Ground state cooling of an optomechanical resonator assisted by a Λ-type atom

Shuo Zhang; Jian-Qi Zhang; Jie Zhang; Chun-Wang Wu; Wei Wu; Ping-Xing Chen

doi:10.1364/OE.22.028118

1. Introduction

Recently, micro- and nano-mechanical resonators (MRs) [1, 2] have attracted a considerable attention in both quantum physics and nanotechnology, due to they take both classical and quantum properties [3]. For this reason, the study of MRs ranges from fundamental physics to applications including entanglement between mesoscopic objects [4, 5], optomechanically induced transparency and absorption [6–8], photon blockade [9, 10], optical Kerr effect [11], ultra-sensitive measurements [3, 12, 13], quantum information processing [14], and biological sensing [15].

It is the fundamental step toward the implementations based on quantum properties of MR to cool the vibration of MRs to their ground state. For this reason, many different MR cooling methods have been proposed. The typical cooling method is known as sideband cooling. In this kind cooling method, the ground state cooling of a MR can be achieved with the assistance of an optical cavity [16–18], in which the radiation pressure is used to increase the effective dissipation of the MR assisted by the driven light in red detuning, and the MR can be theoretically cooled to the final mean vibration number $n_{s s} = {(\frac{κ}{2 ω_{m}})}^{2}$ at zero environment temperature, where 2κ is the cavity linewidth and ω_m is the vibration frequency. As a result, in the condition of the resolved sideband (κ ≪ ω_m), a ground state MR taking a few phonon number is achievable when the red sideband transition for cooling dominant over the cooling process.

However, the resolved sideband condition is hard to fulfill sometimes [19, 20]. In this situation, the final temperature and cooling rate are limited by the residual blue sideband heating transition. To overcome this difficulty, a variety of new cooling methods for the nonresolved sideband regime (κ > ω_m) have been proposed in recent years. In these schemes, the quantum interference effect plays a crucial role, in which the blue sideband heating transitions can be eliminated by the quantum destructive interference. The quantum inference cooling method was originally used to realize the trapped ion and atom coolings [21–30], where the heating transitions are suppressed by the electromagnetically induced transparency (EIT) [31] or EIT-like effects [23,26,32], and then, the EIT cooling is extended to MR electromechanical system in [33], in which the MR couples to a three-level superconduction flux qubit. Subsequently, there are several further approaches based on EIT or EIT-like effects being investigated by coupling the MR to the NV center [34], quantum dots [35], an auxiliary cavity mode in optomechanical system [36, 37], the atomic ensembles [38–41], and a single atom [42–44].

In this work, we propose a cooling scheme with a Λ-type three level atom trapped in an optomechanics. The cooling scheme is based on a single-photon strong coupling optomechanical system [10, 17] that the MR can be effectively moved by a single photon in cavity. In our scheme, when the cavity is weakly excited, the hybrid system can be treated as a four-level system. With the assistance of the EIT-like quantum interference in the hybrid system, the heating transitions can be highly suppressed, and the MR can be cooled to its ground state.

The paper is organized as follows. In Section 2, we present the basic Hamiltonian and master equation of our model, and give the corresponding effective master equation in the parameter regimes we concerned. In Section 3, we analytically calculate the cooling dynamics, and demonstrate the elimination mechanism of the heating transition, followed by discussions and numerical simulations of the cooling dynamics to quantify the efficiency of our scheme in Sec. 4. Finally, a brief conclusion is given.

2. Model

2.1. Basic equations

As it is illustrated in Figs. 1(a) and 1(b), a Λ-type atom is fixed in an optomechanical cavity. The fixed atom takes one excited state |e〉 and two ground states |g〉 and |r〉, and the corresponding energies for these states are h̄ω_e, h̄ω_g and h̄ω_r, respectively. The ground state |g〉 (|r〉) can be excited to the excited state |e〉 by a cavity mode (driven light) with a frequency ω_c (ω_L) and a coupling strength g (driven strength Ω_r). Moreover, the optomechanical cavity is driven by a pump laser in a frequency ω_P with a coupling strength Ω_P, and the MR in the optomechanical cavity can couple to the optical cavity mode by a radiation pressure coupling with a strength λ.

