## Repeated interaction model for diffusion-induced Ramsey narrowing

Optics Express, Vol. 16, Issue 18, pp. 14128-14141 (2008)

http://dx.doi.org/10.1364/OE.16.014128

Acrobat PDF (357 KB)

### Abstract

In a recent paper [Y. Xiao et al., Phys. Rev. Lett. 96, 043601 (2006)] we characterized diffusion-induced Ramsey narrowing as a general phenomenon, in which diffusion of coherence in-and-out of an interaction region such as a laser beam induces spectral narrowing of the associated resonance lineshape. Here we provide a detailed presentation of the repeated interaction model of diffusion-induced Ramsey narrowing, with particular focus on its application to Electromagnetically Induced Transparency (EIT) of atomic vapor in a buffer gas cell. We compare this model both to experimental data and numerical calculations.

© 2008 Optical Society of America

## 1. Introduction

1. W. Happer, “Optical pumping,” Rev. Mod. Phys. **44**, 169–249 (1972). [CrossRef]

2. E. Arimondo, “Relaxation processes in coherent-population trapping,” Phys. Rev. A **54**, 2216–2223 (1996). [CrossRef] [PubMed]

3. M. Erhard and H. Helm, “Buffer-gas effects on dark resonances: theory and experiment,” Phys. Rev. A **63**, 043813 (2001). [CrossRef]

4. Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. **96**, 043601 (2006). [CrossRef] [PubMed]

6. A. S. Zibrov, I. Novikova, and A. B. Matsko, “Observation of Ramsey fringes in an atomic cell with buffer gas,” Opt. Lett. **26**, 1311–1313 (2001). [CrossRef]

7. A. S. Zibrov and A. B. Matsko, “Optical Ramsey fringes induced by Zeeman coherence,” Phys. Rev. A **65**, 013814 (2001). [CrossRef]

8. E. Alipieva, S. Gateva, E. Taskova, and S. Cartaleva, “Narrow structure in the coherent population trapping resonance in rubidium,” Opt. Lett. **28**, 1817–1819 (2003). [CrossRef] [PubMed]

9. G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE , **5830**, 181–185 (2005). [CrossRef]

10. J. Vanier, “Atomic clocks based on Coherent Population Trapping: a review,” Appl. Phys. B **81(4)**, 421–442 (2005). [CrossRef]

1. W. Happer, “Optical pumping,” Rev. Mod. Phys. **44**, 169–249 (1972). [CrossRef]

10. J. Vanier, “Atomic clocks based on Coherent Population Trapping: a review,” Appl. Phys. B **81(4)**, 421–442 (2005). [CrossRef]

11. D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. **74**, 1153–1201 (2002). [CrossRef]

12. I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt. , **52**, 2381–2390 (2005). [CrossRef]

13. M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. **75**, 457–472 (2003). [CrossRef]

4. Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. **96**, 043601 (2006). [CrossRef] [PubMed]

## 2. Repeated interaction model

### 2.1. Atomic density matrix following a Ramsey sequence

4. Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. **96**, 043601 (2006). [CrossRef] [PubMed]

*ω*and

_{p}*ω*and Rabi frequencies (defined in the same way as in [16

_{d}16. M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A **55**, 2272–2282 (1997). [CrossRef]

*(probe beam) and Ω*

_{P}*(drive beam) couple two ground electronic states (typically hyperfine levels) to an electronically excited state: |*

_{D}*a*〉 → |

*e*〉 and |

*b*〉 → |

*e*〉, respectively. The excited state decays to the two ground states with equal rate

*γ*/2. The ground state population difference and coherence each relax to zero with a rate Γ

_{0}(including residual magnetic field inhomogeneity and spin exchange etc., but not including effects of optical pumping and collision with the cell walls). A step-like cylindrical laser profile is assumed (see See Fig. 1a) to ease computation and isolate diffusion-induced spectral reshaping from effects due to the transverse variation of the laser beam profile [17

17. F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D **12**, 53(2000). [CrossRef]

18. A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A **69**, 024501 (2004). [CrossRef]

19. E. Pfleghaar, J. Wurster, S. I. Kanorsky , and A. Weis, “Time of flight effects in nonliear magneto-optical spectroscopy,” Optics Commun. **99**, 303–308 (1993). [CrossRef]

