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

Zhen Yi; Wen-ju Gu; Gao-xiang Li

doi:10.1364/OE.21.003445

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], quantum information processing [2], precision detection of mass [3] and investigations in precision spectral and time and frequency standards [4]. All conventional methods to laser cooling atoms [5–7] 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, 9] 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], 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, 11]. Experimentally the cavity cooling technique has been successfully realized for cooling neutral atoms or ions confined in an intracavity dipole trap [12–15]. This technique also opens the possibility to cool down complex quantum systems with scalable number of degrees of freedom, such as large molecules [16, 17].

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], substantial reduction of classical and quantum phase noise of the beat note of optical oscillators [19], quantum optical transistor [14] and the engineering of Fock state superpositions of flying light pulses [20]. Via exploring properties produced by the combination of cavity-enhanced scattering and EIT, Bienert et al.[21] 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], 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], the scheme based on the existence of a joint dark state in EIT and Stark-shift cooling [23], the dark-state scheme by exploiting the property of standing waves [24]. 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] 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, 28]. 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 |g_i〉(i = 1, 2, 3) with energy frequencies ω_e and ω_gi 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]. Dipole-allowed transitions |e〉 ↔ |g_j〉(j = 1, 2) are irradiated by two laser fields with frequencies ω_Lj and Rabi frequencies Ω_j respectively and the third ground state couples to the excited state through the cavity field with frequency ω_c 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 ω_p and driving strength Ω_p. Thus the Hamiltonian of the system in a frame rotating with the lasers’ frequencies is given by

H = H_{at} + H_{cav} + H_{ext} + H_{L-cav} + H_{at-cav} + H_{L-at},

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

\begin{array}{l} H_{at} = - δ_{c 3} | e 〉 〈 e | + \sum_{j} (δ_{j} - δ_{c 3}) | g_{j} 〉 〈 g_{j} | + Δ | g_{3} 〉 〈 g_{3} |, \\ H_{cav} = - Δ a^{†} a, \\ H_{ext} = ν (b^{†} b + \frac{1}{2}), \end{array}

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 |g_j〉 → |e〉 are respectively expressed as

\begin{array}{l} H_{L-cav} = \frac{Ω_{p}}{2} (a + a^{†}), \\ H_{at-cav} = g (x) (| e 〉 〈 g_{3} | a + | g_{3} 〉 〈 e | a^{†}), \\ H_{L-at} = \sum_{j} \frac{Ω_{j}}{2} (| e 〉 〈 g_{j} | exp (i k_{j} cos θ_{L j} x) + h . c .) . \end{array}

Here δ_c3 = ω_c − (ω_e − ω_g₃) is the detuning of the cavity frequency ω_c to the atomic transition |e〉 ↔ |g₃〉, δ_j = ω_Lj − (ω_e − ω_gj) indicates the detuning of the j-th laser frequency ω_Lj to the corresponding atomic dipole transition |e〉 ↔ |g_j〉 and Δ = ω_p − ω_c 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(k_cx cosθ_c + φ), where k_c(k_j) is the wave number of the cavity mode (the laser fields), θ_c(θ_Lj) 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 = \sqrt{\bar{h} / 2 M ν}

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 Ω_p and decays at the rate κ. θ_lj (θ_c) 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 |g_j〉 → |e〉, respectively, while the cavity field couples to the transition |g₃〉 → |e〉. δ_j and δ_c₃ are the detunings between the photon fields and the corresponding transitions and Δ is the detuning between the cavity field and the probe laser.

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The density operator ρ of the system obeys the master equation

\frac{d}{d t} ρ = - i [H, ρ] + ℒ_{at} ρ + 𝒦 ρ,

with the superoperators ℒ_atρ and 𝒦ρ describing the dipole spontaneous emissions and the cavity decay respectively, which are given in the form

\begin{array}{l} ℒ_{at} ρ = \sum_{i} \frac{γ_{i}}{2} (2 | g_{i} 〉 〈 e | \tilde{ρ} | e 〉 〈 g_{i} | - | e 〉 〈 e | ρ - ρ | e 〉 〈 e |), \\ 𝒦 ρ = \frac{κ}{2} (2 a ρ a^{†} - a^{†} a ρ - ρ a a^{†}), \end{array}

where the damping rates are γ_i, κ and

\tilde{ρ} = \int_{- 1}^{1} 𝒩_{i} (θ) exp (i k_{i} x cos θ) ρ exp (- i k_{i} x cos θ) d cos θ

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

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 ${| \frac{Ω_{p} / 2}{Δ + i κ} |}^{2} ≪ 1$ , which allows us to truncate the Hilbert space of the system. This is practical in experiment, in which the typical value for ${| \frac{Ω_{p} / 2}{Δ + i κ} |}^{2}$ is of the order of 10⁻² to 10⁻⁴[26, 29]. 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 〉, | g_{1}, 0 〉, | g_{2}, 0 〉, | g_{3}, 0 〉, | g_{3}, 1 〉 .

