OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 3 — Feb. 11, 2013
  • pp: 3445–3462
« Show journal navigation

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

Zhen Yi, Wen-ju Gu, and Gao-xiang Li  »View Author Affiliations


Optics Express, Vol. 21, Issue 3, pp. 3445-3462 (2013)
http://dx.doi.org/10.1364/OE.21.003445


View Full Text Article

Acrobat PDF (1007 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Realization of atoms or ions trapped within the vicinity of quantum ground state is the key ingredient in a number of practical applications, such as efficient quantum computation [1

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. 103259–328 (1998). [CrossRef]

], quantum information processing [2

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 62, 042305 (2000). [CrossRef]

], precision detection of mass [3

3. E. Buks and B. Yurke, “Mass detection with a nonlinear nanomechanical resonator,” Phys. Rev. E 74, 046619 (2006). [CrossRef]

] and investigations in precision spectral and time and frequency standards [4

4. J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland, “Laser-Cooled-Atomic Frequency Standard,” Phys. Rev. Lett. 54, 1000–1003 (1985). [CrossRef] [PubMed]

]. All conventional methods to laser cooling atoms [5

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

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]

] rely on repeated cycles of optical pumping and spontaneous emission of a photon by the atom. Spontaneous emission in a random direction provides the dissipative mechanism required to remove entropy from the atom. Thereafter a new competitive cooling technique: cavity cooling [8

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]

, 9

9. V. Vuletić and S. Chu, “Laser Cooling of Atoms, Ions, or Molecules by Coherent Scattering,” Phys. Rev. Lett. 84, 3787–3790 (2000). [CrossRef]

] is proposed to realize efficient cooling dynamics assisted with the optical resonator, especially in the regime of the strong coupling of the atomic transitions to a high-fineness resonator. The resonator gives rise to substantial modifications of atomic spectroscopic properties [10

10. P. R. Berman, Cavity Quantum Electrodynamics (Academic Press, New York) (1994).

], which allows one to alter the atomic scattering cross section, thereby tailoring the mechanical motion of the atomic center of the mass that may not be completed in free space [9

9. V. Vuletić and S. Chu, “Laser Cooling of Atoms, Ions, or Molecules by Coherent Scattering,” Phys. Rev. Lett. 84, 3787–3790 (2000). [CrossRef]

, 11

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

]. Experimentally the cavity cooling technique has been successfully realized for cooling neutral atoms or ions confined in an intracavity dipole trap [12

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]

15

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).

]. This technique also opens the possibility to cool down complex quantum systems with scalable number of degrees of freedom, such as large molecules [16

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

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]

].

Incorporating the electromagnetically induced transparency (EIT) with the optical cavity will boost the capabilities of cavity quantum electrodynamics (CQED), such as dynamical control of the photon statistics of propagating light fields [18

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]

], substantial reduction of classical and quantum phase noise of the beat note of optical oscillators [19

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

], quantum optical transistor [14

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]

] and the engineering of Fock state superpositions of flying light pulses [20

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

]. Via exploring properties produced by the combination of cavity-enhanced scattering and EIT, Bienert 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]

] investigated the cooling dynamics of the trapped atom in the intracavity EIT system, and demonstrated an efficient ground-state cooling with the elimination of carrier transition. Recently, the scheme proposed by Bienert et al. has been experimentally investigated and the results show remarkable agreement with the theoretical predictions in Ref. [26

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).

], 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.

In order to eliminate blue-sideband transition, a variety of cooling schemes in free space have been proposed, for example, the double-EIT scheme via creating two independent EIT-structures [22

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

], the scheme based on the existence of a joint dark state in EIT and Stark-shift cooling [23

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]

], the dark-state scheme by exploiting the property of standing waves [24

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]

]. These approaches offer the possibility to eliminate the blue-sideband transition appeared in the cavity-induced EIT scheme. In addition, in 2005 Zippilli 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]

] have proposed a scheme, where the heating processes due to the blue-sideband transition relevant to atomic spontaneous dissipation is eliminated while that relevant to cavity decay still exists. Thus, in this paper we are especially interested in whether we could construct two independent cavity-induced EIT structures to eliminate the heating processes due to the blue-sideband transitions relevant to the atomic spontaneous dissipation and cavity decay.

In this paper we propose a ground-state cooling scheme in the cavity-induced double EIT system which merges a double-EIT structure system comprised of a four-level trapped atom in tripod configuration with the cavity quantum electrodynamics. This is able to effectively create cavity-induced double EIT for an appropriate choice of parameters in the system. Carrier transition can be eliminated due to the destructive quantum interference effect between excitation paths of one cavity-induced EIT when the involved one cavity photon and two laser photons fulfill the three-photon resonance condition [27

27. M. D. Lukin, S. F. Yelin, M. Fleichhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 603225–3228 (1999). [CrossRef]

, 28

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 65043805 (2002). [CrossRef]

]. The blue-sideband transition, which is mediated by the cavity field, creates a quantum of harmonic center-of-mass motion. It can destructively interfere with the appropriate tuned additional transition of the additional ground state and the excited state, leading to the elimination of heating processes due to the blue-sideband transitions relevant to atomic spontaneous dissipation and cavity decay. As a consequence, the motional ground state for the trapped atom is achieved in the leading order of Lamb-Dicke parameters with the same cooling coefficient as compared with that obtained in the cavity-induced single EIT system. This paper is organized as follows. In Sec. 2, the physical system is introduced. In Sec. 3, under the assumption of the weak cavity-driving laser, the master equation of cooling dynamics is derived and numerical calculations for the heating and cooling coefficients are presented. In Sec. 4, analytical expressions of heating and cooling coefficients are obtained with the perturbation method with respect to the weak driving laser. Physical discussions on the cooling dynamics are presented in Sec. 5 and finally we give a summary.

