## Localized decay of excited atoms in cavities

Optics Express, Vol. 1, Issue 6, pp. 134-140 (1997)

http://dx.doi.org/10.1364/OE.1.000134

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### Abstract

Spontaneous decay of excited cold atoms into cavity can drastically affect their translational dynamics, namely, atomic reflection, transmission or localization in the cavity.

© Optical Society of America

## 1. Transmission of emitting tunneling atoms in cavities

^{1}. Since tunneling is a distinct manifestation of wavelike properties, it is important to raise the basic questions: can spontaneous decay of internal excitations in tunneling atoms be viewed as a decoherence process that is analogous to its counterpart in diffracted atoms? and if so, how would such decoherence manifest itself?

^{5,6}. (ii) Nearly all of the cavity-enhanced spontaneous emission is funneled into the continuum of nearly resonant modes with wave-vectors

**q**≈ (

*ω*/

*c*)

*ẑ*, which are aligned with the cavity axis

*z*, perpendicular to the atomic incidence axis

*x*. This allows us to use the dipole approximation, since

**q∙x**≈ 0, and neglect off-axis photon recoil effects on the atomic wavepacket. Hence, the RWA interaction Hamiltonian of the atom with the cavity-mode continuum becomes effectively one-dimensional,

*H*

_{int}= -ζ(

*x*)∫

*dωρ*(

*ω*)[

*g*

_{ω}

*a*

_{ω}|

*e*〉〉

*g*| +

*h.c*.]. Here ζ(

*x*) = 1 for 0 ≤

*x*≤

*L*and 0 elsewhere, i.e., the interaction is confined to the cavity, whose

*x*-axis extent coincides with that of the barrier;

*ρ*(

*ω*) is a Lorentzian mode-density distribution associated with the cavity-mode linewidth

*η*

^{7};

*g*

_{ω}is the coupling of the atom to the cavity mode at

*ω*and

*a*

_{ω}is the corresponding annihilation operator. The transition frequency

*ω*

_{eg}is shifted (renormalized) by the difference between the AC Stark shifts of |

*e*〉 and |

*g*〉,

^{5,6}, resulting in exponential decay of the excited state, with the Feynman path-integral method, which yields a coherent sum over the atomic classical trajectories contributing to tunneling

^{8}.

*σ*(

*E*

_{k},

*V*) is the transmission amplitude for a structureless particle of kinetic energy

*E*

_{k}through a square potential barrier of height

*V*and length

*L*,

*k*= √2

*mE*

_{k}/

*ħ*and

*p*= √2

*m*(

*E*

_{k}-

*V*)/

*ħ*being the corresponding wavevectors outside and inside the barrier, respectively. The effect of spontaneous emission is to shift the effective potential

*V*by -

*iħ*Γ.

*γ*in both the tunneling (below-barrier) and allowed (above-barrier) regimes of

*E*

_{k}. The corresponding probability

*P*

_{ω}associated with photon emission at

*ω*

*ω*) =

*ρ*(

*ω*)|

*g*

_{ω}|

^{2}|/(Δ

^{2}+

*γ*

^{2}) and

*σ*

_{ω}(

*E*

_{k},

*V*) is a complicated function of

*E*

_{k},

*V*and

*ω*. The most salient effect of spontaneous emission is seen to be (Fig. 1a) the huge enhancement of

*γ*for atoms initially in the deep tunneling regime

*p*

_{L}= √2

*m*(

*V*-

*E*

_{k})

*L*/

*ħ*> 1.

*η*and

*E*

_{k}satisfy the following inequalities

*E*

_{k}and becomes Lorentzian in this range, F(

*ω*) ≈ ℒ

_{γ}(Δ), since the spectral variation of

*ρ*(

*ω*) and |

*g*

_{ω}|

^{2}is slow,

*ρ*(

*ω*)|

*g*

_{ω}|

^{2}≈ 2

*πγ*, in accordance with the WW approximation. The equation for

*σ*

_{ω}can now be simplified to

*E*

_{k}-

*ħ*Δ,

*which can be above the barrier*if Δ < 0. By contrast, the second term in (5) corresponds to atoms that have decayed shortly before exiting the barrier after having effectively been transmitted as excited atoms with the

*initial*kinetic energy

*E*

_{k}, whence this term is exponentially small in the tunneling regime. The use of Eq. (5) in Eq. (3) therefore leads to the enhancement of

*P*

_{ω}(Fig. 1a) and

*ħω*

_{eg}. In the deep tunneling regime, assuming that

*γ*≪ (

*V*-

*E*

_{k}), Eqs. (3)-(5) allow us to roughly estimate that the atoms have probability of order

*E*

_{k}-

*V*< 0 (Fig. 1).

