## Enhanced atom capturing in a high-*Q* cavity by help of several transverse modes

Optics Express, Vol. 10, Issue 21, pp. 1204-1214 (2002)

http://dx.doi.org/10.1364/OE.10.001204

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### Abstract

We predict a strong enhancement of the capture rate and the friction force for atoms crossing a driven high-*Q* cavity field if several near degenerate cavity modes are simultaneously coupled to the atom. In contrast to the case of a single *TEM*_{00} mode, circular orbits are not stable and damping of the angular and radial motion occurs. Depending on the chosen atom-field detuning the atoms phase lock the cavity modes to create a localized field minimum or maximum at their current positions. This corresponds to a local potential minimum which the atom drags along with its motion. The stimulated photon redistribution between the modes then creates a large friction force. The effect is further enhanced if the atom is directly driven by a coherent field from the side. Several atoms in the field interact via the cavity modes, which leads to a strongly correlated motion.

© 2002 Optical Society of America

## 1. Introduction

*Q*cavity, has been at the heart of theoretical and experimental quantum optics research during the last decades. Starting from demonstrating basic consequences as vacuum Rabi splitting [1

1. R. J. Thompson, G. Rempe, and H. J. Kimble, “Observation of normal-mode splitting for an atom in an optical cavity,” Phys. Rev. Lett. **68**, 1132 (1992). [CrossRef] [PubMed]

2. M. Hennrich, T. Legero, A. Kuhn, and G. Rempe, “Vacuum-Stimulated Raman Scattering Based on Adiabatic Passage in a High-Finesse Optical Cavity,” Phys. Rev. Lett. **85**, 4872 (2000). [CrossRef] [PubMed]

3. P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, “Cavity-induced Atom Cooling in the Strong Coupling Regime,” Phys. Rev. Lett. **79**, 4974–4977 (1997). [CrossRef]

4. V. Vuletić, H. W. Chan, and A. T. Black, “Three-dimensional cavity Doppler cooling and cavity sideband cooling by coherent scattering,” Phys. Rev. A **64**, 033405 (2001). [CrossRef]

5. P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, “Trapping an atom with single photons,” Nature (London) **404**, 365 (2000). [CrossRef]

6. C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, “The Atom-Cavity Microscope: Single-Atoms Bound in Orbit by Single Photons,” Science **287**, 1447 (2000). [CrossRef] [PubMed]

*w*

_{0}, transversally. Hence there is much less cavity-induced friction force and confinement in the radial direction, and the atoms cannot be trapped radially or escape after a short time [7

7. A. C. Doherty, T. W. Lynn, C. J. Hood, and H. J. Kimble, “Trapping of single atoms with single photons in cavity QED,” Phys. Rev. A **63**, 013401 (2000). [CrossRef]

8. T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, “Feedback on the Motion of a Single Atom in an Optical Cavity,” Phys. Rev. Lett. **88**, 163002 (2002). [CrossRef] [PubMed]

9. S. J. van Enk, J. McKeever, H. J. Kimble, and J. Ye, “Cooling of a single atom in an optical trap inside a resonator,” Phys. Rev. A **64**, 013407 (2001). [CrossRef]

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

11. P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, “Optical Kaleidoscope Using a Single Atom,” Phys. Rev. Lett. **88**, 043601 (2002). [CrossRef] [PubMed]

## 2. Semiclassical description of an atom coupled to a multimode field

*ω*

_{a}strongly coupled to

*M*modes with nearly degenerate frequencies

*ω*

_{n}≈

*ω*

_{c}of a high-finesse cavity (e.g. in quasi-confocal geometry). The atom is transversally injected into the cavity field. For simplicity, we assume longitudinally very cold atoms (in practice this is automatically guaranteed by spatial filtering of the atoms by a cavity entrance slit), so that we are able to restrict our study to the radial dimensions, i.e. perpendicular to the cavity axis, only. In general, we assume two coherent laser fields pumping the atom-field system, which are set to have the same frequency

*ω*

_{p}. One field part is injected into the cavity through one of the mirrors, yielding an effective pump strengths

*η*

_{n}for the

*n*th mode. The second part is directly driving the atom in the form of a broad standing wave, transverse to the cavity axis. Hence, we can safely reduce its spatial dependence to simple plain standing wave with effective pump strength

*η*

_{t}(x) =

*ζh*(x) =

*ζ*cos(

*k*

_{p}

*y*). A schematic sketch of the system is depicted in Fig. 1. Both the atom and the cavity field are coupled to external reservoirs, which gives rise to spontaneous emission (γ) and cavity decay (

*κ*).