*ρ*, which for an optically thin medium is homogeneous along the longitudinal axis of the atomic ensemble and laser beam. In the appendix we derive the atomic density matrix for the simplest Ramsey sequence in which an atom spends time

_{ea}*t*

_{1}in the laser beam, evolves freely in the the dark for time

*t*

_{2}, and then re-enters the laser beam for time

*t*

_{3}(see Fig. 1b). We assume that the drive field is resonant with the |

*a*〉 → |

*e*〉 transition, the probe field is near resonance with the |

*b*〉 → |

*e*〉 transition, the ratio of probe to drive Rabi frequencies is small, and the overall intensity of all optical fields is weak enough that the excited state can be adiabatically eliminated. Then the imaginary part of

*ρ*following the above Ramsey sequence has a relatively simple form:

_{ea}*α*=Ω

_{D}^{2}

*/2*

_{D}*γ*is the optical pumping rate, Γ=

*α*+Γ

_{D}_{0}, and Δ is the two-photon detuning. This result follows from Eq. (16) of the appendix in the limit

*γ*≫Δ relevant to EIT experiments discussed here. One can extend Eq. (1) to Ramsey sequences consisting of an arbitrary number of dark periods, noticing that terms associated with various time periods simply form a power series.

*t*

_{1}=

*t*

_{2}=0 and

*t*

_{3}>> 1/Γ; (b) “transit-time-limited EIT,” in which atoms leave the beam before equilibrating with the laser fields and do not return before decohering, given by the limit

*t*

_{1}=

*t*

_{2}=0 and

*t*

_{3}< ≈ 1/Γ; and (c) “Ramsey EIT,” in which there is significant return of atomic coherence to the laser beam after evolution in the dark, given for the general case with

*t*

_{1},

*t*

_{2}and

*t*

_{3}all being nonzero. As seen from Eq. (1),

*e*

^{-t3Γ}≈ 0 for equilibrium EIT, which leads to a Lorentzian lineshape with halfwidth of Γ. Transit-time-limited EIT has a sinc-function lineshape (dashed curve of Fig. 4), with a period of 1/

*t*

_{3}. Such lineshapes are observed in atomic beam experiments; however, the distribution of times spent in the laser beam for a steady-state vapor cell measurement washes out all but the central sinc-function lobe, as discussed below. The Ramsey EIT lineshape has a transit-time-limited envelope with underlying fringes, as shown by the solid curve of Fig. 4. The envelope has a width of 1/

*t*and a fringe width of 1/

_{3}*t*

_{2}. The fringes are caused by differential phase evolution of the atomic coherence for times spent inside and outside the laser beam. Similar lineshapes have been observed previously in pulsed-laser-beam vapor cells [14

14. T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. **94**, 193002 (2005). [CrossRef] [PubMed]

20. P. R. Hemmer, M. S. Shahriar, V. D. Natoli, and S. Ezekiel,“AC Stark shifts in a two-zone Raman interaction,” J. Opt. Soc. Am. B **6**, 1519–1528 (1989). [CrossRef]

### 2.2. Probability distributions for Ramsey sequences

*P*(

*t*

_{1},

*t*

_{2},

*t*

_{3}) to denote the probability density of atoms that have an interaction history of being in the beam for a period

*t*

_{1}, then outside for

*t*

_{2}, and then back in the beam for

*t*

_{3}. Distributions of dark and bright times are independent so

*P*

_{in}(

*t*

_{1}) and

*P*

_{out}(

*t*

_{2}) are the probability for an atom to continuously stay in and out of the beam for

*t*

_{1}and

*t*

_{2}, etc. In general, the radial symmetry of both the vapor cell and the laser beam allows for a solution for

*P*

_{in}and

*P*

_{out}in cylindrical coordinates. Atom collisions with the cell walls are assumed to destroy all coherence between the ground states. For simplicity, the longitudinal (z-axis) boundary is assumed to be at infinity, reducing the problem to the radial variable

*r*only.