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

\begin{array}{l} H_{L-cav} = \frac{Ω_{p}}{2} (| g_{3} 〉 〈 1 | + | 1 〉 〈 g_{3} |), \\ H_{at-cav} = g (x) (| e 〉 〈 1 | + | 1 〉 〈 e |), \end{array}

and

𝒦 ρ = \frac{κ}{2} (2 | g_{3} 〉 〈 1 | ρ | 1 〉 | g_{3} 〉 - | 1 〉 〈 1 | ρ - ρ | 1 〉 〈 1 |),

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]. 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, 31]. 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.

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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], and the regime is characterized by the Lamb-Dicke parameter $η = k \sqrt{\bar{h} / 2 M ν}$ . For later convenience we define η_lj = η cosθ_lj and η_c = η cosθ_c, 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, 35]. 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(ik_j cosθ_Ljx) and the coupling function g(x) in Eq. (3). Thus the first-order term is obtained in the form H₁ = V₁(b + b^†), with

V_{1} = \sum_{j} \frac{i Ω_{j} η_{l j}}{2} | e 〉 〈 g_{j} | - η_{c} g sin φ | e 〉 〈 1 | + h . c .,

and the zeroth-order term is given by

H_{0} = H_{at} + H_{cav} + H_{L-cav} + H_{0 L-at} + H_{0 at-cav},

with H_0L-at = g cosφ(|e〉〈1|+h.c.) and

H_{0 at-cav} = \sum_{j} \frac{Ω_{j}}{2} (| e 〉 〈 g_{j} | + h . c .)

. Then Eq. (4) reduces as

\frac{d}{d t} ρ = ℒ_{0} ρ + ℒ_{0 E} ρ + ℒ_{1} ρ,

in which the Liouvillian operator

ℒ_{0} ρ = - i [H_{0}, ρ] + 𝒦 ρ + ℒ_{0 at} ρ,

with

ℒ_{0 at} ρ = \sum_{i} \frac{γ_{i}}{2} (2 | g_{i} 〉 〈 e | ρ | e 〉 〈 g_{i} | - | e 〉 〈 e | ρ - ρ | e 〉 〈 e |)

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

ℒ_{0 E} ρ = - i [H_{ext}, ρ]

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

ℒ_{1} ρ = - i [H_{1}, ρ] .

We now follow the procedure proposed by Cirac et al.[35] 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

\frac{d}{d t} μ = [S (- ν) + D] (b μ b^{†} - b^{†} b μ) + [S (ν) + D] (b^{†} μ b - b b^{†} μ) + h . c .,

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

\begin{matrix} S (ν) = \int_{0}^{\infty} d τ e^{- i ν τ} {〈 V_{1} (τ) V_{1} (0) 〉}_{s}, \\ D = \sum_{i} α_{i} \frac{γ_{i}}{2} η_{i}^{2} Tr {σ_{e g_{i}} σ_{g_{i} e} ρ_{s}}, \end{matrix}

where S(ν) is the two-time correlation function of atomic and cavity operators and D is the diffusion term with

α_{i} = \int_{- 1}^{1} d cos θ {cos}^{2} θ 𝒩_{i} (θ)

indicating the angular dispersion of the atom momentum due to the spontaneous emission of photons, which takes

\frac{2}{5}

for usual dipole dissipation transitions [32]. Here the transition operators σ_mn = |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

〈 \dot{n} 〉 = - (A_{-} - A_{+}) 〈 n 〉 + A_{+},

where

A_{\pm} = 2 Re {S (\pm ν) + D}

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〉_st and the cooling rate W are obtained as

{〈 n 〉}_{s t} = \frac{A_{+}}{A_{-} - A_{+}}, W = A_{-} - A_{+} .

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]. 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 P_1/2(F′ = 1, m_F = 1) and three lower states S_1/2(F = 1, m_F = 1, 0) and S_1/2(F = 0, m_F = 0), it is practical to take the trap frequency ν = 1MHz [33]. When the ion couples to a high-finesse cavity (ℱ = 1.2 × 10⁶), the experimental parameters in units of ν: γ = 10ν, κ = 0.2ν, g = 10ν, Ω₁ = 12ν, Δ = ν, δ₁ = 60ν, δ_c₃ = 59ν, θ_c = θ_li = 0, φ = π/3, Ω_p/2 = 0.1, α₃ = α = 2/5 are also practical [29]. 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]. In addition, the cooling coefficient A₋in 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, Ω_p/2 = 0.1ν, δ₁ = 60ν, δ_c3 = 59ν, Δ = ν, θ_c = θ_li = 0, φ = π/3, α = 2/5. The dash line in A₊ shows the zero heating coefficient.