Fig. 1 (a) The model of the MR cooling. A Λ-type atom is fixed in an optomechanical cavity, and the MR is coupled to the cavity via the radiative pressure. A pump laser is applied to inject photons into the cavity, and the Λ-type atom is coupled to the cavity mode and a laser field. The dissipations of the whole system include the atomic spontaneous emission, the cavity decay and the MR dissipating to the thermal bath. The dynamics of the whole system can be described by the master equation (5). (b) The energy levels and the corresponding transitions in the atom. This atom takes one excited state |e〉 and two ground states |g〉 and |r〉. The states |e〉 and |g〉 couples to the cavity field, and the one|e〉 ↔ |r〉 is driven by an external laser. The detunings are defined by Eq. (3).

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Then the Hamiltonian of our model can be written as [26]

H = H_{0} + H_{I} .

In the rotating frame of the driven frequencies ω_p and ω_L, the free Hamiltonian H₀ for the cavity mode, MR and atom can be given as (h̄ = 1)

H_{0} = - δ_{P} a^{†} a + ω_{m} b^{†} b - δ_{g} | e 〉 〈 e | + δ_{P} | g 〉 〈 g | + (Δ_{r} - δ_{g}) | r 〉 〈 r |,

where ω_m is the fundamental eigen-frequency of the MR; a and b are the annihilation operators for the cavity mode and the phonon of the MR, respectively; and the corresponding detunings are defined by

\begin{array}{l} δ_{P} & = & ω_{P} - ω_{c}, \\ Δ_{r} & = & ω_{L} - (ω_{e} - ω_{r}), \\ δ_{g} & = & ω_{c} - (ω_{e} - ω_{g}) . \end{array}

The corresponding Hamiltonian for the interactions can be given by

H_{I} = Ω_{P} (a + a^{†}) + g (a | e 〉 〈 g | + a^{†} | g 〉 〈 e |) + Ω_{r} (| e 〉 〈 r | + | r 〉 〈 e |) + λ a^{†} a (b^{†} + b),

where the first term describes the cavity is driven by the external light field with a strength Ω_P; the second one shows the transition |g〉 ↔ |e〉 interacts with the cavity mode; the third one presents the transition |r〉 ↔ |e〉 is acted by a laser field with a Rabi frequency Ω_r; and the last one depicts there is a radiation pressure coupling between the cavity mode and the optomechanical resonator.

After the dissipations from the environment are taken into account, the dynamics of the whole system follows the master equation as

\frac{d}{d t} ρ = - i [H, ρ] + ℒ_{γ} ρ + ℒ_{κ} ρ + ℒ_{m} ρ,

where

ℒ_{γ} ρ = \sum_{j = g, r} γ_{j} (2 | j 〉 〈 e | ρ | e 〉 〈 j | - | e 〉 〈 e | ρ - ρ | e 〉 〈 e |),

ℒ_{κ} ρ = κ (2 a ρ a^{†} - a^{†} a ρ - ρ a^{†} a),

and

ℒ_{m} ρ = γ_{m} (n_{th} + 1) (2 b ρ b^{†} - b^{†} b ρ - ρ b^{†} b) + γ_{m} n_{th} (2 b^{†} ρ b - b b^{†} ρ - ρ b b^{†}),

are describe the atomic spontaneous emission from the excited state |e〉 to the ground state |j〉 (j = g, r), the cavity interacting with the environment noise, and the dissipative processes for the MR coupling to the thermal bath, respectively. 2γ_j, 2κ, and 2γ_m = ω_m/Q are corresponding to the decay rates of atom, cavity, and MR with a quality factor Q, respectively. n_th = 1/ [exp(h̄ω_m/k_BT) − 1] is the thermal occupation number of the MR at temperature T with k_B being the Boltzmann constant.

2.2. Basic assumptions

Similar to the cooling schemes of. [42] and [43], our study still focuses on two parameter regimes. (i) the system should work in a Lamb-Dicke regime, which implies that the coupling strength λ between the optical cavity and the MR should be much weaker than the vibrational frequency ω_m (λ/ω_m ≪ 1). A typical value for this ratio (Lamb-Dicke parameter) is about λ/ω_m ∼ 0.1, which means that the interaction between the single-photon and the MR in the optomechanical cavity can be treated as a perturbation. (ii) the cavity is in weakly excited, which implies the steady photon number in the cavity should be much lower than a unity ( ${〈 a^{†} a 〉}_{st} ≪ 1$ ). In the experiment, it can be realized with the optomechanical cavity driven by a weak light.

