## Self ordering threshold and superradiant backscattering to slow a fast gas beam in a ring cavity with counter propagating pump

Optics Express, Vol. 15, Issue 10, pp. 6019-6035 (2007)

http://dx.doi.org/10.1364/OE.15.006019

Acrobat PDF (586 KB)

### 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

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]

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]

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]

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]

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]

## 2. Model

*N*of linearly polarizable point particles interacting with two degenerate counter- propagating linearly polarized plane wave ring cavity modes.

*a*

_{±}(

*t*) gives the field amplitude.

*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

*, and atom-field coupling g the effective mode frequency shift reads:*

_{a}^{∼}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.

*, the molecular beam experiences two different Doppler shifted cavity fields with detunings Δ*

_{c}*= Δ*

_{±}*±*

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

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

*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*

_{±}≈ 0, while an optimally ordered phase gives ∣

*R*

_{±}∣ → 1.

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

*U*

_{0}< 0, i.e. Δ

*< 0, this attracts atoms to maxima of the cavity field. Using the definition of the field in terms of the amplitudes*

_{a}*a*

_{±},

*E*

^{+}(

*x*) = ε(

*a*+

*e*

*+*

^{ikx}*a*

_{-}

*e*

^{-ikx}) we get:

*a*

_{+}

*a**

_{-}. Note that for calculating the force the constant terms can be omitted from the potential and the average force then simply reads:

## 3. Coupled particle field dynamics

### 3.1. Quasistationary field

*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:

*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:

*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*we define the relative shift

^{iπ}*δ*= φ/(2

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

*x*

_{0}by

*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):

*x*

_{0}but are shifted by

*δ*. Hence we obviously cannot find a self consistent time independent configuration of particles and field as for the standing wave case [24

_{x}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]

*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}〉

*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

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]

### 3.2. Quasistationary accelerated atomic distribution

*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.

*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

*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

*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.

*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.

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]

*Z*= ∫ exp[-

*V*(

*x*)/

*k*]

_{B}T*dx*.

*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

*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.

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]

*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

*r*(

*x*) could be in general any periodic function

*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.

*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.

*A*

_{2}. Thus stability requires:

## 4. Results and discussion

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

*v*= 0, i.e Δ

_{+}= Δ

_{-}= Δ

_{c}. In this case the threshold condition reads:

*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}.

*=*

_{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.

*=*

_{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.

*N*particles and the field modes. For this we use a semiclassical dynamical model as developed in ref. [3

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

*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

*NU*

_{0}≥ κ has the biggest potential for a self ordered phase and interesting dynamics, so we will primarily focus on this regime[24

**72**(5) 053417 (2005). [CrossRef]

*=*

_{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*

_{-}∣

^{2},

*NU*

_{0}, where there is a distinct, sharp maxima of ∣

*a*

_{-}∣

^{2}in the numerical results at Δ

*= 0 and Δ*

_{c}*= 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}*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.

*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

^{∼}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.

_{±}= 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.

*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

*kT*=

*h*̄κ but with additional average momentum of

*p*= -2000

_{z}*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}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]

## 5. Conclusions

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]

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

## Acknowledgments

## 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. |

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. |

3. | P. Domokos and H. Ritsch, “Mechanical effects of light in
optical resonators,” J. Opt. Soc. Am. B |

4. | A. Beige, P. L. Knight, and G. Vitiello, “Cooling many particles at
once,” New J. Phys. |

5. | V. Vuletić and S. Chu, “Laser cooling of atoms, ions, or
molecules by coherent scattering,” Phys.
Rev. Lett. |

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. |

7. | P. Maunz, T. Puppe, I. Schuster, N. Syassen, P. W. H. Pinkse, and G. Rempe, “Cavity cooling of a single
atom,” Nature London |

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. |

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. |

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 |

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 |

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 |

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. |

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 |

16. | R. Fulton, A.I. Bishop, and P.F. Barker, “Optical Stark Decelerator for
molecules,” Phys. Rev. Lett. |

17. | R. Bonifacio, C. Pellegrini, and L.M. Narducci, “Collective instabilities and
high-gain regime in a free electron laser,”
Opt. Comm. |

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 |

19. | D. Kruse, C. von Cube, C. Zimmermann, and Ph.W. Courteille, “Observation of lasing mediated by
collective atomic recoil,” Phys. Rev.
Lett. |

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. |

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. |

22. | A.T. Black, J.K. Thompson, and V. Vuletić, “Collective light forces on atoms in
resonators,” J. Phys. B: At. Mol. Opt.
Phys. |

23. | M. Gangl and H. Ritsch, “Cold atoms in a high-Q ring
cavity,” Phys. Rev. A |

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 |

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. |

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 |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(270.6630) Quantum optics : Superradiance, superfluorescence

**ToC Category:**

Trapping

**History**

Original Manuscript: October 23, 2006

Revised Manuscript: April 22, 2007

Manuscript Accepted: April 24, 2007

Published: May 2, 2007

**Citation**

C. Maes, J. K. Asbóth, and H. Ritsch, "Self ordering threshold and superradiant backscattering to slow a fast gas beam in a ring cavity with counter propagating pump," Opt. Express **15**, 6019-6035 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6019

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### References

- 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]
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