## Abstract

We study the dynamics of a fast gaseous beam in a high Q ring cavity counter propagating a strong pump laser with large detuning from any particle optical resonance. As spontaneous emission is strongly suppressed the particles can be treated as polarizable point masses forming a dynamic moving mirror. Above a threshold intensity the particles exhibit spatial periodic ordering enhancing collective coherent backscattering which decelerates the beam. Based on a linear stability analysis in their accelerated rest frame we derive analytic bounds for the intensity threshold of this selforganization as a function of particle number, average velocity, kinetic temperature, pump detuning and resonator linewidth. The analytical results agree well with time dependent simulations of the N-particle motion including field damping and spontaneous emission noise. Our results give conditions which may be easily evaluated for stopping and cooling a fast molecular beam.

© 2007 Optical Society of America

## 1. Introduction

Light forces have become an essential tool to cool, trap and manipulate atomic gases[1]. Ultimately it allowed to reach BEC and thus the lowest temperatures experimentally available[2]. Ever since this success there has been an ongoing quest to extend laser cooling to more atomic species and also molecules. Here the need for a closed optical cycling transition and the preparation of an initial sample of sufficient low temperature and high phase space density to start the cooling process has posed severe limitations. In particular for molecules, spontaneous emission induced by the laser fields populates a great manifold of electronic and rovibrational levels, which hampers the standard laser cooling techniques.

Recently is has been theoretically predicted[3, 4, 5] and experimentally confirmed[6, 7, 8] that using high Q resonators for the cooling light fields, the role of spontaneous emission can be strongly suppressed in laser cooling. As one further decisive experiment superradiant enhancement of cavity cooling was recently demonstrated for Cesium atoms in a confocal cavity[9]. Although the experiments were performed on atoms, this opens very promising prospects on molecules as well. Trapping and cooling of atoms in a far detuned ring cavity field was recently also achieved[10, 11, 12].

As most sources of gaseous cold molecules are fast beams, the second problem to be dealt with is stopping the molecules of such a beam in a very short distance within the vacuum chamber and trap them. A first success in this direction was achieved by using periodic electrostatic fields to stop a molecular beam[13] or by creating the molecules already cold by laser ablation or directly from an atomic magneto optical trap[14]. More closely related to our analysis is the recent success of using a deep periodic optical lattice created by an intense 5.8 ns laser pulse to trap and slow a group of cold molecules[15, 16]. This clearly demonstrates that strong enough non destructive light forces can be applied to molecules. Thus also cooling should be possible in principle.

We study the prospects of using a ring cavity setup as an alternative way to slow down a molecular gaseous beam. The particles’ kinetic energy is converted into light energy using inelastic coherent backscattering. Such a setup, called collective atomic recoil laser(CARL), was historically proposed in analogy to the Free electron laser (FEL) as laser gain medium using atoms[17, 18]. Only recently, a first unambiguous observation of the (inverse-) CARL mechanism to accelerate a cold gas by coherent light scattering in a cavity was achieved in Tübingen using rubidium atoms[19, 20, 21]. Even stronger stopping forces were observed and exploited by Vuletić in a standing wave cavity setup [9, 22]. In these experiments, the particles were atoms with a closed optical cycle which were cold and relatively slow from the start. Nevertheless, these results show great promise for using collective light forces on particles.Hence, we investigate the experimental conditions necessary to stop and trap a fast molecular beam by help of a single side pumped ring cavity geometry. The idea relates closely to the original idea of CARL, but we are interested in stopping and cooling particles and not in laser gain.

In contrast to most previous work we consider the limit of large detuning, where inversion and spontaneous emission from the particles plays no role. This enables one to adiabatically eliminate the excited states from the dynamics, which allows us to apply the results to any point like objects with suitable polarizability and in particular cold molecules. On the other hand we cannot neglect the dynamic refractive index the particles create in the modes and we have to include the full corresponding field dynamics for pump and backscattered mode.

In the far detuning limit the equations for the internal particle dynamics simplify dramatically but, of course, the atom field interaction strength per particle gets very small. Here only cooperative light scattering by a large number of scatters can induce a sufficiently strong force. In the quasi 1D geometry along the cavity axis this requires a periodic particle arrangement, so that they act like a Bragg mirror coupling the light fields within the resonator. For favorable parameters the interference of the injected and backscattered light creates an optical potential suitable to stabilize such a regular distribution or even induce it via selforganization from an initial flat distribution. This behavior strongly depends on the effective detuning of field modes, pump light and Doppler shifted reflected light.

No threshold formula for a ring cavity setup with counter-propagating pump has been published to our knowledge which is derived from a microscopic theory of atomic motion including cavity damping and the frequency shifts induced by the atoms for the modes. In the following we try to answer two closely related questions in this respect: (1) Under what conditions and circumstances is a homogeneous particle distribution within the resonator unstable against small density fluctuations so that the atoms will start to selforganize? (2) How does the self consistent atom field distribution look like in the regime where self ordering happens and what is the corresponding field intensity and average force on the cloud? As a third important question we address the scaling of the system with particle number, volume and required pump intensity to see whether the effect persists on a macroscopic scale.