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In order to gain more physical insight into the cooling scheme, we follow the perturbation treatment adopted in Ref. [25] in the following, where the cavity is weakly driven, namely, Rabi frequency Ω_p 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 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 Ω_p. In the following we denote $ρ_{m n}^{(0)}$ , $ρ_{m n}^{(1)}$ and $ρ_{m n}^{(2)}$ to represent the density matrix elements in the orders of $Ω_{p}^{0}$ , $Ω_{p}^{1}$ and $Ω_{p}^{2}$ respectively. Cooling and heating coefficients are obtained by taking the Fourier transform of two-time correlation functions $〈 σ_{m n}^{(i)} (τ) σ_{k l}^{(i)} (0) 〉$ , where $σ_{m n}^{(i)}$ are the transition operators appeared in V₁ in Eq. (10) and the index i also indicates the order of Ω_p. These two-time correlation functions can be calculated from single-time averages $〈 σ_{m n}^{(i)} (τ) 〉$ by applying the quantum regression theorem [37, 38], and the single-time averages $〈 σ_{m n}^{(i)} (τ) 〉$ obey the same equations as $ρ_{n m}^{(i)} (τ)$ (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

ρ_{n m}^{(i)} (s) = \int_{0}^{\infty} e^{- s τ} ρ_{n m}^{(i)} (τ) d τ .

4.1. The density matrix in the zeroth order of Ω_p

In this subsection, we first calculate the density matrix elements in the zeroth order of Ω_p from the evolution equations in appendix A. Then a separate equation

{\dot{ρ}}_{g_{3} g_{3}}^{(0)} = γ ρ_{e e}^{(0)}

is obtained. Steady-state solutions for the density matrix elements are obvious that

ρ_{g_{3} g_{3}}^{(0)} (\infty) = 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)} (\infty) = | g_{3} 〉 〈 g_{3} | .

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

ρ_{g_{3} g_{3}}^{(0)} (τ)

is predicted as elastic Rayleigh scattering, which does not contribute to the two-time correlation functions in Eq. (18) [39].

4.2. The corrections for density matrix in the first order of Ω_p

To obtain the corrections of density matrix elements 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

{\dot{ρ}}_{e g_{3}}^{(1)} = [i (δ_{c 3} + Δ) - \frac{γ}{2}] ρ_{e g_{3}}^{(1)} - i \sum_{j} \frac{Ω_{j}}{2} ρ_{g_{j} g_{3}}^{(1)} - i g cos φ ρ_{1 g_{3}}^{(1)},

{\dot{ρ}}_{g_{1} g_{3}}^{(1)} = i [Δ - (δ_{1} - δ_{c 3})] ρ_{g_{1} g_{3}}^{(1)} - i \frac{Ω_{1}}{2} ρ_{e g_{3}}^{(1)},

{\dot{ρ}}_{g_{2} g_{3}}^{(1)} = i [Δ - (δ_{2} - δ_{c 3})] ρ_{g_{2} g_{3}}^{(1)} - i \frac{Ω_{2}}{2} ρ_{e g_{3}}^{(1)},

{\dot{ρ}}_{1 g_{3}}^{(1)} = (i Δ - \frac{κ}{2}) ρ_{1 g_{3}}^{(1)} - i g cos φ ρ_{e g_{3}}^{(1)} - i \frac{Ω_{p}}{2} ρ_{g_{3} g_{3}}^{(0)},

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

ρ_{g_{3} g_{3}}^{(0)} (\infty)

into the equations, which are shown in the following

ρ_{e g_{3}}^{(1)} (\infty) = \frac{Ω_{p} / 2}{f (Δ)} g cos φ (δ_{c 3} + Δ - δ_{1}) (δ_{c 3} + Δ - δ_{2}),

ρ_{g_{1} g_{3}}^{(1)} (\infty) = \frac{Ω_{p} / 2}{f (Δ)} g cos φ \frac{Ω_{1}}{2} (δ_{c 3} + Δ - δ_{2}),

