High-order corrections on the laser cooling limit in the Lamb-Dicke regime

Zhen Yi; Wen-ju Gu

doi:10.1364/OE.25.001314

1. Introduction

Laser cooling and atom trapping techniques have been intensively exploited to exhibit the exotic and fascinating effects governed by quantum mechanics, such as Bose-Einstein Condensation (BEC) [1, 2], study of ultracold collisions [3, 4], atomic interferometry [5], as well as development of microwave and optical atomic clocks [6–8], and also provide a heuristic insight in comprehensive control of large and complex mechanical systems, such as molecules [9, 10], mechanical resonators [11,12], etc. The implementation of ultra-low temperatures is the requisite for quantum applications, and a number of achievements have been made in approaching the zero-point energy of motion in theories and experiments [13,14]. Of many variants of the laser cooling, the ion or atom trap is undeniably the workhorse [15]. The cooling behaviors of the trapped ion or atom are well-described in the Lamb-Dicke (LD) regime in which the coupling between the ion’s internal and external degrees of freedom induced by an external laser field is sufficiently small [16], where the perturbation method is feasible to describe the cooling dynamics. To our knowledge, the treatment on the laser cooling dynamics is mainly focused on the first-order perturbation expansion of LD parameters, i.e. single-phonon transitions, where the heating mechanism is mainly caused by carrier and blue-sideband excitations.

Nowadays a variety of schemes are proposed to realize the efficient cooling by removing carrier and blue-sideband excitations via employing quantum interferences, known as double-dark-state cooling scheme [17] to eliminate the single-phonon heating processes. The scheme includes double electromagnetically induced transparency (EIT) cooling via two independent Λ structures [18, 19], joint of EIT and stark-shift cooling in Δ structure [20], and Λ structure coupled by four lasers [21] or using standing waves [22] etc. All of the systems in these schemes finally evolve into a state

ρ = | dark 〉 〈 dark | \otimes | 0 〉 〈 0 | + O (η^{2}),

where |dark〉 refers to the internal atomic degree of freedom, |0〉 is the vanishing phonon excitation of motional state, and η is the LD parameter. Thus the final phonon occupation is 0 in the leading order expansion of LD parameters without the consideration of phonon thermal noise and laser fluctuation effects [23]. At the moment, it is natural to take into account of two-phonon heating processes to achieve a more accurate result of the final phonon occupation, because all the numerical simulations in these schemes show nonzero occupations.

In this work, we focus on high-order corrections on the “zero” cooling limit in the first double-dark-state cooling scheme, i.e., double EIT cooling [18], via employing the scattering method similar to cooling approaches in Refs. [17,22,24,25] in the LD regime. The double-EIT scheme can lower the cooling limit by a factor of order of the squared LD parameter (η²) as compared to single-EIT cooling, and here we derive the explicit expression of cooling limit in η². With the elimination of single-phonon heating processes, the system evolves into a double dark state, from which we will achieve the occupation on the single-phonon excitation state. Moreover, two-phonon heating processes become an important heating mechanism, which include a direct two-phonon excitation from the dark state governed by the second-order expansion of LD parameters and the further excitation from the single-phonon excited state, and also increase the final phonon occupations. Here we ignore the modification on the single-phonon cooling rate introduced by the two-phonon cooling rate because it is two order of LD parameters smaller in the LD regime. By adding up two parts of corrections, we achieve a good match with numerical simulations to validate our predictions. In addition, we find that these two two-phonon heating pathways can destructively interfere with each other, leading to the elimination of two-phonon heating mechanism, and a lower cooling limit is achievable.

This paper is organized as follows. In section 2, the physical system is introduced. The theoretical derivation of high-order corrections on the cooling limit and two-phonon heating transitions in the LD regime are presented in section 3, and the comparison with numerical simulations is discussed in section 4. Finally, the conclusion is given.

2. Description of the double EIT system in the LD regime

The double EIT cooling configuration is realizable by a Ca⁺ or ¹⁹⁹Hg⁺ ion confined in a harmonic potential with trap frequency ν [26]. The center-of-mass of the ion oscillates along the x-axis, while degrees of freedom of the transverse motion have been traced out by assuming that the transverse confinement is much steeper. Then the ion’s position operator is given by x = x_zpf(b + b^†), where $x_{zpf} = \sqrt{ℏ / 2 M ν}$ is the zero-point fluctuation, M is the mass of ion and operators b and b^† describe the annihilation and creation of per phonon excitation. For the double EIT scheme, the ion is in the tripod configuration comprised of one excited state |e〉 and three ground states |g〉 (j = 1, 2, 3) with energy frequencies ω_e and ω_{g_j} respectively, as shown in Fig. 1. The dipole transitions |e〉 ↔ |g〉 are irradiated by three laser fields with Rabi frequencies Ω_j. Then the Hamiltonian in the rotating frame of laser frequencies is given by (ħ = 1)