Considering this hybrid system can be divided into two parts: the internal degrees of freedom (DOF) for the hybrid states of the atom-cavity system, and the external one for the vibrational state of the MR. The Lamb-Dicke regime ensures that the external DOF are weakly coupled to the internal one, and the evolution of the external DOF can be much slower than the internal one. As a result, the rapidly change of the internal DOF can be adiabatically eliminated, and the equation for the external DOF of MR can be obtained in this way. Moreover, the second condition guarantees that the steady photon occupation of the cavity should be much smaller than a single photon. Therefore, we can suppose that the steady state of internal DOF is almost in the state |g, 0_c〉, where |m_c〉 (m = 0, 1) denotes photon number in the cavity. When the cavity only can be excited to its lowest excitation, the scattering processes of the internal DOF take place within the subspace spanned by

{| g, 0_{c} 〉, | g, 1_{c} 〉, | e, 0_{c} 〉, | r, 0_{c} 〉} .

For simplicity, we use the abbreviated notation for the internal states as |g,0_c〉 ≡ |g〉, |g,1_c〉 ≡ |1〉, |e,0_c〉 ≡ |e〉, |r,0_c〉 ≡ |r〉 in the following. Therefore, the internal DOF are equivalent to a four-level system, which can be divided into two weakly coupled subspaces: the subspace for the ground state |g〉 and the excited subspace spanned by {|1〉, |e〉, |r〉}.

Then the master equation (5) for the four-level system (9) can be reduced to

\frac{d}{d t} ρ = - i [H^{'}, ρ] + ℒ_{γ} ρ + ℒ_{κ}^{'} ρ + ℒ_{m} ρ,

where

\begin{array}{l} H^{'} & = & H_{0}^{'} + V_{0} + V_{1}, \\ H_{0}^{'} & = & - δ_{P} | 1 〉 〈 1 | + ω_{m} b^{†} b - Δ_{g} | e 〉 〈 e | + (Δ_{r} - Δ_{g}) | r 〉 〈 r | \\ + Ω_{r} (| e 〉 〈 r | + | r 〉 〈 e |) + g (| e 〉 〈 1 | + | 1 〉 〈 e |), \\ V_{0} & = & Ω_{P} (| g 〉 〈 1 | + | 1 〉 〈 g |), \\ V_{1} & = & λ | 1 〉 〈 1 | (b^{†} + b); \end{array}

and

ℒ_{κ}^{'} ρ = κ [2 | g 〉 〈 1 | ρ | 1 〉 〈 g | - | 1 〉 〈 1 | ρ - ρ | 1 〉 〈 1 |] .

Here, H′₀ is the free Hamiltonian for the motional DOF and excited subspace of internal DOF; V₀ describes the interaction between the two subspaces of internal DOF; V₁ shows the perturbative interaction between the internal DOF and motional DOF. In the Hamiltonian (11), we have set the energy of internal state |g〉 to zero, and defined

Δ_{g} = δ_{P} + δ_{g} = ω_{P} - (ω_{e} - ω_{g}) .

3. Analytical results

3.1. Heating and cooling coefficients

With the reduced master equation (10) obtained, the cooling dynamics of our model can be analyzed by employing the perturbative method [25, 42, 45]. This method is first developed in [45], and then applied to calculate the cooling dynamics of trapped atom [25] and MR systems [42].

After the internal DOF in (10) is eliminated adiabatically and the perturbation method is applied, one can get the rate equation for the population of the n-th vibrational state p_n:

\begin{array}{l} \frac{d}{d t} p_{n} & = & (n + 1) (A_{-} + 2 (n_{th} + 1) γ_{m}) p_{n + 1} + n (A_{+} + 2 n_{th} γ_{m}) p_{n - 1} \\ - [(n + 1) (A_{+} + 2 n_{th} γ_{m}) + n (A_{-} + 2 (n_{th} + 1) γ_{m})] p_{n} . \end{array}

where A_± are heating/cooling coefficients.

