Introduction

Cool a mechanical resonator to its ground state is of fundamental importance in many quantum information processing and metrological experiments: ultrahigh sensitive detection, the observation of quantum behavior in a mechanical oscillator and so forth. The subject has therefore attracted much attention from researchers in recent years1,2. In many of these schemes, an auxiliary system is introduced to cool the mechanical resonator. The ground state cooling of a mechanical resonator is realized by coupling the resonator with a driven cavity via radiation pressure (a dispersively coupled optomechanics)3, e.g. backaction cooling via a detuning cavity, cold-damping quantum feedback cooling4,5,6,7 and other methods8,9,10,11,12,13,14,15,16. By employing a chirped pulse, the phonon occupation of the mechanical resonator is reduced to a very small value quickly11,12,13. Experimental realization of cooling scheme for a mechanical resonator has been implemented17,18,19,20,21,22,23,24,25,26. By combining sideband cooling technique with cryogenic cooling, the position of the cooled resonator has been measured20,21. Recently, a quantum coherent exchange between the mechanical system and the micro/optical wave is achieved25,26 and the phonon occupation number of the cooled resonator is reduced to single-phonon level.

In several recent microwave optomechanical experiments27,28,29,30,31,32, it is shown that dissipative coupling, in which the cavity damping strength is modulated by the motion of mechanical resonator, should be considered for the cooling of resonator, aside from dispersive coupling. Theoretical studies also show that the coexistence of the dispersive coupling and dissipative coupling leads to the phenomenon of the quantum destructive interference in the mechanical resonator cooling33,34,35,36. In such a dispersively and dissipatively coupled system, the optimal cavity driven frequency depends significantly on the ratio of the dispersive coupling strength to the dissipative coupling strength. Moreover, in the microdisk-waveguide optomechanical system experiment37, it has been demonstrated that the force from the dissipative coupling dominates the total force applied to the waveguide in such dispersively and dissipatively coupled system.

In this paper, we study the rapid cooling of mechanical resonator for systems that are subject to both dispersive and dissipative coupling. With a chirped pulse, the fast cooling of a mechanical resonator can be realized within a short time. We show that, by modulating the incident pulses, the phonon occupation of the resonator can be reduced to single-phonon level. When the chirped pulse is applied in a pure dissipatively coupled optomechanical system, the cooling performance of the resonator becomes better with increasing cavity damping strength. In comparison, this is different from the optomechanical system that posseses only dispersive coupling, where it is shown that the cooling of the mechanical resonator improves with a smaller cavity damping strength, as shown by Liao and Law13. We also explore the cooling performance of our scheme with different frequency-sweep strengths. Moreover, we show that our scheme is effective for a wide range of dispersive and dissipative coupling strengths.

Results

The description of the system and its solution

We consider an optomechanical system consisting of a mechanical resonator and a cavity mode. Taking into account both dispersive and dissipative coupling, the system is described by the Hamiltonian33,

Here, we have used the following notations: ωc is the cavity frequency, ωm is the resonance frequency of the mechanical oscillator, a (a†) is the annihilation (creation) operator of the cavity mode, b (b†) is the annihilation (creation) operator of the mechanical resonator, Hκ and Hγ are the damping of the cavity and mechanical resonator, respectively, A (B) is the dispersive (dissipative) coupling strength, κ is the cavity damping strength, bq () is the annihilation (creation) operator of the optical bath coupled to the cavity mode and ρ is the density of state for the optical bath. Using the Heisenberg equation, we obtain36,38,

the input-output relation is

where ain is the input mode. The Hamiltonian is linearized and each operator is separated into a mean amplitude and a fluctuation component. Using , b = 〈b〉 + δb and , ωd is the frequency of the driven field, ξin is the noise induced by the optical bath. The detailed expressions of dynamics of the fluctuation components δa, δb as well as the mean amplitudes 〈a〉, 〈b〉, are described in Methods section.

Our aim is to reduce the value of fluctuation components 〈δb†δb〉. A covariance matrix notation is introduced. vl is the lth element of , the vector is constructed from such four fluctuation components,

The dynamics of R is formulated as

The explicit forms of the above parameters, i.e. G(t), Z(t), are given in details in the Methods section. We assume that the initial optical bath is a zero-temperature bath, the initial condition R(0) has three nonzero elements, R13(0) = 1, R24(0) = Nth + 1 and R42(0) = Nth. Based on the definition of the covariance matrix , we then obtain the time-dependent mean displaced phonon number as 〈δb†δb〉 = R42(t) and the mean displaced photo number 〈δa†δa〉 = R31(t).

