^{1}and Gao-xiang Li

^{1,a)}

### Abstract

A scheme for ground-state cooling of a mechanical resonator by single- and two-phonon processes is analyzed. The mechanical resonator is coupled to two coupled quantum dots forming an effective Λ-type three-level structure and connected with two normal metal leads. The quantum dots are driven by two light fields; by choosing appropriate parameters, the electron can be trapped in the dark state of the system, a superposition of the two ground states. When the single-phonon absorption and emission processes are dominant, under the weak (strong) driving field circumstances, the mechanical resonator is cooled through absorbing a phonon when the electron jumps from dark state to bright state (one of the dressed states) and then tunnels out of the two coupled dots. Net cooling of the resonator to its ground state is possible in the absence of the electron-phonon dephasing via single-phonon processes. When the two-phonon processes are tuned to be stronger than the single-phonon processes, the mechanical resonator can be cooled to its nonclassical state.

This work is supported by the National Natural Science Foundation of China (Grant Nos. 60878004 and 11074087), the Natural Science Foundation of Hubei Province (Grant No. 2010CDA075), and the Nature Science Foundation of Wuhan City (Grant No. 201150530149).

I. INTRODUCTION

II. MODEL AND EQUATIONS OF MOTION

III. GROUND-STATE COOLING OF MECHANICAL RESONATOR BY SINGLE-PHONON PROCESSES

A. Weak driving field regime

B. Strong driving field regime

IV. NON-THERMAL STATE COOLING OF MECHANICAL RESONATOR

V. CONCLUSIONS

### Key Topics

- Quantum dots
- 63.0
- Phonons
- 47.0
- Vibration resonance
- 42.0
- Dark states
- 27.0
- Ground states
- 25.0

## Figures

Schematic diagram of a Λ-type three-level system that consists of the ground state |*l*⟩ in the left dot, and the ground state |*r*⟩ and the first excited state |*e*⟩ in the right dot in a double quantum dot device. The right dot is driven by a light field of the Rabi frequency Ω_{2} (with frequency ω_{2}) and a light field of the Rabi frequency Ω_{1} (with frequency ω_{1}) is applied to drive the transition between |*l*⟩ and |*e*⟩. The chemical potential μ_{ L } is well above the level |*l*⟩ and μ_{ R } between the levels |*r*⟩ and |*e*⟩, i.e. μ_{ L } ≫ *E* _{ l } and *E* _{ r } ≪ μ_{ R } ≪ *E* _{ e }. The excited state of the left dot is far-off-resonance with the excited state of the right one, thus electrons are allowed to transfer from left dot to right dot through the transition |*l*⟩↔|*e*⟩ and the tunneling between the two ground states.

Schematic diagram of a Λ-type three-level system that consists of the ground state |*l*⟩ in the left dot, and the ground state |*r*⟩ and the first excited state |*e*⟩ in the right dot in a double quantum dot device. The right dot is driven by a light field of the Rabi frequency Ω_{2} (with frequency ω_{2}) and a light field of the Rabi frequency Ω_{1} (with frequency ω_{1}) is applied to drive the transition between |*l*⟩ and |*e*⟩. The chemical potential μ_{ L } is well above the level |*l*⟩ and μ_{ R } between the levels |*r*⟩ and |*e*⟩, i.e. μ_{ L } ≫ *E* _{ l } and *E* _{ r } ≪ μ_{ R } ≪ *E* _{ e }. The excited state of the left dot is far-off-resonance with the excited state of the right one, thus electrons are allowed to transfer from left dot to right dot through the transition |*l*⟩↔|*e*⟩ and the tunneling between the two ground states.

The population occupations ρ_{ BB } and ρ_{ DD } in steady state are plotted as a function of Ω in different dephasings and as a function of β (inset) under different light strengths. For simplicity, we set β_{ i } = β, I = 1, 2, 3, and Ω_{1} = Ω_{2} = Ω. The other parameters are *T* = 50Γ, δ = 0, Γ_{ L } = 10Γ, Γ_{ R } = 3Γ, and γ_{1} = γ_{2} = 0.1Γ. All parameters are scaled in unit of Γ.

The population occupations ρ_{ BB } and ρ_{ DD } in steady state are plotted as a function of Ω in different dephasings and as a function of β (inset) under different light strengths. For simplicity, we set β_{ i } = β, I = 1, 2, 3, and Ω_{1} = Ω_{2} = Ω. The other parameters are *T* = 50Γ, δ = 0, Γ_{ L } = 10Γ, Γ_{ R } = 3Γ, and γ_{1} = γ_{2} = 0.1Γ. All parameters are scaled in unit of Γ.

The total cooling rate γ_{ tot } and the mean phonon number of the resonator ⟨*n*⟩_{ st } as functions of dephasings and Ω (a) or ω_{ m } (b). In (a) (including inset), the resonator frequency ω_{ m } = 100.2Γ; in (b) (including inset), Ω = 4.48Γ. We set Ω_{1} = Ω_{2} = Ω and β_{ i } = β, I = 1, 2, 3. Here α = Γ, , and γ_{ p } = 1 × 10^{−5}Γ. The other parameters are same with that in Fig. 2.

The total cooling rate γ_{ tot } and the mean phonon number of the resonator ⟨*n*⟩_{ st } as functions of dephasings and Ω (a) or ω_{ m } (b). In (a) (including inset), the resonator frequency ω_{ m } = 100.2Γ; in (b) (including inset), Ω = 4.48Γ. We set Ω_{1} = Ω_{2} = Ω and β_{ i } = β, I = 1, 2, 3. Here α = Γ, , and γ_{ p } = 1 × 10^{−5}Γ. The other parameters are same with that in Fig. 2.

Similar to Figs. 3(a) and 3(b) except ω_{ m } = 120Γ in (a) and Ω = 53Γ in (b) (including inset). In (b), the symbol star is plotted for the case β = 10^{−4}Γ and the solid line for the case β = 0.

Similar to Figs. 3(a) and 3(b) except ω_{ m } = 120Γ in (a) and Ω = 53Γ in (b) (including inset). In (b), the symbol star is plotted for the case β = 10^{−4}Γ and the solid line for the case β = 0.

(a) and (b) illustrate the mean phonon number of the resonator and the Mandel’s parameter *Q* as the function of Ω, respectively, when 2ω_{ m } = *E* _{+} − *E* _{ D }. The other parameters are Γ_{ L } = 10Γ, Γ_{ R } = 0.5Γ, *T* = 50Γ, γ_{1} = γ_{2} = 0.1Γ, γ_{ p } = 10^{−5}Γ, α = 10Γ and *n* _{ p } = 400.

(a) and (b) illustrate the mean phonon number of the resonator and the Mandel’s parameter *Q* as the function of Ω, respectively, when 2ω_{ m } = *E* _{+} − *E* _{ D }. The other parameters are Γ_{ L } = 10Γ, Γ_{ R } = 0.5Γ, *T* = 50Γ, γ_{1} = γ_{2} = 0.1Γ, γ_{ p } = 10^{−5}Γ, α = 10Γ and *n* _{ p } = 400.

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