we can use the property of wave packets staying lo-
calized on resonances of a classical mixed phase space to simplify the complicated quantum
dynamics in phase space. In this case we can describe the quantum dynamics of the wave
packet by the classical motion in a modified potential. This is not only valid for the cosine
potential investigated, but also as already mentioned in [11] to polynomial potentials of
arbitrary high order and to other systems that has been topic of investigations of the rela-
tionship of classical chaotic motion and the corresponding quantum dynamics. First there
is the atomic bouncer in an evanescent field [14]
V (q, t) = λq + κ(1 +  cos t) exp(−q) (8)
This setup of evanescent light waves can be modified to get a Morse potential [15] which
serves as an atomic trap. In these two cases it is also very straightforward to find the
modified potential and to come to similar conclusions to those in this paper.
9REFERENCES
[1] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics. Springer-Verlag, New
York, 1990.
[2] M. V. Berry, Phil. Trans. R. Soc. A 287, 237 (1977) and M. V. Berry and N. L. Balazs,
J. Phys. A : Math. Gen. 12, 625 (1979).
[3] see the review articles M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner,
Phys. Rep. 106, 121 (1984) and H.-W. Lee Phys. Rep. 259, 147 (1995).
[4] C. Leichtle, I. Sh. Averbukh, and W. P. Schleich, Phys. Rev. Let. 77, 3999 (1996) and
references therein.
[5] for a recent review on driven tunneling see M. Grifoni and P. Ha¨nggi Phys. Rep. 304,
229 (1998).
[6] S. Dyrting and G. J. Milburn, Phys. Rev. A 51, 3136 (1995).
[7] R. Graham, M. Schlautmann, and P. Zoller, Phys. Rev. A 45, R19 (1992).
[8] S. Dyrting, G. J. Milburn, and C. A. Holmes, Phys. Rev. E 48, 969 (1993).
[9] M. D. Feit, J. A. Fleck Jr., and A. Steiger, J. Comp. Phys. 47, 412 (1982) and S. M.
Tan and D. F. Walls, J. Phys. II(France) 4, 1897 (1994).
[10] A. G. Truscott, W. K. Hensinger, M. Hug, M. E. J. Friese, H. Rubinsztein-Dunlop,
N. R. Heckenberg, and G. J. Milburn, to be submitted to Phys. Rev. Let.
[11] N. E. Henriksen, G. D. Billing, and F. Y. Hansen, Chem. Phys. Let. 149, 397 (1988).
[12] E. Wigner, Phys. Rev. 40, 749 (1932).
[13] W. H. Zurek and J. P. Paz, Phys. Rev. Let. 72, 2508 (1994).
[14] Wen-Yu Chen and G. J. Milburn, Phys. Rev. E 56, 351 (1997).
[15] Yu. B. Ovchinnikov, S. V. Shul’ga, and V. I. Balykin J. Phys. B 24, 3173 (1991).
10
FIGURES
FIG. 1. Left: Stroboscopic phase space portrait of the classical motion described by the
Hamiltonian Eq. 1 for κ = 1.2 and  = 0.2. Middle and right: Stroboscopic phase space portraits
of the corresponding effective potentials Eq. 6 with k = 0.25 and k = 0.35
FIG. 2. Left: Mean momentum 〈p〉 of the quantum mechanical simulation of the dynamics of
two wave packets in dependence of number of cycles s. The first (straight line) is initially sitting
on the classical resonance (pm = 1.03), the second (dashed line) on the modified at pm = 0.84.
Here the parameters are k = 0.25, κ = 1.2, and  = 0.2 Right: Corresponding variance V [p].
FIG. 3. Mean momentums 〈p〉 of the quantum mechanical simulation of the dynamics of
several wave packets in dependence on number of cycles s with κ = 1.2 and  = 0.2. Here k takes
on the values 0.15, 0.2, 0.25, 0.3 (from top).
FIG. 4. Left: Quantum, modified classical, and purely classical simulation (from top) of
momentum distributions P [p] of snapshots after 2.25 modulation cycles of the Hamiltonian Eq.7
with κ = 1.2,  = 0.2, and k = 0.35. Right: The same simulations but for the unmodulated case
 = 0.
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Quantum Slow Motion

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Quantum slow motion

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