2. The description of the model

The system under consideration consists of an atom of mass M confined inside a high-finesse optical cavity by a harmonic trap of frequency ν shown in Fig. 1(a). The atom is in tripod configuration, comprised of one excited state |e〉 and three ground states |gi〉(i = 1, 2, 3) with energy frequencies ωe and ωgi respectively shown in Fig. 1(b). It is noted that the ground state |g2〉 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]

]. Dipole-allowed transitions |e〉 ↔ |gj〉(j = 1, 2) are irradiated by two laser fields with frequencies ωLj and Rabi frequencies Ωj respectively and the third ground state couples to the excited state through the cavity field with frequency ωc and coupling strength 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 ωp and driving strength Ωp. Thus the Hamiltonian of the system in a frame rotating with the lasers’ frequencies is given by
H=Hat+Hcav+Hext+HL-cav+Hat-cav+HL-at,
(1)
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
Hat=δc3|ee|+j(δjδc3)|gjgj|+Δ|g3g3|,Hcav=Δaa,Hext=ν(bb+12),
(2)
and the coupling interactions composed of the drive of the cavity field, Jaynes-Cummings interaction between the cavity mode and the atomic transition |g3〉 → |e〉 and the drive of laser fields on the transitions |gj〉 → |e〉 are respectively expressed as
HL-cav=Ωp2(a+a),Hat-cav=g(x)(|eg3|a+|g3e|a),HL-at=jΩj2(|egj|exp(ikjcosθLjx)+h.c.).
(3)
Here δc3 = ωc − (ωeωg3) is the detuning of the cavity frequency ωc to the atomic transition |e〉 ↔ |g3〉, δj = ωLj − (ωeωgj) indicates the detuning of the j-th laser frequency ωLj to the corresponding atomic dipole transition |e〉 ↔ |gj〉 and Δ = ωpωc describes the detuning between the cavity and the probe fields. The operators a and b are annihilation operators of cavity and vibrational phonon fields respectively. The cavity-atom coupling coefficient is g(x) = gcos(kcx cosθc + φ), where kc(kj) is the wave number of the cavity mode (the laser fields), θc(θLj) is the angle determined by orientation of the cavity wave vector (laser field vectors) with respect to the axis of motion and 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 ξ=h¯/2Mν denotes the size of the ground-state wave packet.

Fig. 1 (a) The setup of the cooling system. An atom confined inside a high-finesse optical resonator by a harmonic trap with frequency ν is transversally driven by two laser fields with Rabi frequencies Ω1 and Ω2 and couples to the cavity mode with strength g(x). The cavity field is weakly pumped by a laser with coupling strength Ωp and decays at the rate κ. θlj (θc) denotes the angle between the axis of the motion and the j-th laser (cavity) wave vector. (b)The relevant electronic level transitions. The two transverse laser fields drive transitions |gj〉 → |e〉, respectively, while the cavity field couples to the transition |g3〉 → |e〉. δj and δc3 are the detunings between the photon fields and the corresponding transitions and Δ is the detuning between the cavity field and the probe laser.

The density operator ρ of the system obeys the master equation
ddtρ=i[H,ρ]+atρ+𝒦ρ,
(4)
with the superoperators atρ and 𝒦ρ describing the dipole spontaneous emissions and the cavity decay respectively, which are given in the form
atρ=iγi2(2|gie|ρ˜|egi||ee|ρρ|ee|),𝒦ρ=κ2(2aρaaaρρaa),
(5)
where the damping rates are γi, κ and
ρ˜=11𝒩i(θ)exp(ikixcosθ)ρexp(ikixcosθ)dcosθ
(6)
describes the atomic motional recoil by the emission of a photon, with 𝒩i(θ) evaluated by taking into account of the geometry of the setup.

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

We are only interested in the case that the cavity field is sufficiently weakly driven, which means that the average photon number of the cavity mode is much smaller than unity. Namely we should take the parameters to satisfy |Ωp/2Δ+iκ|21, which allows us to truncate the Hilbert space of the system. This is practical in experiment, in which the typical value for |Ωp/2Δ+iκ|2 is of the order of 10−2 to 10−4[26

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).