For simplicity, only the dissipations of the atom and cavity are taken into account in this section, and the dissipation of the MR will be enclosed in the simulations of the next section.

Then, the dynamics of mean phonon number 〈n〉 = ∑_nnp_n can be given by

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

In the case of A₋ > A₊, the solution of 〈n〉 is

〈 n 〉 = (n_{0} - n_{s s}) e^{- W t} + n_{s s},

where n₀ is the initial mean phonon number;

W = A_{-} - A_{+},

is the cooling rate; and

n_{s s} = \frac{A_{+}}{A_{-} - A_{+}} .

is the final phonon number.

The heating/cooling coefficients A_± can be achieved by calculating the scattering amplitudes of all possible heating/cooling processes [46], as depicted in Figs. 2(a) and 2(b): suppose our system is initially in the state |g〉|n〉 with n being n-th Fock state of the MR, it can be excited to state |1〉|n〉 by the pump laser at first, and then the system can transfer to state |1〉|n ± 1〉 via the perturbation interaction (radiation pressure coupling) V₁, following by another internal scattering process in the excited subspace of n ± 1–th vibrational state, the whole system will dissipate to the state |g〉|n ± 1〉 by atomic dissipation or cavity decay. According to the calculations in Appendix, one can obtain heating/cooling coefficients:

A_{\pm} = A_{\pm}^{κ} + A_{\pm}^{γ},

where

A_{\pm}^{κ}

and

A_{\pm}^{γ}

denote the heating/cooling processes due to cavity decay and atomic dissipation, respectively:

A_{\pm}^{κ} = 2 κ 𝒮 {| λ \frac{(\mp ω_{m} + Δ_{g}) (\mp ω_{m} + Δ_{g} - Δ_{r}) - Ω_{r}^{2} + i γ (\mp ω_{m} + Δ_{g} - Δ_{r})}{f (\mp ω_{m})} |}^{2},

A_{\pm}^{γ} = 2 γ 𝒮 {| λ \frac{- g (\mp ω_{m} + Δ_{g} - Δ_{r})}{f (\mp ω_{m})} |}^{2},

where 𝒮 is the saturation parameter which is the steady population of the state |1〉〈1|:

𝒮 = {| Ω_{P} \frac{Δ_{g} (Δ_{g} - Δ_{r}) - Ω_{r}^{2} + i γ (Δ_{g} - Δ_{r})}{f (0)} |}^{2},

and

f (x) = - Ω_{r}^{2} (x + δ_{P}) - i κ Ω_{r}^{2} + (x + Δ_{g} - Δ_{r}) [- g^{2} + (i κ + x + δ_{P}) (i γ + x + Δ_{g})] .

Fig. 2 The schematic illustration of the cooling and heating processes. Suppose the system is initially in the state |g〉|n〉, it would be excited to state |1〉|n〉 by the pump laser, and then it will evolve to the state |1〉|n ± 1〉 via the interaction V₁. After that, with the assistance of the internal scattering process, the whole system will decay to the state |g〉|n±1〉 via the (a) cavity decay or the (b) atomic dissipation.

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3.2. Optimal cooling condition and cooling dynamics

Now, we will investigate the cooling dynamics in strong atom-cavity coupling regime, which corresponds to the case of large cooperativity C:

C = \frac{g^{2}}{κ γ} ≫ 1 .

Our goal is to find the optimal cooling condition that A₋ is maximum while A₊ is minimum in such regime.

By inspection of the analytical results in Eqs. (19) (20) and (21), the condition for the minimum of A₊ should satisfy

(- ω_{m} + Δ_{g}) (- ω_{m} + Δ_{g} - Δ_{r}) - Ω_{r}^{2} = 0 .

The physical explanation for the above equation can be understood with dressed states in Fig. 3(a). Under the action of the external light field with a Rabi frequency Ω_r and a detuning Δ_r, two states |e〉 and |r〉 will be rewritten as two dressed states

\begin{array}{l} | D_{1} 〉 & = & sin ϑ | r 〉 + cos ϑ | e 〉, \\ | D_{2} 〉 & = & cos ϑ | r 〉 - sin ϑ | e 〉, \end{array}

with

ϑ = arctan \frac{Ω_{r}}{\frac{1}{2} (\sqrt{\frac{Δ_{r}^{2}}{4} + Ω_{r}^{2}} - Δ_{r})} .