ρ_{g_{2} g_{3}}^{(1)} (\infty) = \frac{Ω_{p} / 2}{f (Δ)} g cos φ \frac{Ω_{2}}{2} (δ_{c 3} + Δ - δ_{1}),

\begin{array}{l} ρ_{1 g_{3}}^{(1)} (\infty) & = \frac{Ω_{p} / 2}{f (Δ)} [(δ_{c 3} + Δ - δ_{1}) (δ_{c 3} + Δ + i \frac{γ}{2}) (δ_{c 3} + Δ - δ_{2}) - (δ_{c 3} + Δ - δ_{2}) Ω_{1}^{2} / 4 \\ - (δ_{c 3} + Δ - δ_{1}) Ω_{2}^{2} / 4], \end{array}

proportional to Ω_p and inversely proportional to

\begin{matrix} f (Δ) = (i \frac{κ}{2} + Δ) [(δ_{c 3} + Δ - δ_{1}) (δ_{c 3} + Δ + i \frac{γ}{2}) (δ_{c 3} + Δ - δ_{2}) - (δ_{c 3} + Δ - δ_{2}) Ω_{1}^{2} / 4 \\ - (δ_{c 3} + Δ - δ_{1}) Ω_{2}^{2} / 4] - g^{2} {cos}^{2} φ (δ_{c 3} + Δ - δ_{1}) (δ_{c 3} + Δ - δ_{2}) . \end{matrix}

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 S^κ and S^γ,

S^{κ} \propto {| ρ_{1 g_{3}}^{(1)} (\infty) |}^{2},

S^{γ} \propto {| ρ_{e g_{3}}^{(1)} (\infty) |}^{2} .

It can be verified from Eq. (27a) that the atomic excitation spectrum exhibits zero value when the detuning Δ satisfies the relation

δ_{c 3} + Δ - δ_{1} = 0,

or

δ_{c 3} + Δ - δ_{2} = 0 .

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 δ_c₃ + 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 $ρ_{e g_{3}}^{(1)} (s)$ , $ρ_{g_{1} g_{3}}^{(1)} (s)$ , $ρ_{g_{2} g_{3}}^{(1)} (s)$ , $ρ_{1 g_{3}}^{(1)} (s)$ , we apply the procedure of Laplace transform given in Eq. (23) on the evolution Eqs. (26a)–(26d), which are given from

ρ^{(1)} (s) = \frac{1}{s - ℒ^{(1)}} ρ^{(1)} (\infty),

with the Liouvillian operator 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

To obtain the corrections on density matrix elements of 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

{\dot{ρ}}_{g_{j} g_{j}}^{(2)} = γ_{j} δ_{j 3} ρ_{e e}^{(2)} - i \frac{Ω_{j}}{2} (ρ_{e g_{j}}^{(2)} - ρ_{g_{j} e}^{(2)}),

{\dot{ρ}}_{11}^{(2)} = - κ ρ_{11}^{(2)} + i (\frac{Ω_{p}}{2} ρ_{1 g_{3}}^{(1)} - g cos φ ρ_{e 1}^{(2)}) + c . c,

{\dot{ρ}}_{e e}^{(2)} = - γ ρ_{e e}^{(2)} + i (\frac{Ω_{1}}{2} ρ_{e g_{1}}^{(2)} + \frac{Ω_{2}}{2} ρ_{e g_{2}}^{(2)} + g cos φ ρ_{e 1}^{(2)}) + c . c,

{\dot{ρ}}_{g_{1} g_{2}}^{(2)} = i (δ_{2} - δ_{1}) ρ_{g_{1} g_{2}}^{(2)} - i \frac{Ω_{1}}{2} ρ_{e g_{2}}^{(2)} + i \frac{Ω_{2}}{2} ρ_{g_{1} e}^{(2)},

{\dot{ρ}}_{e g_{1}}^{(2)} = (i δ_{1} - \frac{γ}{2}) ρ_{e g_{1}}^{(2)} + i \frac{Ω_{1}}{2} (ρ_{e e}^{(2)} - ρ_{g_{1} g_{1}}^{(2)}) - i \frac{Ω_{2}}{2} ρ_{g_{2} g_{1}}^{(2)} - i g cos φ ρ_{1 g_{1}}^{(2)},

{\dot{ρ}}_{e g_{2}}^{(2)} = (i δ_{2} - \frac{γ}{2}) ρ_{e g_{2}}^{(2)} + i \frac{Ω_{2}}{2} (ρ_{e e}^{(2)} - ρ_{g_{2} g_{2}}^{(2)}) - i \frac{Ω_{1}}{2} ρ_{g_{1} g_{2}}^{(2)} - i g cos φ ρ_{1 g_{2}}^{(2)},

{\dot{ρ}}_{e 1}^{(2)} = (i δ_{c 3} - \frac{γ + κ}{2}) ρ_{e 1}^{(2)} + i g cos φ (ρ_{e e}^{(2)} - ρ_{11}^{(2)}) - i \sum_{j} \frac{Ω_{j}}{2} ρ_{g_{j} 1}^{(2)} + i \frac{Ω_{p}}{2} ρ_{e g_{3}}^{(1)},