Methodically we will adapt an idea already successfully applied to selforganization of atoms in standing wave cavity fields. For this, one iteratively determines the self consistent steady state of the cavity field with particles ordered in a way just to create this very same field. The iteration is done assuming fast atomic thermalization and field equilibration but only slow change of the average atomic temperature. In order to adapt the method to particles in a ring cavity we have to change to the accelerated rest frame of the atomic clouds center of mass and additionally assume that the equilibration is faster than the center of mass acceleration. This allows us to find an accelerating position distribution with a largely time independent shape. Investigating the stability and self consistence of this distribution will then give us limiting conditions for which the proposed method should work. In order to check the predictions we then will directly simulate the microscopic equations of motions for finite ensembles of particles and the fields.

## 2. Model

As a model system let us consider a large number *N* of linearly
polarizable point particles interacting with two degenerate counter- propagating
linearly polarized plane wave ring cavity modes.

The electric field inside the cavity is then

where the electric field per photon is
$\epsilon =\sqrt{\hslash {\omega}_{P}/2{\epsilon}_{0}V}$
and *a*
_{±}(*t*) gives the field
amplitude.

The pump and cavity frequency are assumed to be sufficiently detuned from any optical
resonance so that the particles are only weakly excited and the polarization can be
adiabatically eliminated. Thus they experience a Stark shift linear in the local
intensity and act like a dynamic refractive index in the field. Let the parameter
*U*
_{0} denote the light shift per photon in the field.
From energy conservation one sees that *U*
_{0} then also has
to give the frequency shift of the cavity mode resonances per atom. In terms of the
two level linewidth Γ, detuning Δ* _{a}*, and atom-field coupling g the effective mode frequency shift reads:

In a typical setup one sends the molecular beam at average velocity v0 into the ring
cavity nearly antiparallel to the driven circulating field. The molecules might be
moving quite fast (up to ^{∼}100 m/s) without some pre-slowing so
that the ratio of the Doppler shift to the linewidth
*kv*/κ, can be larger than unity but not too large. At
much larger velocities, the pump and or the backscattered fields are far off
resonance, which requires stronger and stronger pump amplitudes to generate a
noticeable force.

In the following we will shift our reference frame to coincide with the particles
center of mass motion. Hence, while a particle at rest in the cavity would see a
single cavity detuning Δ* _{c}*, the molecular beam experiences two different Doppler shifted cavity
fields with detunings Δ

*= Δ*

_{±}*±*

_{c}*kv*. Thus the equation of motions for the field mode amplitudes are[23]:

Here *N* denotes the particle number and
*η*
_{±} gives the cavity pump
amplitudes. We see that the particles density distribution couples the two counter
propagating modes via the spatial average *R*
_{±}
=
〈*e*
^{∓2ikx}〉,
which is called the particle bunching parameter and one has
${R}_{+}={R}_{-}^{*}$
. A
uniform distribution gives *R*
_{±} ≈ 0,
while an optimally ordered phase gives
∣*R*
_{±}∣ →
1.

As second part in our model we include the force of the light field on the particles. Again assuming the large detuning limit this force is dominated by the dipole force[23]. This force can be derived from the effective optical potential, which is determined by the local field intensity simply by:

For *U*
_{0} < 0, i.e. Δ* _{a}* < 0, this attracts atoms to maxima of the cavity field. Using
the definition of the field in terms of the amplitudes

*a*

_{±},

*E*

^{+}(

*x*) = ε(

*a*+

*e*

*+*

^{ikx}*a*

_{-}

*e*

^{-ikx}) we get:

with α being the argument of
*a*
_{+}
*a**_{-}. Note
that for calculating the force the constant terms can be omitted from the potential
and the average force then simply reads:

This force will then acts on the atomic distribution and so forth.

## 3. Coupled particle field dynamics

#### 3.1. Quasistationary field

Lets us now get some first qualitative insight in the ongoing dynamics for a
thermal cloud in a single side pumped ring cavity
(*η*
_{+} =
*η*,*η*
_{-} =0).
While for an empty cavity the field intensity and thus the optical potential is
constant along the axis, the particles will coherently backscatter some light
and create a periodic intensity modulation. In a typical experimental setup this
field redistribution will happen much faster than the typical time on which the
particle distribution evolves. Hence the field will almost instantaneously
reflect the momentary particle distribution. As the only relevant quantity of
the particle distribution entering the field equations is the parameter
*R*
_{-} =
〈*e*
^{2ikx}〉,
the field amplitudes will relax towards:

To get some more insight in the physical meaning of
*R*
_{-} let us consider the prototype case of a
periodically modulated gas density with symmetric maxima at *x* =
*x*
_{0}+*nλ*/2.*R*
_{-}
than can be written as *R*
_{-} =
∣*R*
_{-}∣*e*
^{2ikx0},
where the modulus ∣*R*
_{-}∣
measures the modulation depth and the location of the maximum density determines
the phase of *R*
_{-}. Inserting these expressions into
the field equations, we then get:

Here we see that the better the atomic localization
∣*R*
_{-}∣ the more
backscattering and stronger modulation of the intracavity intensity we get. As a
second important piece of information this result allows us now to determine the
spatial shift α of the cavity field intensity maxima with respect to
*x*
_{0} by help of Eq.7. Since *U*
_{0} =
-∣*U*
_{0}∣ =
∣*U*
_{0}∣*e ^{iπ}*
we define the relative shift

*δ*= φ/(2

_{x}*k*) of the field intensity maxima with respect to

*x*

_{0}by

This gives *α*
_{±} =
*φ*-2*kx*
_{0} =
-2*k*(*x*
_{0}
-*φ*/(2*k*)) =
-2*k*(*x*
_{0} -
*δ*
*x*), where the value of
*φ* is determined by (11):

As the maxima of the light field intensity determine the minima of the optical
potential Eq. 6 (stable equilibrium points), we see immediately that
field minima induced by the density distribution of the gas do not coincide with
the peak density positions *x*
_{0} but are shifted by
*δ _{x}*. Hence we obviously cannot
find a self consistent time independent configuration of particles and field as
for the standing wave case [24]. It is very interesting to note though, that the
magnitude of the shift only depends on the total particle number via

*NU*

_{0}and cavity parameters, but not on the pump strength η or the precise form of the atomic distribution beyond the average

*e*

^{2ikx0}〉

Obviously the above calculated shift of the potential implies an average force on
the cloud and an acceleration of the particles center of mass position. This
result can also be derived from basic momentum conservation arguments as
follows: since all photons in the backscattered mode have lead to a deposition
of 2*h*̄*k* of momentum in the cloud.
The steady state average force is directly proportional to the output intensity
of the backscattered, counter-propagating mode; the gas will always have an
acceleration in the pump direction. As a consequence of this acceleration,
*R*
_{-} will be time dependent in general. Hence in
contrast to atomic selforganization in a standing wave cavity field[25], we cannot expect the system to reach a time
independent self consistent atom-field steady state. Nevertheless, as we will
see below, quasistationary self consistent atom field distributions can still
form.

#### 3.2. Quasistationary accelerated atomic distribution

As argued above the coupled atom field dynamics produce an acceleration of the
particles center of mass. As long as this acceleration is not too fast the
corresponding periodic field intensity distribution will almost instantaneously
follow the particles center of mass motion. Hence the particles see only a
slowly time varying optical potential relative to their center of mass position.
This should allow the particle distribution to adjust to this accelerating
potential. In order to get some quantitative insight here we will now transform
our equations to a suitable *accelerated* reference. Primarily
this introduces an *inertial force* (acceleration):
*F _{in}* =

*mg*into the atomic equations of motion, which can be accounted by adding a term

*V*= -

_{in}*mgx*to the effective atomic potential. Obviously a suitable choice of g requires to shift the periodic local potential minima to coincide with the peaks of the atomic position distribution. This allows subsequently to find an approximate stationary atomic distribution with respect to the accelerated potential.

Of course the transformation to this frame is reflected in the field equations, where the de-tunings are now time-dependent quantities induced by the time dependent Doppler shifts of the cavity modes(5). Nevertheless, as long as the acceleration is not too fast on the timescale of the field evolution κ, the fields will still adiabatically follow changes of atomic velocity and distribution. Lets consider this in a bit more detail now. Omitting the constants parts, the potential in the accelerated frame reads:

where we choose *g* in a way, that the peaks of the atomic density
distribution *x*
_{0} +
*nλ*/2 coincide with local potential minima, i.e. we
want
(*d*/*dx*)*V*(*x*) =
0 at *x* = *x*
_{0} +
*nλ*/2. This requires

Substituting the steady state expressions for sinφ and for ${a}_{+}={a}_{-}^{*}$ , we then get

The resulting acceleration *g* thus is *negative*
and depends on the magnitude of the atomic localization parameter
*R*
_{-}. This is consistent with our expectations,
since the mode *e*
^{2ikx} is pumped, and
the scattering converts photons traveling the negative
*x*-direction to photons in the positive direction and we have

Besides providing the basis for the choice of an accelerated frame, Eqs.15, 16 are quite interesting and useful results on their own.
While for small *N* the acceleration grows linearly with
*N* showing the superradiant *N*
^{2}
enhancement of backscattering, for large enough *N* it decays
like 1/*N*. Similarly for large κ(bad cavity) and a
nearly flat distribution *R* ≈ 0 acceleration
vanishes. In these limits we can expect the distribution to have enough time to
equilibrate to a quasistationary distribution in the accelerated frame.