H = ν b^{†} b + \sum_{j} [Δ_{j} σ_{g_{j} g_{j}} + \frac{Ω_{j}}{2} (e^{- i k_{j} cos ϕ_{j} x} σ_{e g_{j}} + H . c .)],

where σ_mn = |m〉〈n| are atomic operators,

Δ_{j} = ω_{j}^{L} - (ω_{e} - ω_{g_{j}})

are detunings between the j-th laser frequency

ω_{j}^{L}

and the corresponding atomic transition, k_j are the wave numbers, and ϕ_j are the angles between laser traveling directions and the x-axis.

Fig. 1 The level configuration of double EIT system. The excited state |e〉 couples to the ground states |g〉 (j = 1, 2, 3) by three lasers with strengths Ω_j and detunings Δ_j, and γ_j are the spontaneous decay rates.

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The density operator ρ of the system obeys the master equation

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

where Lindblad operators ℒ_j describe the spontaneous decay of dipole transitions |e〉 → |g〉 and are written in the form

ℒ_{j} ρ = \frac{γ_{j}}{2} (2 σ_{g_{j} e} \tilde{ρ} σ_{e g_{j}} - σ_{e e} ρ - ρ σ_{e e}),

with damping rates γ_j. We assume the symmetry condition γ₁ = γ₂ = γ₃ = γ/3 for simplification, which does not qualitatively affect the cooling dynamics explored here. To include the effect of the recoil of spontaneously emitted photons on the mechanical motion, the density operator ρ̃ is related to the angular distribution 𝒩_j (cos θ) by the expression [27]

\tilde{ρ} = \frac{1}{2} \int_{- 1}^{1} 𝒩_{j} (cos θ) e^{i k_{j} x cos θ} ρ e^{i k_{j} x cos θ} d cos θ,

which takes the form 𝒩_j (cos θ) = 3(1 + cos² θ)/4 for the pattern of spontaneous emission of dipole transitions.

For the ongoing experiments of efficient ground-state cooling of the single trapped ion, LD approximation well describes the dynamics of cooling processes. Lamb-Dicke parameters are characterized by η_j = k_j x_zpf for the j-th laser field, which scale on mechanical effects of light on the ion motion [16]. In the LD regime, the perturbation theorem is adoptable for weak couplings between the external and internal degrees of freedom [28]. To achieve high-order corrections on the cooling limits of LD parameters, we expand Hamiltonian of Eq. (2) in Taylor series of η_j up to the second order

H = H_{0} + V_{1} + V_{2},

where H₀ in the zeroth order of LD parameters indicates the free evolution of the external and internal degrees of freedom without mutual couplings and is given by

H_{0} = ν b^{†} b + \sum_{j} [Δ_{j} σ_{g_{j} g_{j}} + \frac{Ω_{j}}{2} (σ_{e g_{j}} + H . c .)],

V₁ in the first order of LD parameters indicates the linear coupling to mechanical motion corresponding to the annihilation or creation of one phonon once a time during atomic transitions and is given by

V_{1} = - i \sum_{j} \frac{Ω_{j}}{2} {\tilde{η}}_{j} (σ_{e g_{j}} - σ_{g_{j} e}) (b + b^{†}),

and V₂ in the second order of LD parameters indicates the quadratic coupling corresponding to two phonons involved during atomic transitions and is given by

V_{2} = - \frac{1}{2} \sum_{j} \frac{Ω_{j}}{2} {\tilde{η}}_{j}^{2} (σ_{e g_{j}} + σ_{g_{j} e}) {(b + b^{†})}^{2} .