, 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. 105153603 (2010). [CrossRef]

]. Hence it is feasible to investigate the cooling dynamics in the subspace comprised of at most one excitation of the cavity mode, i.e. the relevant Hilbert space spanned by the states
|e,0,|g1,0,|g2,0,|g3,0,|g3,1.
(7)
For later convenience we denote |e, 0〉 ≡ |e〉, |g1, 0〉 ≡ |g1〉, |g2, 0〉 ≡ |g2〉, |g3, 0〉 ≡ |g3〉 and |g3, 1〉 ≡ |1〉. Therefore, the Hamiltonian in Eq. (1) and the superoperators in Eq. (5) should be rewritten in the relevant space as
HL-cav=Ωp2(|g31|+|1g3|),Hat-cav=g(x)(|e1|+|1e|),
(8)
and
𝒦ρ=κ2(2|g31|ρ|1|g3|11|ρρ|11|),
(9)
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 74033816 (2006). [CrossRef]

]. The carrier transition can be eliminated via the three-photon process |g3〉 ↔ |1〉 ↔ |e〉 ↔ |g1〉, which simultaneously involves a cavity photon and two laser photons. This is because under the condition of three-photon resonance, the transition |g3〉 → |1〉 → |e〉 destructively interferes with the transition |g1〉 → |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

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]

]. In addition, the blue-sideband transition, which creates a quantum of harmonic center-of-mass motion, can also be eliminated via another three-photon process with the help of the additional transition |g2〉 → |e〉. Finally, the ground-state cooling of the trapped atom can be achieved. Detailed discussions are presented in Sec. 5.

Fig. 2 Sketch of the level configuration described by the states in Eq. (7). The ground state |g2〉 is the additional level with respect to transition paths |g3〉 ↔ |1〉 ↔ |e〉 ↔ |g1〉 presented in the cavity-induced single EIT system, which is designed to cancel the carrier transition.

Here we focus on the Lamb-Dicke regime, where the atom is localized on a length scale much smaller than the light wave length [32

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

], and the regime is characterized by the Lamb-Dicke parameter η=kh¯/2Mν. For later convenience we define ηlj = η cosθlj and ηc = η cosθc, 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

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

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]

]. We expand the total Hamiltonian up to the first order of the Lamb-Dicke parameters by performing Taylor expansion in η of the exponential terms exp(ikj cosθLjx) and the coupling function g(x) in Eq. (3). Thus the first-order term is obtained in the form H1 = V1(b + b), with
V1=jiΩjηlj2|egj|ηcgsinφ|e1|+h.c.,
(10)
and the zeroth-order term is given by
H0=Hat+Hcav+HL-cav+H0L-at+H0at-cav,
(11)
with H0L-at = g cosφ(|e〉〈1|+h.c.) and H0at-cav=jΩj2(|egj|+h.c.). Then Eq. (4) reduces as
ddtρ=0ρ+0Eρ+1ρ,
(12)
in which the Liouvillian operator
0ρ=i[H0,ρ]+𝒦ρ+0atρ,
(13)
with
0atρ=iγi2(2|gie|ρ|egi||ee|ρρ|ee|)
(14)
describes the laser-atom and cavity-atom interactions and dissipations from the cavity field and the excited atomic state in absence of the coupling to the phonon filed, and
0Eρ=i[Hext,ρ]
(15)
describes the (undamped) vibrational motion of the atom in the trap. The interaction between internal atomic states and cavity field and the phonon mode is described by the Liouvillian operator
1ρ=i[H1,ρ].
(16)

We now follow the procedure proposed by Cirac 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]

] by use of the second-order perturbation theory with respect to the Lamb-Dicke parameters to obtain the master equation of the phonon mode from Eq. (12). Straightforward calculations yield the following equation
ddtμ=[S(ν)+D](bμbbbμ)+[S(ν)+D](bμbbbμ)+h.c.,
(17)
where 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
S(ν)=0dτeiντV1(τ)V1(0)s,D=iαiγi2ηi2Tr{σegiσgieρs},
(18)
where S(ν) is the two-time correlation function of atomic and cavity operators and D is the diffusion term with
αi=11dcosθcos2θ𝒩i(θ)
(19)
indicating the angular dispersion of the atom momentum due to the spontaneous emission of photons, which takes 25 for usual dipole dissipation transitions [32

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

]. Here the transition operators σmn = |m〉〈n| are expressed in terms of a complete set of states {|m〉} = {|e〉, |g1〉, |g2〉, |g3〉, |1〉}. From Eq. (17) we can directly derive the rate equation for the average phonon number 〈n〉, namely
n˙=(AA+)n+A+,
(20)
where
A±=2Re{S(±ν)+D}
(21)
are the heating (A+) and cooling (A) coefficients. By using the quantum regression theorem [36

36. J. S. Peng and G. X. Li, Introduction to Modern Quantum Optics (Singapore: World Scientific) (1998).

], 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 〈nst and the cooling rate W are obtained as
nst=A+AA+,W=AA+.
(22)

For the dipole dissipation transitions we assume that there only exists the spontaneous decay along the transition |e〉 → |g3〉, 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

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

]. Hence in the following we denote γ3 = γ.

In the experiment of the cooling for a trapped 199Hg+-ion, which is in tripod configuration with upper state P1/2(F′ = 1, mF = 1) and three lower states S1/2(F = 1, mF = 1, 0) and S1/2(F = 0, mF = 0), it is practical to take the trap frequency 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]

]. When the ion couples to a high-finesse cavity ( = 1.2 × 106), the experimental parameters in units of ν: γ = 10ν, κ = 0.2ν, g = 10ν, Ω1 = 12ν, Δ = ν, δ1 = 60ν, δc3 = 59ν, θc = θli = 0, φ = π/3, Ωp/2 = 0.1, α3 = 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. 105153603 (2010). [CrossRef]

]. We numerically plot heating and cooling coefficients A±/η2 as functions of δ2 and Ω2 in Fig. 3, where the heating coefficient is pretty close to zero as indicated by the dot line at δ2 = 59ν regardless of the value of Ω2. 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

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

]. In addition, the cooling coefficient Ain the plot is comparable with that obtained in the cavity-induced single EIT scheme.