*P*(

*r*,

*t*) for atoms not to have diffused beyond radius

*r*in time

*t*. We then calculated

*P*

_{in}(

*t*)=∫

_{beam}

*P*(

*r*,

*t*)2

*πr*d

*r*with the initial condition of a uniform distribution,

*P*(

*r*,0)=1/

*(πa*

^{2}) for

*r*<

*a*where

*a*is the laser beam diameter; and the boundary condition

*P*(

*r*=

*a*,

*t*)=0 to eliminate atoms that diffuse out of the beam. Integrating over the atoms remaining in the beam at time

*t*gives the probability density,

*τ*=

_{D}*a*

^{2}/4

*D*(the lowest order diffusion mode in two dimensions),

*D*is the atomic diffusion coefficient, and

*χ*is the

_{m}*mth*zero of

*J*

_{0}, the zeroth Bessel function of the first kind. For

*t*> ≈

*τ*, the lowest order mode dominates this distribution. Note that the distribution of times inside the beam with times measured in units of the lowest order diffusion mode

_{D}*t*̃

_{in}=

*t*/

*τ*is independent of any geometric or diffusion parameters. The dashed line in Fig. 5a shows

_{D}*P*

_{in}(

*t*̃

_{in}) as determined from Eq. (3).

*P*

_{out}(

*t*), the distribution of times spent out of the beam, can be found using the “small beam approximation” in which we assume that the laser beam cross-section is much smaller than the cell diameter. In this approximation, an atom spends negligible time in the laser beam.

*P*

_{out}(

*t*) is then approximated as the probability that an atom, starting in the laser beam at time zero, is again in the laser beam at

*t*. Because atoms spend little time in the beam, we assume that it was outside the beam for the entire period. This approximation leads to

*J*

_{1}is the first Bessel function of the first kind. The dashed line in Figure 5b shows

*P*

_{out}(

*t*̃

_{out}) for

*d*/

*a*=30, where

*t*̃

_{out}=

*t*/

*τ*for the small beam approximation. Note that

_{D}*P*

_{out}(

*t*̃

_{out}) is a function of

*d*/

*a*, the ratio of the cell radius to the beam radius (Fig. 1a). Larger ratios of

*d*/

*a*(smaller beams) lead to larger values of

*P*

_{out}(

*t*̃

_{out}) and thus to larger mean times outside the beam. For beam sizes small compared to the vapor cell size, atoms spend a majority of their time outside of the laser beam, which can accentuate the spectral narrowing effects of diffusion-induced Ramsey narrowing. This trend saturates when the mean diffusion time to the cell wall becomes comparable to 1/Γ

_{0}— the ground state coherence time due to mechanisms other than wall collisions — since in this limit, many atoms return to the beam after such long periods in the dark that they have lost ground state coherence.

*P*

_{in}(

*t*̃

_{in}) and

*P*

_{out}(

*t*̃

_{out}) numerically for random walk atomic motion and initial and final conditions that have atoms start and end on the laser boundary. A two-dimensional random walk was evaluated by placing a particle on a grid and repeatedly moving it in random directions on the grid. The time of each crossing of the boundary between the laser beam and the dark region of the vapor cell by the particle was tagged. By histogramming the time differences corresponding to the times during which the particle was in or out of the laser beam, the distributions were assembled. The results of these lengthy numerical calculations for

*P*

_{in}(

*t*̃

_{in}) and

*P*

_{out}(

*t*̃

_{out}) are given by the solid lines of Fig. 5. Further detail is provided in Sec. 4.

*P*

_{in}(

*t*̃

_{in}) [Eq. (3)] and

*P*

_{out}(

*t*̃

_{out}) [Eq. (4)] underestimate short time departures from and returns to the beam, respectively. Additionally, we assume a step-like laser beam profile, rather than a more realistic profile such as a Gaussian. Therefore, the behavior of atoms close to the boundary is not well modeled. However, at low laser intensities and/or small beam diameters, such that the optical pumping time is long compared to the time to diffuse through the beam (

*α*

^{-1}

*τ*<< 1), we find that the detail of the beam profile is unimportant and may be accounted for in the repeated interaction model through small, consistent adjustments of beam diameter and intensity. With these small adjustments we find equivalent calculated EIT lineshapes for the analytical and numerical versions of the

_{D}*P*

_{in}(

*t*̃

_{in}) and

*P*

_{out}(

*t*̃

_{out}) distributions. See Secs. 3 and 4.

### 2.3. Integration over all Ramsey sequences

*P*

_{in}and

*P*

_{out}given by Eqs. (3) and (4)]. To simplify evaluation of this integration we approximate the infinite sums in Eqs. (3) and (4) by fitting these functions to a finite sum of exponentials.

*τ*

^{-1}

*); low buffer gas pressure and small laser beam size such that atomic diffusion out of the laser beam is fast compared to ground state decoherence (Γ*

_{D}_{0}<<

*τ*

^{-1}

_{D}); and small laser beam radius compared to cell size (

*a*<<

*d*) such that atomic coherence can have long evolution in the dark without wall collisions. In these limits (see Fig. 6a) the sharp central peak is largely insensitive to power broadening because it results from the long evolution of atomic coherence in the dark. However, at high laser intensity the calculated central peak loses contrast relative to the broad pedestal because a sufficiently large optical pumping rate

*α*[see eq. (11)] drives to zero all terms in Eq. (1) proportional to exp(-

*αt*), leaving a Lorentzian lineshape due to the lowest-order diffusion mode with a power-broadened width of Γ=

_{i}*α*+Γ

_{0}. At moderate laser intensities, reduced but non-zero Ramsey fringe contrast results in moderate narrowing of the EIT lineshape center (see Fig. 6b). Similarly, as shown in Fig. 6c, the calculated contrast of the sharp central peak is reduced at higher buffer gas pressure (and hence slower atomic diffusion and greater

*τ*) for two reasons: (i) longer residence time in the beam compared to a fixed optical pumping rate takes the system toward the regime of equilibrium EIT, in which the atomic ground state populations and coherence approach equilibrium with the optical fields (and the associated Lorentzian EIT lineshape) during a single atomic residence time in the laser beam; and (ii) slower diffusion decreases the fraction of atoms that undergo coherent evolution in the dark and return to the laser beam before decohering.

_{D}## 3. Comparison with experiment

**96**, 043601 (2006). [CrossRef] [PubMed]

21. M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. **79**, 2959–2962 (1997). [CrossRef]

12. I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt. , **52**, 2381–2390 (2005). [CrossRef]

**96**, 043601 (2006). [CrossRef] [PubMed]

22. J. Vanier, M. W. Levine, D. Janssen, and M. Delaney, “Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb : Effect of optical pumping,” Phys. Rev. A. **67**, 065801 (2003). [CrossRef]

*and Ω*

_{D}*with an empirical fitting-parameter that varied only with buffer gas pressure. Also, as noted above, we employed small, consistent adjustments of the laser beam diameter and intensity to correct for effects of the step-like beam profile assumed in the repeated interaction model. For example, the EIT data shown in Figs. 2 and 6 was acquired with a laser beam diameter measured to be 0.8 mm, as determined from the 1/e intensity positions across the approximately Gaussian laser profile. We found that an effective beam diameter fixed at 0.96 mm optimized fits of the repeated interaction model for all these EIT lineshapes.*

_{P}17. F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D **12**, 53(2000). [CrossRef]

18. A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A **69**, 024501 (2004). [CrossRef]

## 4. Comparison with numerical calculations

*R*⃗, represents the populations and coherence of the two ground states of the three level Λ-system as an effective two-level system [see appendix and esp. Eq. (9)]. The equations of motion in the presence of diffusion may be written as

^{2}(

*r*)=Ω

^{2}

*(*

_{P}*r*)+Ω

^{2}

*(*

_{D}*r*), sin

*θ*=Ω

*P*/Ω [see Eq. (9)], and

*D*is the diffusion coefficient. The optical coherence and thus the transmitted intensity can then be calculated from Eq. (15) of the appendix.