Here we have defined η̃_j = η_j cos ϕ_j which is related to the geometry of the physical setup. The term V₂ accounting for two-phonon transitions is usually neglected for weak effects on the cooling limit compared to single-phonon transitions in the LD regime. However, in this work we should take it into account for high-order corrections due to the elimination of single-phonon heating transitions in the double EIT cooling scheme. Moreover, Lindblad operators given in Eq. (4) are split into

ℒ_{j}^{(0)} ρ = \frac{γ_{j}}{2} (2 σ_{g_{j} e} ρ σ_{e g_{j}} - σ_{e e} ρ - ρ σ_{e e}),

referring to the common damping forms, and

ℒ_{j}^{(2)} ρ = \frac{γ_{j}}{2} α_{j} η_{j}^{2} [2 (b + b^{†}) ϱ_{j} (b + b^{†}) - {(b + b^{†})}^{2} ϱ_{j} - ϱ_{j} {(b + b^{†})}^{2}]

denoting the diffusion caused by the recoil of spontaneous emitted photons, where

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

is equal to

\frac{2}{5}

for the usual dipole transition [28] and ϱ_j = σ_{g_je}ρσ_{eg_j} depends on the population of the excited state |e〉. Via utilizing the EIT effect, the atomic population of zeroth order η̃_j is trapped in the dark state, and the excitation population ρ_ee = 0 [29]. Therefore, the diffusion in Eq. (11) is eliminated by the quantum interference, only the normal form of atomic dissipations in Eq. (10) is considered.

Under the condition Δ₁ = Δ₃ = Δ in Fig. 1, the two-photon resonance in the Λ configuration formed by levels |g₁〉, |g₃〉, |e〉 occurs, and it is convenient to investigate the cooling dynamics by introducing bright and dark states

\begin{array}{l} | c 〉 = \frac{1}{Ω} (Ω_{1} | g_{1} 〉 + Ω_{3} | g_{3} 〉), \\ | d 〉 = \frac{1}{Ω} (Ω_{3} | g_{1} 〉 - Ω_{1} | g_{3} 〉), \end{array}

with

Ω = \sqrt{Ω_{1}^{2} + Ω_{3}^{2}}

. In this dressed representation, the free Hamiltonian becomes

H_{0} = v b^{†} b + Δ (σ_{c c} + σ_{d d}) + Δ_{2} σ_{g_{2} g_{2}} + \frac{Ω}{2} (σ_{e c} + σ_{c e}) + \frac{Ω_{2}}{2} (σ_{e g_{2}} + σ_{g_{2} e}),

and interactions of single- and two-phonon transitions become

\begin{array}{l} V_{1} = & - \frac{i}{2} [Ω_{2} {\tilde{η}}_{2} (σ_{e g_{2}} - σ_{g_{2} e}) + \frac{Ω_{1}^{2} {\tilde{η}}_{1} + Ω_{3}^{2} {\tilde{η}}_{3}}{Ω} (σ_{e c} - σ_{c e}) \\ + \frac{Ω_{1} Ω_{3}}{Ω} ({\tilde{η}}_{1} - {\tilde{η}}_{3}) (σ_{e d} - σ_{d e})] (b + b^{†}), \\ V_{2} = & - [\frac{Ω_{2} {\tilde{η}}_{2}^{2}}{4} (σ_{e g_{2}} + σ_{g_{2} e}) + \frac{Ω_{1}^{2} {\tilde{η}}_{1}^{2} + Ω_{3}^{2} {\tilde{η}}_{3}^{2}}{4 Ω} (σ_{e c} + σ_{c e}) \\ + \frac{Ω_{1} Ω_{3} ({\tilde{η}}_{1}^{2} - {\tilde{η}}_{3}^{2})}{4 Ω} (σ_{e d} + σ_{d e})] {(b + b^{†})}^{2} . \end{array}

With these interactions the evaluation of optically induced phonon transitions can be performed to achieve high-order corrections on the cooling limit.

3. High-order corrections on the cooling limit

When detunings fulfill the condition Δ₁ = Δ₃, the atomic population is trapped in the dark state |d〉 due to the quantum destructive interference, and taking into account of phononic excitation state |n〉 in the mechanical motion, we can suppose the initial state of the system is |ψ₀〉 = |d, n〉. The subsequent transition amplitudes will determine the final phonon excitations, i.e. the cooling limit.

3.1. The correction introduced by the double dark state

The first part of corrections is introduced by the double dark state, which is related to the laser induced single-phonon transitions determined by V₁ in Eq. (14). Heating and cooling processes starting from the dark state |d, n〉 are connected to two separate manifolds M_n+1 = {|e, n + 1〉, |g₂, n + 1〉, |c, n + 1〉} and M_n−1 = {|e, n − 1〉, |g₂, n − 1〉, |c, n − 1〉} respectively, as shown in Fig. 2. To calculate transition amplitudes, we apply the effective Hamiltonian H_eff = H₀ − iγ |e〉〈e| by phenomenologically adding the atomic dissipation as in Refs. [17,24], and employ the state function method which is equivalent to the resolvent method therein. Beginning with the initial state |d, n〉, the first-order perturbation state function can be written in the form

| ψ_{1} 〉 = | ψ_{1}^{n + 1} 〉 + | ψ_{1}^{n - 1} 〉,

where

| ψ_{1}^{n \pm 1} 〉

describe single-phonon heating and cooling processes respectively and are expressed as

| ψ_{1}^{n \pm 1} 〉 = C_{e}^{n \pm 1} | e, n \pm 1 〉 + C_{c}^{n \pm 1} | c, n \pm 1 〉 + C_{g_{2}}^{n \pm 1} | g_{2}, n \pm 1 〉 .

