Phase Transition in a Radiation-Matter Interaction with Recoil ... - ISC

VOLUME 86, NUMBER 20 PHYSICAL REVIEW LETTERS 14MAY 2001
Phase Transition in a Radiation-Matter Interaction with Recoil and Collisions
M. Perrin, 1 G. L. Lippi, 1 and A. Politi1,2 1Institut Non Linéaire de Nice, UMR 6618 CNRS, Université de Nice-Sophia Antipolis,
1361 Route des Lucioles, F-06560 Valbonne, France
2Istituto Nazionale di Ottica Applicata, Largo E. Fermi 6, 50125 Firenze, Italy
(Received 19 January 2001)
The standard model introduced to describe the collective atomic recoil of an ensemble of atoms interacting
with a strong electromagnetic field has been here extended by the inclusion of collisions with a
buffer gas. As a result, we find that in the thermodynamic limit the coherent emission of radiation exhibits
a continuous phase transition upon increasing the pump intensity. The output laser field is strictly
larger than 0 only above a critical value. We find that the transition is not associated with the onset of
spatial ordering but rather with the onset of a synchronization between the polarization phase and spatial
position. A coherence parameter is introduced to characterize the phase transition.
DOI: 10.1103/PhysRevLett.86.4520 PACS numbers: 42.50.Vk, 05.45.Xt, 05.65.+b, 42.65.Sf
A few years ago, a theoretical model, describing the
interaction between a strong electromagnetic field and
quasiresonant two-level atoms, predicted the occurrence of
an instability. Under appropriate conditions, the initially
disordered atomic sample would organize itself along the
direction of propagation of the (pump) field and form
a periodic structure in space. As a result, part of the
incident field would be reflected back, giving rise to what
has been called a collective atomic recoil laser (CARL)
[1]. The new ingredient contained in that model is the
self-consistent treatment of the momentum transfer in
each atom-field interaction to account for its effect on the
system’s global behavior.
Experimental indications suggesting the existence
of such an ordering transition have been reported in
Refs. [2,3]. However, the simplifications of the model are
so strong that neither experiment [2,3] could be conducted
under conditions that matched the theory [1].
An alternative interpretation of the experimental results
was proposed, which did not make use of atomic recoil
[4]. A simple polarization grating, as demonstrated experimentally
in potassium vapor [5], could produce some of
the features observed in the experiments [2,3]. However,
applying those same ideas to one of the experimental systems
showed that removing recoil from the interaction [6]
produced results that were no longer in agreement with the
original observations [2].
Very recently, the collective interaction between matter
and radiation has been experimentally demonstrated on a
Bose-Einstein condensate [7]. This result renews the interest
in collective atom-field problems, in spite of the fact
that the physical picture is much simpler in this case. Indeed,
since the matter is already in a collective state, the
appearance of a spatial modulation at the optical wavelength
is not too surprising.
The aim of this Letter is to reexamine the problem for
noncondensed atoms by removing one of the simplifications
of the model which is at the basis of a strong disagreement
with the experimental conditions. In [1] the
atoms were considered to move with (nearly) the same
(vectorial) velocity and to be free of collisions. While
such a representation holds well in an accelerated beam
(e.g., for the free electron laser [8] from which the idea
for [1] came), the atomic density values required for the
instability to occur are so high that a supersonic beam [3]
or —even better — a cell [2,3] is necessary.
The cell, however, presents the two following features
that strongly contrast with the model’s approximations:
(1) the atomic motion is of thermal nature and does not
occur with the same (vectorial) velocity for all atoms;
(2) collisions are present in the vapor (and become important
when a buffer gas is used [2,3]). Both effects
are expected to destroy any spatial ordering that may appear
in the system. Thermal motion will displace the
atoms so that a “density grating” should be prevented from
appearing (or would be washed out, if it existed somehow).
On the other hand, even if a periodic structure
were created, collisions would bump the atoms out of
their privileged positions, homogenizing the atomic distribution
in space [9]. It is therefore legitimate to harbor
doubts as to whether the CARL effect should still exist
in the presence of collisions and thermally distributed
atoms. A positive answer was provided in [10], where
both Doppler broadening and collisions have been taken
into account. However, the further simplifications and assumptions
introduced to derive analytic expressions do not
yet allow clear-cut conclusions. A deeper understanding of
CARL passes necessarily through the investigation of more
realistic regimes.