{\dot{ρ}}_{g_{1} 1}^{(2)} = [i (δ_{c 3} - δ_{1}) - \frac{κ}{2}] ρ_{g_{1} 1}^{(2)} - i \frac{Ω_{1}}{2} ρ_{e 1}^{(2)} + i g cos φ ρ_{g_{1} e}^{(2)} + i \frac{Ω_{p}}{2} ρ_{g_{1} g_{3}}^{(1)},

{\dot{ρ}}_{g_{2} 1}^{(2)} = [i (δ_{c 3} - δ_{2}) - \frac{κ}{2}] ρ_{g_{2} 1}^{(2)} - i \frac{Ω_{2}}{2} ρ_{e 1}^{(2)} + i g cos φ ρ_{g_{2} e}^{(2)} + i \frac{Ω_{p}}{2} ρ_{g_{2} g_{3}}^{(1)} .

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 $Ω_{p}^{2}$ .

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) = \frac{1}{s - ℒ^{(2)}} [ρ^{(2)} (\infty) + \frac{1}{s - ℒ^{(1)}} ρ^{(1)} (\infty)],

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

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 = \frac{γ}{2} η_{3}^{2} α 〈 σ_{e e}^{(2)} (\infty) 〉 .

It is obvious when the parameters fulfill the three-photon resonance condition δ_c₃ + Δ − δ₁ = 0 as mentioned in Sec. 4.2, the steady-state expression of

〈 σ_{e e}^{(2)} (\infty) 〉

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 iν. Therefore, by applying the quantum regression theorem, S(ν) can be derived from the single-time average ${〈 V_{1} (s) 〉}_{s} = \sum_{j} \frac{i Ω_{j} η_{l j}}{2} [ρ_{g_{j} e}^{(2)} (s) - ρ_{e g_{j}}^{(2)} (s)] - η_{c} g sin φ [ρ_{1 e}^{(2)} (s) + ρ_{e 1}^{(2)} (s)]$ , which can be obtained from Eq. (34). After some calculations the analytical expressions for heating and cooling coefficients are given by

A_{\pm} = \frac{{| ε |}^{2} {\tilde{η}}^{2} g^{2} ν^{2}}{{| \tilde{f} (Δ \mp ν) |}^{2}} {γ {| Δ \mp ν + i κ / 2 |}^{2} + κ g^{2} {cos}^{2} φ},

where the small parameter

{| ε |}^{2} = {| \frac{Ω_{p} / 2}{Δ + i κ / 2} |}^{2}

indicates the mean number of intracavity photons when no atom is presented, and

{\tilde{η}}^{2} = η_{l 1}^{2} {cos}^{2} φ + η_{c}^{2} {sin}^{2} φ

is the compound Lamb-Dicke parameter combining geometries of the laser 1 and cavity field. The function f̃(Δ ∓ ν) takes the form

\tilde{f} (Δ \mp ν) = \pm ν g^{2} {cos}^{2} φ + (Δ \mp ν + i κ / 2) [\mp ν (δ_{1} \mp ν + i γ / 2) - {(Ω_{1} / 2)}^{2} + ε_{\pm}],

with

ε_{\pm} = \pm \frac{ν {(Ω_{2} / 2)}^{2}}{δ_{1} \mp ν - δ_{2}} .

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

\begin{array}{l} A_{\pm} & = \frac{{| ε |}^{2} {\tilde{η}}^{2} g^{2} γ (1 + C_{\pm}) {(δ_{1} \mp ν - δ_{2})}^{2}}{\frac{γ^{2}}{4} {(1 + C_{\pm})}^{2} {(δ_{1} \mp ν - δ_{2})}^{2} + {\frac{Ω_{2}^{2}}{4} + [\frac{γ}{κ} C_{\pm} (Δ \mp ν) \pm ν - δ_{1} \mp \frac{Ω_{1}^{2}}{4 ν}] (δ_{1} \mp ν - δ_{2})}^{2}} \\ = \frac{{| ε |}^{2} {\tilde{η}}^{2} g^{2} γ (1 + C_{\pm})}{\frac{γ^{2}}{4} {(1 + C_{\pm})}^{2} + {[\mp \frac{1}{ν} (\frac{Ω_{1}^{2}}{4} - ε_{\pm}) \pm ν - δ_{1} + \frac{γ}{κ} C_{\pm} (Δ \mp ν)]}^{2}}, \end{array}

where the parameter

C_{\pm} = C \frac{{(κ / 2)}^{2}}{{(Δ \mp ν)}^{2} + {(κ / 2)}^{2}}

is proportional to the single-atom cooperativity

C = \frac{g^{2} {cos}^{2} φ}{κ γ / 4}

[40]. 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].

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, Ω_p/2 = 0.1ν, δ₁ = 60ν, δ_c₃ = 59ν, Δ = ν, θ_c = θ_li = 0, φ = π/3. The dot line shows the zero heating coefficient.

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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.

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5.1. The cancellation of heating processes

Generally there exist two kinds of heating sources: carrier- and blue-sideband transitions [22]. 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 η_cg 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 δ_c3 + 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 (δ₁ − ν − δ₂)². It is clear that if we take the relation δ₂ = δ₁ − 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 |g₂, n + 1〉 becomes equal to that for the state |g₁, n〉 when the parameters satisfy the relation δ₂ = δ₁ − ν. 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], 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

ρ_{dark} = | Ψ 〉 〈 Ψ |,

with

| Ψ 〉 = | g_{3}, 0 〉 + \frac{Ω_{p} / 2}{Δ + i κ / 2} | 1, 0 〉 - \frac{Ω_{p}}{Ω_{1}} \frac{g cos φ}{Δ + i κ / 2} | g_{1}, 0 〉 + \frac{Ω_{p} g / Ω_{2}}{Δ + i κ / 2} (i {\tilde{η}}_{l 1} cos φ - {\tilde{η}}_{c} sin φ) | g_{2}, 1 〉,

is invariant up to the order

Ω_{p}^{2} η^{2}

as it satisfies the relation

(ℒ_{0} + ℒ_{0 E} + ℒ_{1}) ρ_{dark} = 0 .

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 = δ₁ − ν, the term ε₋ that characterizes the mechanical effect caused by the additional transition |g₂〉 ↔ |e〉 becomes $- \frac{Ω_{2}^{2}}{8}$ . The cooling coefficient A₋ changes into

A_{-} = \frac{{| ε |}^{2} {\tilde{η}}^{2} g^{2} γ (1 + C_{-})}{\frac{γ^{2}}{4} {(1 + C_{-})}^{2} + {[\frac{1}{ν} (\frac{Ω_{1}^{2}}{4} + \frac{Ω_{2}^{2}}{8}) - ν - δ_{1} + \frac{γ}{κ} C_{-} (Δ + ν)]}^{2}} .

Cooling is enhanced when the denominator of rate A₋ in Eq. (45) becomes minimal, i.e. the parameters satisfy the following condition

\frac{Ω_{2}^{2}}{8 ν} = δ_{1} - \frac{Ω_{1}^{2}}{4 ν} + ν - (Δ + ν) \frac{g^{2} {cos}^{2} φ}{{(Δ + ν)}^{2} + {(κ / 2)}^{2}} .

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]. Then the maximal cooling coefficient A₋ becomes

A_{-} = \frac{{| ε |}^{2} {\tilde{η}}^{2} g^{2}}{\frac{γ}{4} (1 + C_{-})},

which is the same as the maximal cooling coefficient of the cavity-induced single EIT system in Ref. [21]. 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, Ω_p/2 = 0.1ν, δ₁ = 60ν, δ₂ = 59ν, δ_c₃ = 59ν, Δ = ν, θ_c = θ_li = 0, φ = π/3.

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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], the two-time correlation functions 〈σ_mn(t)σ_kl(0)〉_s obey the same equation as ρ_nm(t), which are given in the Eq. (A1), with the initial values 〈σ_mn(0)σ_kl(0)〉_s = δ_nkρ_lm(∞) and ρ_lm(∞) are the steady-state solutions. The evolution equations for time-dependent density matrix elements ρ_mn(t) are given by