The coefficients

C_{j}^{n \pm 1}

(j = e, c, g₂) are transition amplitudes, from which we will achieve heating and cooling rates.

Fig. 2 Single-phonon heating and cooling transitions governed by the interaction V₁ in Eq. (14). Starting from the atomic dark state with nth-vibration |d, n〉, heating and cooling processes follow by |d, n〉 → |e, n ± 1〉 and subsequent laser-mediated transitions.

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From the first-order perturbation schrödinger equation, the state function is governed by

i \frac{d}{d t} | ψ_{1} 〉 = H_{eff} | ψ_{1} 〉 + V_{1} | ψ_{0} 〉,

and the evolution equations for transition amplitudes are obtained as follows:

\begin{array}{l} i {\dot{C}}_{e}^{n \pm 1} = & [(n \pm 1) ν - i \frac{γ}{2}] C_{e}^{n \pm 1} + \frac{Ω}{2} C_{c}^{n \pm 1} + \frac{Ω_{2}}{2} C_{g_{2}}^{n \pm 1} \\ - i \frac{Ω_{1} Ω_{3}}{2 Ω} ({\tilde{η}}_{1} - {\tilde{η}}_{3}) e^{- i (Δ + n ν) t} \sqrt{n + δ_{\pm}}, \\ i {\dot{C}}_{g_{2}}^{n \pm 1} = & [Δ_{2} + (n \pm 1) ν] C_{g_{2}}^{n \pm 1} + \frac{Ω_{2}}{2} C_{e}^{n \pm 1}, \\ i {\dot{C}}_{c}^{n \pm 1} = & [Δ + (n \pm 1) ν] C_{c}^{n \pm 1} + \frac{Ω}{2} C_{e}^{n \pm 1}, \end{array}

where the abbreviation δ_± yields 1 for ‘+’ and 0 for ‘−’. Applying the rotating transformation

C_{j}^{n \pm 1} = {\tilde{C}}_{j}^{n \pm 1} e^{- i (Δ + n ν) t}

, above equations are changed into the form

\begin{array}{l} i {\tilde{C}}_{e}^{n \pm 1} = & [- Δ + ν - i \frac{γ}{2}] {\tilde{C}}_{e}^{n \pm 1} + \frac{Ω}{2} {\tilde{C}}_{c}^{n \pm 1} + \frac{Ω_{2}}{2} {\tilde{C}}_{g_{2}}^{n \pm 1} \\ - i \frac{Ω_{1} Ω_{3}}{2 Ω} ({\tilde{η}}_{1} - {\tilde{η}}_{3}) \sqrt{n + δ_{\pm}}, \\ i {\dot{\tilde{C}}}_{g_{2}}^{n \pm 1} = & [Δ_{2} - Δ \pm ν] {\tilde{C}}_{g_{2}}^{n \pm 1} + \frac{Ω_{2}}{2} {\tilde{C}}_{e}^{n \pm 1}, \\ i {\dot{\tilde{C}}}_{c}^{n \pm 1} = & \pm ν {\tilde{C}}_{c}^{n \pm 1} + \frac{Ω}{2} {\tilde{C}}_{e}^{n \pm 1} . \end{array}

Transition rates of heating and cooling processes, which are denoted by Γ_n→n±1, are connected to the population on |e, n ± 1〉 and the atomic dissipation rate [17,24] by the relation

Γ_{n \to n \pm 1} = γ {| {\tilde{C}}_{e}^{n \pm 1} |}^{2} .