Note that the acceleration for not too large *N* depends on
*U*
_{0}
^{2} and thus on the inverse square of
the particle-field detuning as it is the case for near resonant radiation
pressure and spontaneous emission. Hence at first sight this does not look
favorable for molecules. However, the collective enhancement factor
*N* in the force will not show up in spontaneous emission, which
allows to strongly increase the force per spontaneously scattered photon in the
resonators.

Having now found a proper frame as a next step, we will calculate a corresponding atomic distribution. For this we adapt the ideas developed for the standing wave cavity case [24] and simply use a thermodynamic equilibrium canonical density distribution for the particles. Hence the particle distribution is approximated by

where the normalization is given by the partition function *Z* =
∫
exp[-*V*(*x*)/*k _{B}T*]

*dx*.

Note that the effective potential now contains the acceleration terms
*mgx* and this procedure only makes sense as long as
*V*(*x*) has periodic local minima of
sufficient depth. This is the case for small *g* as it occurs for
an initial nearly homogeneous particle distribution, where
*R*
_{-} is close to zero. Hence we can at least use
this formalism to study the initial phase and threshold conditions of the
selforganization process.

#### 3.3. Selforganization threshold

As we have seen above the cavity field and inertial force from the acceleration
*g* determine an effective combined potential
*V*(*x*) for the accelerated particles for
which we get a corresponding stationary distribution from Eq. 17. This modified density distribution corresponds to a new
bunching parameter R and a new optical potential. By subsequent application of
Eqs. 11, 13 this procedure can be iterated and in many cases a self
consistent solution is obtained for ρ(*x*) and
*V*(*x*) after a few iterations. Note that in
order to get a meaningful result, we have to restrict ourselves to a single
potential well (local potential minimum) to determine the density distribution
ρ(*x*) and assume that all other wells to have a
similar number of particles. This can be physically motivated from the fact that
redistribution between different wells will occur on a much longer time scale as
internal thermalization. Note the we do not account for direct interparticle
interactions here, which would occur for very high local densities.

There are now several decisive aspects that can be learned about the physics of our problem from this mathematical iteration procedure. We will start here with the most simple but central question, whether a homogeneous initial particle distribution at rest within the resonator is unstable against small perturbations. This amounts to a CARL type setup, where particles, which are originally evenly distributed are collectively accelerated by superradiant scattering[26]. In the following we will first determine the minimal pumping threshold required to initiate such superradiant backscattering via self ordering. In a second step we will then check, whether a quasistationary accelerated ordered phase can develop out of these initial fluctuations.

Starting from an initial flat particle distribution, we have
*R*
_{-} = 0 and the atoms generate no coherent
backscattered field. Hence the corresponding average acceleration is zero at the
beginning and we can start in laboratory rest frame for the dynamics.
Superradiant backscattering and collective forces in this case can only build
up, if this solution is unstable against small density fluctuations with period
*λ*/2. Hence we add an infinitesimal perturbation
to our uniform distribution $\rho \left(x\right)=\frac{1}{\lambda}$ in the form:

*r*(*x*) could be in general any periodic function

and we see whether this perturbation is amplified during an iteration cycle. Note
that within the ring cavity geometry the only perturbation that contributes to
*R*
_{-} is that for *m* = 2, i.e. we
can limit ourselves to *r*(*x*) =
*A*
_{2} cos(2*k*(*x* -
*x*
_{0}))+*B*
_{2}
sin(2*k*(*x* -
*x*
_{0})) as a perturbation. Due to translational
invariance of the problem we can also set *B*
_{2} = 0.
Hence the perturbation simply leads to *R*
_{-} =
ε*A*
_{2}/2 for the mode coupling
parameter.

As a first step in the iteration we will now use this
*R*
_{-} to determine the corresponding fields and the
optical potential. As shown above in Eq.13 the optical potential is shifted with respect to
*x*
_{0} by an amount independent of
*A*
_{2}. In the second step we now choose the
acceleration *g* in a way described above to pin the effective
potential minimum to *x*
_{0} and apply Eqs. 13 and 17 to find the corresponding thermal density distribution
for this potential. Of course this distribution now shows a modulation induced
by the optical potential which we have to compare with the magnitude of our
initial perturbation.

As *ε* is small we do this analytically by expanding
ρ(*x*) in Eq.17 to lowest order in *ε* to find:

The central question now is, whether the density perturbation has been enlarged
or diminished in this iteration cycle. For a flat distribution to be stable, the
first order correction term in ε must be less than
*A*
_{2}. Thus stability requires:

This formula is one of the central results of this work, which we’ll discuss in more detail in different examples that follow. It allows to clearly determine the experimental conditions for the cavity, initial particle phase space density and laser power required for such setup. We will later also substantiate the key properties of this formula by direct multi particle simulations.