*ω*

_{p}, the quantum master equation for this system is given by

_{a}=

*ω*

_{p}-

*ω*

_{a}and Δ

_{c}=

*ω*

_{p}-

*ω*

_{c}are the atomic and cavity-field detunings respectively. Further

*σ*

^{+}and

*σ*

^{-}denote the atomic raising and lowering operators, respectively,

*a*

_{n}the field creation and annihilation operators for the

*n*th mode. The coupling between the atom and the

*n*th mode is given by

*d*is the atomic dipole moment and

*V*

_{n}the effective mode volume. The second term in Eq. (2b) contains the momentum recoil due to spontaneous emission.

12. P. Domokos, P. Horak, and H. Ritsch, “Semiclassical theory of cavity-assisted atom cooling,” J. Phys. B: At. Mol. Opt. Phys. **34**, 187–198 (2001). [CrossRef]

^{±}evolve on a fast timescale due to a large detuning Δ

_{a}or a large damping rate γ.

*W*(x,

**p**,

*α*

_{1}...

*α*

_{M},

12. P. Domokos, P. Horak, and H. Ritsch, “Semiclassical theory of cavity-assisted atom cooling,” J. Phys. B: At. Mol. Opt. Phys. **34**, 187–198 (2001). [CrossRef]

*U*

_{0}=

*g*

^{2}Δ

_{a}/(

^{2}) is the light shift per photon and Γ

_{0}=

*g*

^{2}γ/(

^{2}) the photon scattering rate. As a shortcut in Eq.(4), we have introduced the field amplitude at the position of the atom,

*η*

_{eff}=

*U*

_{0}

*ζ*/

*g*, which describes photon scattering into the cavity via the atom. Further, there appears another radiative force characterized by the scattering rate γ

_{eff}= Γ

_{0}

*ζ/g*. The last term is simply the free-space dipole force acting on an atom in a standing wave. It is clear that these last three terms vanish if only the cavity is driven directly. Similarly, Eq. (4c) points up the effect of the laser pumping the atom on the cavity field. Two additional terms appear describing the coherent dynamics (

*η*

_{eff}) and the decay (γ

_{eff}) of the cavity field. Since these effects are only provided by the presence of the atom, they strongly depend on the atomic position.

13. P. Domokos, T. Salzburger, and H. Ritsch “Dissipative motion of an atom with transverse coherent driving in a cavity with many degenerate modes,” Phys. Rev. A **66**, 043406 (2002). [CrossRef]

*E*= (Δ

**p**)

^{2}/2

*m*≈

*D*

_{rec}τ/2

*m*, with the depth of the potential

*ħU*

_{0}|ε|

^{2}. Here we use only

*D*

_{rec}, the momentum diffusion coefficient due to recoil heating, because the other contribution arising from the fluctuations of the dipole force is of the same order of magnitude in the longitudinal and much less in the transverse directions. This consideration leads to the condition

*ω*

_{rec}is the recoil frequency of the atom. With sufficiently large detunings (Δ

_{A}/γ > 50), τ can attain values of the order of milliseconds for a Rubidium atom, which is in agreement with the numerical simulations of the full dynamics [12

12. P. Domokos, P. Horak, and H. Ritsch, “Semiclassical theory of cavity-assisted atom cooling,” J. Phys. B: At. Mol. Opt. Phys. **34**, 187–198 (2001). [CrossRef]

*x*and

*y*. In order to avoid confusion, we will hold this notation in the following.

