Cooling a mechanical resonator via coupling to a tunable double quantum dot

Shi-Hua Ouyang, J. Q. You, and Franco Nori

Phys. Rev. B 79, 075304 – Published 6 February 2009

Abstract

We study the cooling of a mechanical resonator (MR) that is capacitively coupled to a double quantum dot (DQD). The MR is cooled by the dynamical backaction induced by the capacitive coupling between the DQD and the MR. The state transition between the two dots of the DQD is excited by an ac field and afterward a tunneling event results in the decay of the excited state of the DQD. An important advantage of this system is that both the energy-level splitting and the decay rate of the DQD can be well tuned by varying the gate voltage. We find that the steady average occupancy, below unity, of the MR can be achieved by changing both the decay rate of the excited state and the red-detuning between the transition frequency of the DQD and the microwave frequency, in analogy to the laser sideband cooling of an atom or trapped ion in atomic physics. Our results show that the cooling of the MR to the ground state is experimentally implementable. Also, the MR can be heated and the steady-state average occupancy becomes infinite when the microwave frequency is larger than the transition frequency of the DQD (i.e., the blue-detuning).

Received 25 August 2008

DOI:https://doi.org/10.1103/PhysRevB.79.075304

Authors & Affiliations

Shi-Hua Ouyang^1,2, J. Q. You^1,2, and Franco Nori^2,3

¹Department of Physics and Surface Physics Laboratory (National Key Laboratory), Fudan University, Shanghai 200433, China
²Advanced Science Institute, The Institute of Physical and Chemical Research (RIKEN), Wako-shi 351-0198, Japan
³Department of Physics, Center for Theoretical Physics, and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 79, Iss. 7 — 15 February 2009

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(Color online) (a) Schematic diagram of a DQD connected to an electron source S and drain D via tunneling barriers. An oscillating plate (the MR) and another static plate form a capacitor on the left dot, which provides a capacitive coupling between the oscillating MR and the DQD. The energy level of each dot is tunable by varying the gate voltage $V_{g}$ applied to the dot through the capacitor. (b) Transport process of an electron through a DQD: First, an electron tunnels from the source to the left dot, and then a microwave field drives it to the right dot. Finally, it tunnels to the drain on the right side.Reuse & Permissions
Figure 2
(Color online) Schematic diagram of the cooling (heating) process in the coupled MR-DQD system. When the DQD is excited by a red-detuned microwave field, i.e., $Δ = ω_{d} - ω_{0} < 0$ , the anti-Stokes process $(| 1 ⟩ | n ⟩ \to | 2 ⟩ | n - 1 ⟩)$ is resonantly enhanced. A subsequent decay from the excited state $| 2 ⟩$ to the ground state $| 1 ⟩$ reduces the energy of the MR by one quanta. This cools the MR because the emitted (blue) photon has more energy than the absorbed (green) photon. Due to the off-resonance, the Stokes process $(| 1 ⟩ | n ⟩ \to | 2 ⟩ | n + 1 ⟩)$ is suppressed. However, in the resonant case, the Stokes process dominates and the cycle heats the system. The blue (red) vertical downward arrow shows the cooling (heating) process, which decreases (increases) the phonon occupancy $n$ of the MR.Reuse & Permissions
Figure 3
(Color online) (a) The rate $W / η^{2}$ as a function of the driving detuning $Δ = ω_{d} - ω_{0}$ for different driving strengths $Ω_{0}$ . For red-detuning $Δ < 0$ , the rate $W > 0$ and the MR can be cooled. In contrast, the heating process $(W < 0)$ dominates the dynamics of the MR for blue-detuning $Δ > 0$ . (b) The steady-state phonon occupancy $n_{f}$ as a function of the normalized driving detuning $Δ / Γ$ . The parameters are $Γ = 1$ , $ω_{m} = Γ$ , and $Ω_{0} = Γ$ (black curve), $Ω_{0} = 2 Γ$ (red dashed curve), and $Ω_{0} = 5 Γ$ (green curve).Reuse & Permissions
Figure 4
(Color online) (a) Microwave-induced steady-state average phonon occupancy $n_{f}$ , shown by colors or shades, as a function of both the normalized driving detuning $Δ / Γ$ and the normalized oscillation frequency $ω_{m} / Γ$ . (b) Three contour curves of the microwave-induced steady-state average phonon occupancy for $n_{f} = 1$ , 0.1, and 0.01. The parameters are $Γ = 1$ and $Ω_{0} = 2 Γ$ .Reuse & Permissions
Figure 5
(Color online) Comparison of the steady-state average phonon occupancy $n_{f}$ between the exact [Eq. (28)] and approximate expressions [Eq. (34)] as a function of the normalized driving detuning $Δ / ω_{m}$ . The parameters are $Γ = 1$ and $Ω_{0} = 2 Γ$ , as well as (a) $ω_{m} / Γ = 100$ and (b) $ω_{m} / Γ = 1$ .Reuse & Permissions

Physical Review B

covering condensed matter and materials physics