^{2}/

*γ*is slow compared to the rate associated with the lowest order diffusion mode, 1/

*τ*=4

_{D}*D*/

*a*

^{2}, the lineshapes are identical for the two types of beam profiles (Fig. 8a). This result is unsurprising as atoms under this condition typically sample the full profile of the laser beam before reaching equilibrium. However, as the beam power is increased such that optical pumping is fast compared to the rate of diffusion out of the laser beam, the laser profile begins to affect the resonance lineshape (Fig. 8b). Additionally, the lineshapes change from the low-power lineshapes and begin to resemble the peaked arctangent shape found previously in a model designed to treat this fast optical pumping limit [18

18. A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A **69**, 024501 (2004). [CrossRef]

## 5. Conclusions

25. W. Zhang and D. G. Cory, “First Direct Measurement of the Spin Diffusion Rate in a Homogenous Solid,” Phys. Rev. Lett. **80**, 1324–1327 (1998). [CrossRef]

27. J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science **309**, 2180–2184 (2005). [CrossRef] [PubMed]

12. I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt. , **52**, 2381–2390 (2005). [CrossRef]

## Appendix : Evolution of atomic density matrix for a Ramsey sequence

*e*〉} (see Fig. 3) can be written as

*θ*=Ω

*P*/Ω, cos

*θ*=Ω

*/Ω, Ω*

_{D}^{2}=Ω

^{2}

*+Ω*

_{P}^{2}

*, and |*

_{D}*ã*〉=|

*a*〉

*e*and |

^{-iωpt}*b*̃〉=|

*b*〉

*e*are the ground state basis in the rotating frame. The density matrix equation in this dark state basis is [16

^{-iωdt}16. M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A **55**, 2272–2282 (1997). [CrossRef]

*H*, is given by

*C*

_{2}=cos2

*θ*,

*S*=sin2

_{2}*θ*, Δ the two-photon detuning, and

*δ*the one-photon detuning of both fields (see Fig. 3). If

*S*Δ and Ω ≪

_{2}*γ*, then the excited state adiabatically follows the ground state, reducing the system to a two-level system with new basis {|-〉, |+〉

*d*} where

16. M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A **55**, 2272–2282 (1997). [CrossRef]

*γ*allows us to represent the system with a Bloch vector

*R*⃗ given by [16

**55**, 2272–2282 (1997). [CrossRef]

*R*

_{1}=

*ρ*+-+

*ρ*-+,

*R*

_{2}=

*i*(

*ρ*+--

*ρ*-+), and

*R*

_{3}=

*ρ*---

*ρ*++, and an equation of motion:

*α*also acts as a decoherence mechanism for the two-photon resonance, leading to power broadening. As pointed out in [16

**55**, 2272–2282 (1997). [CrossRef]

*R*⃗, about the axis

*Q*⃗=-

*S*

_{2}Δ

*e*⃗

_{1}+

*β*′

*e*⃗

_{3}at a rate

*R*⃗

*is the steady state solution of Eq. (9),*

_{s}*R*⃗(0) is the initial value (which is zero if the system starts without coherence and with equal populations in the ground states), and

*α*+Γ

_{0}) [see Eq. (12)] which is also the inverse of the steady state EIT linewidth when not limited by transit time broadening.

*t*

_{1},

*t*

_{2},

*t*

_{3}] shown in Fig. 1b, and noting that Ω

_{eff}=Δ, when atoms are outside of the laser beam,

*R*⃗ at the end of the sequence is

*t*

_{3}; and the second row, which is responsible for the Ramsey fringes, describes the polarization that is obtained during

*t*

_{1}and then under phase evolution and amplitude loss during

*t*

_{2}and

*t*

_{3}. The decay rate of the Bloch vector associated with a time in the beam is

*α*+Γ

*0*, while outside the beam the rate is Γ

_{0}.

*R*⃗ as

*n*is atomic density,

*λ*the optical wavelength, and

*γr*the radiative decay rate of the excited state.

*θ*≈

*θ*), the imaginary part of

*ρ*has a relatively simple form:

_{ẽã}*α*+

_{D}_{Γ}ο,

*α*=Ω

_{D}^{2}

*/2*

_{D}*γ*, and “⋯” represents terms accounting for more than one diffusive return to the laser beam. Each return adds two terms similar to those in the parentheses. Further interpretation of this equation is provided in sec. 2.