Fig. 3 The numerical calculations of heating and cooling coefficients A±/η2 as functions of δ2 and Ω2 with the parameters in units of ν: γ = 10ν, κ = 0.2ν, g = 10ν, Ω1 = 12ν, Ωp/2 = 0.1ν, δ1 = 60ν, δc3 = 59ν, Δ = ν, θc = θli = 0, φ = π/3, α = 2/5. The dash line in A+ shows the zero heating coefficient.

In order to gain more physical insight into the cooling scheme, we follow the perturbation treatment adopted in 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]

] in the following, where the cavity is weakly driven, namely, Rabi frequency Ωp 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 Ωp and η by neglecting the terms of the order η4Ω2, η2Ω4 and higher.

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

To obtain the analytical expressions of heating and cooling coefficients A± in Eq. (21), we will make use of the perturbation method with respect to Ωp. In the following we denote ρmn(0), ρmn(1) and ρmn(2) to represent the density matrix elements in the orders of Ωp0, Ωp1 and Ωp2 respectively. Cooling and heating coefficients are obtained by taking the Fourier transform of two-time correlation functions σmn(i)(τ)σkl(i)(0), where σmn(i) are the transition operators appeared in V1 in Eq. (10) and the index i also indicates the order of Ωp. These two-time correlation functions can be calculated from single-time averages σmn(i)(τ) by applying the quantum regression theorem [37

37. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge1997).

, 38

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]

], and the single-time averages σmn(i)(τ) obey the same equations as ρnm(i)(τ) (see Appendix A). In this paper we use the method of Laplace transform to the evolution equations to solve the time-dependent density matrix elements and Laplace transform expressions are defined as
ρnm(i)(s)=0esτρnm(i)(τ)dτ.
(23)

4.1. The density matrix in the zeroth order of Ωp

In this subsection, we first calculate the density matrix elements in the zeroth order of Ωp from the evolution equations in appendix A. Then a separate equation
ρ˙g3g3(0)=γρee(0)
(24)
is obtained. Steady-state solutions for the density matrix elements are obvious that ρg3g3(0)()=1 while all the other terms are equal to zero. Thus the steady state of the atomic states and cavity field is described by the density operator
ρ(0)()=|g3g3|.
(25)
Physically when there is no laser driving on the cavity mode, the cavity field is in vacuum and the atom stays in the ground state in the long time limit. Note that the Laplace transform expression for ρg3g3(0)(τ) is predicted as elastic Rayleigh scattering, which does not contribute to the two-time correlation functions in Eq. (18) [39

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 Ωp

To obtain the corrections of density matrix elements in the first order of Ωp, 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
ρ˙eg3(1)=[i(δc3+Δ)γ2]ρeg3(1)ijΩj2ρgjg3(1)igcosφρ1g3(1),
(26a)
ρ˙g1g3(1)=i[Δ(δ1δc3)]ρg1g3(1)iΩ12ρeg3(1),
(26b)
ρ˙g2g3(1)=i[Δ(δ2δc3)]ρg2g3(1)iΩ22ρeg3(1),
(26c)
ρ˙1g3(1)=(iΔκ2)ρ1g3(1)igcosφρeg3(1)iΩp2ρg3g3(0),
(26d)
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 ρg3g3(0)() into the equations, which are shown in the following
ρeg3(1)()=Ωp/2f(Δ)gcosφ(δc3+Δδ1)(δc3+Δδ2),
(27a)
ρg1g3(1)()=Ωp/2f(Δ)gcosφΩ12(δc3+Δδ2),
(27b)
ρg2g3(1)()=Ωp/2f(Δ)gcosφΩ22(δc3+Δδ1),
(27c)
ρ1g3(1)()=Ωp/2f(Δ)[(δc3+Δδ1)(δc3+Δ+iγ2)(δc3+Δδ2)(δc3+Δδ2)Ω12/4(δc3+Δδ1)Ω22/4],
(27d)
proportional to Ωp and inversely proportional to
f(Δ)=(iκ2+Δ)[(δc3+Δδ1)(δc3+Δ+iγ2)(δc3+Δδ2)(δc3+Δδ2)Ω12/4(δc3+Δδ1)Ω22/4]g2cos2φ(δc3+Δδ1)(δc3+Δδ2).
(28)

The excitation spectra of the cavity and the atom, i.e. the rate of the photon emission respectively from the cavity and atom as functions of the detuning Δ, are evaluated for the stationary state of the internal atom-cavity system. They are respectively proportional to the probability that the cavity contains one photon and the atom stays at excited state and denoted as Sκ and Sγ,
Sκ|ρ1g3(1)()|2,
(29a)
Sγ|ρeg3(1)()|2.
(29b)
It can be verified from Eq. (27a) that the atomic excitation spectrum exhibits zero value when the detuning Δ satisfies the relation
δc3+Δδ1=0,
(30)
or
δc3+Δδ2=0.
(31)
In fact the two relations above correspond to two independent three-photon resonance conditions, under which the transition |g3〉 → |1〉 → |e〉 mediated by the cavity field can destructively interfere with either the transition |g1〉 → |e〉 or |g2〉 → |e〉. The destructive interference can prohibit the population in state |e〉. Without loss of generality, we take the relation δc3 + Δ − δ1 = 0 and leave the additional transition |g2〉 ↔ |e〉 to be appropriately tuned so that the heating processes due to blue-sideband transitions can be eliminated further.