## 4. Results and discussion

#### 4.1. Threshold condition and acceleration of a gas at rest

To investigate the underlying physics in more detail let us first consider a
cloud at rest *v* = 0, i.e Δ_{+} =
Δ_{-} = Δ_{c}. In this case the threshold condition reads:

Obviously a flat distribution always gets unstable for sufficient pump power. One
needs less pump intensity the more particles N one has and the stronger they are
coupled to cavity mode; i.e., a larger
∣*U*
_{0}∣. The required pump
intensity scales linearly with the initial temperature, which gives some
important restriction for useful molecular sources in particular. Of course the
validity of the threshold formula has its limitations as the weak saturation and
two level approximation for the particles eventually will break down at certain
pump intensities. Note that this stability threshold for a flat distribution
attains a minimum value, when Δ* _{c}* =

*NU*

_{0}.

Of course Δ* _{c}* =

*NU*

_{0}is just the resonance condition for the cavity mode shifted by the refractive index of the particles and hence gives a maximum of the intracavity photon number. As we will see later from Eq.3.1 this is a rather tricky parameter choice. In this case the corresponding shift of the optical potential minima with respect to the atomic center of mass just corresponds to

*π*/4. Here the particles sit on the position of maximum slope and the optical dipole force has maximum as well. However, as the curvature of the potential is zero at this point the effective potential in the accelerated frame has no local minimum and one cannot expect the formation of a self consistent atomic distribution of the form of Eq.17.

In the following we will investigate the validity of this threshold condition
numerically by iterating Eqs. 13,17 for a selfconsistent atom-field configuration. We see
that the iteration results agree nicely with the analytic condition. As shown in Figs. 1, 2, and 3 we find a numerical instability in agreement with Eq.23 with the minimum threshold at Δ* _{c}* =

*NU*

_{0}. The onset of self ordering contour is readily seen to closely follow the red dashed line representing the analytic threshold condition Eq.23.

Let us now further substantiate these predictions for some special parameter
choices by a real space dynamical solution of the equation of motion of
*N* particles and the field modes. For this we use a
semiclassical dynamical model as developed in ref. [3], where the atoms are modeled by polarizable point
particles and the fields are amplitudes are represented by coherent states. This
model has already been successfully applied to simulate cavity enhanced laser
cooling and atomic selforganization in a standing wave cavity geometry[24]. Starting from a flat thermal distribution atomic
distribution and with zero field amplitude, we then simply monitor the coupled
atom-field evolution for a set of typical parameters. Due to the large number of
variables we have to restrict ourselves to monitor only some characteristic
observable like the evolution of the total kinetic energy, the magnitude of the
bunching or the time evolution of the center of mass of the cloud. This is shown
in Fig. 3 where we compare contour lines the kinetic energy
gain of the gas after a time of κ*t* = 60 with the
analytic threshold formula from above. Indeed selfordering and significant
acceleration of the particles is found only above the analytic threshold line.
This is also consistent with the plot of maximum particle density distribution
shown in Fig. 4 that is similarly bounded by the analytic
threshold formula (blue crosses).

#### 4.2. Quasistationary accelerated particle distribution

As we now found a condition for the breakdown of stability of a homogeneous distribution, the next big question now is what happens in this instable regime? What kind of dynamics can we expect in general and under what conditions can we expect selforganization and an ordered accelerated particle distribution?

Obviously the backscattered field amplitude gets nonzero and the center of mass of the cloud gets accelerated. As a first guess for the further evolution we will now simply iterate the above loop further and see under which conditions a self consistent accelerated atom field distribution could possibly form. As we have no reliable time scales for the atomic thermalization process available, we will simply assume that it is faster than the time scale of significant acceleration of the cloud so that we keep the average velocity constant in these iterations.

Experimentally and in numerical simulations it has been found that in particular
the regime *NU*
_{0} ≥ κ has the
biggest potential for a self ordered phase and interesting dynamics, so we will
primarily focus on this regime[24].

In general for a wide range of parameters a self consistent accelerated atom field distribution can be found at a sufficient pump strength. A plot of the density distribution below, at, and above pump threshold is shown in Fig 5. Actually the iterations show, that the onset of self ordering depends non trivially on all the parameters and clear indications can already be seen even after only about five iterations. Typical numerical results can be visualized by plotting the peak atomic density versus pump strength for 5, 10 and 100 iterations as in Fig 6. There are no significant differences in convergence between 50 and 100 iterations. This peak density is a good indicator of atomic ordering, which due to translational invariance may form at any location between the mirrors. Alternatively the self consistent backscattered intensity turns out to be a good indicator for the onset of self ordering as a function of pump strength. From this quantity we can also determine the average force on the cloud.