## References and links

1. | W. Happer, “Optical pumping,” Rev. Mod. Phys. |

2. | E. Arimondo, “Relaxation processes in coherent-population trapping,” Phys. Rev. A |

3. | M. Erhard and H. Helm, “Buffer-gas effects on dark resonances: theory and experiment,” Phys. Rev. A |

4. | Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. |

5. | N. F. Ramsey, |

6. | A. S. Zibrov, I. Novikova, and A. B. Matsko, “Observation of Ramsey fringes in an atomic cell with buffer gas,” Opt. Lett. |

7. | A. S. Zibrov and A. B. Matsko, “Optical Ramsey fringes induced by Zeeman coherence,” Phys. Rev. A |

8. | E. Alipieva, S. Gateva, E. Taskova, and S. Cartaleva, “Narrow structure in the coherent population trapping resonance in rubidium,” Opt. Lett. |

9. | G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE , |

10. | J. Vanier, “Atomic clocks based on Coherent Population Trapping: a review,” Appl. Phys. B |

11. | D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. |

12. | I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt. , |

13. | M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. |

14. | T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. |

15. | M. O. Scully and M. S. Zubairy, |

16. | M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A |

17. | F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D |

18. | A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A |

19. | E. Pfleghaar, J. Wurster, S. I. Kanorsky , and A. Weis, “Time of flight effects in nonliear magneto-optical spectroscopy,” Optics Commun. |

20. | P. R. Hemmer, M. S. Shahriar, V. D. Natoli, and S. Ezekiel,“AC Stark shifts in a two-zone Raman interaction,” J. Opt. Soc. Am. B |

21. | M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. |

22. | J. Vanier, M. W. Levine, D. Janssen, and M. Delaney, “Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb : Effect of optical pumping,” Phys. Rev. A. |

23. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Cambridge University Press, New York (1992) Ch. 19. |

24. | A typical EIT lineshape, such as that shown in figure 7, takes approximately 12 hours to calculate on a 1.25 GHz PowerPC G4 whereas the repeated interaction model can be evaluated immediately. |

25. | W. Zhang and D. G. Cory, “First Direct Measurement of the Spin Diffusion Rate in a Homogenous Solid,” Phys. Rev. Lett. |

26. | J. Taylor. |

27. | J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(020.3690) Atomic and molecular physics : Line shapes and shifts

(030.1640) Coherence and statistical optics : Coherence

(270.1670) Quantum optics : Coherent optical effects

(300.3700) Spectroscopy : Linewidth

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: June 6, 2008

Revised Manuscript: August 1, 2008

Manuscript Accepted: August 4, 2008

Published: August 26, 2008

**Citation**

Yanhong Xiao, Irina Novikova, David F. Phillips, and Ronald L. Walsworth, "Repeated interaction model for
diffusion-induced Ramsey narrowing," Opt. Express **16**, 14128-14141 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14128

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### References

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- M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg and M. O. Scully, "Spectroscopy in dense coherent media: Line narrowing and interference effects," Phys. Rev. Lett. 79, 2959-2962 (1997). [CrossRef]
- J. Vanier, M. W. Levine, D. Janssen, M. Delaney, "Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb : Effect of optical pumping," Phys. Rev. A. 67, 065801 (2003). [CrossRef]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical recipies in C: the art of scientific computing, second edition, , (Cambridge University Press, New York, 1992) Ch. 19.
- A typical EIT lineshape, such as that shown in figure 7, takes approximately 12 hours to calculate on a 1.25 GHz PowerPC G4 whereas the repeated interaction model can be evaluated immediately.
- W. Zhang and D. G. Cory, "First Direct Measurement of the Spin Diffusion Rate in a Homogenous Solid," Phys. Rev. Lett. 80, 1324-1327 (1998). [CrossRef]
- For example, the interplay of spin diffusion with electron-spin-mediated nuclear-spin coherence may play a critical role in recent measurements in quantum dots; private communication, J. Taylor.
- J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, A. C. Gossard, "Coherent manipulation of coupled electron spins in semiconductor quantum dots," Science 309, 2180-2184 (2005). [CrossRef] [PubMed]

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