To obtain the Laplace transform expressions for these first-order density matrix elements ρeg3(1)(s), ρg1g3(1)(s), ρg2g3(1)(s), ρ1g3(1)(s), we apply the procedure of Laplace transform given in Eq. (23) on the evolution Eqs. (26a)(26d), which are given from
ρ(1)(s)=1s(1)ρ(1)(),
(32)
with the Liouvillian operator (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 Ωp

To obtain the corrections on density matrix elements of the second order of Ωp, 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
ρ˙gjgj(2)=γjδj3ρee(2)iΩj2(ρegj(2)ρgje(2)),
(33a)
ρ˙11(2)=κρ11(2)+i(Ωp2ρ1g3(1)gcosφρe1(2))+c.c,
(33b)
ρ˙ee(2)=γρee(2)+i(Ω12ρeg1(2)+Ω22ρeg2(2)+gcosφρe1(2))+c.c,
(33c)
ρ˙g1g2(2)=i(δ2δ1)ρg1g2(2)iΩ12ρeg2(2)+iΩ22ρg1e(2),
(33d)
ρ˙eg1(2)=(iδ1γ2)ρeg1(2)+iΩ12(ρee(2)ρg1g1(2))iΩ22ρg2g1(2)igcosφρ1g1(2),
(33e)
ρ˙eg2(2)=(iδ2γ2)ρeg2(2)+iΩ22(ρee(2)ρg2g2(2))iΩ12ρg1g2(2)igcosφρ1g2(2),
(33f)
ρ˙e1(2)=(iδc3γ+κ2)ρe1(2)+igcosφ(ρee(2)ρ11(2))ijΩj2ρgj1(2)+iΩp2ρeg3(1),
(33g)
ρ˙g11(2)=[i(δc3δ1)κ2]ρg11(2)iΩ12ρe1(2)+igcosφρg1e(2)+iΩp2ρg1g3(1),
(33h)
ρ˙g21(2)=[i(δc3δ2)κ2]ρg21(2)iΩ22ρe1(2)+igcosφρg2e(2)+iΩp2ρg2g3(1).
(33i)
It is obvious that the average value of the operators that appear in V1 in Eq. (10) are included in this group of elements.

For the second-order steady-state solution 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 Ωp2.

The Laplace transform expression for second-order density matrix elements ρ(2)(s) is also obtained by Laplace transforming the Eqs. (33a)(33i) and given by
ρ(2)(s)=1s(2)[ρ(2)()+1s(1)ρ(1)()],
(34)
where (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

The 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
D=γ2η32ασee(2)().
(35)
It is obvious when the parameters fulfill the three-photon resonance condition δc3 + Δ − δ1 = 0 as mentioned in Sec. 4.2, the steady-state expression of σee(2)() shown in Eq. (B1) becomes zero due to the destructive interference between the transition paths |g3〉 → |1〉 → |e〉 and |g1〉 → |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 |g2〉 ↔ |e〉.

Two-time correlation functions S(ν) given in Eq. (18) can be calculated from 〈V1(s)V1(0)〉s by substituting s with . Therefore, by applying the quantum regression theorem, S(ν) can be derived from the single-time average V1(s)s=jiΩjηlj2[ρgje(2)(s)ρegj(2)(s)]ηcgsinφ[ρ1e(2)(s)+ρe1(2)(s)], which can be obtained from Eq. (34). After some calculations the analytical expressions for heating and cooling coefficients are given by
A±=|ε|2η˜2g2ν2|f˜(Δν)|2{γ|Δν+iκ/2|2+κg2cos2φ},
(36)
where the small parameter |ε|2=|Ωp/2Δ+iκ/2|2 indicates the mean number of intracavity photons when no atom is presented, and
η˜2=ηl12cos2φ+ηc2sin2φ
(37)
is the compound Lamb-Dicke parameter combining geometries of the laser 1 and cavity field. The function f̃(Δ ∓ ν) takes the form
f˜(Δν)=±νg2cos2φ+(Δν+iκ/2)[ν(δ1ν+iγ/2)(Ω1/2)2+ε±],
(38)
with
ε±=±ν(Ω2/2)2δ1νδ2.
(39)

The increase and decrease of phonon number are accompanied by either spontaneous decay of the excited state |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.