For the single-phonon transitions, the heating mechanism is caused by carrier and blue-sideband excitations. The carrier excitation is eliminated due to the atomic population trapped in the dark state. To cancel the blue-sideband heating transition, i.e. Γ_n→n+1 = 0, from Eq. (19) we can choose

Δ_{2} - Δ + ν = 0,

and then the amplitude of heating transition becomes

{\tilde{C}}_{e}^{n + 1} = 0

. In addition,

\begin{array}{l} {\tilde{C}}_{c}^{n + 1} = 0, \\ {\tilde{C}}_{g_{2}}^{n + 1} = i \frac{Ω_{1} Ω_{3}}{Ω_{2} Ω} ({\tilde{η}}_{1} - {\tilde{η}}_{3}) \sqrt{n + 1} . \end{array}

Under the condition in Eq. (21), after some calculations we can achieve the cooling transition rate

Γ_{n \to n - 1} = n A_{-} = \frac{n γ {({\tilde{η}}_{1} - {\tilde{η}}_{3})}^{2} ν^{2} Ω_{1}^{2} Ω_{3}^{2} / Ω^{2}}{γ^{2} ν^{2} + 4 {[- ν (Δ + ν) + \frac{Ω_{1}^{2}}{4} + \frac{Ω_{2}^{2}}{8} + \frac{Ω_{3}^{2}}{4}]}^{2}},

where A₋ is the cooling rate which is identical to the result in the double EIT cooling scheme [18]. To achieve the optimal cooling rate, we should minimize the denominator with the parameters fulfilling the condition

ν = \frac{1}{2} (\sqrt{Δ^{2} + Ω_{1}^{2} + Ω_{2}^{2} / 2 + Ω_{3}^{2}} - Δ),

and the single-phonon cooling rate becomes

A_{-} = {({\tilde{η}}_{1} - {\tilde{η}}_{3})}^{2} \frac{Ω_{1}^{2}}{Ω^{2}} \frac{Ω_{3}^{2}}{γ},

which is independent of the trap frequency ν that is usually ∼ 1MHz and hardly to adjust for a given experimental setup.

In such a cooling scheme, optically induced single-phonon heating effects vanish, and with the strong laser induced cooling rate, the motion of the trapped ion will be cooled to the ground state, i.e. n → 0. Finally, the system will evolve into a double dark state which is the superposition of ground state |0〉 and single-phonon excitation state |1〉,

| Ψ_{f} 〉 = | d, 0 〉 + i \frac{Ω_{1} Ω_{3}}{Ω_{2} Ω} ({\tilde{η}}_{1} - {\tilde{η}}_{3}) | g_{2}, 1 〉 + O (η^{2}) .

Therefore, the occupation on the single-phonon excitation state is

P_{1} = \frac{Ω_{1}^{2} Ω_{3}^{2}}{Ω_{2}^{2} Ω^{2}} {({\tilde{η}}_{1} - {\tilde{η}}_{3})}^{2},

which is the first part of corrections on the cooling limit in the second-order LD parameters. To achieve more accurate descriptions of cooling processes, we should consider the laser induced two-phonon transitions for two reasons: one is that due to the elimination of single-phonon heating transitions it is natural to take into account of two-phonon transitions; the other is that besides the second-order correction on the cooling limit in Eq. (27), the fourth-order heating rate induced by two-phonon transitions divided by the second-order cooling rate in Eq. (25) can result in another part of corrections in the second-order LD parameters. Therefore, for the completeness and consistency, two-phonon transitions should be investigated. Next, we will focus on the modification of two-phonon heating transitions Γ_n→n+2 on the cooling limit. However, two-phonon cooling transitions are much weaker compared to single-phonon cooling transitions in the LD regime, it is feasible to assume that the cooling transition rate is equal to Γ_n→n−1.

3.2. The correction introduced by two-phonon heating transitions

There exist two pathways for two-phonon heating transitions: starting from the initial dark state |d, n〉 governed by the interaction V₂ in Eq. (14), and the further one-phonon transitions from the first-order perturbation state |g₂, n + 1〉, which are shown in Fig. 3. Two-phonon excitations are confined within the manifold M_n+2 = {|e, n + 2〉, |g₂, n + 2〉, |c, n + 2〉}, and the second-order perturbation state function can be written in the form

| ψ_{2}^{n + 2} 〉 = C_{e}^{n + 2} | e, n + 2 〉 + C_{g_{2}}^{n + 2} | g_{2}, n + 2 〉 + C_{c}^{n + 2} | c, n + 2 〉,

where the coefficients

C_{j}^{n + 2}

are two-phonon transition amplitudes, from which we will achieve the two-phonon heating rate.

Fig. 3 Two pathways of two-phonon heating transitions: (a) two-phonon excitation that starts from the initial dark states |d, n〉 governed by the interaction V₂ in Eq. (14); (b) the further one-phonon transitions from the first-order perturbation state |g₂, n + 1〉.