The corresponding eigenenergies are

\begin{array}{l} E_{D_{1}} & = & \frac{1}{2} (- 2 Δ_{g} + Δ_{r} + \sqrt{Δ_{r}^{2} + 4 Ω_{r}^{2}}), \\ E_{D_{2}} & = & \frac{1}{2} (- 2 Δ_{g} + Δ_{r} - \sqrt{Δ_{r}^{2} + 4 Δ_{r}^{2}}), \end{array}

respectively. The Hamiltonian H′₀ in Eq. (11) can be rewritten in the dressed state representation of {|D₁〉, |D₂〉, |1〉, |g〉} as

\begin{array}{l} H_{0}^{'} & = & - δ_{P} | 1 〉 〈 1 | + ω_{m} b^{†} b + E_{D_{1}} | D_{1} 〉 〈 D_{1} | + E_{D_{2}} | D_{2} 〉 〈 D_{2} | \\ + g (cos ϑ | D_{1} 〉 - sin ϑ | D_{2} 〉) 〈 1 | + H . c . \end{array}

Fig. 3 (a) The EIT-like interference in the dressed state representation. The external light field dresses two states |e〉 and |r〉 into two dressed states |D₁〉 and |D₂〉. Then the coupling between |1〉 and |e〉 can be effectively treated as two transition channels |1〉 ↔ |D₁〉 and |1〉 ↔ |D₂〉. There is a EIT-like structure among the states {|D₂〉, |g〉, |1〉}, which include two transitions |g〉 ↔ |1〉 and |D₂〉 ↔ |1〉 and their corresponding detunings −δ_P and −δ_P − E_D₂. When tune the two detunings to the same, i.e. E_D₂ = 0, the EIT-like effect will arise, and therefore the two transitions are suppressed. (b) The schematic illustration for the suppression mechanism of the heating transitions. A heating transition starts from internal steady state |g〉|n〉. Then the system is excited to |1〉|n〉, and heated to |1〉|n + 1〉 via the interaction V₁. It can be suppressed by the transition |D₂〉|n + 1〉 ↔ |1〉|n + 1〉 due to the EIT-like effect. When the two detunings are the same, the EIT-like dark resonance will prohibit the two paths.

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As it is depicted in Fig. 3(a), there is a EIT-like structure among the internal DOF {|D₂〉, |g〉, |1〉}, which include two transitions |g〉 ↔ |1〉 and |D₂〉 ↔ |1〉 with the corresponding detunings −δ_P and −δ_P − E_D₂, respectively. When these two detunings are tuned to the same, i.e. E_D₂ = 0, the EIT-like effect will arise, and the two transitions can be suppressed.

As it is illustrated in Fig. 3(b), in our scheme, we exploit such EIT-like effect to suppress the heating transitions. The heating scattering process |g〉|n〉 ↔ |1〉|n〉 ↔ |1〉|n + 1〉 can be suppressed by the transition |D₂〉|n + 1〉 ↔ |1〉|n + 1〉 due to the destructive interference. When the two detunings are tuned to the same

- δ_{P} + ω_{m} = - δ_{P} - E_{D_{2}},

the EIT-like dark resonance will prohibit these two paths. One can find that Eq. (30) is equivalent to the Eq. (25), which indicates that heating path is highly suppressed.

On the other hand, a large cooling coefficient A₋ can be achieved by decreasing the denominator in (23) [27, 42]. Note that the real part of f (ω_m) (Re f (ω_m)) includes g², while the imagine one (Im f (ω_m)) is only related to κ and γ. For this reason, we can set

Re f (ω_{m}) = 0,

which corresponds to

- Ω_{r}^{2} (ω_{m} + δ_{P}) + (ω_{m} + Δ_{g} - Δ_{r}) [- g^{2} - κ γ + (ω_{m} + δ_{P}) (ω_{m} + Δ_{g})] = 0 .