The self ordering threshold can be located in Fig. 6 or from the contour plots
such as Fig 1 or 2 via the transition from the flat homogeneously
distributed background to the exponential increase of local particle density.
The transition is easy to see by eye. Numerically we have to select some value
of particle density larger than the homogenous density and above the
”noise background” as an indicator of selforganization. In
this way we can generate contour lines of the maximum density of the self
consistent particle density. This allows a good comparison with the analytic
form for the pumping threshold as a function of other parameters such as
detuning. The key feature to note in Figs. 1 and 2 is that in accordance with the analytical results
derived above, the minimum threshold which occurs, when Δ* _{c}* =

*NU*

_{0}, with a symmetric increase around this minimum. In seems that for a large range of parameters the instability of the flat distribution is connected with the existence of an ordered accelerated phase which is particularly stable for Δ

*<*

_{c}*NU*

_{0}.

A decrease of cavity damping leads to a sharper resonance of backscattered light
at certain values of cavity detuning. We can estimate the value of cavity
detuning which leads to the maximum backscattering (and average force) while
holding other parameters fixed by taking the derivative of
∣*a*
_{-}∣^{2},

Recall the bunching parameter approaches unity here. This result is only useful
for small damping, κ < *NU*
_{0},
where there is a distinct, sharp maxima of
∣*a*
_{-}∣^{2} in the
numerical results at Δ* _{c}* = 0 and Δ

*= 2*

_{c}*NU*

_{0}(with very little backscattering at Δ

*=*

_{c}*NU*

_{0}). Thus as seen in Fig. 7 there are effectively these two resonant frequencies of the cavity, with a larger Q leading to more backscattered light, but over a narrower Δ

*frequency interval.*

_{c}Alternatively, for larger values of damping, κ >
*NU*
_{0}, the numerical results for backscattering
show a maximum value at Δ* _{c}* =

*NU*

_{0}which decreases slowly away from this value of detuning. The limit on the overall range of cavity detuning which still permits self ordering of the particles is set by the pump. The threshold pump value given in Eq. 23 should be exceeded to observe these backscattering dependencies on detuning.

To get some quantitative comparison with the selfconsistent atomic distributions
calculated by the above iteration procedure and direct numerical simulations, we
now also depict a contour plot the maximum of the atomic density distribution of
a cloud of *N* = 1000 atoms *NU*
_{0} =
-2κ after a given time interval Δ*t* =
60/κ. We see that clear signs of selforganization appear when we
cross the theoretically found threshold condition depicted as the line with
crosses as a function of detuning and pump strength. We also see that crossing
the threshold and ordering is accompanied by acceleration as expected.

#### 4.3. Threshold conditions and deceleration prospects for a moving thermal cloud

Having investigated the dynamics of a gas at rest in some detail, we now come
back to one of our original central goals and check the possibility of trapping
and cooling molecules from a beam by superradiant scattering. In a typical setup
one would send a molecular beam through the cavity in a small angle almost
parallel to the ring cavity axis and inject a strong counter propagating
coherent field. In such a setup the molecules are initially moving quite fast
(up to ^{∼}100 m/s) with constant spatial density. The
central question now is for which phase space densities, velocities and pump
amplitudes superradiant scattering will appear and can be used slow down and
trap the molecules.

In general using the rest frame of the beam the equations are the same as above
except for different detunings Δ_{±} = Δ* _{c}*±

*kv*. for both cavity modes. Hence one can expect a similar threshold behavior. Naturally for a large value of the ratio of

*kv*/κ the pump and the backscattered field cannot be on resonance simultaneously. This will increase the threshold or even make self ordering not be possible. In principle on has the two possibilities of resonant pumping but a non resonant backscattered field or off resonant pumping to get resonant enhanced of the back-scattered field. The threshold for self ordering is first calculated assuming the cloud velocity is constant throughout the iteration. We can compare this result to the threshold obtained using the accelerated frame where the velocity is updated throughout the iteration.

The most notable feature of the numerical results as seen in Fig. 8 is the asymmetry in the pumping threshold with detuning as compared to the case with particles at rest. Additionally, the analytic stability analysis leads to a first order stability equation for optical pumping threshold given by Eq. 22.

For *NU*
_{0} > κ there is once again
excellent agreement between the numerical and analytic results for pumping
threshold as shown in Fig. 8 with only about a two percent difference in
threshold values. Trapping molecular beams with relatively large velocities is
possible with a large pump and large cavity decay rate. In this region where
*NU*
_{0} < κ, the analytic and
numerical results for threshold are less in agreement, although the results have
the same functional form as *kv* increases.The backscattered
intensity as a function of pumping versus velocity of the moving cloud is shown
in Fig. 7.

#### 4.4. Numerical multiparticle simulations of cold beam stopping

As a final check and demonstration of the efficiency of the setup we will now
present some results on the possibility to stop a fast beam in such a setup
again using real space simulations. In this case we start with beam of N=3000
particles in a thermal distribution of temperature *kT* =
*h*̄κ but with additional average
momentum of *p _{z}* =
-2000

*h*̄

*k*per particle in the negative

*z*-direction counter propagating the pump. As a measure of the efficiency of the scheme we first plot in Fig. 9.(a) the amount of average kinetic energy extracted from the atoms by the fields in a time interval κ

*t*= 60 as a function of pump strength η and pump detuning Δ

*.*

_{c}Obviously we get efficient deceleration of the cloud when either the pump mode directly or the reflected light field is in resonance with the effective cavity eigenfrequencies[10]. When we plot the time averaged bunching parameter in Fig. 9.(b) and 10. we see that efficient energy transfer is associated with atomic selfordering as analytically predicted in Eq.22 (line with crosses). As the atomic bunching is more efficient for negative detunings we get a more efficient energy extraction in this limit.

## 5. Conclusions

We derived the analytic threshold for the laser pump power to stop and maybe cool a fast molecular beam by help of a single side pumped high Q ring cavity geometry in a CARL setup. The analytical results for kinetic energy extraction from a self consistent stationary solution of the coupled particle field equations were checked by numerical simulations. In the limit of large detuning, inversion is negligible and coherent light scattering dominates. As the atom field interaction strength per particle is small in this regime significant forces can only be generated by collective enhancement of the Raleigh scattering blue shifted with respect to the excitation light. This can be achieved in a regime where the particles arrange in a periodic density lattice and form a moving Bragg mirror.

Finding the effective optical potential in an accelerated frame and applying a linear stability analysis allows to derive the analytic threshold for the laser pump power, above which an initial homogenous particle distribution gets unstable against small density fluctuations. We see that the available phase space density at the beginning determines the required threshold laser intensity. Above this threshold a suitable choice of detunings causes the interference of the injected and collectively backscattered light fields to establish a periodic particle density acting like a moving Bragg mirror, which yields a time dependent coupling of the counter propagating fields within the resonator. By using a mean-field analysis the self consistent atom-field distribution is found by iteration which allows to quantitatively predict the maximal acceleration and localization of the cloud.

In a second step we checked these predictions by simulating the microscopic equations of motions of the particles for finite ensembles. These simulations show surprisingly good qualitative and quantitative agreement with the analytic threshold result and acceleration predictions within the regime, where the analytical model predicts selforganization. Outside this regime, where the effective potential in the accelerated frame shows no periodic local minima the simulations differ from the analytic predictions. This can be traced to cavity induced heating processes in this regime as well as remnant radiation pressure not accounted for in the analytic model. In any case the analytically found threshold formula for the selforganization process provides a solid criterion for the prospects of a corresponding experimental setup.

As a bottom line trapping and stopping of a fast molecular beam in a ring cavity seems experimentally feasible with current technology, provided one can find a strong enough pump laser at a frequency where the particles are polarizable but sufficiently transparent as shown in ref.[15]. This might require operation in a pulsed regime. The higher the initial phase space density of the particles, the faster this process will happen and the stronger are the deceleration forces and the suppression of spontaneous emission. In favorable cases this could be combined with cavity cooling in the particles rest frame as well[3].

Future work will be to include a loss level for the particles to simulate molecules as well as to refine the numerical iteration to produce real time dependent results.

## Acknowledgments

This work was supported by the Air Force Office of Scientific Research, the Institute for Information Technology Applications and the Austrian Science Fund (FWF) by grants S1512 and P17709.

## References and links

**1. **S. Chu, “Nobel Lecture: The manipulation of
neutral particles,” C. Cohen-Tannoudji, “Nobel Lecture: Manipulating atoms
with photons,” and W. D. Phillips, “Nobel Lecture: Laser cooling and
trapping of neutral atoms,” Rev. Mod.
Phys. **70**, 685–741
(1998). [CrossRef]

**2. **E.A. Cornell and C. E. Wieman, “Nobel Lecture: Bose-Einstein
condensation in a dilute gas, the first 70 years and some recent
experiments,” Rev. Mod. Phys. **74**, 875–893
(2002). [CrossRef]

**3. **P. Domokos and H. Ritsch, “Mechanical effects of light in
optical resonators,” J. Opt. Soc. Am. B **20**,1098–1130
(2003). [CrossRef]

**4. **A. Beige, P. L. Knight, and G. Vitiello, “Cooling many particles at
once,” New J. Phys. **7**, 96 (2005),
http://www.iop.org/EJ/article/1367-2630/7/1/096/njp5-1-096.html. [CrossRef]

**5. **V. Vuletić and S. Chu, “Laser cooling of atoms, ions, or
molecules by coherent scattering,” Phys.
Rev. Lett. **84**, 3787–3790
(2000). [CrossRef] [PubMed]

**6. **H. W. Chan,, A. T. Black, and V. Vuletić, “Observation of
collective-emission-induced cooling of atoms in an optical
cavity,” Phys. Rev. Lett. **90**, 063003 (2003). [CrossRef]

**7. **P. Maunz, T. Puppe, I. Schuster, N. Syassen, P. W. H. Pinkse, and G. Rempe, “Cavity cooling of a single
atom,” Nature London **428**, 50–52
(2004). [CrossRef] [PubMed]

**8. **S. Nussmann, K. Murr, M. Hijlkema, B. Weber, A. Kuhn, and G. Rempe, “Vacuum-stimulated cooling of single
atoms in three dimensions,” e-print
quant-ph/0506067.