\begin{array}{l} {\dot{ρ}}_{e e} = - γ ρ_{e e} + i (\frac{Ω_{1}}{2} ρ_{e g_{1}} + \frac{Ω_{2}}{2} ρ_{e g_{2}} + g cos φ ρ_{e 1}) + c . c ., \\ {\dot{ρ}}_{g_{j} g_{j}} = γ_{j} ρ_{e e} - i \frac{Ω_{j}}{2} (ρ_{e g_{j}} - ρ_{g_{j} e}), \\ {\dot{ρ}}_{11} = - κ ρ_{11} + i (\frac{Ω_{p}}{2} ρ_{1 g_{3}} - g cos φ ρ_{e 1}) + c . c ., \\ {\dot{ρ}}_{e g_{1}} = (i δ_{1} - \frac{γ}{2}) ρ_{e g_{1}} + i \frac{Ω_{1}}{2} (ρ_{e e} - ρ_{g_{1} g_{1}}) - i \frac{Ω_{2}}{2} ρ_{g_{2} g_{1}} - i g cos φ ρ_{1 g_{1}}, \\ {\dot{ρ}}_{e g_{2}} = (i δ_{2} - \frac{γ}{2}) ρ_{e g_{2}} + i \frac{Ω_{2}}{2} (ρ_{e e} - ρ_{g_{2} g_{2}}) - i \frac{Ω_{1}}{2} ρ_{g_{1} g_{2}} - i g cos φ ρ_{1 g_{2}}, \\ {\dot{ρ}}_{e 1} = (i δ_{c 3} - \frac{γ + κ}{2}) ρ_{e 1} + i g cos φ (ρ_{e e} - ρ_{11} - i \sum_{j} \frac{Ω_{j}}{2} ρ_{g_{j} 1} + i \frac{Ω_{p}}{2} ρ_{e g_{3}}), \\ {\dot{ρ}}_{g_{1} g_{2}} = i (δ_{2} - δ_{1}) ρ_{g_{1} g_{2}} - i \frac{Ω_{1}}{2} ρ_{e g_{2}} + i \frac{Ω_{2}}{2} ρ_{g_{1} e}, \\ {\dot{ρ}}_{g_{1} 1} = [i (δ_{c 3} - δ_{1}) - \frac{κ}{2}] ρ_{g_{1} 1} - i \frac{Ω_{1}}{2} ρ_{e 1} + i g cos φ ρ_{g_{1} e} + i \frac{Ω_{p}}{2} ρ_{g_{1} g_{3}}, \\ {\dot{ρ}}_{g_{2} 1} = [i (δ_{c 3} - δ_{2}) - \frac{κ}{2}] ρ_{g_{2} 1} - i \frac{Ω_{2}}{2} ρ_{e 1} + i g cos φ ρ_{g_{2} e} + i \frac{Ω_{p}}{2} ρ_{g_{2} g_{3}}, \\ {\dot{ρ}}_{e g_{3}} = [i (δ_{c 3} + Δ) - \frac{γ}{2}] ρ_{e g_{3}} - i \sum_{j} \frac{Ω_{j}}{2} ρ_{g_{j} g_{3}} - i g cos φ ρ_{1 g_{3}} + i \frac{Ω_{p}}{2} ρ_{e 1}, \\ {\dot{ρ}}_{g_{1} g_{3}} = i [Δ - (δ_{1} - δ_{c 3})] ρ_{g_{1} g_{3}} - i \frac{Ω_{1}}{2} ρ_{e g_{3}} + i \frac{Ω_{p}}{2} ρ_{g_{1} 1}, \\ {\dot{ρ}}_{g_{2} g_{3}} = i [Δ - (δ_{2} - δ_{c 3})] ρ_{g_{2} g_{3}} - i \frac{Ω_{2}}{2} ρ_{e g_{3}} + i \frac{Ω_{p}}{2} ρ_{g_{2} 1}, \\ {\dot{ρ}}_{1 g_{3}} = (i Δ - \frac{κ}{2}) ρ_{1 g_{3}} - i g cos φ ρ_{e g_{3}} - i \frac{Ω_{p}}{2} (ρ_{g_{3} g_{3}} - ρ_{11}) . \end{array}

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

Steady-state solutions of the nonzero density matrix elements $ρ_{m n}^{(2)} (\infty)$ in the second-order Ω_p, which are calculated by substituting steady-state solutions $ρ_{m n}^{(1)} (\infty)$ given in Eqs. (27a)–(27d) into the equations given in (33a)–(33i), are obtained as

\begin{array}{l} ρ_{e e}^{(2)} (\infty) & = \frac{{(Ω_{p} / 2)}^{2}}{{| f (Δ) |}^{2}} g^{2} {cos}^{2} φ {(δ_{c 3} + Δ - δ_{1})}^{2} {(δ_{c 3} + Δ - δ_{2})}^{2}, \\ ρ_{g_{1} g_{1}}^{(2)} (\infty) & = \frac{{(Ω_{p} / 2)}^{2}}{{| f (Δ) |}^{2}} g^{2} {cos}^{2} φ {(Ω_{1} / 2)}^{2} {(δ_{c 3} + Δ - δ_{2})}^{2}, \\ ρ_{g_{2} g_{2}}^{(2)} (\infty) & = \frac{{(Ω_{p} / 2)}^{2}}{{| f (Δ) |}^{2}} g^{2} {cos}^{2} φ {(Ω_{2} / 2)}^{2} {(δ_{c 3} + Δ - δ_{1})}^{2}, \\ ρ_{11}^{(2)} (\infty) & = \frac{{(Ω_{p} / 2)}^{2}}{{| f (Δ) |}^{2}} {{(γ / 2)}^{2} {(δ_{c 3} + Δ - δ_{1})}^{2} {(δ_{c 3} + Δ - δ_{2})}^{2} + [(δ_{c 3} + Δ) (δ_{c 3} + Δ - δ_{1}) (δ_{c 3} + Δ - δ_{2}) \\ {- {(Ω_{1} / 2)}^{2} (δ_{c 3} + Δ - δ_{2}) - (δ_{c 3} + Δ - δ_{1}) {(Ω_{2} / 2)}^{2}]}^{2}}, \\ ρ_{e g_{1}}^{(2)} (\infty) & = \frac{{(Ω_{p} / 2)}^{2}}{{| f (Δ) |}^{2}} g^{2} {cos}^{2} φ (Ω_{1} / 2) (δ_{c 3} + Δ - δ_{1}) {(δ_{c 3} + Δ - δ_{2})}^{2}, \\ ρ_{e g_{2}}^{(2)} (\infty) & = \frac{{(Ω_{p} / 2)}^{2}}{{| f (Δ) |}^{2}} g^{2} {cos}^{2} φ (Ω_{2} / 2) {(δ_{c 3} + Δ - δ_{1})}^{2} (δ_{c 3} + Δ - δ_{2}), \\ ρ_{e 1}^{(2)} (\infty) & = \frac{{(Ω_{p} / 2)}^{2}}{{| f (Δ) |}^{2}} g cos φ (δ_{c 3} + Δ - δ_{1}) (δ_{c 3} + Δ - δ_{2}) {(δ_{c 3} + Δ - i \frac{γ}{2}) (δ_{c 3} + Δ - δ_{1}) (δ_{c 3} + Δ - δ_{2}) \\ - {(Ω_{1} / 2)}^{2} (δ_{c 3} + Δ - δ_{2}) - (δ_{c 3} + Δ - δ_{1}) {(Ω_{2} / 2)}^{2}}, \\ ρ_{g_{1} g_{2}}^{(2)} (\infty) & = \frac{{(Ω_{p} / 2)}^{2}}{{| f (Δ) |}^{2}} g^{2} {cos}^{2} φ (Ω_{1} / 2) (Ω_{2} / 2) (δ_{c 3} + Δ - δ_{1}) (δ_{c 3} + Δ - δ_{2}), \\ ρ_{g_{1} 1}^{(2)} (\infty) & = \frac{{(Ω_{p} / 2)}^{2}}{{| f (Δ) |}^{2}} g cos φ (Ω_{1} / 2) (δ_{c 3} + Δ - δ_{2}) {(δ_{c 3} + Δ - i \frac{γ}{2}) (δ_{c 3} + Δ - δ_{1}) (δ_{c 3} + Δ - δ_{2}) \\ - {(Ω_{1} / 2)}^{2} (δ_{c 3} + Δ - δ_{2}) - (δ_{c 3} + Δ - δ_{1}) {(Ω_{2} / 2)}^{2}}, \\ ρ_{g_{2} 1}^{(2)} (\infty) & = \frac{{(Ω_{p} / 2)}^{2}}{{| f (Δ) |}^{2}} g cos φ (Ω_{2} / 2) (δ_{c 3} + Δ - δ_{1}) {(δ_{c 3} + Δ - i \frac{γ}{2}) (δ_{c 3} + Δ - δ_{1}) (δ_{c 3} + Δ - δ_{2}) \\ - {(Ω_{1} / 2)}^{2} (δ_{c 3} + Δ - δ_{2}) - (δ_{c 3} + Δ - δ_{1}) {(Ω_{2} / 2)}^{2}}, \end{array}

which are proportional to

Ω_{p}^{2}

.

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.

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

Abstract

1. Introduction

2. The description of the model

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

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

4.1. The density matrix in the zeroth order of Ω_p

4.2. The corrections for density matrix in the first order of Ω_p

4.3. The corrections for density matrix in the second order of Ω_p

4.4. Calculations of heating and cooling coefficients

5. Cooling behavior induced by cavity-induced double EIT

5.1. The cancellation of heating processes

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

6. Conclusions

A. The evolution of density matrix elements

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

Acknowledgments

References and links

Cited By

Figures (6)

Equations (64)

Optics Express

Abstract

1. Introduction

2. The description of the model

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

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

4.1. The density matrix in the zeroth order of Ωp

4.2. The corrections for density matrix in the first order of Ωp

4.3. The corrections for density matrix in the second order of Ωp

4.4. Calculations of heating and cooling coefficients

5. Cooling behavior induced by cavity-induced double EIT

5.1. The cancellation of heating processes

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

6. Conclusions

A. The evolution of density matrix elements

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

Acknowledgments

References and links

Cited By

Figures (6)

Equations (64)

Optics Express

4.1. The density matrix in the zeroth order of Ω_p

4.2. The corrections for density matrix in the first order of Ω_p

4.3. The corrections for density matrix in the second order of Ω_p

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

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