To explore quantum interference effects in this cooling scheme, we substitute the explicit expression of f̃(Δ ∓ ν) into heating and cooling coefficients A± and rewrite A± into a much simpler form
A±=|ε|2η˜2g2γ(1+C±)(δ1νδ2)2γ24(1+C±)2(δ1νδ2)2+{Ω224+[γκC±(Δν)±νδ1Ω124ν](δ1νδ2)}2=|ε|2η˜2g2γ(1+C±)γ24(1+C±)2+[1ν(Ω124ε±)±νδ1+γκC±(Δν)]2,
(40)
where the parameter
C±=C(κ/2)2(Δν)2+(κ/2)2
(41)
is proportional to the single-atom cooperativity C=g2cos2φκγ/4[40

40. H. J. Kimble, in Cavity Quantum Electrodynamics, ed. P. R. Berman, (Academic Press, New York) (1994).

]. To reflect the mechanical effects on the atomic motion induced by the cavity field, the system is in the strong coupling regime, i.e. 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

For the heating and cooling coefficients in Eq. (40), ε± indicate mechanical effects caused by the additional transition |g2〉 ↔ |e〉. If the laser field that drives this additional transition is turned off, i.e. Ω2 = 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.

In order to look into the mechanical effects induced by the additional transition, we plot the analytical expressions of heating and cooling coefficients A±/η2 given by Eq. (40) as functions of Ω2 and δ2 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.

Fig. 4 The analytical plots of heating and cooling coefficients A±/η2 as functions of δ2 and Ω2 with the parameters in units of ν: γ = 10ν, κ = 0.2ν, g = 10ν, Ω1 = 12ν, Ωp/2 = 0.1ν, δ1 = 60ν, δc3 = 59ν, Δ = ν, θc = θli = 0, φ = π/3. The dot line shows the zero heating coefficient.
Fig. 5 The sketch of heating processes. The states are at phonon state |n〉 in (a) and |n + 1〉 in (b). By using cavity-induced double EIT, i.e. the quantum destructive interferences between excitation paths |g3, n〉 → |1, n〉 → |e, n〉 and |g1, n〉 → |e, n〉, and between excitation paths |g3, n〉 → |1, n〉 → |e, n + 1〉 and |g2, n + 1〉 → |e, n + 1〉 respectively, carrier- and blue-sideband transitions are eliminated.

5.1. The cancellation of heating processes

Generally there exist two kinds of heating sources: carrier- and blue-sideband transitions [22

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

]. In Fig. 5, the carrier transition |g3, 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 ηcg sinφ 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 |g1, n〉 → |e, n〉 when the three-photon resonance condition δc3 + Δ − δ1 = 0 is fulfilled. The process is shown by the red lines in Fig. 5, which confirms the first cavity-induced EIT.

Meanwhile, the blue-sideband transition |g3, 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 (δ1νδ2)2. It is clear that if we take the relation δ2 = δ1ν, 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 |g2, n + 1〉 becomes equal to that for the state |g1, n〉 when the parameters satisfy the relation δ2 = δ1ν. Therefore, with the help of the additional transition |g2, 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]

], in which blue-sideband heating relevant to 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.

In addition, the physics of the scheme can be interpreted by finding out an invariant dark state. It can be verified that the density operator
ρdark=|ΨΨ|,
(42)
with
|Ψ=|g3,0+Ωp/2Δ+iκ/2|1,0ΩpΩ1gcosφΔ+iκ/2|g1,0+Ωpg/Ω2Δ+iκ/2(iη˜l1cosφη˜csinφ)|g2,1,
(43)
is invariant up to the order Ωp2η2 as it satisfies the relation
(0+0E+1)ρdark=0.
(44)
Here we omit normalization coefficient in state |Ψ〉. As discussed above, by using the phenomenon of cavity-induced double EIT, carrier- and blue-sideband transitions can be eliminated, and finally the motional ground state for trapped atom can be achieved in the leading-order Lamb-Dicke parameters.

5.2. Analysis of the cooling coefficient A while heating processes are eliminated

The cooling efficiency is also concerned with the cooling rate besides the cooling limit, therefore one has to analyze the cooling coefficient A given in Eq. (40) under the condition A+ = 0. In the case δ2 = δ1ν, the term ε that characterizes the mechanical effect caused by the additional transition |g2〉 ↔ |e〉 becomes Ω228. The cooling coefficient A changes into
A=|ε|2η˜2g2γ(1+C)γ24(1+C)2+[1ν(Ω124+Ω228)νδ1+γκC(Δ+ν)]2.
(45)
Cooling is enhanced when the denominator of rate A in Eq. (45) becomes minimal, i.e. the parameters satisfy the following condition
Ω228ν=δ1Ω124ν+ν(Δ+ν)g2cos2φ(Δ+ν)2+(κ/2)2.
(46)
By now the coupling strength Ω2 and detuning δ2 for the additional transition |g2〉 ↔ |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. 105153603 (2010). [CrossRef]

]. Then the maximal cooling coefficient A becomes
A=|ε|2η˜2g2γ4(1+C),
(47)
which is the same as the maximal cooling coefficient of 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]

]. The cooling rate 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 |g2〉 ↔ |e〉. In Fig. 6 we plot the cooling coefficient A given in Eq. (45) as the function of Ω2 with the parameters shown in depiction, which have already satisfied the condition A+ = 0. When A/η2 obtains the maximum value 0.35ν, the corresponding driving strength Ω2 = 10ν, which is exactly the value calculated from the Eq. (46).

Fig. 6 The cooling coefficient A/η2 as a function of Ω2 with the parameters in units of ν: γ = 10ν, κ = 0.2ν, g = 10ν, Ω1 = 12ν, Ωp/2 = 0.1ν, δ1 = 60ν, δ2 = 59ν, δc3 = 59ν, Δ = ν, θc = θli = 0, φ = π/3.