Cooling of the resonator with chirped pulse

Chirped pulse form

The chirped pulse has been employed to convert the population of a two-level-system efficiently11,12. In our dispersively and dissipatively coupled optomechanics, the interaction between the cavity mode and the mechanical resonator mode (δa and δb) has a form similar to the optical-matter interaction11,12,13. The incident chirped pulse is designed as follows: , χ(t) = χ0 · sech(α(t − t0)), the frequency sweep , the amplitude of the coherent laser drive Ω(t) as shown in Methods section. We present the performance of such a chirped pulse in the cooling of mechanical resonator below for our optomechanical system with two couplings.

We numerically obtain the residual phonon number of the mechanical resonator (〈δb†δb〉) as a function of time. For comparison, two different schemes are provided: with chirped pulse (chirped pulse scheme) and without the chirped pulse (no-chirped pulse scheme). The different driven frequencies ωd are discussed below. Here, the cavity detuning is Δ = ωd − ωc. From Fig. 1(a), when Δ/ωm = −1, we find that at time ωmt = 80 or longer, the phonon number of the mechanical resonator can be reduced to a relatively low value and kept stable for the chirped pulse cooling scheme. The figure inset shows the amplitude of the laser drive Ω(t) with time. By employing the Fourier transform, we find that the full width at half maximum (FWHM) of such a pulse Ω(t) is about 0.05ωm. So for the case of ωmt0 = 40, the effect of pulse bandwidth is neglectable. While, for the no-chirped pulse scheme, the cooling effect of the resonator is not well-behaved and the phonon number changes with time drastically. Although the small phonon occupation number of the resonator is achieved in the no-chirped pulse case for some time interval, the phonon occupation quickly raises to a high value at later time. It means that we do not achieve perfect cooling of resonator with the no-chirped pulse scheme. Next, another driven frequency is applied (Δ/ωm = 0.5). In our parameters settings (A = 0, B ≠ 0, a purely dissipative optomechanics), this value of driven frequency is the optimal one for the steady state cooling of resonator33,35,36. The cooling behaviors with different cavity damping strengths are shown in Fig. 1(b) and (c). Compared with the cooling results of the no-chirped pulse scheme, an improved cooling performance of the resonator is acheived with the chirped pulse. Taking the cavity damping strength κ/ωm = 0.5 as an example, in the chirped pulse scheme, at time ωmt = 70, the mechanical resonator number (〈δb†δb〉) reaches the value 1.6, which is much smaller than the case with the no-chirped pulse scheme.

Figure 1
figure 1

The time evolution of the phonon number 〈nosc〉 = 〈δb†δb〉.

(a) Blue solid one denotes the chirped pulse form, red dashed one is the no-chirped form. Other parameters: A = 0, B = 2 * 10−4, κ/ωm = 0.01, γ/ωm = 10−5, Nth = 100, 〈a(0)〉 = 200, Δ/ωm = −1. For the chirped pulse form, α/ωm = 0.14, β/ωm = 0.04, , ωmt0 = 40. (b) (chirped pulse form) and (c) (no-chirped pulse form), A = 0, B = 2 * 10−4, γ/ωm = 10−6, Nth = 50, 〈a(0)〉 = 103, Δ/ωm = 0.5; for the chirped pulse form (b), α/ωm = 0.15, β/ωm = 0.05, , ωmt0 = 30.

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We explore the relation between the cooling of resonator (〈nosc〉) and the cavity damping strength (κ) for the chirped pulse scheme. Fig. 2(a) and Fig. 2(b) illustrate the situation when the cavity driven frequency satisfies Δ/ωm = 0.5. When the cavity damping strength increases, we achieve a smaller phonon occupation of the resonator. From Eq. 7 in Methods section, we show that the time evolution of cavity mode operator δa is affected by the intrinsic damping term F1(t), the interaction term F2(t) associated with the mechanical mode δb and the noise induced from the optical bath F3(t). The detailed expression of F1(t), F2(t) and F3(t) is given by Eq. 11, 12, 13 in Methods section. From Fig. 2(b), we find that the value of |F1(t)|, |F2(t)| and |F3(t)| rise with the increase of the cavity damping strength κ. This means that the damping of cavity mode δa and interaction between the cavity mode δa and the mechanical resonator mode δb both increase. The energy of the resonator is quickly transferred to the zero-temperature cavity mode and then is dissipated into the bath rapidly. This is why we can obtain the smaller phonon occupation of the resonator with the larger cavity damping strength.

