## Ground-state cooling for a trapped atom using cavity-induced double electromagnetically induced transparency |

Optics Express, Vol. 21, Issue 3, pp. 3445-3462 (2013)

http://dx.doi.org/10.1364/OE.21.003445

Acrobat PDF (1007 KB)

### Abstract

We propose a cooling scheme for a trapped atom using the phenomenon of cavity-induced double electromagnetically induced transparency (EIT), where the atom comprising of four levels in tripod configuration is confined inside a high-finesse optical cavity. By exploiting one cavity-induced EIT, which involves one cavity photon and two laser photons, carrier transition can be eliminated due to the quantum destructive interference of excitation paths. Heating process originated from blue-sideband transition mediated by cavity field can also be prohibited due to the destructive quantum interference with the additional transition between the additional ground state and the excited state. As a consequence, the trapped atom can be cooled to the motional ground state in the leading order of the Lamb-Dicke parameters. In addition, the cooling rate is of the same order of magnitude as that obtained in the cavity-induced single EIT scheme.

© 2013 OSA

## 1. Introduction

1. D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl Inst. Stand. Technol. **103**259–328 (1998). [CrossRef]

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5. F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser Cooling to the Zero-Point Energy of Motion,” Phys. Rev. Lett. **62**, 403–406 (1989). [CrossRef] [PubMed]

7. G. Morigi, J. Eschner, and C. H. Keitel, “Ground State Laser Cooling Using Electromagnetically Induced Transparency,” Phys. Rev. Lett. **85**, 4458–4461 (2000). [CrossRef] [PubMed]

8. 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]

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

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

16. G. Morigi, P. W. H. Pinkse, M. Kowalewski, and R. de Vivie-Riedle, “Cavity Cooling of Internal Molecular Motion,” Phys. Rev. Lett. **99**, 073001 (2007). [CrossRef] [PubMed]

17. M. Kowalewski, G. Morigi, P. W. H. Pinkse, and R. de Vivie-Riedle, “Cavity cooling of translational and ro-vibrational motion of molecules: ab initio-based simulations for OH and NO,” Appl. Phys. B **89**, 459–467 (2007). [CrossRef]

18. S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A **65**, 063804 (2002). [CrossRef]

19. M. D. Lukin, M. Fleischhauer, M. O. Scully, and V. L. Velichansky, “Intracavity electromagnetically induced transparency,” Opt. Lett. **23**, 295–297 (1998). [CrossRef]

14. M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature **465**, 755–758 (2010). [CrossRef] [PubMed]

20. G. Nikoghosyan and M. Fleischhauer, “Photon-Number Selective Group Delay in Cavity Induced Transparency,” Phys. Rev. Lett. **105**, 013601 (2010). [CrossRef] [PubMed]

*et al.*[21

21. M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New J. Phys. **14**, 023002 (2012). [CrossRef]

*et al.*has been experimentally investigated and the results show remarkable agreement with the theoretical predictions in Ref. [26], in which the superiority to EIT cooling outside the cavity is demonstrated. However the nonvanishing blue-sideband transition prohibits further cooling to realize the zero phonon number state in the leading order of Lamb-Dikce parameters for the trapped atom, which leads to a natural question of whether and how one can eliminate blue-sideband transition while carrier transition is eliminated.

22. J. Evers and C. H. Keitel, “Double-EIT ground-state laser coupling without bue-sideband heating,” Europhys. Lett. **68**, 370–376 (2004). [CrossRef]

23. J. Cerrillo, A. Retzker, and M. B. Plenio, “Fast and Robust Laser Cooling of Trapped Systems,” Phys. Rev. Lett. **104**, 043003 (2010). [CrossRef] [PubMed]

24. S. Zhang, C. W. Wu, and P. X. Chen, “Dark-state laser cooling of a trapped ion using standing waves,” Phys. Rev. A **85**, 053420 (2012). [CrossRef]

*et al.*[25

25. S. Zippilli and G. Morigi, “Mechanical effects of optical resonators on driven trapped atoms: Ground-state cooling in a high-finesse cavity,” Phys. Rev. A **72**, 053408 (2005). [CrossRef]

27. M. D. Lukin, S. F. Yelin, M. Fleichhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A **60**3225–3228 (1999). [CrossRef]

