Fast optical cooling of a nanomechanical cantilever by a dynamical Stark-shift gate

The efficient cooling of nanomechanical resonators is essential to exploration of quantum properties of the macroscopic or mesoscopic systems. We propose such a laser-cooling scheme for a nanomechanical cantilever, which works even for the low-frequency mechanical mode and under weak cooling lasers. The cantilever is coupled by a diamond nitrogen-vacancy center under a strong magnetic field gradient and the cooling is assisted by a dynamical Stark-shift gate. Our scheme can effectively enhance the desired cooling efficiency by avoiding the off-resonant and undesired carrier transitions, and thereby cool the cantilever down to the vicinity of the vibrational ground state in a fast fashion.

Two-photon Raman process. The ground state sublevels |0⟩ and |1⟩ cannot be coupled directly unless the external magnetic field applied is of an angle with respect to the NV crystal axis. Unfortunately, our model requires the external magnetic field to be applied exactly along the axis of the NV crystal. As a result, we have to consider the effective coupling between |0⟩ and |1⟩ by a two-photon Raman process, where two additional lasers with large detunings from the |A 2 ⟩ are employed to couple, respectively, |0⟩ and |1⟩ to the exited state |A 2 ⟩ with frequency ω ′ 0 (ω ′ 1 ) and Rabi frequency Ω ′ 0 (Ω ′ 1 ). In this context, the original model for cooling plotted in Supplementary Figure 1(a) includes other two laser fields applied to drive the transition from the ground state |0⟩ (|1⟩) to the excited one |A 2 ⟩ with frequencȳ ω 0 (ω 1 ) and Rabi frequencyΩ 0 (Ω 1 ). Please note that, for the same parameters, we define different symbols from in the main text due to slight difference of parameter values after the two-photon Raman process is applied (see equation (4) below).
Thus our system is governed by the Hamiltonian In the rotating frame with R ≡ω 1 |1⟩ ⟨1| +ω 0 |0⟩ ⟨0|, the whole system is described as Under the large detuning ∆ ′ and utilizing the method proposed in [1], we obtain an effective Hamiltonian where ∆ ef f =∆ − 3Ω ′ 2 4∆ ′ and ∆ ′ < 0. Note that the energy shifts caused by the Raman process have been included in this effective Hamiltonian. In the case of (3) here is unitarily equivalent to equation (2) plus equation (3) in the main text.
The corresponding parameters in equation (2) of the main text and in above equations (1, 2, 3) are as follows, (4) To justify the effective Hamiltonian in equation (2), we numerically compare equation (2) with equation (3) in Supplementary Figure 1(b), which shows a good fitting in the case of ∆ ′ /2π = −400 MHz.
The cooling and heating rates. In what follows, we derive the cooling and heating rates by the non-equilibrium fluctuation-dissipation relation. Using X = x 0 (a † + a), we rewrite the interaction Hamiltonian in equation (3) in the main text as As a result, the Heisenberg operator F (t) is given by where we have defined σ m,n The steady state ρ ss for the NV center can be obtained from following Bloch equations of H rot [2], where Γ d (= Γ b ) = (γ 0 + γ 1 )/2 is the decay rate from the excited state |A 2 ⟩ to the state |d⟩ (|b⟩), Γ = γ 0 + γ 1 is the total decay rate with γ 1 and γ 0 being the decay rates, respectively, from the excited state |A 2 ⟩ to the states |1⟩ and |0⟩, ρ bb = |b⟩⟨b|, ρ dd = |d⟩⟨d| and σ m,n y = −i(|m⟩⟨n| − |n⟩⟨m|). The steady state for these Bloch equations can be solved by making the left side of equation 7 equal to zero (i.e., d⟨···⟩ dt = 0) and it is ρ ss = ρ dd , which means a dark state regarding the NV center.
When the NV center is in the dark state, the fluctuation spectrum is written as where η = λ/ω. According to the quantum regression theorem [2], the differential equations of the correlation functions can be written as Defining a Fourier transformation we obtain Applying the transformation to the differential equations and solving the corresponding equations, we have and the corresponding heating (cooling) coefficient We discuss by equation (12) whether the cooling efficiency can be optimal at the work point Ω L /ω k = 1 of the Stark-shift gate. A straightforward evaluation can be done by ∂ ∂Ω L A − = 0. As comparison in Supplementary  Figure 2(a), the optimal cooling efficiency occurs at Ω L /ω k =0.984, slightly different from the work point of the Stark-shift gate. The difference is resulted from the rotating wave approximation we made for obtaining the work point of the Stark-shift gate and also from the energy correction caused by the external light fields. Specifically, if we consider a second-order approximation of Γ under the condition Γ ≫ max[∆, Ω, ω], we may have an analytical expression of the difference Ω L − ω k ∝ (ω k + 2∆)[5(ω k + 2∆) 2 − 16Ω 2 ]/Γ 2 . Since the difference is tiny, we may still employ the work point of the Stark-shift gate as the approximately optimal cooling point, which is physically clear and easily understood.
To be more clarified, we plot Re[S(ω)] for various Ω L in Supplementary Figure 2(b). In fact, we know from equation (12) that, different from the fluctuation spectra of EIT obtained in Ref. [3], the Stark-shift fluctuation spectra maximize at different values conditional on Rabi frequencies Ω, Ω L and detuning ∆.