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The state function $| ψ_{2}^{n + 2} 〉$ obeys the second-order perturbation Schrödinger equation

i \frac{d}{d t} | ψ_{2}^{n + 2} 〉 = H_{eff} | ψ_{2}^{n + 2} 〉 + V_{1} | ψ_{1}^{n + 1} 〉 + V_{2} | ψ_{0} 〉 .

In the rotating frame of

C_{j}^{n + 2} = {\tilde{C}}_{j}^{n + 2} e^{- i (Δ + n ν) t}

and under the condition of elimination of single-phonon heating transtions in Eq. (21), transition amplitudes fulfill the evolution equations

\begin{array}{l} i {\dot{\tilde{C}}}_{e}^{n + 2} = & [- Δ + 2 ν - i \frac{γ}{2}] {\tilde{C}}_{e}^{n + 2} + \frac{Ω}{2} {\tilde{C}}_{c}^{n + 2} + \frac{Ω_{2}}{2} {\tilde{C}}_{g_{2}}^{n + 2} \\ - \frac{Ω_{1} Ω_{3}}{4 Ω} ({\tilde{η}}_{1} - {\tilde{η}}_{3}) {\tilde{η}}^{'} \sqrt{(n + 1) (n + 2)}, \\ i {\dot{\tilde{C}}}_{g_{2}}^{n + 2} = & ν {\tilde{C}}_{g_{2}}^{n + 2} + \frac{Ω_{2}}{2} {\tilde{C}}_{e}^{n + 2}, \\ i {\dot{\tilde{C}}}_{c}^{n + 2} = & 2 ν {\tilde{C}}_{c}^{n + 2} + \frac{Ω}{2} {\tilde{C}}_{e}^{n + 2}, \end{array}

where η̃′ = η̃₁ + η̃₃ − 2η̃₂ is produced by the interference between two pathways for two-phonon heating transitions. The steady-state solution is achieved as

{\tilde{C}}_{e}^{n + 2} = - \frac{ν Ω_{1} Ω_{3} ({\tilde{η}}_{1} - {\tilde{η}}_{3}) {\tilde{η}}^{'} \sqrt{(n + 1) (n + 2)}}{4 Ω [i \frac{γ}{2} ν + ν (Δ - 2 ν) + \frac{Ω^{2}}{8} + \frac{Ω_{2}^{2}}{4}]},

and the two-phonon transition rate which is also accompanied by the atomic dissipation is determined by

Γ_{n \to n + 2} = (n + 1) (n + 2) A_{+}^{(2)} = γ {| {\tilde{C}}_{e}^{n + 2} |}^{2},

where

A_{+}^{(2)} = \frac{γ ν^{2} Ω_{1}^{2} Ω_{3}^{2} {({\tilde{η}}_{1} - {\tilde{η}}_{3})}^{2} {\tilde{η}}^{' 2}}{16 Ω^{2} {\frac{γ^{2}}{4} ν^{2} + {[ν (Δ - 2 ν) + \frac{Ω^{2}}{8} + \frac{Ω_{2}^{2}}{4}]}^{2}}}

is the two-phonon heating rate and proportional to the fourth-order LD parameters.

In the LD regime, by taking into account of two-phonon heating transitions and single-phonon cooling transitions, the rate equation for the mechanical motion is given by

\begin{array}{l} \frac{d}{d t} P_{n} = Γ_{n - 2 \to n} P_{n - 2} - Γ_{n \to n + 2} P_{n} \\ + Γ_{n + 1 \to n} P_{n + 1} - Γ_{n \to n - 1} P_{n}, \end{array}

where P_n is the population of the n-th excitation state. Then the average phonon number

〈 n 〉 = \sum_{n} n P_{n}

obeys the equation

\frac{d}{d t} 〈 n 〉 = - A_{-} 〈 n 〉 + 2 A_{+}^{(2)} (〈 n^{2} 〉 + 3 〈 n 〉 + 2) .

For two-phonon heating rate lower than single-phonon cooling rate by a factor of two orders of LD parameters in the LD regime, the mechanical motion will be cooled to the ground state, i.e. 〈n〉 ≪ 1, and thus the correction on the cooling limit produced by two-phonon heating transitions denoted by 〈n^c〉 is approximately

〈 n^{c} 〉 \approx 4 A_{+}^{(2)} / A_{-},

which is in the second order of LD parameters. As a whole, the final cooling limit corrected by second-order LD parameters in the LD regime for the mechanical motion is the sum of two parts

n_{s s} = P_{1} + 〈 n^{c} 〉 .