28. C. Y. Ye, A. S. Zibrov, Yu. V. Rostovtsev, and M. O. Scully, “Unexpected Doppler-free resonance in generalized double dark states,” Phys. Rev. A **65**043805 (2002). [CrossRef]

## 2. The description of the model

*ν*shown in Fig. 1(a). The atom is in tripod configuration, comprised of one excited state |

*e*〉 and three ground states |

*g*〉(

_{i}*i*= 1, 2, 3) with energy frequencies

*ω*and

_{e}*ω*respectively shown in Fig. 1(b). It is noted that the ground state |

_{gi}*g*

_{2}〉 is the additional state as compared with the cavity-induced single EIT system in Ref. [21

21. M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New J. Phys. **14**, 023002 (2012). [CrossRef]

*e*〉 ↔ |

*g*〉(

_{j}*j*= 1, 2) are irradiated by two laser fields with frequencies

*ω*and Rabi frequencies Ω

_{Lj}*respectively and the third ground state couples to the excited state through the cavity field with frequency*

_{j}*ω*and coupling strength

_{c}*g*(

*x*). Here we have assumed that the atomic center-of-mass motion is treated in one dimension along the

*x*-axis and

*x*indicates the position of the atom. The cavity field is weakly pumped by the third laser with frequency

*ω*and driving strength Ω

_{p}*. Thus the Hamiltonian of the system in a frame rotating with the lasers’ frequencies is given by where the Hamiltonians describing the internal atomic states, the cavity field and the external center-of-mass motion of the atom are respectively expressed as and the coupling interactions composed of the drive of the cavity field, Jaynes-Cummings interaction between the cavity mode and the atomic transition |*

_{p}*g*

_{3}〉 → |

*e*〉 and the drive of laser fields on the transitions |

*g*〉 → |

_{j}*e*〉 are respectively expressed as Here

*δ*

_{c3}=

*ω*− (

_{c}*ω*−

_{e}*ω*

_{g}_{3}) is the detuning of the cavity frequency

*ω*to the atomic transition |

_{c}*e*〉 ↔ |

*g*

_{3}〉,

*δ*=

_{j}*ω*− (

_{Lj}*ω*−

_{e}*ω*) indicates the detuning of the

_{gj}*j*-th laser frequency

*ω*to the corresponding atomic dipole transition |

_{Lj}*e*〉 ↔ |

*g*〉 and Δ =

_{j}*ω*−

_{p}*ω*describes the detuning between the cavity and the probe fields. The operators

_{c}*a*and

*b*are annihilation operators of cavity and vibrational phonon fields respectively. The cavity-atom coupling coefficient is

*g*(

*x*) =

*g*cos(

*k*cos

_{c}x*θ*+

_{c}*φ*), where

*k*(

_{c}*k*) is the wave number of the cavity mode (the laser fields),

_{j}*θ*(

_{c}*θ*) is the angle determined by orientation of the cavity wave vector (laser field vectors) with respect to the axis of motion and

_{Lj}*φ*is the phase determined by the equilibrium position of the atom in the trapping potential. The atom’s position operator

*x*connects with the phonon operators by the relation

*x*=

*ξ*(

*b*+

*b*

^{†}), where

*ρ*of the system obeys the master equation with the superoperators

*ℒ*and

_{at}ρ*𝒦ρ*describing the dipole spontaneous emissions and the cavity decay respectively, which are given in the form where the damping rates are

*γ*,

_{i}*κ*and describes the atomic motional recoil by the emission of a photon, with

*𝒩*(

_{i}*θ*) evaluated by taking into account of the geometry of the setup.

## 3. The derivation of master equation and numerical analysis for cooling dynamics

^{−2}to 10

^{−4}[26, 29

29. T. Kampschulte, W. Alt, S. Brakhane, M. Eckstein, R. Reimann, A. Widera, and D. Meschede, “Optical Control of the Refractive Index of a Single Atom,” Phys. Rev. Lett. **105**153603 (2010). [CrossRef]

*e*, 0〉 ≡ |

*e*〉, |

*g*

_{1}, 0〉 ≡ |

*g*

_{1}〉, |

*g*

_{2}, 0〉 ≡ |

*g*

_{2}〉, |

*g*

_{3}, 0〉 ≡ |

*g*

_{3}〉 and |

*g*

_{3}, 1〉 ≡ |1〉. Therefore, the Hamiltonian in Eq. (1) and the superoperators in Eq. (5) should be rewritten in the relevant space as and with the form of the other terms unchanged. Actually the system under consideration is treated as an effective five-level atomic system depicted in Fig. 2. Such a coherently coupled five-level atomic system possesses potential applications, e.g. multiple spontaneously generated coherence shown in Ref. [30

30. J.-H. Li, J.-B. Liu, A.-X. Chen, and Ch.-Ch. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A **74**033816 (2006). [CrossRef]

*g*

_{3}〉 ↔ |1〉 ↔ |

*e*〉 ↔ |

*g*

_{1}〉, which simultaneously involves a cavity photon and two laser photons. This is because under the condition of three-photon resonance, the transition |

*g*

_{3}〉 → |1〉 → |

*e*〉 destructively interferes with the transition |

*g*

_{1}〉 → |

*e*〉, which results in the elimination of the carrier transition [21

21. M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New J. Phys. **14**, 023002 (2012). [CrossRef]

31. Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A **67**, 013811 (2003). [CrossRef]

*g*

_{2}〉 → |

*e*〉. Finally, the ground-state cooling of the trapped atom can be achieved. Detailed discussions are presented in Sec. 5.

32. S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. **58**, 699–739 (1986). [CrossRef]

*η*=

_{lj}*η*cos

*θ*and

_{lj}*η*=

_{c}*η*cos

*θ*, which show the mechanical effects influenced by the angles between the motional axis and the wave vectors. The regime allows for a perturbation treatment due to the weak coupling between external and internal atomic degrees of freedom [34

_{c}34. J. Javanainen, M. Lindberg, and S. Stenholm, “Laser cooling of trapped ions: dynamics of the final stages,” J. Opt. Soc. Am. B **1**, 111–115 (1984). [CrossRef]

35. J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A **46**, 2668–2681 (1992). [CrossRef] [PubMed]

*η*of the exponential terms exp(

*ik*cos

_{j}*θ*) and the coupling function

_{Lj}x*g*(

*x*) in Eq. (3). Thus the first-order term is obtained in the form

*H*

_{1}=

*V*

_{1}(

*b*+

*b*

^{†}), with and the zeroth-order term is given by with

*H*

_{0L-at}=

*g*cos

*φ*(|

*e*〉〈1|+h.c.) and

*et al.*[35

35. J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A **46**, 2668–2681 (1992). [CrossRef] [PubMed]

*μ*is the density matrix for the phonon mode, obtained by tracing over the internal atomic and cavity degrees of freedom. And the coefficients

*S*(

*ν*) and

*D*are given by where

*S*(

*ν*) is the two-time correlation function of atomic and cavity operators and

*D*is the diffusion term with indicating the angular dispersion of the atom momentum due to the spontaneous emission of photons, which takes

32. S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. **58**, 699–739 (1986). [CrossRef]

*σ*= |

_{mn}*m*〉〈

*n*| are expressed in terms of a complete set of states {|

*m*〉} = {|

*e*〉, |

*g*

_{1}〉, |

*g*

_{2}〉, |

*g*

_{3}〉, |1〉}. From Eq. (17) we can directly derive the rate equation for the average phonon number 〈

*n*〉, namely where are the heating (

*A*

_{+}) and cooling (

*A*

_{−}) coefficients. By using the quantum regression theorem [36], these coefficients of the two-time correlation function can be calculated from the single-time average function expressed in the master equation by ignoring the coupling between the internal atomic states and cavity field and the phonon field (see appendix A). The steady-state average phonon number 〈

*n*〉

*and the cooling rate*

_{st}*W*are obtained as

*e*〉 → |

*g*

_{3}〉, which will lead to a considerable simplification of the analytical expressions for heating and cooling coefficients in the following section, and even when the other dissipations are nonzero, they will not qualitatively affect the cooling dynamics [21

**14**, 023002 (2012). [CrossRef]

*γ*

_{3}=

*γ*.