*ħ*Δ ≪

*E*

_{k}, we can obtain a simplified expression for the total transmission probability

*(*σ ^

*t*,

*V*), the Fourier transform of

*σ*(

*E*,

*V*), is the impulse response (to a temporal

*δ*-function) for transmission of a structureless particle. We thus obtain the following important result: the total transmission probability

*γ*[Eq. (4)], with the transmission probability of a partially incoherent wavepacket of a structureless particle with coherence time

*γ*

^{-1}(See Ref. 9).

*E*

_{k}-

*ħ*Δ above the barrier, whereas their counterparts below the barrier only contribute an exponentially small tail to this distribution. (b) The fact that fast atoms emerging from the barrier are almost always unexcited means that the barrier acts as a “filter” that transmits almost only atoms that have already decayed.

*γ*→ 0, which is realizable by detuning the cavity off resonance with

*ω*

_{eg}, the excited atomic wavepacket with

*E*

_{k}<

*V*exhibits tunneling, which is a result of interference between many classical trajectories, and is characterized by exponentially low transmission

*γ*is appreciable, the wavepacket is dominated by the portion that has decohered by decay into the field-mode continuum and has thereby lost its tunneling properties: its energy spread becomes classical (statistical), giving rise to a Lorentzian tail into the above-barrier energy range, thereby allowing for enhancement of the transmission [Eqs. (3),(7)]. The effects of this decoherence on barrier traversal times will be discussed elsewhere.

*e*〉 → |

*g*〉 transition should preferably be long, above 10

^{-6}sec. A confocal cavity whose finesse is ~ 10

^{5}and subtends a solid angle of ~ 0.1 steradians can enhance spontaneous emission rate

*γ*by a factor of ~ 30. The cavity linewidth

*η*should be much larger than

*γ*, i.e., preferably above 10MHz. Correspondingly, the potential energy

*V*and the kinetic energy

*E*

_{k}must be above 0.1GHz, which requires the laser Rabi frequency Ω

_{e(g)}and detuning

*δ*

_{e,(g)}to be well within the GHz range. This implies that the transition frequency

*ω*

_{eg}can lie anywhere between the GHz and the optical ranges.

## 2. Atomic reflection and localization at cavity interfaces

*x*< 0) to a region where spontaneous emission is strongly enhanced (

*x*> 0), due to the high density of the electromagnetic field modes. The wavefunction of the total system (atom plus field) can be written in the following general form in the rotating wave approximation

*e*, {0}) denotes the atom in the excited state with no photons in the field, whereas |

*g*, {

**q**}) corresponds to the ground state of the atom with a photon emitted at a mode

**q**, and

_{e}(

**q**) are the corresponding amplitudes. One obtains coupled Schrödinger equations for the envelopes of these states given an atom with initial energy

*E*and transition frequency

*ω*

_{0},

*ψ*

_{e}(

**r**) and

*ψ*

_{q}(

**r**) by assuming

ψ ¯

_{e,q}(

**r**,t) =

*ψ*

_{e,q}(

**r**)

*e*

^{-i(E+ħω0/2)t}Far from the interaction region the solution describes propagation of the atomic wavepacket. The total energy of the incident excited atom

*E*+

*ħ*

*ω*

_{0}is then equal to the kinetic energy of the ground state atom plus the emitted photon energy

*ħω*

_{q}.

*ψ*

_{e}and

*ψ*

_{q}yield a complicated integro-differential wave equation for

*ψ*

_{e}(

**r**

_{e}), with Γ(

**r**,

**ŕ**) acting as a non-local complex potential whose shape and strength are determined by the confined mode eigenfunctions ε

_{q}(

**r**). If the linewidth of the spatially confined modes

*ħη*

_{c}is much larger than the atomic energy

*E*, the recoil energy

*E*

_{rec}≡

*ħ*

^{2}

*mc*

^{2}and the spontaneous linewidth in the confined reservoir,

*ħγ*

_{c}, then the correlation length of the interaction of the emitted photon with the atom is much shorter than the spontaneous decay length and the deBroglie wavelength λ

_{DB}. Such an atom effectively moves in a

*local*complex potential

*μ*is the atomic dipole matrix element,

*ε*

_{q}(

**r**) are the field mode amplitudes and Δ

_{c}is the detuning of the atomic transition frequency

*ω*

_{0}from the center of the spectral line of the reservoir.

