arXiv:1001.1286v1 [cond-mat.mes-hall] 8 Jan 2010 Cooling a mechanical resonator by quantum interference in a triple quantum dot Shi-Hua Ouyang, 1, 2 Chi-Hang Lam, 1 and J. Q. You 2 $^{1}$Department of Applied Physics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China $^{2}$Department of Physics and Surface Physics Laboratory (National Key Laboratory), Fudan University, Shanghai 200433, China (Dated: November 8, 2018) We propose an approach to cool a mechanical resonator (MR) via quantum interference in a triple quantum dot (TQD) capacitively coupled to the MR. The TQD connected to three electrodes is an electronic analog of a three-level atom in Λ configuration. The electrons can tunnel from the left electrode into one of the two dots with lower-energy states, but can only tunnel out from the higher-energy state at the third dot to the right electrode. When the two lower-energy states are tuned to be degenerate, an electron in the TQD can be trapped in a superposition of the degenerate states called the dark state. This effect is caused by the destructive quantum interference between tunneling from the two lower-energy states to the higher-energy state. Under this condition, an electron in the dark state readily absorbs an energy quantum from the MR. Repeating this process, the MR can be cooled to its ground state. Moreover, we propose a scheme for verifying the cooling result by measuring the current spectrum of a charge detector adjacent to a double quantum dot coupled to the MR. PACS numbers: 03.65.Ta, 85.35.Be, 42.50.Gy Mechanical resonators (MRs) with a high resonant fre- quency and a small mass have wide applications and are attracting considerable recent attentions [1, 2]. Tech- nically, these MRs can be used as ultrasensitive sen- sors in high-precision displacement measurements [3], de- tection of gravitational waves [4] or mass detection [5]. Also, quantized MRs can be useful in quantum infor- mation processing. Indeed, quantized motion of buck- ling nanoscale bars has been proposed for qubit imple- mentation [6, 7] and also for creating quantum entangle- ment [8–10]. However, for all these applications, a basic prerequirement is that the dynamics of the MRs must approach the quantum regime. Quantum behaviors of a MR are usually suppressed by the coupling to its environment. One way to approach the quantum regime is to increase its resonant frequency so that an energy quantum of the MR becomes larger than the thermal energy. Recently, MRs based on metal- lic beams [11] and carbon nanotubes [12] have been de- veloped, which have resonance frequencies of several hun- dred megahertz. However, for a MR with a frequency of 200 MHz, a temperature lower than 10 mK (below the present dilution refrigerator temperature) is required to maintain the MR at the quantum regime. To attain the quantum regime, one needs to cool the MR further via coupling to an optical or an on-chip electronic system. Numerous experiments on cooling a single MR via radia- tion pressure or dynamical backaction have been reported (see, e.g., [13–21]). Theoretically, cooling by coupling to a Cooper pair box [22] or to a three-level flux qubit [23] via periodic resonant coupling have also been proposed. In these schemes, a strong resonant coupling between the MR and the qubit is required to cool the MR to its ground state. For laser cooling of atoms, an alternative approach [33] based on quantum interference in the internal degrees of freedom of the atoms without the need to follow the resolved-sideband cooling condition has been proposed. B. Cooling atomic motion via quantum interference in a three-level atom In the weak coupling regime, a conventional method for cooling the MR is the sideband cooling approach (see, e.g., Ref. [24–31]). In this case, a MR is coupled to a two- level system (TLS) in which the two states can be elec- tronic states in quantum dots [24–26], photonic states in a cavity [27–30], or charge states in superconduct- ing qubits [32]. In order to achieve ground-state cooling in the sideband cooling approach, the resolved-sideband cooling condition ω$_{m}$ ≫ Γ (with ω$_{m}$ denoting the os- cillating frequency of the MR and Γ the decay rate of the TLS) must be followed in order to selectively drive the lowest sideband of the TLS. Then, the excitation of the TLS from the ground state | g 〉 to the excited state | e 〉 and the subsequent decay from this excited state to the ground state will, on average, decrease an energy quantum in the MR. This process can be described by | g, n 〉 → | e, n − 1 〉 → | g, n − 1 〉 , where n denotes to the state with n phonons. However, the frequency of a typ- ical MR is about 100 MHz [11, 12]. It is in general of the same order of the decay rate of the two-level system. This indicates that the resolved-sideband cooling condi- tion is not easy to fulfill. Violating the condition means that the processes of a carrier transition and a subsequent sideband transition (i.e., | g, n 〉 → | e, n 〉 → | g, n + 1 〉 ) will occur. This will heat up the MR instead and suppress any ground-state cooling [29]. A. Sideband cooling of a MR I. INTRODUCTION