Figure 2
figure 2

(a), the residual phonon number 〈nosc〉 = 〈δb†δb〉 at time ωmt = 70, with the increase of cavity damping strength. Parameters set, Δ/ωm = 0.5, A = 0, B = 2 * 10−4, γ/ωm = 10−6, Nth = 50, 〈a(0)〉 = 103. For the chirped pulse form (blue cross term), α/ωm = 0.15, β/ωm = 0.05, , ωmt0 = 30. (b), the amplitude of F1, F2 and F3 with time evolution. Solid, κ/ωm = 0.1; dashed, κ/ωm = 0.3; dotted, κ/ωm = 0.5. Other parameters are same as in (a).

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Moreover, we study the relation between the cavity damping strength (κ) and the steady-state phonon number of the mechanical resonator in the no-chirped pulse scheme. Our aim is to reveal the different cooling behaviors between the purely dispersively and purely dissipatively coupled optomechanics, when the cavity driven frequency is chosen optimal in steady-state cooling. For the no-chirped pulse scheme, the mechanical intrinsic damping (Σeq) and quantum backaction force from the driven cavity (Sbac) contribute to the final phonon occupation of resonator at steady-state

where Scc(ω) is the mechanical spectrum, σth(ω) is a quantity related to the mechanical thermal bath and optomechanical self-energy, σopt(ω) combines the intracavity amplitude and quantum backaction force spectrum. N(ω) comprises the response function of the mechanical resonator and optomechanical self-energy. κ and γ are the cavity damping strength and mechanical intrinsic damping strength, respectively. The pure dispersive coupling case is shown in Fig. 3(a) and the cavity driven frequency is optimal, Δ/ωm = −1. In such a case, the phonon occupation of resonator decreases with cavity damping strength κ first, then rises up quickly. Fig. 3(b) and (c) show the pure dissipative coupling case with cavity detunings Δ/ωm = −1 and Δ/ωm = 0.5, respectively. When the cavity detuning is optimal (Δ/ωm = 0.5, see Fig. 3(c)), the residual phonon number of the resonator decreases with an increase in the cavity damping strength. It is noted that, for such a dissipatively coupled system, the relation between the residual phonon number of the steady-state cooling and the cavity damping strength is similar to the relation for the chirped pulse cooling as shown in Fig. 2(a).

Figure 3
figure 3

The steady-state phonon number 〈nosc〉 = 〈δb†δb〉, the intrinsic oscillator damping Σeq, the backaction from the cavity Sbac, with the cavity damping strength κ.

(a), pure dispersive coupling case, A = 2 * 10−4, B = 0, Δ/ωm = −1; (b), pure dissipative coupling case, A = 0, B = 2 * 10−4, Δ/ωm = −1; (c), pure dissipative coupling case, A = 0, B = 2 * 10−4, Δ/ωm = 0.5. For all of three plots, γ/ωm = 10−6, Nth = 50, intra-cavity amplitude at the steady state, 〈a〉 = 103.

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Next, we show the advantage of the chirped pulse scheme explicitly. We compare the cooling behaviors of the mechanical resonator with and without chirped pulses. When considering such a purely dissipatively coupled optomechanics, we allow the cavity driven frequency in the no-chirped pulse scheme to satisfy Δ/ωm = 0.5. At such an optimal detuning, the cavity acts as an effective zero-temperature bath33. From Fig. 4, we find that by employing the chirped pulse, the mean resonator number is reduced to less than one quickly and kept such a low population for a long time. We need to mention that, the rise of the phonon number of the resonator after the pulse duration is due mainly to the heating from the bath of mechanical resonator13. In our discussion, the phonon number increases from 0.14 to 1.08 with the time varying from ωmt = 80 to ωmt = 2000. Considering the fact that the initial phonon occupation of the resonator is Nth = 50, such a rise of phonon number from 0.14 to 1.08 after the pulse duration is actually rather small. In contrast, for the no-chirped pulse scheme, we do not even achieve the ground state of resonator at time ωmt = 2000. So, the chirped pulse scheme provides a new way to cool the resonator in finite time duration and this is helpful for controlling the state of the resonator.

Figure 4
figure 4

The time evolution of the phonon number 〈nosc〉 = 〈δb†δb〉.

Red dashed, the chirped pulse form, Δ/ωm = −1, α/ωm = 0.14, β/ωm = 0.04, , ωmt0 = 40; blue solid, the no-chirped form, Δ/ωm = 0.5. Other parameters: A = 0, B = 2 * 10−4, γ/ωm = 10−5, Nth = 50, 〈a(0)〉 = 103.

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We now discuss the universality of the chirped pulse scheme in resonator cooling. The evolution time is ωmt = 70 ≥ 2t0. Firstly, the effect of the different strengths of frequency sweeping field and optomechanical coupling strength ratios (dispersive coupling strength/dissipative coupling strength) is shown in Fig. 5(a). In the region of 0.08 ≤ |β/ωm| ≤ 0.18, the mean residual occupation is lower than one. This result is insensitive to the strength of sweeping field. It is well known that the dispersive coupling and dissipative coupling strength can be modulated in the designed scheme30,31. So we can achieve the ground state cooling of the mechanical resonator with different dispersive and dissipative coupling strengths. Secondly, we explore the effects of the cavity detunings. The residual phonon number of the mechanical resonator with the chirped pulse scheme and no-chirped pulse scheme are compared in Fig. 5(b) and (c). For the chirped pulse case (b), we can reduce the phonon occupation of the resonator to one in the vicinity of Δ/ωm = −1, with the coupling strength ratio A/B ≤ 0.2. The resonator is cooled to around one phonon with the coupling strength ratio A/B ≥ 5 in the no-chirped pulse scheme (c). When we choose the ratio A/B ≥ 5 and in the chirped pulse case, the mean phonon occupation of the resonator is smaller than three. Although it might not be easy to achieve the same effective cooling performance as the no-chirped pulse scheme in the region A/B ≥ 5, the chirped pulse scheme can cool the resonator efficiently for a much wider region, 5 * 10−3 ≤ A/B ≤ 10. Finally, in the pure dissipative coupling case, the influence of the frequency sweep and cavity detuning on the phonon occupation is shown in Fig. 5(d) and (e). A better cooling effect appears when the frequency sweep value |β/ωm| is relatively large. In the region around Δ/ωm = −1 and Δ/ωm = 0.4, we reduce the residual phonon occupation of the resonator below one when the strength of sweeping field is chosen as β/ωm = 0.125.

Figure 5
figure 5

The chirped pulse form, .

(a), the effect of the optomechanical coupling strength ratio (A/B) and the strength of frequency sweeping field β/ωm, α/ωm = 0.15, Δ/ωm = −1. (b), the effect of the optomechanical coupling strength ratio (A/B) and cavity detuning Δ/ωm, α/ωm = 0.15, β/ωm = 0.05. (c), a no-chirped pulse scheme, to compares with (b). (d) and (e), the strength of frequency sweeping field β/ωm and cavity detuning Δ/ωm to the phonon number 〈nosc〉 = 〈δb†δb〉, A = 0, α/ωm = 0.15. For all of the plots, the time is set ωmt = 70, B = 2 * 10−4, γ/ωm = 10−6, κ/ωm = 0.3, Nth = 50, 〈a(0)〉 = 200, ωmt0 = 30.

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Discussion and conclusion

In summary, we have extensively studied the cooling of a mechanical resonator for the system consisting of both dispersive coupling and dissipative coupling in finite time. As stated in Ref. 13, the authors employ a chirped pulse to cool the mechanical resonator efficiently in a dispersively coupled optomechanics. The pulse shape is found in Fig. 2(b) of Ref. 13. There, a better cooling result is obtained with a smaller cavity damping strength. Here, we show that a smaller phonon occupation of the resonator is obtained with a larger cavity damping strength given our parameter settings. This is different from Ref. 13. This result is due to the collective effect of the cavity dissipation and the energy transfer between the cavity mode and mechanical resonator mode. By employing our chirped pulse scheme, the mechanical resonator is cooled quickly and the phonon occupation of the resonator remains stable for relatively long time. Our proposal with a chirped pulse can be used to cool resonators efficiently in finite time in a large range of system parameter settings.

Methods

Applying linearization, we obtain four physical quantities δa, δb, 〈a〉 and 〈b〉 representing the operators of one cavity mode a and mechanical resonator b. To illustrate the dynamics of fluctuation components δa and δb, we use the following coupled differential equations of in rotating frame at the given driven frequency36,

Here, Δ = ωd − ωc is the detuning of the cavity drive, δa (δb) is the fluctuation component of the cavity mode (mechanical mode), 〈a〉 is the intra-cavity amplitude, is the amplitude of the coherent laser drive, η is the noise influencing the mechanical resonator, γ is the mechanical intrinsic damping strength and is the real part of 〈b(t)〉. The time evolution equations for 〈a〉 and 〈b〉 are

The convariance matrix method is applied to get the phonon number of the resonator. For simplicity, we use the following expressions

We set , the dynamics of is satisfied by

with

and the noise related terms,

We introduce a covariance matrix notation (l, l′ = 1, 2, 3, 4), vl is the lth element of . The dynamics of R is obtained as

here, Z(t) is,

with G(t) being governed by , G(0) is an identity matrix, (l, l′ = 1, 2, 3, 4), Nl is the lth element of . By considering a Markovian bath, we write C(τ, τ′) as C(τ)δ(τ − τ′) and the matrix expression of C(τ)

Here, Nth is the distribution of the thermal bath surrounding the mechanical resonator.