The model introduced in Ref. [1] involves four variables
to describe each atom: the complex polarization Sj, the
population inversion Dj, the position uj, and the momentum
Pj. Additionally, there is a single equation for the
output field A1, while the dynamics of the input (pump)
field A2 is neglected, as the model equations are derived
under the approximation of a small response. With reference
to the standard adimensional variables introduced in
Ref. [1], the equations read as
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VOLUME 86, NUMBER 20 PHYSICAL REVIEW LETTERS 14MAY 2001
uj Pj ,
Pj 2 Re2A 1e 2iuj 1 A 2Sj ,
Sj i
2 Pj 1 2D20Sj
2rDjA1e iuj 1 A2 2GSj ,
Dj 4rReA 1e 2iuj 1 A 2Sj 2GDj 2 Deq ,
A1 iD21A1 1 1
N
NX
Sje 2iuj ,
j1
where r is a function of the density of atoms and G is
the atomic decay rate. D20 and D21 are the detunings of
the input field frequency with atomic and output field frequencies,
respectively. Deq is the equilibrium population
inversion.
The main drawback of this model is the absence of any
mechanism for dissipating the kinetic energy gained by the
atoms through the recoil effect. As a result, one can employ
the model only for studying transient regimes. In order
to overcome this difficulty, it has been proposed to add
a dissipation also to the momentum equations [10] (see also
Ref. [11] for some critical remarks); however, in a realistic
experimental setup, each atom is not truly subject to a drag
force, but it is rather thermalized through energy exchanges
with, e.g., the atoms of the buffer gas (as in the experiment
described in Ref. [2]). For this reason, we have decided
to include the interaction with a heat bath in analogy to
molecular-dynamics simulations: besides the smooth evolution
generated by Eq. (1), we assume that each atom independently
undergoes random collisions whose effect is
to reset its momentum to a Gaussian distributed value and
its polarization phase to a uniformly distributed value in
the interval 0, 2p.
Preliminary simulations made with parameters chosen
so as to closely reproduce the experimental conditions of
Ref. [2] have revealed serious stiffness problems in the
numerical integration besides an extremely slow convergence
towards a stationary regime. Accordingly, we have
preferred to stay closer to the parameter values chosen in
Ref. [1], aiming at a clear understanding of the implications
of model (1).
More precisely, by referring to the rescaled time t
vrrt (vr being the single-photon recoil frequency shift),
we have fixed G D21 1 and D20 215; additionally,
we have chosen Deq 1 and r 10. This implies
that G 6.3 ms 21 in absolute scales. For what concerns
the collision parameters, we have assumed a Poisson distribution
with an average collision time tc 45 (corresponding
to 7.2 ms), while the variance of the momentum
distribution has been fixed equal to s 2 33.3, which,
with the above normalizations, corresponds to a temperature
of a few mK.
As a first step, we have verified that the competition of
the ordering due to the input field and the randomization
(1)
due to collisions can give rise to a regime characterized either
by an incoherent emission of the various atoms (and,
correspondingly, by a vanishing output field), or by a partial
synchronization of the atomic dynamics (and, correspondingly,
by a finite output field). This is, in fact, the
message contained in Fig. 1, where we can see that for a
fixed A2 value, the output intensity Io A1A 1 decreases as
a power law with N (notice that the data reported in the figure
have been obtained by averaging over at least 2 3 10 4
time units after having discarded a transient of approximately
10 3 units). Although the best fit toN 2a for the
decay yields an exponent a 0.91, the residual curvature
suggests that the asymptotic decay rate might approach 1
for N ! `. In the same figure, we can also see that for a
larger value of A2, Io is almost independent of N. This is
a clear indication of a qualitatively different behavior.
It is thus natural to study the transition from the former
to the latter regime. This is the subject of the analysis summarized
in Fig. 2, where we report the gain factor IoIi
as a function of the input pump intensity Ii for an increasing
number of atoms. We have preferred to plot the gain
rather than the output intensity for two reasons: (i) this
helps checking directly that it remains much smaller than
1 [in fact, model (1) has been derived under the assumption
of a small retroreflected field compared to the pump
intensity]; (ii) it amplifies the transition region, showing
more clearly the onset of a singular behavior for N ! `.
The data sets confirm that for sufficiently small Ii, the output
intensity decreases to 0, while above a critical value
I c i , it saturates to a finite value. A careful analysis of the
results for the largest number of atoms that we have analyzed
(N 2048) suggests that I c i 1.17 6 0.02. With
such data, we have also attempted a scaling analysis in the
vicinity of the critical point, but we can state only that
the growth of the output intensity is compatible with a
FIG. 1. Output intensity Io versus the number N of atoms for
A2 0.1 (diamonds —the corresponding scale is on the left),
and A2 2 (full circles —the corresponding scale is on the
right). The straight line with slope equal to 20.91 is the result
of a best fit with a power law.
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VOLUME 86, NUMBER 20 PHYSICAL REVIEW LETTERS 14MAY 2001
FIG. 2. Gain factor versus the input intensity I1 for different
numbers of atoms: crosses, circles, squares, and diamonds refer
to N 256, 512, 1024, and 2048, respectively. The vertical
dashed line is placed in correspondence with the estimated
critical intensity I c i 1.17.
square root behavior: i.e., the typical law expected when
a mean-field approach applies.