To obtain the efficient cooling of the motion of trapped ions, besides the elimination of single-phonon heating transitions, we should try to suppress multi-phonon heating transitions, such as two-phonon transitions explored here. For the two-phonon heating rate in Eq. (33), if we choose LD parameters fulfill

{\tilde{η}}^{'} = {\tilde{η}}_{1} + {\tilde{η}}_{3} - 2 {\tilde{η}}_{2} = 0,

the two-phonon heating rate

A_{+}^{(2)} = 0

, which is caused by the destructive interference between two pathways of two-phonon excitations shown in Fig. 3. The correction induced by two-phonon heating transitions vanishes and the cooling limit is minimized to n_ss = P₁.

4. Numerical results and discussions

In this section we present the numerical simulation of the full master equation (3) of the system by using the quantum toolbox [30] to confirm our analytical predictions. For the efficient laser cooling, we should enlarge the cooling transition rate Γ_n→n−1 in Eq. (23), allowing for a faster preparation of ground state and robustness to noise. Thus it is better for η̃₁ and η̃₃ in the opposite sign. Meanwhile, the heating rate should be minimized, where single-phonon heating mechanism is inhibited by double EIT phenomena [18], and the further suppression of two-phonon heating transitions requires that LD parameters fulfill Eq. (38). One possible case is η̃₂ = 0 under the condition η̃₁ = − η̃₃ via employing the lasers from two opposite directions as the arrangement of single EIT cooling, which means the ignorance of mechanical effects induced by the laser 2 via tuning the angle ϕ₂ = π/2. Moreover, the cooling rate in Eq. (25) is not affected by the mechanical effect of η̃₂. In Fig. 4 we give numerical simulations of the cooling dynamics for parameters realizable in setup of a ¹⁹⁹Hg⁺-ion [31], which fulfill the optimal conditions of single-phonon transitions in Eqs. (21) and (24). The results verify that when we choose η̃₂ = 0 with η̃₁ = − η̃₃ to inhibit the two-phonon heating transitions, the trapped ion can be cooled to a lower occupation compared with taking into account of mechanical effects of the laser 2, i.e. η̃₂ = 0.13.

Fig. 4 Numerical simulations of the cooling dynamics with different LD parameters of the second laser η̃₂ = 0 (red-dotted line) and η̃₂ = 0.13 (blue-solide line). The other parameters are γ = 69MHz, γ₁ = γ₂ = γ₃ = γ/3, ν = 1.5MHz, Δ₁ = Δ₃ = Δ = 80MHz, η̃₁ = 0.13, η̃₃ = −0.13, Ω₁ = 21MHz, Ω₂ = 8MHz, Ω₃ = 4MHz, and Δ₂ = Δ − ν.

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To verify analytical results of high-order corrections on the cooling limit, in table 1 we numerically show phonon occupations in the zeroth, first and second excitation states, i.e. P₀, P₁ and P₂. It clearly displays that in the double EIT cooling scheme the phonon population is mainly in the ground state, but there is still a little population in excitation states indicated by small values of P₁ and P₂. By substituting the parameters into the one-phonon occupation P₁ in Eq. (27), the analytical result is P₁ = 0.0163, which matches well with numerical results. The second part of correction produced by two-phonon transitions 〈n^c〉 = 0 when η̃₂ = 0, due to the destructive interference between two pathways of two-phonon transitions: (1) starting from the initial dark states |d, n〉 governed by the interaction V₂ in Eq. (14) and (2) the further one-phonon transitions from the first-order perturbation state |g₂, n + 1〉, as shown in Fig. 3. For the case of η̃₂ ≠ 0, two-phonon heating transitions begin to take effect and will increase the final phonon occupation by a value of 〈n^c〉. With the parameters shown in table I, 〈n^c〉 = 0.0047, which also matches well with the numerical result of 2P₂ = 0.0046. The two-phonon excitation accounts for about 20 percent of the total excitations, and thus it is an essential component of high-order corrections on the cooling limit. In addition, when the two-phonon transition is inhibited, i.e., η̃₂ = 0, the further multi-phonon transitions are also suppressed, leading to the excitation mainly in the double dark state. Now the theoretical prediction 0.0163 agrees much better with the numerical result 0.0160, which indicates the necessity of two-phonon transition from another perspective.

Table 1. Numerical simulations for the occupation of zero-, one-, and two-phonon excitation states with different values of LD parameter η̃₂. The other parameters are γ = 69MHz, γ₁ = γ₂ = γ₃ = γ/3, ν = 1.5MHz, Δ₁ = Δ₃ = Δ = 80MHz, η̃₁ = 0.13, η̃₃ = −0.13, Ω₁ = 21MHz, Ω₂ = 8MHz, Ω₃ = 4MHz, and Δ₂ = Δ − ν.

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Furthermore, to more generally validate the analytical predictions, especially the correction produced by two-phonon heating transitions, we compare with the numerical results for another group of LD parameters with η̃₁ = 0.2, η̃₃ = −0.1 and the changing η̃₂ in table 2. Analytical results also show a good match with numerical results for the changing parameters. Moreover, when the destructive interference of two-phonon heating transitions occurs, i.e. fulfilling the condition in Eq. (38), the cooling limit is lower because the excitation is mainly in the double dark state and the further multi-phonon excitations are also suppressed due to inhibition of two-phonon transition. Otherwise, two-phonon heating processes will increase the cooling limit. Therefore, two-phonon transition is an essential component of corrections on cooling limit. In order to obtain the more efficient mechanical cooling, we should utilize the quantum interference to suppress multi-phonon transitions. In addition, the approach is also extensible to calculate high-order corrections on the cooling limit in the other double-dark-state schemes.

Table 2. Numerical simulations of phonon occupations versus analytical results with different values of η̃₂. The other parameters are γ = 69MHz, γ₁ = γ₂ = γ₃ = γ/3, ν = 1.5MHz, Δ₁ = Δ₃ = Δ = 80MHz, η̃₁ = 0.2, η̃₃ = −0.1, Ω₁ = 21MHz, Ω₂ = 8MHz, Ω₃ = 4MHz, and Δ₂ = Δ − ν. Footnotes ^(a) and ^(b) denote numerical and analytical results.

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In our approach, we ignore the diffusion introduced by the recoil of spontaneously emitted photons due to the vanishing population in the excited state by using the EIT effect [29]. Therefore, the form of atomic dissipation is in a normal Lindblad form, making it appropriate to employ the effective Hamiltonian H_eff by phenomenologically adding the dissipation. Due to the strong cooling transition rate in double EIT scheme, we ignore the phonon heating caused by the thermal reservoir and here mainly focus on the laser induced heating mechanism.

5. Conclusion

In conclusion, we have discussed high-order corrections on the cooling limit of the double EIT cooling in LD regime. In this double-dark-state cooling scheme, the system will evolve into a double dark state with the elimination of single-phonon heating transitions, from which we will obtain the first part of corrections on cooling limit in second order of LD parameters. Moreover, the other part of corrections is produced by two-phonon heating transitions. There exist two pathways of two-phonon heating transitions: direct two-phonon excitation from the dark state and further single-phonon excitation from the double dark state. By adding up these two parts of corrections, analytical results display a good match with numerical simulations. Moreover, we find that two pathways can destructively interfere with each other, leading to the elimination of two-phonon heating transitions and achieving a lower cooling limit. In addition, the approach can be extended to investigate high-order corrections on the cooling limit of the other double-dark-state cooling schemes.

Funding

National Natural Science Foundation of China (NSFC) (61505014, 11504031); Yangtze Funds for Youth Teams of Science and Technology Innovation (2015cqt03).

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η̃₂	$P_{1}^{(a)}$	$2 P_{2}^{(a)}$	$P_{1}^{(b)}$	〈n^c〉^(b)
−0.05	0.0227	0.0024	0.0217	0.0028
0.05	0.0214	0	0.0217	0
0.1	0.0216	0.00064	0.0217	0.0007
0.15	0.0223	0.0026	0.0217	0.0028
0.2	0.0238	0.006	0.0217	0.0063

η̃₂	$P_{1}^{(a)}$	$2 P_{2}^{(a)}$	$P_{1}^{(b)}$	〈n^c〉^(b)
−0.05	0.0227	0.0024	0.0217	0.0028
0.05	0.0214	0	0.0217	0
0.1	0.0216	0.00064	0.0217	0.0007
0.15	0.0223	0.0026	0.0217	0.0028
0.2	0.0238	0.006	0.0217	0.0063

High-order corrections on the laser cooling limit in the Lamb-Dicke regime

Abstract

1. Introduction

2. Description of the double EIT system in the LD regime

3. High-order corrections on the cooling limit

3.1. The correction introduced by the double dark state

3.2. The correction introduced by two-phonon heating transitions

4. Numerical results and discussions

5. Conclusion

Funding

References and links

Cited By

Figures (4)

Tables (2)

Equations (38)

Optics Express

	P₀	P₁	P₂	〈n〉
η̃₂ = 0	0.9840	0.0160	2.2 × 10⁻¹²	0.0160
η̃₂ = 0.13	0.9786	0.0186	0.0023	0.0249