*x*and avoid diffraction effects caused by the local potential in the directions perpendicular to

*x*, we consider a multimode confocal cavity where the many degenerate modes contributing to Γ(

**r**render it approximately uniform in the directions perpendicular to

*x*. We assume that the transition frequency

*ω*

_{0}is resonant with the Lorentzian center of the degenerate modes. Then the real part of Γ(

*x*) is much less than the imaginary part

*γ*

_{c}(

*x*) = Im{Γ(

*x*)}. We then obtain

*γ*

_{c}(

*x*) =

*γ*

_{c}Θ(

*x*), where Θ(

*x*) is the Heaviside step function, the probability to detect an excited atom decreases as

*eik*

_{γ}

*x*, where

*k*

_{γ}= √2

*m*(

*E*+

*iħγ*

_{c})/

*ħ*, so that only the fraction |

*r*|

^{2}of excited atoms remains at large negative

*x*(to the left of the interface). This reflection increases with the spontaneous emission rate

*γ*

_{c}. The atomic interaction with the confined vacuum reservoir for

*ħ*

_{γ}>

*E*is thus analogous to the

*skin effect*of light reflection from metals. If the energy of the incident atom is comparable to

*E*

_{rec}, the width Δ

*x*of the interface should satisfy Δ

*x*≈ λ

_{BD}(

*E*) ~ λ

_{opt}. A realistic description of the atomic entry into a confocal cavity shows a much lower reflection probability, even for subrecoil energies. However, when the real part of Γ(

*x*) contributes too, for

*ω*

_{0}well off the center of the Lorentzian spectrum (large Δ

_{c}), the cavity can be strongly reflective. This spectral dependence of the reflectivity on the detuning is characteristic of the atomic skin effect.

**q**-mode amplitude in Eq. (8) can be estimated for a strong decay

*ħγ*

_{c}≪

*E*and incidence energy well above the recoil limit. Then

*ψ*

_{q}∞

*e*

^{±ikqx}, where

*ħk*

_{q}= √2

*m*(

*E*-

*ħ*Δ

_{q}) and Δ

_{q}=

*ω*

_{q}-

*ω*

_{q}. Whenever

*E*>

*ħ*Δ

_{q},

*k*

_{q}becomes imaginary and

*ψ*

_{q}(

**r**) is exponentially localized at the interface between free space and the confined-field region. A solution with imaginary

*disappears*after the incident atomic wavepacket decays or leaves the interface, and is accompanied by a

*transient bound photon*, which eventually disappears with it, after the time ~

*ħ*/Δ

*E*, the inverse of the energy bandwidth Δ

*E*of the incident atom. If such a photon is detected, then a localized atomic state is formed. The subsequent evolution of the atomic wavepacket is governed by the free-space Schrodinger equations with the localized atomic distribution serving as the initial condition.

12. Excited state and total amplitude of ground state http://www.weizmann.ac.il/ cfyoni/movie.mpg

*γ*(

*x*) and reproduces the qualitative features of the atomic skin effect and localization at the interface.

10. B. G. Englert, J. Schwinger, A. O. Barut, and M. O. Scully, “Reflecting slow atoms from a micromaser field,” Europhys. Lett. **14**, 25 (1991) [CrossRef]

11. M.O. Scully, G.M. Meyer, and H. Walther, “Induced emission due to the quantized motion of ultracold atoms passing through a micromaser cavity,” Phys. Rev. Lett. **76**, 4144 (1996) [CrossRef] [PubMed]

## References and links

1. | Y. Japha and G. Kurizki, “Spontaneous emission from tunneling two-level atoms,” Phys. Rev. Lett. |

2. | D. Sokolovski and J. N. L. Connor, “Quantum interference and determination of the traversal time,” Phys. Rev. A |