In the previous studies of this phenomenon, the so-called
bunching parameter [1]
Ç NX 1
b e
N j1
2iuj
Ç
(2)
was introduced as a way of measuring the amount of spatial
order. In fact, it was assumed that the collective behavior
is associated with the onset of a spatial grating. However,
in our simulations, b 0 both below and above threshold:
in fact, the instantaneous distribution of u values appears
to be totally flat. This confirms the naive expectation that
random collisions do wash out any spatial structure. Thus,
the ordering source for the onset of a macroscopic field
must be looked for somewhere else. A qualitative analysis
of the distribution of the atomic variables has revealed
the appearance, above threshold, of polarization-phase and
population-inversion gratings. Moreover, from Eq. (1), it
is reasonable to conjecture that the growth of A1 depends
on the distribution of the phase differences between the
atomic polarization Sj and the spatial position uj. Accordingly,
a direct way to measure the amount of atomic
coherence is represented by the evaluation of
c 1
NX
Sje
N j1
2iuj , (3)
which we indeed call coherence parameter. Physically,
c represents the correlation function between the polarization
and the position-dependent phase factor e2iuj . At variance
with the bunching parameter, c is able to capture the
essence of the phase transition. This can be seen in Fig. 3a,
where we have plotted the real and imaginary parts of c for
two different numbers of atoms above the critical intensity.
Upon increasing N, the data tend to cluster along a smooth
curve suggesting that eventually (in the limit N ! `), the
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FIG. 3. (a) Values of the coherence parameter c : cr 1 ici
[see Eq. (3)] for N 256 and 2048 (for the sake of clarity, only
points with positive and, respectively, negative cr are plotted)
and the same A2 2; (b) time evolution of cr (filled circles), for
N 2048 after a suitable transient compared with a sinusoidal
fit (solid curve).
dynamics reduce to a limit cycle. To further check this hypothesis,
we have computed the standard deviation of jcj,
verifying that it decreases from 7.5 3 10 23 for N 256
to 2.9 3 10 23 for N 2048: the almost 1 p N dependence
is fully compatible with a statistical explanation of
such fluctuations.
In order to refine the analysis, we have reported in
Fig. 3b the temporal behavior of (the real part of) c. The
nice fit with a sinusoidal function confirms once more the
(almost) periodic behavior with a frequency v 1.046.
The onset of correlations between the spatial position
and the polarization of each single atom provides,
a posteriori, a possible explanation for the qualitative difference
with Ref. [10]. In fact, the adiabatic elimination
of the polarization of the atoms (contained in [10]) allows
one to express Sj as a deterministic function of uj and
thus automatically introduces some degree of correlation
for any input intensity.
Yet another way of detecting the onset of some degree
of atomic coherence is from the probability distribution of
momenta Qp. Below the transition, Qp does not differ
significantly from a Gaussian (see the dashed curve in
Fig. 4). The only relevant deviation, due to the radiationmatter
interaction, is a shift to the left due to the atomic
recoil resulting from the photon emission. The small hole
present for slightly negative p values is, in fact, a finitesize
effect that disappears upon increasing N. This is
at variance with the hole and the sharp peak observed
above threshold, which survive in the N ! ` limit. Their
presence is certainly connected with the above-mentioned
onset of a polarization grating and with the resulting formation
of moving potential wells which trap the atoms.
This would explain the existence of both the peak and the
hole only above threshold. Nevertheless, a quantitative explanation
is not straightforward and goes beyond the scope
of the present Letter.
As mentioned in the introduction, we have mainly studied
model (1) with collisions for a rather small temperature:

VOLUME 86, NUMBER 20 PHYSICAL REVIEW LETTERS 14MAY 2001
FIG. 4. Probability distribution Qp of the momenta below
and above threshold: the dashed curve refers to A2 0.5, while
the solid curve corresponds to A2 5. Both simulations have
been performed with N 256 atoms.
this has allowed us to keep the integration time within an
affordable range. Since performing simulations at room
temperature appears utterly difficult, it would be at least
desirable to develop some scaling arguments to determine
how the critical intensity I c i depends on both r and T
and eventually conclude whether the experimental observation
of the transition is compatible with the accessible
laser powers. This is the task we plan to undertake in the
near future.
We conclude by commenting about our results from a
dynamical-system point of view. Model (1) basically describes
a system of globally coupled oscillators in the presence
of noise [12,13]. In fact, the position of each atom can
be interpreted as the phase of a suitable rotator, while the
input and output field provide a global mean-field type of
coupling; finally, the random resetting of the “frequency”
Pj plays the role of a noise term. Accordingly, it would
be interesting to discover to what extent model (1) can be
simplified and reduced to one of the prototypic models introduced
to investigate synchronization of oscillators.
We warmly thank L. M. Narducci for fruitful discussions.
G. L. L. acknowledges useful exchanges of opinion
with W. Gawlik and J. R. Tredicce. A. P. is grateful to
the CNRS for supporting part of his work as a Chercheur
Associe.
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