6. Conclusions

By using cavity-induced double EIT, we have presented an efficient ground-state laser cooling scheme for a trapped atom comprised of four levels in the tripod configuration, which is confined inside a high-finesse cavity. Carrier transition |g3, n〉 → |1, n〉 → |e, n〉 mediated by a cavity photon can be eliminated due to the quantum destructive interference with the transition |g1, 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 |g3, 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 |g2, 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

Following the procedure of quantum regression theorem [36

36. J. S. Peng and G. X. Li, Introduction to Modern Quantum Optics (Singapore: World Scientific) (1998).

], the two-time correlation functions 〈σmn(t)σkl(0)〉s obey the same equation as ρnm(t), which are given in the Eq. (A1), with the initial values 〈σmn(0)σkl(0)〉s = δnkρlm(∞) and ρlm(∞) are the steady-state solutions. The evolution equations for time-dependent density matrix elements ρmn(t) are given by
ρ˙ee=γρee+i(Ω12ρeg1+Ω22ρeg2+gcosφρe1)+c.c.,ρ˙gjgj=γjρeeiΩj2(ρegjρgje),ρ˙11=κρ11+i(Ωp2ρ1g3gcosφρe1)+c.c.,ρ˙eg1=(iδ1γ2)ρeg1+iΩ12(ρeeρg1g1)iΩ22ρg2g1igcosφρ1g1,ρ˙eg2=(iδ2γ2)ρeg2+iΩ22(ρeeρg2g2)iΩ12ρg1g2igcosφρ1g2,ρ˙e1=(iδc3γ+κ2)ρe1+igcosφ(ρeeρ11ijΩj2ρgj1+iΩp2ρeg3),ρ˙g1g2=i(δ2δ1)ρg1g2iΩ12ρeg2+iΩ22ρg1e,ρ˙g11=[i(δc3δ1)κ2]ρg11iΩ12ρe1+igcosφρg1e+iΩp2ρg1g3,ρ˙g21=[i(δc3δ2)κ2]ρg21iΩ22ρe1+igcosφρg2e+iΩp2ρg2g3,ρ˙eg3=[i(δc3+Δ)γ2]ρeg3ijΩj2ρgjg3igcosφρ1g3+iΩp2ρe1,ρ˙g1g3=i[Δ(δ1δc3)]ρg1g3iΩ12ρeg3+iΩp2ρg11,ρ˙g2g3=i[Δ(δ2δc3)]ρg2g3iΩ22ρeg3+iΩp2ρg21,ρ˙1g3=(iΔκ2)ρ1g3igcosφρeg3iΩp2(ρg3g3ρ11).
(A1)

B. Steady-state solutions of the second-order Ωp

Steady-state solutions of the nonzero density matrix elements ρmn(2)() in the second-order Ωp, which are calculated by substituting steady-state solutions ρmn(1)() given in Eqs. (27a)(27d) into the equations given in (33a)(33i), are obtained as
ρee(2)()=(Ωp/2)2|f(Δ)|2g2cos2φ(δc3+Δδ1)2(δc3+Δδ2)2,ρg1g1(2)()=(Ωp/2)2|f(Δ)|2g2cos2φ(Ω1/2)2(δc3+Δδ2)2,ρg2g2(2)()=(Ωp/2)2|f(Δ)|2g2cos2φ(Ω2/2)2(δc3+Δδ1)2,ρ11(2)()=(Ωp/2)2|f(Δ)|2{(γ/2)2(δc3+Δδ1)2(δc3+Δδ2)2+[(δc3+Δ)(δc3+Δδ1)(δc3+Δδ2)(Ω1/2)2(δc3+Δδ2)(δc3+Δδ1)(Ω2/2)2]2},ρeg1(2)()=(Ωp/2)2|f(Δ)|2g2cos2φ(Ω1/2)(δc3+Δδ1)(δc3+Δδ2)2,ρeg2(2)()=(Ωp/2)2|f(Δ)|2g2cos2φ(Ω2/2)(δc3+Δδ1)2(δc3+Δδ2),ρe1(2)()=(Ωp/2)2|f(Δ)|2gcosφ(δc3+Δδ1)(δc3+Δδ2){(δc3+Δiγ2)(δc3+Δδ1)(δc3+Δδ2)(Ω1/2)2(δc3+Δδ2)(δc3+Δδ1)(Ω2/2)2},ρg1g2(2)()=(Ωp/2)2|f(Δ)|2g2cos2φ(Ω1/2)(Ω2/2)(δc3+Δδ1)(δc3+Δδ2),ρg11(2)()=(Ωp/2)2|f(Δ)|2gcosφ(Ω1/2)(δc3+Δδ2){(δc3+Δiγ2)(δc3+Δδ1)(δc3+Δδ2)(Ω1/2)2(δc3+Δδ2)(δc3+Δδ1)(Ω2/2)2},ρg21(2)()=(Ωp/2)2|f(Δ)|2gcosφ(Ω2/2)(δc3+Δδ1){(δc3+Δiγ2)(δc3+Δδ1)(δc3+Δδ2)(Ω1/2)2(δc3+Δδ2)(δc3+Δδ1)(Ω2/2)2},
(B1)
which are proportional to Ωp2.