3. | T. Pfau, S. Spälter, Ch. Kurtsiefer, C. R. Ekstrom, and J. Mlynek, “Loss of spatial coherence by a single spontaneous emission,” Phys. Rev. Lett. |

4. | A. Stern, Y. Aharonov, and Y. Imry, “Phase uncertainty and loss of interference: a general picture,” Phys. Rev. A |

5. | C. Cohen-Tannoudji et. al., |

6. | G. S. Agarwal, |

7. | D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. |

8. | R. P. Feynman and A. R. Hibbs, |

9. | Y. Japha, V. M. Akulin, and G. Kurizki, “Localized decoherence of two-level wavepackets: Atomic binding and skin effects,” Phys. Rev. Lett. (submitted) |

10. | B. G. Englert, J. Schwinger, A. O. Barut, and M. O. Scully, “Reflecting slow atoms from a micromaser field,” Europhys. Lett. |

11. | M.O. Scully, G.M. Meyer, and H. Walther, “Induced emission due to the quantized motion of ultracold atoms passing through a micromaser cavity,” Phys. Rev. Lett. |

12. | Excited state and total amplitude of ground state http://www.weizmann.ac.il/ cfyoni/movie.mpg |

13. | Ground state entangled with resonant emission http://www.weizmann.ac.il/ cfyoni/movie1.mpg |

14. | Ground state entangled with positive detuning http://www.weizmann.ac.il/ cfyoni/movie2.mpg |

15. | Ground state entangled with negative detuning http://www.weizmann.ac.il/ cfyoni/movie3.mpg |

16. | Ground state entangled with forbidden emission http://www.weizmann.ac.il/ cfyoni/movie4.mpg |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Focus Issue: Local field effects

**History**

Original Manuscript: August 21, 1997

Published: September 15, 1997

**Citation**

Yonathan Japha, Gershon Kurizki, and V. Akulin, "Localized decay of excited atoms in
cavities," Opt. Express **1**, 134-140 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-6-134

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### References

- Y. Japha and G. Kurizki, "Spontaneous emission from tunneling two-level atoms," Phys. Rev. Lett. 77, 2909 (1996) [CrossRef] [PubMed]
- D. Sokolovski and J. N. L. Connor, "Quantum interference and determination of the traversal time," Phys. Rev. A 47, 4677 (1993) note the connection between traversal-time measurement in tunneling and path information. [CrossRef] [PubMed]
- T. Pfau, S. Spalter, Ch. Kurtsiefer, C. R. Ekstrom and J. Mlynek, "Loss of spatial coherence by a single spontaneous emission," Phys. Rev. Lett. 73, 1223 (1994) [CrossRef] [PubMed]
- A. Stern, Y. Aharonov and Y. Imry, "Phase uncertainty and loss of interference: a general picture," Phys. Rev. A 41, 3436 (1990). [CrossRef] [PubMed]
- C. Cohen-Tannoudji et. al., Atom-Field Interactions (Wiley, New-York,1992);
- G. S. Agarwal, Quantum Statistical Theories of Spontaneous Emission (Springer, Berlin, 1974).
- D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, "Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator," Phys. Rev. Lett. 58, 1320 (1987) [CrossRef] [PubMed]
- R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, New York, 1965).
- Y. Japha, V. M. Akulin and G. Kurizki, "Localized decoherence of two-level wavepackets: Atomic binding and skin effects," Phys. Rev. Lett. (submitted)
- B. G. Englert, J. Schwinger, A. O. Barut and M. O. Scully, "Reflecting slow atoms from a micromaser field," Europhys. Lett.14, 25 (1991) [CrossRef]
- M.O. Scully, G.M. Meyer and H. Walther, "Induced emission due to the quantized motion of ultracold atoms passing through a micromaser cavity," Phys. Rev. Lett. 76, 4144 (1996) [CrossRef] [PubMed]
- Excited state and total amplitude of ground state http://www.weizmann.ac.il/ cfyoni/movie.mpg
- Ground state entangled with resonant emission http://www.weizmann.ac.il/ cfyoni/movie1.mpg
- Ground state entangled with positive detuning http://www.weizmann.ac.il/ cfyoni/movie2.mpg
- Ground state entangled with negative detuning http://www.weizmann.ac.il/ cfyoni/movie3.mpg
- Ground state entangled with forbidden emission http://www.weizmann.ac.il/ cfyoni/movie4.mpg

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