*n*and

*m*indicate the Hermite polynomials and

*w*

_{0}is the spot size. As mentioned above, we consider only very cold atoms in the

*z*-direction as they are spatially filtered from the source to the cavity entrance. Hence, we can set cos(

*kz*) ≈ 1 in Eq. (9) and consider the transverse motion only. Of course, on a longer time scale when momentum diffusion heats up the

*z*-motion, this would not be valid and some spatial averaging on the

*z*-motion should be included. However, we expect our results not to be qualitatively changed by this assumption.

## 3. Steady-state intensity distributions

*α*

_{n}can be easily calculated by setting

*= 0 in Eq. (4c) and solving the linear system*α ˙

_{n}*f*

_{n}(x) large. The relative phases are locked and, accordingly, the cavity field exhibits a localized peak at the position of the atom. This behavior is illustrated in Fig. 2, where we took the first four sets of modes into account, i.e.

*n*+

*m*≤ 3 and

*M*= 10. If Δ

_{c}is of the order of

*U*

_{0}the atom shifts the cavity field into resonance and the total intensity becomes maximum while it decreases for a large detuning Δ

_{a}.

*TEM*

_{00-}mode is driven through the mirrors. Interestingly, a similar effect occurs as above. The atom raises the local field drastically for Δ

_{c}≃

*U*

_{0}and creates a maximum close to its position. This can be seen in Fig. 3a, where we considered the first ten modes and the driving laser only coupled to the ground mode (i.e.

*η*

_{ij}= 0, ∀

*i*,

*j*≠ 0). However, a big drop appears at the atomic position if |Δ

_{a}| ≃ |Δ

_{c}|. The atom pushes the field maximum away and the intensity decreases. Fig. 3b shows the sharp intensity decline at the atomic position. In order to get similar values for the saturation parameter and thus for the photon number as in the red detuned case, the pumping strength has to be chosen significantly higher here. The other parameters are left the same. In the case if |Δ

_{a}| ≫ |Δ

_{c}|, the effect of the atom becomes small. The rise (for equal signs of the detunings) and the drop (opposite signs) of the cavity field is does not play any important role anymore.

*f*

_{n}(x) =

*f*

_{n}(x

_{1}) +

*f*

_{n}(x

_{2}). This is completely different if several modes are involved as the whole field intensity distribution is changed in this case. We demonstrate this at an example in Fig. 4, where we plot the steady-state field intensity distributions for two atoms simultaneously in the field and parameters analogous to Fig. 2. Clearly, two peaks are visible at the positions of the two atoms. From Fig. 4 and the fact that the field intensity is directly proportional to the optical potential for the atoms, it is also clear, that one gets a strongly cavity-enhanced atom-atom interaction which was observed in Ref. [14

14. P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe, “Observation of Cavity-Mediated Long-Range Light Forces between Strongly Coupled Atoms,” Phys. Rev. Lett. **84**, 4068–4071 (2000). [CrossRef] [PubMed]

## 4. Dynamic capturing and trapping of an atom

*TEM*mode function

*f*

_{00}(

*x, y*). This is the standard setup of most previous theoretical and experimental treatments and we will use it as a reference here. The atom is put at a random position inside the cavity with a small initial velocity and we assume that the interaction starts at a given initial time

*t*= 0. In Fig. 5a we have plotted the trajectory for a Rubidium atom with initial velocity

*v*= 12 cm/s. The blue curve shows how the atom moves for the first two milliseconds (0 <

*t*< 2 ms) showing elliptic orbits around the cavity axis avoiding the region of maximal coupling due to the angular momentum barrier. As the field is angularly symmetric, the dynamic cooling will only influence the radial motion which is damped in this case. This is shown by the red circle, which represents the atomic trajectory for some time after 5 ms. In this steady-state,

*r*is fixed and hence the cavity field remains constant and constitutes a conservative potential. Orbiting in such a large radius state, diffusion is very likely to kick the atom out of the potential.

*n*+

*m*≤ 2. We even decreased the pump strength by 30 percent in this case to keep the saturation of the atom low. Again, the blue curve represents the atomic motion for 0 <

*t*< 2 ms, which is more extended and irregular now. However, the atom is slowed down and captured to a position very close to the cavity axis with velocity

*v*= 0.3 cm/s after about 5 ms. Its trajectory reduces to the red point shown in the center.

*y*-direction the trajectories showa qualitatively similar behavior as above. Figs. 6a and b correspond to the case of only the ground mode and the lowest six modes considered, respectively. As mentioned above, we increased the pumping strength

*ζ*while the other parameters are the ones as in Fig. 2. Clearly, the

*x*as well as the

*y*-motion are much more strongly damped in the multi-mode case. This plot also demonstrates that cavity cooling and trapping can be applied in an efficient way for atoms trapped by an external potential. Here it is the dipole potential created by the pump laser, which, at least conceptually, further simplifies the setup.

*v*= 12 cm/s (blue curve) where a second atom is already trapped and is at rest (green curve). We use the case of atom driving where both atoms are trapped and confined in the potential wells of the transverse pump laser from the beginning. The interesting part of the motion, hence, will take place along the

*x*-direction, which we will consider in the following. The incoming atom shifts the field maximum in the resonator created by the already present atom towards its position. This causes a field gradient at the position of the trapped atom and an effective attractive force between the two particles, making them approach each other. At the point where the atoms get closest, they have the maximum relative velocity and also the local field reaches its maximum value. When the atoms move apart again, the field maximum is now behind both of the moving atoms, leading to a deceleration. As the field is now stronger than in the phase where they approach each other, the relative attractive force is enhanced when they separate. Hence, in addition to the cavity induced damping, motional energy is transferred from the fast to the slow atom. This strongly enhances the capturing probability for the second atom. After a certain time the kinetic energy is evenly distributed among the atoms and the roles of the two atoms interchange. We get a periodic energy exchange between the two atoms. This can be seen in Fig. 7b where the green curve corresponds to the atom initial at rest. Note that the oscillations have nowthe same amplitudes and are damped simultaneously.

## 5. Conclusions

*Q*cavity. This increases the probability of the atoms to be captured. Apart from the field intensity, also the relative mode phases and hence the field shape is now a dynamic quantity. Monitoring the fields would allow highly precise tracking of the motion of even several particles simultaneously. In addition by properly choosing the parameters, one has a new handle of the effective atom-atom interaction. Besides a buildup of correlations as it happens for two-level atoms, this could be used for controlled entanglement in the case of atoms with more complex internal structure.

## References and links

1. | R. J. Thompson, G. Rempe, and H. J. Kimble, “Observation of normal-mode splitting for an atom in an optical cavity,” Phys. Rev. Lett. |

2. | M. Hennrich, T. Legero, A. Kuhn, and G. Rempe, “Vacuum-Stimulated Raman Scattering Based on Adiabatic Passage in a High-Finesse Optical Cavity,” Phys. Rev. Lett. |

3. | P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, “Cavity-induced Atom Cooling in the Strong Coupling Regime,” Phys. Rev. Lett. |

4. | V. Vuletić, H. W. Chan, and A. T. Black, “Three-dimensional cavity Doppler cooling and cavity sideband cooling by coherent scattering,” Phys. Rev. A |

5. | P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, “Trapping an atom with single photons,” Nature (London) |

6. | C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, “The Atom-Cavity Microscope: Single-Atoms Bound in Orbit by Single Photons,” Science |

7. | A. C. Doherty, T. W. Lynn, C. J. Hood, and H. J. Kimble, “Trapping of single atoms with single photons in cavity QED,” Phys. Rev. A |

8. | T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, “Feedback on the Motion of a Single Atom in an Optical Cavity,” Phys. Rev. Lett. |

9. | S. J. van Enk, J. McKeever, H. J. Kimble, and J. Ye, “Cooling of a single atom in an optical trap inside a resonator,” Phys. Rev. A |

10. | M. Gangl and H. Ritsch, “Cold atoms in a high- |

11. | P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, “Optical Kaleidoscope Using a Single Atom,” Phys. Rev. Lett. |

12. | P. Domokos, P. Horak, and H. Ritsch, “Semiclassical theory of cavity-assisted atom cooling,” J. Phys. B: At. Mol. Opt. Phys. |

13. | P. Domokos, T. Salzburger, and H. Ritsch “Dissipative motion of an atom with transverse coherent driving in a cavity with many degenerate modes,” Phys. Rev. A |

14. | P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe, “Observation of Cavity-Mediated Long-Range Light Forces between Strongly Coupled Atoms,” Phys. Rev. Lett. |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(140.3320) Lasers and laser optics : Laser cooling

(270.5580) Quantum optics : Quantum electrodynamics

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 27, 2002

Revised Manuscript: October 14, 2002

Published: October 21, 2002

**Citation**

Thomas Salzburger, P. Domokos, and H. Ritsch, "Enhanced atom capturing in a high-Q cavity by help of several transverse modes," Opt. Express **10**, 1204-1214 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-21-1204

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### References

- R. J. Thompson, G. Rempe, and H. J. Kimble, �??Observation of normal-mode splitting for an atom in an optical cavity,�?? Phys. Rev. Lett. 68, 1132 (1992). [CrossRef] [PubMed]
- M. Hennrich, T. Legero, A. Kuhn and G. Rempe, �??Vacuum-Stimulated Raman Scattering Based on Adiabatic Passage in a High-Finesse Optical Cavity,�?? Phys. Rev. Lett. 85, 4872 (2000). [CrossRef] [PubMed]
- P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, �??Cavity-induced Atom Cooling in the Strong Coupling Regime,�?? Phys. Rev. Lett. 79, 4974�??4977 (1997). [CrossRef]
- V. Vuleti´c, H. W. Chan, and A. T. Black, �??Three-dimensional cavity Doppler cooling and cavity sideband cooling by coherent scattering,�?? Phys. Rev. A 64, 033405 (2001). [CrossRef]
- P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, �??Trapping an atom with single photons,�?? Nature (London) 404, 365 (2000). [CrossRef]
- C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, �??The Atom-Cavity Microscope: Single-Atoms Bound in Orbit by Single Photons,�?? Science 287, 1447 (2000). [CrossRef] [PubMed]
- A. C. Doherty, T. W. Lynn, C. J. Hood, and H. J. Kimble, �??Trapping of single atoms with single photons in cavity QED,�?? Phys. Rev. A 63, 013401 (2000). [CrossRef]
- T. Fischer, P. Maunz, P. W. H. Pinkse, T . Puppe, and G. Rempe, �??Feedback on the Motion of a Single Atom in an Optical Cavity,�?? Phys. Rev. Lett. 88, 163002 (2002). [CrossRef] [PubMed]
- S. J. van Enk, J. McKeever, H. J. Kimble, and J. Ye, �??Cooling of a single atom in an optical trap inside a resonator,�?? Phys. Rev. A 64, 013407 (2001). [CrossRef]
- M. Gangl and H. Ritsch, �??Cold atoms in a high-Q ring cavity,�?? Phys. Rev. A 61, 043405 (2000). [CrossRef]
- P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, �??Optical Kaleidoscope Using a Single Atom,�?? Phys. Rev. Lett. 88, 043601 (2002). [CrossRef] [PubMed]
- P. Domokos, P. Horak, and H. Ritsch, �??Semiclassical theory of cavity-assisted atom cooling,�?? J. Phys. B: At. Mol. Opt. Phys. 34, 187�??198 (2001). [CrossRef]
- P. Domokos, T. Salzburger, and H. Ritsch �??Dissipative motion of an atom with transverse coherent driving in a cavity with many degenerate modes,�?? Phys. Rev. A 66, 043406 (2002). [CrossRef]
- P. M¨unstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe, �??Observation of Cavity-Mediated Long-Range Light Forces between Strongly Coupled Atoms,�?? Phys. Rev. Lett. 84, 4068�??4071 (2000). [CrossRef] [PubMed]

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