Then, the heating/cooling coefficients A_± can be obtained under the conditions (25) and (31):

A_{+} \approx 2 λ^{2} 𝒮 γ {[\frac{ω_{m}}{g (- ω_{m} + Δ_{g} - Δ_{r})}]}^{2} ~ 2 λ^{2} 𝒮 γ \frac{1}{g^{2}},

A_{-} \approx 2 λ^{2} 𝒮 \frac{1}{[κ + \frac{γ}{g^{2}} {(ω_{m} + δ_{P})}^{2}]} .

For A₋ ≫ A₊, the final mean phonon number is

n_{s s} \approx \frac{A_{+}}{A_{-}} = \frac{1}{C} + \frac{γ^{2} {(ω_{m} + δ_{P})}^{2}}{g^{4}} .

The second term of (35) is less than

\frac{1}{C^{2}}

by tuning δ_P that |ω_m + δ_P| ⩽ κ, and then it can be neglected in the regime C ≫ 1. As a result, the final mean phonon number in our scheme is in the order of

\frac{1}{C}

, which is the same order of the one in [43], and lower than the one in [42].

4. Discussion

In this section, we will give some discussions of our scheme with the comparisons and simulations.

At first, we would compare our scheme to related works [38, 42, 43].

It is a pioneer work that cooling an optomechanical MR with an atom ensemble in [38], where the cooling of MR can be achieved by tailoring the cavity response via the EIT effect caused by the atom ensemble. Compared with this work, both of the cooling schemes are proposed by indirectly coupling the MR to the Λ–type three level atoms. But our scheme is different from the one in [38]. It is because that different EIT configurations are exploited. In [38], C. Genes et al exploit the EIT interference effect among the excited subspace to tailor the cavity linewidth to a narrow absorption under the condition of g ≫ Ω_r; while in our scheme, we exploit the EIT-like dark resonance in dressed state representation, which can highly suppress the heating transition |g,0_c〉|n〉 → |g,1_c〉|n〉 → |g,1_c〉|n + 1〉 in strong atom-cavity coupling regime C ≫ 1.

Compared with the MR cooling methods in [42] and [43], all the schemes are carried out by coupling the optomechanical system to a single atom, work in the same regimes as the weak cavity excitation regime, the Lamb-Dicke regime, and the strong atom-cavity coupling regime that the single atom cooperativity C ≫ 1. However, there are some significant differences between our scheme and these two cooling schemes. Firstly, the atomic level configurations for cooling are different, for example, the atom in our work is in Λ-type, while the ones in [42] and [43] are in a two level and a tripod configurations, respectively. Therefore, the quantum interference effects for cooling are different for the different energy levels. Secondly, the final mean phonon numbers in these proposals are different. For instance, the heating effects in [42] can be not completely eliminated so that the final phonon number is higher than the order of $O (\frac{1}{C})$ ; while in our scheme as well as in [43], the heating transitions can be highly suppressed, and the mean phonon number can reach the order of $O (\frac{1}{C})$ . Finally, different from [42] which requires that the atomic dissipation rate should be much smaller than the cavity decay and in [43] which follows that the atomic dissipation should be much larger than the cavity decay, in our scheme, there are no specific requirements on the relationship between γ and κ. Therefore, our scheme is more suitable to be realized in the experiment than the previous ones [42, 43].

The analytical results of log₁₀ n_ss and log₁₀ (W/ω_m) as a function of the detunings Δ_g and Δ_g − Δ_r are plotted in Figs. 4(a) and 4(b), respectively. The solid and dashed lines are corresponding to the maximum cooling coefficient A₋ and the minimum heating one A₊, respectively. One can find that the lowest phonon number and fast cooling rate can be achieved around intersection of the two optimal conditions as the analytical predictions. As it is shown in Fig. 4(b), the cooling rate in our scheme can be higher than the order of O(10⁻³ ω_m), which is the same as the one in [43].

Fig. 4 The analytical results of (a) log₁₀ n_ss (b) log₁₀ (W/ω_m) as a function of the detunings Δ_g and Δ_g − Δ_r(in units of ω_m). The solid lines correspond to the condition (31), while the dashed lines correspond to the condition (25). The other parameters are γ = κ = ω_m, λ = 0.1ω_m, δ_P = −2ω_m, g = 7ω_m, ω_P = 0.8ω_m, Ω_r = 5ω_m.

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After the dissipation rate γ_m (γ_m > 0) and thermal occupation number n_th are taken into Eq. (14), the cooling rate W′ and the final mean phonon number n′_ss can be given as

W^{'} = W + 2 γ_{m}; n_{s s}^{'} = \frac{A_{+} + 2 γ_{m} n_{th}}{W^{'}} .

Figure 5 shows the final mean phonon number depends on the quality Q of the MR and the environment temperature T. To the experimental achievable parameters Q > 10⁵ and T ∼ 20mK [47–49], the MR can be cooled to a final mean phonon number lower than 1 (n′_ss < 1).

Fig. 5 The analytically predicted final mean phonon number log₁₀ n_ss as a function of the environment temperature T and the quality of the MR. The parameters are ω_m = 2π × 1MHz, γ = κ = ω_m, λ = 0.1ω_m, δ_P = −2ω_m, g = 7ω_m, ω_P = 0.8ω_m, Ω_r = 5ω_m, Δ_g = −20ω_m, Δ_r is chosen under the condition (25).

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To verify the analytical treatment, we directly solve the original master equation (5) via the quantum optics toolbox [50]. The numerical results are shown in Fig. 6: the cooling dynamics of the numerical simulation is in good agreement with the corresponding analytical prediction. Here, we choose the dissipation rates γ = κ = ω_m, which is out of the regime for working in [42,43], and the MR can be successfully cooled to its ground state with a vibrational occupation in the order of O(10⁻²). Figure 6 also shows the numerical result with the MR coupled to the thermal bath. In this situation, when the parameters are ω_m = 2π × 1MHz, γ_m = 10Hz and T ∼ 20mK, the MR still can be cooled to the vibrational number lower than 1 successfully.

Fig. 6 Numerical simulations of cooling dynamics. The parameters are ω_m = 2π × 1MHz, γ = κ = ω_m, λ = 0.1ω_m, δ_P = −2ω_m, g = 7ω_m, ω_P = 0.8ω_m, Ω_r = 5ω_m, Δ_g = −20ω_m, Δ_r is chosen under the condition (25). The blue line denotes the case of γ_m = 0 and the dashed line is the corresponding analytical prediction. The red line denotes the case of γ_m = 10Hz, T = 20mK.

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In addition, similar to [42], in our scheme, the pump laser that couples to the cavity mode can be replaced by a laser beam which couples the atomic transition |g〉 ↔ |e〉 directly. By calculating the rate equation of the new model, the similar form of cooling and heating coefficients A_± and lowest final phonon number can be obtained, as it is predicted in [42]. It can provide an optional choice for experimental realization.

5. Conclusion

In conclusion, we have presented a new cooling scheme with a Λ-type atom trapped in a weak excited optomechanical system. In the Lamb-Dicke regime, the dynamics of the mean phonon number of the MR is analytically calculated by using the perturbative method. Moreover, we find that the heating transitions can be effectively suppressed by the EIT-like quantum coherent effect. Therefore, the final mean phonon number can reach the order of $\frac{1}{C}$ , and the numerical simulations also demonstrate that the final mean phonon number of the MR can be less than 1 with the current experimental technology. In addition, compared with previous MR cooling methods with a single-atom assisted, there is no requirement on the linewidths of the cavity and atom. As a result, this method can be very suitable to be achieved in the experiment.

Appendix: The derivation of A_±

When the internal DOF can be eliminated adiabatically, with the Eq. (10) applied, the equation of the population of the n-th phonon state p_n can be given as

\frac{d}{d t} p_{n} = \sum_{m = n \pm 1} (Γ_{m \to n} p_{m} - Γ_{n \to m} p_{n}),

where Γ_i→j is the transition rate from the initial state with a phonon number i to the final with a phonon number j.

According to the method in [25, 46], the transition rates can be calculated by considering all the possible heating and cooling scattering processes. Then, Γ_n→n±1 is the sum of two incoherent scattering processes:

Γ_{n \to n \pm 1} = Γ_{n \to n \pm 1}^{κ} + Γ_{n \to n \pm 1}^{γ},

where

Γ_{n \to n \pm 1}^{κ}

and

Γ_{n \to n \pm 1}^{γ}

are the transition rates from n-th vibration steady state |ψ|g〉|n〉 to n ± 1-th vibration steady state |ψ_n±1〉 = |g〉|n ± 1〉, respectively:

Γ_{n \to n \pm 1}^{κ} = 2 κ {| 𝒯_{n}^{κ, \pm} |}^{2}, Γ_{n \to n \pm 1}^{γ} = 2 γ {| 𝒯_{n}^{γ, \pm} |}^{2} .

𝒯_{n}^{κ, \pm}

and

𝒯_{n}^{γ, \pm}

are the corresponding scattering amplitudes in the form of

\begin{array}{l} 𝒯_{n}^{κ, \pm} & = & 〈 ψ_{n \pm 1} | W_{κ} G (E_{n}) V_{1} G (E_{n}) V_{0} | ψ_{n} 〉, \\ 𝒯_{n}^{γ, \pm} & = & 〈 ψ_{n \pm 1} | W_{γ} G (E_{n}) V_{1} G (E_{n}) V_{0} | ψ_{n} 〉, \end{array}

G (z) = \frac{1}{z - H_{eff}},

with an effective Hamiltonian H_eff = H′₀ − iγ |e〉〈e| − iκ |1〉〈1|.

With the Eq. (40) calculated, one can get the

\begin{array}{l} 𝒯_{n}^{κ, \pm} & = & \sqrt{n + δ_{\pm}} Ω_{P} \frac{Δ_{g} (Δ_{g} - Δ_{r}) - Ω_{r}^{2} + i γ (Δ_{g} - Δ_{r})}{f (0)} \\ \cdot λ \frac{(\mp ω_{m} + Δ_{g}) (\mp ω_{m} + Δ_{g} - Δ_{r}) - Ω_{r}^{2} + i γ (\mp ω_{m} + Δ_{g} - Δ_{r})}{f (\mp ω_{m})}, \\ 𝒯_{n}^{γ, \pm} & = & \sqrt{n + δ_{\pm}} Ω_{P} \frac{Δ_{g} (Δ_{g} - Δ_{r}) - Ω_{r}^{2} + i γ (Δ_{g} - Δ_{r})}{f (0)} λ \frac{- g (\mp ω_{m} + Δ_{g} - Δ_{r})}{f (\mp ω_{m})}, \end{array}

where δ_± equals to 1 for ‘+’ and equals to 0 for ‘−’;

f (x) = - Ω_{r}^{2} (x + δ_{P}) - i κ Ω_{r}^{2} + (x + Δ_{g} - Δ_{r}) [- g^{2} + (i κ + x + δ_{P}) (i γ + x + Δ_{g})] .

For any n, it can be found that

Γ_{n \to n \pm 1} = (n + δ_{\pm}) A_{\pm};

and

Γ_{n \to n \pm 1}^{κ} = (n + δ_{\pm}) A_{\pm}^{κ}, Γ_{n \to n \pm 1}^{γ} = (n + δ_{\pm}) A_{\pm}^{γ} .

As a result, the explicit expressions in Eqs. (19), (20) and (21) can be obtained in this way.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11174370, 11304366, 61205108, 10974225 and 11304387), the Open Project Program of the State Key Laboratory of Mathematical Engineering and Advanced Computing (Grant No. 2013A14), and the China Postdoctoral Science Foundation (Grant No. 2013M531771 and 2014T70760).

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Ground state cooling of an optomechanical resonator assisted by a Λ-type atom

Abstract

1. Introduction

2. Model

2.1. Basic equations

2.2. Basic assumptions

3. Analytical results

3.1. Heating and cooling coefficients

3.2. Optimal cooling condition and cooling dynamics

4. Discussion

5. Conclusion

Appendix: The derivation of A_±

Acknowledgments

References and links

Cited By

Figures (6)

Equations (45)

Optics Express

Abstract

1. Introduction

2. Model

2.1. Basic equations

2.2. Basic assumptions

3. Analytical results

3.1. Heating and cooling coefficients

3.2. Optimal cooling condition and cooling dynamics

4. Discussion

5. Conclusion

Appendix: The derivation of A±

Acknowledgments

References and links

Cited By

Figures (6)

Equations (45)

Optics Express

Appendix: The derivation of A_±