^{199}Hg

^{+}-ion, which is in tripod configuration with upper state

*P*

_{1/2}(

*F*′ = 1,

*m*= 1) and three lower states

_{F}*S*

_{1/2}(

*F*= 1,

*m*= 1, 0) and

_{F}*S*

_{1/2}(

*F*= 0,

*m*= 0), it is practical to take the trap frequency

_{F}*ν*= 1MHz [33

33. F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: Application to trapped samples of ions or neutral atoms,” Appl. Phys. B **73**, 807–814 (2001). [CrossRef]

*ℱ*= 1.2 × 10

^{6}), the experimental parameters in units of

*ν: γ*= 10

*ν*,

*κ*= 0.2

*ν*,

*g*= 10

*ν*, Ω

_{1}= 12

*ν*, Δ =

*ν*,

*δ*

_{1}= 60

*ν*,

*δ*

_{c}_{3}= 59

*ν*,

*θ*=

_{c}*θ*= 0,

_{li}*φ*=

*π*/3, Ω

*/2 = 0.1,*

_{p}*α*

_{3}=

*α*= 2/5 are also practical [29

29. T. Kampschulte, W. Alt, S. Brakhane, M. Eckstein, R. Reimann, A. Widera, and D. Meschede, “Optical Control of the Refractive Index of a Single Atom,” Phys. Rev. Lett. **105**153603 (2010). [CrossRef]

*A*

_{±}/

*η*

^{2}as functions of

*δ*

_{2}and Ω

_{2}in Fig. 3, where the heating coefficient is pretty close to zero as indicated by the dot line at

*δ*

_{2}= 59

*ν*regardless of the value of Ω

_{2}. Hence the motional ground state for the trapped atom is achieved in the leading-order expansion of the Lamb-Dicke parameters in the present scheme, which indicates that the heating processes due to the carrier and blue-sideband transitions are suppressed as compared with those in the scheme for a trapped atom using cavity-induced single EIT [21

**14**, 023002 (2012). [CrossRef]

*A*

_{−}in the plot is comparable with that obtained in the cavity-induced single EIT scheme.

25. S. Zippilli and G. Morigi, “Mechanical effects of optical resonators on driven trapped atoms: Ground-state cooling in a high-finesse cavity,” Phys. Rev. A **72**, 053408 (2005). [CrossRef]

*is much smaller than all the other physical parameters that characterize the internal dynamics. Dynamics of cooling processes are studied with the perturbation method in the second order of Ω*

_{p}*and*

_{p}*η*by neglecting the terms of the order

*η*

^{4}Ω

^{2},

*η*

^{2}Ω

^{4}and higher.

## 4. Calculations of heating and cooling coefficients with the perturbation method

*A*

_{±}in Eq. (21), we will make use of the perturbation method with respect to Ω

*. In the following we denote*

_{p}*V*

_{1}in Eq. (10) and the index

*i*also indicates the order of Ω

*. These two-time correlation functions can be calculated from single-time averages*

_{p}38. Z. Yi, W. J. Gu, and G. X. Li, “Sideband cooling of atoms with the help of an auxiliary transition,” Phys. Rev. A **86**, 055401 (2012). [CrossRef]

### 4.1. The density matrix in the zeroth order of Ω_{p}

*from the evolution equations in appendix A. Then a separate equation is obtained. Steady-state solutions for the density matrix elements are obvious that*

_{p}39. P. Rabl, “Cooling of mechanical motion with a two-level system: The high-temperature regime,” Phys. Rev. B **82**, 165320 (2010). [CrossRef]

### 4.2. The corrections for density matrix in the first order of Ω_{p}

*, we substitute the zeroth-order terms into the right hand side (RHS) of the evolution equations in appendix A. We find out that the evolution for nonzero elements of the density matrix is described by the following complete set of equations while the other elements are equal to zero in the long time limit. The steady-state solutions for the density matrix elements governed by Eqs. (26a)–(26d) can be obtained by substituting*

_{p}*and inversely proportional to*

_{p}*S*and

^{κ}*S*, It can be verified from Eq. (27a) that the atomic excitation spectrum exhibits zero value when the detuning Δ satisfies the relation or In fact the two relations above correspond to two independent three-photon resonance conditions, under which the transition |

^{γ}*g*

_{3}〉 → |1〉 → |

*e*〉 mediated by the cavity field can destructively interfere with either the transition |

*g*

_{1}〉 → |

*e*〉 or |

*g*

_{2}〉 → |

*e*〉. The destructive interference can prohibit the population in state |

*e*〉. Without loss of generality, we take the relation

*δ*

_{c}_{3}+ Δ −

*δ*

_{1}= 0 and leave the additional transition |

*g*

_{2}〉 ↔ |

*e*〉 to be appropriately tuned so that the heating processes due to blue-sideband transitions can be eliminated further.

*ℒ*

^{(1)}describing the evolution of

*ρ*

^{(1)}that is determined by Eqs. (26a)–(26d).

### 4.3. The corrections for density matrix in the second order of Ω_{p}

*, we follow the same procedure as presented in the last subsection. We substitute the first-order terms into the RHS of the evolution equations in appendix A. Then we find out that a group of nonzero second-order density matrix elements forming a complete set of evolution equations, which are given as It is obvious that the average value of the operators that appear in*

_{p}*V*

_{1}in Eq. (10) are included in this group of elements.

*ρ*

^{(2)}(∞), we substitute the first-order steady-state terms given in Eqs.(27a)–(27d) into the Eqs. (33a)–(33i), and are able to obtain the steady-state solution

*ρ*

^{(2)}(∞) shown in appendix B, which is proportional to

*ρ*

^{(2)}(

*s*) is also obtained by Laplace transforming the Eqs. (33a)–(33i) and given by where

*ℒ*

^{(2)}is the Liouvillian operator that describes the evolution of

*ρ*

^{(2)}governed by Eqs. (33a)–(33i).

### 4.4. Calculations of heating and cooling coefficients

*A*

_{±}in Eq. (21) can be obtained by applying the quantum regression theorem. Before further calculations we first consider the diffusion term

*D*due to spontaneous emission, which is written in the form It is obvious when the parameters fulfill the three-photon resonance condition

*δ*

_{c}_{3}+ Δ −

*δ*

_{1}= 0 as mentioned in Sec. 4.2, the steady-state expression of

*g*

_{3}〉 → |1〉 → |

*e*〉 and |

*g*

_{1}〉 → |

*e*〉, which coincides with the zero value in the atomic excitation spectrum. In the following we derive heating and cooling coefficients under this three-photon resonance condition to verify that we are able to eliminate heating processes besides term

*D*with the help of the additional transition |

*g*

_{2}〉 ↔ |

*e*〉.

*S*(

*ν*) given in Eq. (18) can be calculated from 〈

*V*

_{1}(

*s*)

*V*

_{1}(0)〉

*by substituting*

_{s}*s*with

*iν*. Therefore, by applying the quantum regression theorem,

*S*(

*ν*) can be derived from the single-time average

*f*̃(Δ ∓

*ν*) takes the form with

*e*〉 with damping rate

*γ*or the cavity decay with rate

*κ*. This can be verified from the expressions of heating and cooling coefficients in Eq. (36), where the rates are split into two independent parts: one is multiplied by

*γ*and the other is multiplied by

*κ*.

*f*̃(Δ ∓

*ν*) into heating and cooling coefficients

*A*

_{±}and rewrite

*A*

_{±}into a much simpler form

*C*≫ 1, which has many applications and is demonstrated in experiment [41

41. P. F. Zhang, Y. Q. Guo, Zh. H. Li, Y. C. Zhang, Y. F. Zhang, J. J. Du, G. Li, J. M. Wang, and T. C. Zhang, “Elimination of the degenerate trajectory of a single atom strongly coupled to a tilted TEM10 cavity mode,” Phys. Rev. A **83**, 031804(R) (2011). [CrossRef]

## 5. Cooling behavior induced by cavity-induced double EIT

*ε*

_{±}indicate mechanical effects caused by the additional transition |

*g*

_{2}〉 ↔ |

*e*〉. If the laser field that drives this additional transition is turned off, i.e. Ω

_{2}= 0,

*ε*

_{±}become zero. Heating and cooling coefficients are reduced to the same form as those obtained in cavity-induced single EIT scheme under the three-photon resonance condition.

*A*

_{±}/

*η*

^{2}given by Eq. (40) as functions of Ω

_{2}and

*δ*

_{2}in Fig. 4 with the same parameters as those taken in Fig. 2. The plots show a good match with numerical calculations in Fig. 2, thus it identities the feasibility of the perturbation method. To acquire the physical insight into the cooling scheme, especially the cancellation of heating processes that are indicated by dot lines in numerical and analytical plots in Figs. 3 and 4, we sketch the heating processes in Fig. 5 by taking into account of the vibrational center-of-mass motion of the atom.

### 5.1. The cancellation of heating processes

22. J. Evers and C. H. Keitel, “Double-EIT ground-state laser coupling without bue-sideband heating,” Europhys. Lett. **68**, 370–376 (2004). [CrossRef]

*g*

_{3},

*n*〉 → |1,

*n*〉 → |

*e*,

*n*〉 mediated by a cavity photon, where |

*n*〉 represents the phonon number state, will produce two potential heating mechanisms: one is caused by the recoils of spontaneous emission photons from the excited state, which is measured by the diffusion term

*D*in Eq. (35); the other is via the transition |

*e*,

*n*〉 → |1,

*n*+ 1〉 with the coupling strength

*η*sin

_{c}g*φ*into the state |1,

*n*+ 1〉, which leads to the heating through the cavity damping. However the carrier transition can be prohibited due to the quantum destructive interference with the transition |

*g*

_{1},

*n*〉 → |

*e*,

*n*〉 when the three-photon resonance condition

*δ*

_{c3}+ Δ −

*δ*

_{1}= 0 is fulfilled. The process is shown by the red lines in Fig. 5, which confirms the first cavity-induced EIT.

*g*

_{3},

*n*〉 → |1,

*n*〉 → |

*e*,

*n*+ 1〉 also mediated by a cavity photon, can create a quantum of the harmonic motion and lead to heating via the dissipation of the excited state. The subsequent transition |

*e*,

*n*+ 1〉 → |1,

*n*+ 1〉 can also heat the motion via the cavity decay channel. In order to eliminate the blue-sdieband transition, we first look into the expression of the heating coefficient

*A*

_{+}in Eq. (40), which is proportional to (

*δ*

_{1}−

*ν*−

*δ*

_{2})

^{2}. It is clear that if we take the relation

*δ*

_{2}=

*δ*

_{1}−

*ν*,

*A*

_{+}becomes zero, which means that the blue-sideband transition is eliminated. This is because by taking into account of the phonon number state, the energy for the state |

*g*

_{2},

*n*+ 1〉 becomes equal to that for the state |

*g*

_{1},

*n*〉 when the parameters satisfy the relation

*δ*

_{2}=

*δ*

_{1}−

*ν*. Therefore, with the help of the additional transition |

*g*

_{2},

*n*+ 1〉 → |

*e*,

*n*+ 1〉, another three-photon process arises. Via utilizing its destructive interference with the blue-sideband transition, none atomic population injects into the states |

*e*,

*n*+ 1〉 and |1,

*n*+ 1〉. As a consequence, the heating processes relevant to atomic dissipation rate

*γ*and cavity decay rate

*κ*are cancelled simultaneously as compared with Ref. [25

25. S. Zippilli and G. Morigi, “Mechanical effects of optical resonators on driven trapped atoms: Ground-state cooling in a high-finesse cavity,” Phys. Rev. A **72**, 053408 (2005). [CrossRef]

*κ*is still alive. The three-photon process is shown by the green lines in Fig. 5, which indicate the second independent cavity-induced EIT.

### 5.2. Analysis of the cooling coefficient A_{−} while heating processes are eliminated

*A*

_{−}given in Eq. (40) under the condition

*A*

_{+}= 0. In the case

*δ*

_{2}=

*δ*

_{1}−

*ν*, the term

*ε*

_{−}that characterizes the mechanical effect caused by the additional transition |

*g*

_{2}〉 ↔ |

*e*〉 becomes

*A*

_{−}changes into Cooling is enhanced when the denominator of rate

*A*

_{−}in Eq. (45) becomes minimal, i.e. the parameters satisfy the following condition By now the coupling strength Ω

_{2}and detuning

*δ*

_{2}for the additional transition |

*g*

_{2}〉 ↔ |

*e*〉 are determined by parameters that are practical in the experiment of cavity EIT system [29

29. T. Kampschulte, W. Alt, S. Brakhane, M. Eckstein, R. Reimann, A. Widera, and D. Meschede, “Optical Control of the Refractive Index of a Single Atom,” Phys. Rev. Lett. **105**153603 (2010). [CrossRef]

*A*

_{−}becomes which is the same as the maximal cooling coefficient of the cavity-induced single EIT system in Ref. [21

**14**, 023002 (2012). [CrossRef]

*W*given in Eq. (22) reads

*W*=

*A*

_{−}, which is of the same order of magnitude as that obtained in the cavity-induced single EIT scheme without the help of the additional transition |

*g*

_{2}〉 ↔ |

*e*〉. In Fig. 6 we plot the cooling coefficient

*A*

_{−}given in Eq. (45) as the function of Ω

_{2}with the parameters shown in depiction, which have already satisfied the condition

*A*

_{+}= 0. When

*A*

_{−}/

*η*

^{2}obtains the maximum value 0.35

*ν*, the corresponding driving strength Ω

_{2}= 10

*ν*, which is exactly the value calculated from the Eq. (46).

## 6. Conclusions

*g*

_{3},

*n*〉 → |1,

*n*〉 → |

*e*,

*n*〉 mediated by a cavity photon can be eliminated due to the quantum destructive interference with the transition |

*g*

_{1},

*n*〉 → |

*e*,

*n*〉 when three-photon resonance condition is fulfilled, where |

*n*〉 the phonon number state. This is the first cavity-induced EIT. The blue-sideband transition |

*g*

_{3},

*n*〉 → |1,

*n*〉 → |

*e*,

*n*+ 1〉 also mediated by a cavity photon, which heats the atom by a quantum of the harmonic center-of-mass motion, can be simultaneously eliminated due to its destructive interference with the additional transition |

*g*

_{2},

*n*+ 1〉 → |

*e*,

*n*+ 1〉. This is the second independent cavity-induced EIT. Finally the motional ground state for the trapped atom can be theoretically achieved in the leading-order Lamb-Dicke parameters. In addition, the cooling rate is of the same order of magnitude as that obtained in the cooling scheme using cavity-induced single EIT.

## A. The evolution of density matrix elements

*σ*(

_{mn}*t*)

*σ*(0)〉

_{kl}*obey the same equation as*

_{s}*ρ*(

_{nm}*t*), which are given in the Eq. (A1), with the initial values 〈

*σ*(0)

_{mn}*σ*(0)〉

_{kl}*=*

_{s}*δ*(∞) and

_{nk}ρ_{lm}*ρ*(∞) are the steady-state solutions. The evolution equations for time-dependent density matrix elements

_{lm}*ρ*(

_{mn}*t*) are given by

## B. Steady-state solutions of the second-order Ω_{p}

_{p}

*, which are calculated by substituting steady-state solutions*

_{p}## Acknowledgments

## References and links

1. | D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl Inst. Stand. Technol. |

2. | A. Steane, C. F. Roos, D. Stevens, A. Mundt, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Speed of ion-trap quantum-information processors,” Phys. Rev. A |

3. | E. Buks and B. Yurke, “Mass detection with a nonlinear nanomechanical resonator,” Phys. Rev. E |

4. | J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland, “Laser-Cooled-Atomic Frequency Standard,” Phys. Rev. Lett. |

5. | F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser Cooling to the Zero-Point Energy of Motion,” Phys. Rev. Lett. |

6. | C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. Gould, “Resolved-Sideband Raman Cooling of a Bound Atom to the 3D Zero-Point Energy,” Phys. Rev. Lett. |

7. | G. Morigi, J. Eschner, and C. H. Keitel, “Ground State Laser Cooling Using Electromagnetically Induced Transparency,” Phys. Rev. Lett. |

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

9. | V. Vuletić and S. Chu, “Laser Cooling of Atoms, Ions, or Molecules by Coherent Scattering,” Phys. Rev. Lett. |

10. | P. R. Berman, |

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

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

13. | D. R. Leibrandt, J. Labaziewicz, V. Vuletić, and I. L. Chuang, “Cavity Sideband Cooling of a Single Trapped Ion,” Phys. Rev. Lett. |

14. | M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature |

15. | A. Reiserer, C. Nölleke, S. Ritter, and G. Rempe, “Ground-state cooling of a single atom at the center of an optical cavity,” arXiv:1212.5295v1 (2012). |

16. | G. Morigi, P. W. H. Pinkse, M. Kowalewski, and R. de Vivie-Riedle, “Cavity Cooling of Internal Molecular Motion,” Phys. Rev. Lett. |

17. | M. Kowalewski, G. Morigi, P. W. H. Pinkse, and R. de Vivie-Riedle, “Cavity cooling of translational and ro-vibrational motion of molecules: ab initio-based simulations for OH and NO,” Appl. Phys. B |

18. | S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A |

19. | M. D. Lukin, M. Fleischhauer, M. O. Scully, and V. L. Velichansky, “Intracavity electromagnetically induced transparency,” Opt. Lett. |

20. | G. Nikoghosyan and M. Fleischhauer, “Photon-Number Selective Group Delay in Cavity Induced Transparency,” Phys. Rev. Lett. |

21. | M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New J. Phys. |

22. | J. Evers and C. H. Keitel, “Double-EIT ground-state laser coupling without bue-sideband heating,” Europhys. Lett. |

23. | J. Cerrillo, A. Retzker, and M. B. Plenio, “Fast and Robust Laser Cooling of Trapped Systems,” Phys. Rev. Lett. |

24. | S. Zhang, C. W. Wu, and P. X. Chen, “Dark-state laser cooling of a trapped ion using standing waves,” Phys. Rev. A |

25. | S. Zippilli and G. Morigi, “Mechanical effects of optical resonators on driven trapped atoms: Ground-state cooling in a high-finesse cavity,” Phys. Rev. A |

26. | T. Kampschulte, W. Alt, S. Manz, M. Martinez-Dorantes, R. Reimann, S. Yoon, D. Meschede, M. Bienert, and G. Morigi, “EIT-control of single-atom motion in an optical cavity,” arXiv:1212.3814v1 (2012). |

27. | M. D. Lukin, S. F. Yelin, M. Fleichhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A |

28. | C. Y. Ye, A. S. Zibrov, Yu. V. Rostovtsev, and M. O. Scully, “Unexpected Doppler-free resonance in generalized double dark states,” Phys. Rev. A |

29. | T. Kampschulte, W. Alt, S. Brakhane, M. Eckstein, R. Reimann, A. Widera, and D. Meschede, “Optical Control of the Refractive Index of a Single Atom,” Phys. Rev. Lett. |

30. | J.-H. Li, J.-B. Liu, A.-X. Chen, and Ch.-Ch. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A |

31. | Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A |

32. | S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. |

33. | F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: Application to trapped samples of ions or neutral atoms,” Appl. Phys. B |

34. | J. Javanainen, M. Lindberg, and S. Stenholm, “Laser cooling of trapped ions: dynamics of the final stages,” J. Opt. Soc. Am. B |

35. | J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A |

36. | J. S. Peng and G. X. Li, |

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

38. | Z. Yi, W. J. Gu, and G. X. Li, “Sideband cooling of atoms with the help of an auxiliary transition,” Phys. Rev. A |

39. | P. Rabl, “Cooling of mechanical motion with a two-level system: The high-temperature regime,” Phys. Rev. B |

40. | H. J. Kimble, in |

41. | P. F. Zhang, Y. Q. Guo, Zh. H. Li, Y. C. Zhang, Y. F. Zhang, J. J. Du, G. Li, J. M. Wang, and T. C. Zhang, “Elimination of the degenerate trajectory of a single atom strongly coupled to a tilted TEM10 cavity mode,” Phys. Rev. A |

**OCIS Codes**

(270.4180) Quantum optics : Multiphoton processes

(020.3320) Atomic and molecular physics : Laser cooling

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: November 27, 2012

Revised Manuscript: January 16, 2013

Manuscript Accepted: January 16, 2013

Published: February 4, 2013

**Citation**

Zhen Yi, Wen-ju Gu, and Gao-xiang Li, "Ground-state cooling for a trapped atom using cavity-induced double electromagnetically induced transparency," Opt. Express **21**, 3445-3462 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3445

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### References

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