**9. **A. T. Black, H. W. Chan, and V. Vuletić, “Observation of collective friction
forces due to spatial self-organization of atoms: from rayleigh to bragg
scattering,” Phys. Rev. Lett. **91**, 203001 (2003). [CrossRef] [PubMed]

**10. **J. Klinner, M. Lindholdt, B. Nagorny, and A. Hemmerich, “Normal mode splitting and mechanical
effects of an optical lattice in a ring cavity,”
Phys. Rev. Lett. **96**, 023002 (2006). [CrossRef] [PubMed]

**11. **B. Nagorny, T. Elsasser, H. Richter, A. Hemmerich, D. Kruse, C. Zimmermann, and P. Courteille, “Optical lattice in a high-finesse
ring resonator,” Phys. Rev. A **67**, 031401(R) (2003). [CrossRef]

**12. **Th. Elsässer, B. Nagorny, and A. Hemmerich, “Optical bistability and collective
behavior of atoms trapped in a high-Q ring
cavity,” Phys. Rev. A **69**, 033403 (2004). [CrossRef]

**13. **H.L. Bethlem, G. Berden, F.M.H. Crompvoets, R.T. Jongma, A.J.A. van Roij, and G. Meijer, “Electrostatic trapping of ammonia
molecules,” Nature London **406**491–494 (2000). [CrossRef] [PubMed]

**14. **N. Vanhaecke, W.D. Melo, B.L. Tolra, D. Comparat, and P. Pillet, “Accumulation of cold cesium
molecules via photoassociation in a mixed atomic and molecular
trap,” Phys. Rev. Lett. **89**, 063001 (2002). [CrossRef] [PubMed]

**15. **R. Fulton, A.I. Bishop, M.N. Shneider,, and P.F. Barker, “Controlling the motion of cold
molecules with deep periodic optical potentials,”
Nature Physics **2**465–468 (2006). [CrossRef]

**16. **R. Fulton, A.I. Bishop, and P.F. Barker, “Optical Stark Decelerator for
molecules,” Phys. Rev. Lett. **93**, 243004 (2004). [CrossRef]

**17. **R. Bonifacio, C. Pellegrini, and L.M. Narducci, “Collective instabilities and
high-gain regime in a free electron laser,”
Opt. Comm. **50**, 373–378
(1984). [CrossRef]

**18. **R. Bonifacio, l. De Salvo, L.M. Narducci, and E.J. D’Angelo, “Exponential gain and self-bunching
in a collective atomic recoil laser,”
Phys. Rev. A **50**, 1716–1724
(1994). [CrossRef] [PubMed]

**19. **D. Kruse, C. von Cube, C. Zimmermann, and Ph.W. Courteille, “Observation of lasing mediated by
collective atomic recoil,” Phys. Rev.
Lett. **91**, 183601 (2003). [CrossRef] [PubMed]

**20. **S. Slama, C. von Cube, B. Deh, A. Ludewig, C. Zimmermann, and Ph. W. Courteille, “Phase-sensitive detection of bragg
scattering at 1D optical lattices,” Phys.
Rev. Lett. **94**, 193901 (2005). [CrossRef] [PubMed]

**21. **C. von Cube, S. Slama, D. Kruse, C. Zimmermann, Ph. W. Courteille, G. R. M. Robb, N. Piovella, and R. Bonifacio, “Self-synchronization and
dissipation-induced threshold in Collective Atomic Recoil
Lasing,” Phys. Rev. Lett. **93**, 083601 (2004) [CrossRef] [PubMed]

**22. **A.T. Black, J.K. Thompson, and V. Vuletić, “Collective light forces on atoms in
resonators,” J. Phys. B: At. Mol. Opt.
Phys. **38**, (2005). [CrossRef]

**23. **M. Gangl and H. Ritsch, “Cold atoms in a high-Q ring
cavity,” Phys. Rev. A **61**, 043405 (2000). [CrossRef]

**24. **J. K. Asbóth, P. Domokos, H. Ritsch, and A. Vukics, “Self-organization of atoms in a
cavity field: Threshold, bistability, and scaling
laws,” Phys. Rev A **72**(5) 053417 (2005). [CrossRef]

**25. **D. Nagy, J. K. Asbóth, P. Domokos, and H. Ritsch, “Self-organization of a laser-driven
cold gas in a ring cavity,” EuroPhys
Lett. **74**(2), 254 (2006). [CrossRef]

**26. **G.R.M. Robb, N. Piovella, A. Ferraro, R. Bonifacio, Ph. W. Courteille, and C. Zimmermann, “Collective atomic recoil lasing
including friction and diffusion effects,”
Phys. Rev. A **69**, 041403(R) (2004). [CrossRef]