Acknowledgments

This work is supported by the National Basic Research Program of China (Grant No. 2012CB921602), the National Natural Science Foundation of China (Grant Nos. 11074087 and 61275123), the Natural Science Foundation of Hubei Province (Grant No. 2010CDA075), the Nature Science Foundation of Wuhan City (Grant No. 201150530149), and the Key Laboratory of Advanced Micro-Structure Materials of Tongji University.

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. 103259–328 (1998). [CrossRef]

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 62, 042305 (2000). [CrossRef]

3.

E. Buks and B. Yurke, “Mass detection with a nonlinear nanomechanical resonator,” Phys. Rev. E 74, 046619 (2006). [CrossRef]

4.

J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland, “Laser-Cooled-Atomic Frequency Standard,” Phys. Rev. Lett. 54, 1000–1003 (1985). [CrossRef] [PubMed]

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]

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. 75, 4011–4014 (1995). [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]

9.

V. Vuletić and S. Chu, “Laser Cooling of Atoms, Ions, or Molecules by Coherent Scattering,” Phys. Rev. Lett. 84, 3787–3790 (2000). [CrossRef]

10.

P. R. Berman, Cavity Quantum Electrodynamics (Academic Press, New York) (1994).

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]

13.

D. R. Leibrandt, J. Labaziewicz, V. Vuletić, and I. L. Chuang, “Cavity Sideband Cooling of a Single Trapped Ion,” Phys. Rev. Lett. 103, 103001 (2009). [CrossRef] [PubMed]

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]

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. 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]

20.

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

21.

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

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]

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]

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 603225–3228 (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 65043805 (2002). [CrossRef]

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. 105153603 (2010). [CrossRef]

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 74033816 (2006). [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]

32.

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

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]

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]

36.

J. S. Peng and G. X. Li, Introduction to Modern Quantum Optics (Singapore: World Scientific) (1998).

37.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge1997).

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]

39.

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

40.

H. J. Kimble, in Cavity Quantum Electrodynamics, ed. P. R. Berman, (Academic Press, New York) (1994).

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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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.103259–328 (1998). [CrossRef]
  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. A62, 042305 (2000). [CrossRef]
  3. E. Buks and B. Yurke, “Mass detection with a nonlinear nanomechanical resonator,” Phys. Rev. E74, 046619 (2006). [CrossRef]
  4. J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland, “Laser-Cooled-Atomic Frequency Standard,” Phys. Rev. Lett.54, 1000–1003 (1985). [CrossRef] [PubMed]
  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]
  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.75, 4011–4014 (1995). [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]
  9. V. Vuletić and S. Chu, “Laser Cooling of Atoms, Ions, or Molecules by Coherent Scattering,” Phys. Rev. Lett.84, 3787–3790 (2000). [CrossRef]
  10. P. R. Berman, Cavity Quantum Electrodynamics (Academic Press, New York) (1994).
  11. P. Domokos and H. Ritsch, “Mechanical effects of light in optical resonators,” J. Opt. Soc. Am. B20, 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,” Nature428, 50–52 (2004). [CrossRef] [PubMed]
  13. D. R. Leibrandt, J. Labaziewicz, V. Vuletić, and I. L. Chuang, “Cavity Sideband Cooling of a Single Trapped Ion,” Phys. Rev. Lett.103, 103001 (2009). [CrossRef] [PubMed]
  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,” Nature465, 755–758 (2010). [CrossRef] [PubMed]
  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.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. B89, 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. A65, 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]
  20. G. Nikoghosyan and M. Fleischhauer, “Photon-Number Selective Group Delay in Cavity Induced Transparency,” Phys. Rev. Lett.105, 013601 (2010). [CrossRef] [PubMed]
  21. M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New J. Phys.14, 023002 (2012). [CrossRef]
  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. A85, 053420 (2012). [CrossRef]
  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. A72, 053408 (2005). [CrossRef]
  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. A603225–3228 (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. A65043805 (2002). [CrossRef]
  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.105153603 (2010). [CrossRef]
  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. A74033816 (2006). [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. A67, 013811 (2003). [CrossRef]
  32. S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys.58, 699–739 (1986). [CrossRef]
  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. B73, 807–814 (2001). [CrossRef]
  34. J. Javanainen, M. Lindberg, and S. Stenholm, “Laser cooling of trapped ions: dynamics of the final stages,” J. Opt. Soc. Am. B1, 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. A46, 2668–2681 (1992). [CrossRef] [PubMed]
  36. J. S. Peng and G. X. Li, Introduction to Modern Quantum Optics (Singapore: World Scientific) (1998).
  37. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge1997).
  38. Z. Yi, W. J. Gu, and G. X. Li, “Sideband cooling of atoms with the help of an auxiliary transition,” Phys. Rev. A86, 055401 (2012). [CrossRef]
  39. P. Rabl, “Cooling of mechanical motion with a two-level system: The high-temperature regime,” Phys. Rev. B82, 165320 (2010). [CrossRef]
  40. H. J. Kimble, in Cavity Quantum Electrodynamics, ed. P. R. Berman, (Academic Press, New York) (1994).
  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. A83, 031804(R) (2011). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited