1. INTRODUCTION

Quantum refrigeration of large-mass oscillators is an essential prerequisite for revealing genuine effects of quantum mechanics at large-mass scales, observing significant quantum signatures of mechanical motion, and probing quantum gravity on massive mechanical systems [1–15]. This is because thermal noise in these oscillators destroys quantum coherence signatures [1–3], precludes the exploration of macroscopic quantum phenomena [4–6], and limits the sensitivity of massive mechanical transducers in metrology and sensing applications [7–15].

Over the past few decades, significant developments have been accomplished in cooling the mechanical resonator to its motional quantum ground state, which has been widely reported both theoretically [16–19] and experimentally [20–27], using cavity optomechanical platforms [28–31]. However, these proposals and experiments still inherently suffer from both large-mass and high-temperature limitations [32–35], which are a major challenge for the preparation of such extremely fragile quantum ground states. The physical origin behind these obstacles is that a large optical power, which is required to cool a large-mass resonator, introduces extraneous excessive heating and intricate instabilities [30]. Especially, a high-temperature environment in the large-mass regime accelerates intrinsic thermal motion and blocks efficient light-motion refrigeration. Besides these issues, quantum ground-state refrigeration of vibrational networks in the large-mass and high-temperature regimes is ultimately required, in terms of the practical applicability of modern quantum technologies for quantum networks [36–45].

Here we propose a simple use of an exceptional point (EP) to overcome these cooling obstacles originating from both resonator-mass and environmental-temperature limitations, and to achieve an exceptional refrigeration of motional networks. We recall that EPs correspond to non-Hermitian spectral degeneracies featuring the simultaneous coalescence of both eigenstates and associated eigenvalues [46–55]. Quantum systems with EPs show a variety of counterintuitive phenomena and intriguing physical effects, e.g., chiral perfect absorption [56], exceptional single-photon sources [57,58], loss-induced lasing [59–61], and giant-enhanced sensing [62–67], which have no correspondence to their Hermitian counterparts.

 

Fig. 1. (a) EP-assisted optomechanical networks, where a passive optical mode $c$ (with decay rate ${\kappa _c}$ and driving-laser frequency ${\omega _L}$) is both coupled to an active optical mode $a$ (with gain rate ${\kappa _a}$) via a photon-tunneling interaction $J$, and coupled to $N$ vibrational modes ${b_{j \in [1,N]}}$ (with mechanical damping rate ${\gamma _j}$) by optomechanical-coupling rates ${g_j}$. (b) Real ${\text{Re}}[{\omega _ \pm} - {\omega _0}]/{\kappa _c}$ and (c) imaginary ${\text{Im}}[{\omega _ \pm} - {\omega _0}]/{\kappa _c}$ parts of the eigenfrequencies versus $J/{\kappa _c}$ and ${\kappa _a}/{\kappa _c}$ reveal spectral properties of the ${\cal P}{\cal T}$ symmetry, when ${\omega _c} = {\omega _a} \equiv {\omega _0}$. The white solid curves depict the case when gain and loss are balanced: ${\kappa _a} = {\kappa _c}$. (d) Net cooling rate ${\gamma _{j,{C}}}$ versus $J$ when ${\kappa _a} = {\kappa _c}$. Note that in (d), the two-peak points denote the maximal net cooling rates at the EP (i.e., $J/{\kappa _c} \approx 1$), and the term “Gain-cavity-off (on)” refers to the case when the gain cavity is turned off (on), i.e., $J/{\kappa _c} = 0$ ($J/{\kappa _c} \ne 0$). (b)–(d) correspond to the two-motional-mode case ($N = 2$). Other parameters are: ${\omega _1}/(2\pi) = 20\,{\text{MHz}}$, ${\omega _2}/{\omega _1} = 0.7$, ${\gamma _j}/{\omega _1} = {10^{- 5}}$, ${m_j} = 100\,{\text{ng}}$, ${\kappa _c}/{\omega _1} = 1/(5\pi)$, ${G_j}/{\kappa _c} = 0.05$, and $\omega /{\omega _1} = 0.99$ (corresponding to an optimal cooling rate [93]).

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We show that a giant enhancement can be achieved for optomechanical refrigeration of the large-mass motion at an EP, which is induced in parity-time (${\cal P}{\cal T}$) symmetric setups consisting of gain and loss cavities. Note that ${\cal P}{\cal T}$-symmetry quantum devices exhibit an EP when switching the unbroken to broken ${\cal P}{\cal T}$-symmetric regimes [46,47,68–82]. We reveal that the contamination-tolerant robustness against a high-temperature environment for the EP-cooling mechanism can be observed to be more than three orders of magnitude stronger than that in the standard-cooling (i.e., resolved-sideband-cooling) protocols [16–18]. Physically, the application of EP technologies to optomechanical networks induces a strong field-localization effect [59,75–78], which dramatically amplifies the net-refrigeration rate of the vibration. Specifically, the excitations concentrate in the active cavity leading to a faster depletion of the excitations in the original passive cavity and thus to an enhanced cooling rate for the vibration. Note that our EP refrigeration is an intensive cooling based on the sideband-cooling technique, and this occurs because of the synergy of the field-localization effect and the sideband-cooling mechanism.

Optomechanical refrigeration in the standard-cooling schemes becomes much worse with increasing resonator mass [16–24,83], while, surprisingly, in our EP cooling, it is nearly independent of the mass. Physically, the refrigeration rate significantly reduces, due to the decrease of the light-motion coupling strength with increasing resonator mass, while it can be completely compensated for or even greatly amplified by employing the field-localization effect [59,75–78]. These results indicate that our proposed ultrahigh-efficiency optomechanical cooling, with immunity against both environmental noise and resonator mass, can be realized by simply utilizing an EP, without the need of any high-cost and colossal LIGO gravitational-wave detectors [84] or magnomechanical materials [85]. In a broader view, our study sheds new light on the synergy of optomechanical-network and EP technologies, and offers exciting opportunities of cooling large-mass and high-temperature quantum devices to their motional quantum ground states. This is usually impossible with standard-cooling protocols [16–31,86–92], while our method can lead to a wide range of applications, e.g., both mass-immune and noise-tolerant quantum sensing, quantum processing, and quantum networks.

2. RESULTS

A. System and Its Exceptional Point

We consider an EP-assisted optomechanical network, where $N$ vibrations are optomechanically coupled to a passive cavity, linked to an active cavity [see Fig. 1(a)]. An EP supported in the loss-gain optical system occurs via the transition from the ${\cal P}{\cal T}$-symmetric to symmetry-broken regimes. A driving laser (with amplitude $|{\varepsilon _L}| = \sqrt {2{\kappa _c}{P_L}/\hbar {\omega _L}}$, frequency ${\omega _L}$, and laser power ${P_L}$) is applied to the loss cavity. In a rotating frame, defined by the unitary transformation operator $\exp [{- i{\omega _L}({{c^{\dagger}}c + {a^{\dagger}}a})t}]$, the Hamiltonian reads ($\hbar = 1$)

To demonstrate the EP of the system, first we only consider a coupled passive-active optical construction, and then its effective Hamiltonian, taking both dissipation (${\kappa _c}$) and gain (${\kappa _a}$) rates into consideration, can be expressed as

Specifically, we see from Figs. 1(b) and 1(c) that when $J \gt {\kappa _c}$ ($J \lt {\kappa _c}$), the eigenfrequencies have an identical imaginary (real) part and two different real (imaginary) parts. This indicates that the system possesses the ${\cal P}{\cal T}$ (broken-${\cal P}{\cal T}$) symmetry with an identical linewidth (frequency) and two different frequencies (linewidths). In particular, the EP (e.g., at $J \approx {\kappa _c}$ when ${\kappa _a} = {\kappa _c}$; see white solid curves), where the eigenstates coalesce, emerges due to the switching between the broken and unbroken ${\cal P}{\cal T}$-symmetric regimes. At this EP, a more than three-orders-of-magnitude amplification of the net cooling rate can be achieved, which is otherwise unattainable in conventional optomechanical devices, as shown in Fig. 1(d). Physically, the phonons concentrate in the gain cavity leading to a faster depletion of the phonons in the loss cavity and thus to an enhanced refrigeration rate for the vibration. In our numerical simulations, the maximum achievable cooling rate can be observed when $J$ is very close to (but slightly smaller than) ${\kappa _c}$. During an experiment, if the value of $J/{\kappa _c}$ fluctuates, it could be possible to push the system from experiencing strong cooling ($J \lt {\kappa _c}$) to cooling suppression ($J \gt {\kappa _c}$) [93]. This would constitute a limiting instability for realizable cooling.

B. Numerical and Analytical Results

By applying a strong optical driving, the dynamics of the system can be linearized by expanding all the operators as sums of their steady-state mean values and quantum fluctuations, i.e., $a(c) = {\alpha _{1(2)}} + \delta a(c)$ and ${b_j} = {\beta _j} + \delta {b_j}$. After ignoring higher-order terms of quantum fluctuations, we obtain the linearized equations of motion for quantum fluctuations: ${\boldsymbol {\dot u}}(t) = {\bf Au}(t) + {\bf N}(t)$, where we introduce the fluctuation operator vector ${\bf u}(t) = {({\delta c,\delta {c^{\dagger}},\delta a,\delta {a^{\dagger}},\delta {b_1},\delta b_1^{\dagger} , \ldots ,\delta {b_N},\delta b_N^{\dagger}})^T}$, the noise operator vector ${\bf N}(t) = \sqrt 2 (\sqrt {{\kappa _c}} {c_{{\text{in}}}},\sqrt {{\kappa _c}} c_{{\text{in}}}^{\dagger} , \sqrt {{\kappa _a}} {a_{{\text{in}}}},\def\LDeqbreak{}\sqrt {{\kappa _a}} a_{{\text{in}}}^{\dagger} ,\sqrt {{\gamma _1}} {b_{1,{\text{in}}}},\sqrt {{\gamma _1}} b_{1,{\text{in}}}^{\dagger} , \ldots ,\sqrt {{\gamma _N}} {b_{N,{\text{in}}}},\sqrt {{\gamma _N}} b_{N,{\text{in}}}^{\dagger})^T$, and the coefficient matrix ${\bf A}$ given in [93], with the effective driving detuning for the loss (gain) cavity ${\Delta _1} = {\Delta _c} + \sum\nolimits_{j = 1}^N {g_j}({\beta _j^ * + {\beta _j}})$ (${\Delta _2} = {\Delta _a}$), and the effective optomechanical-coupling strength ${G_j} = {g_j}{\alpha _2}$ for the $j$th vibrational mode [93]. Without loss of generality, hereafter, we assume $\Delta \equiv {\Delta _1} \approx {\Delta _2}$. All the parameters satisfy the stability conditions derived from the Routh-Hurwitz criterion [94].

The numerical results of the steady-state mean phonon numbers in these vibrations can be obtained via the Lyapunov equation ${\bf AV} + {\bf V}{{\bf A}^T} = - {\bf Q}$, where the covariance matrix ${\bf V}$ is defined by the matrix elements ${{\bf V}_{\textit{ij}}} = {{[{\langle {{{\bf u}_i}(\infty){{\bf u}_j}(\infty)}\rangle + \langle {{{\bf u}_j}(\infty){{\bf u}_i}(\infty)} \rangle}]} / 2}$, for $i,j = 1, \ldots ,(2N + 4)$, and the matrix ${\bf Q} = ({\bf C} + {{\bf C}^T})/2$, with the noise correlation matrix ${\bf C}$ defined by the matrix elements $\langle {{{\bf N}_k}(s){{\bf N}_l}({s^\prime})} \rangle = {{\bf C}_{k,l}}\delta ({s - {s^\prime}})$. Then we derive the final mean phonon numbers of the $j$th vibration [93]:

The analytical net refrigeration rate of the $j$th motional mode, when considering the case of $N = 2$ and ${\kappa _{c(a)}} = {\kappa _0}$, is derived as

C. EP Cooling Immune to Both Temperature and Mass

To demonstrate the exceptional refrigeration of this complex system, we first consider the case of $N = 2$. In Figs. 2(a) and 2(b), the net cooling rate ${\gamma _{j,{ C}}}$ is plotted versus the effective driving detuning $\Delta$, when the system operates in both standard cooling (blue curves) and EP cooling (red curves). It shows that the maximum net cooling rate emerges around the red-sideband resonance ($\Delta \approx {\omega _1}$), and especially, the net refrigeration rate in the EP-cooling case can be three orders of magnitude larger than that for the standard cooling. Physically, the strong field-localization effect [59,75–78] induced by our EP system dramatically enhances the generation rate of the anti-Stokes photons; and meanwhile it extremely reduces the excitation-exchange rate in the swap-heating processes. These results indicate that, in general, by simply employing an EP, the cooling rate of the vibration can be greatly amplified and engineered.

 

Fig. 2. Net cooling rates (a) ${\gamma _{1,{C}}}$ and (b) ${\gamma _{2,{C}}}$ versus the effective driving detuning $\Delta$ in both standard cooling (blue curves, $J/{\kappa _c} = 0$) and EP cooling (red curves, $J/{\kappa _c} = 0.999$). Final mean occupation numbers (c) $n_1^f$ and (d) $n_2^f$ versus the reservoir temperature ${T_j}$ when $\Delta = {\omega _j}$. Other parameters are the same as those in Fig. 1. (e) Energy-level diagram of the EP-cooling mechanism [78]. The number state is described by $|{N_a},{N_c},{n_j}\rangle$, with ${N_a}$ (${N_c}$) being the photon number of the gain (loss) optical mode, and ${n_j}$ denoting the thermal phonon number of the $j$th vibrational mode. The dashed curves depict the heating processes: backaction heating (BH), swap heating (SH), and thermal heating (TH). The solid curves denote the refrigeration processes: swap cooling (SC), counter-wave cooling (CC), dissipation cooling (DC) by the loss-optical mode $c$, and the field-localization cooling by the gain-optical mode $a$.

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In our system, a passive optical mode, connected to an active optical mode, is simultaneously coupled to multiple vibrational modes (yielding in-parallel optomechanical interactions), each of which is not coupled to each other. For convenience, we here explain the physical mechanism of the EP cooling for the $j$th vibrational mode. Physically, the thermal phonons stored in the vibrational mode are first extracted into the loss cavity and then these thermal excitations flow into the gain cavity and finally localize there, which results in energy localization [59,75–78]. In this case, the cooling rate of the motional mode can be greatly amplified by the energy localization in the active cavity $a$. Specifically, we can see from Fig. 2(e) [78] that the absorption rate of anti-Stokes photons [corresponding to the field-localization-cooling process] of the passive cavity $c$ can be dramatically enhanced, compared with the standard resolved-sideband optomechanical cooling for which there is only cavity dissipation [i.e., the DC process]. Meanwhile, the heat-exchange rate [i.e., the SH process] is greatly reduced, owing to the joint effect of the field localization and the decrease of anti-Stokes photons [77,78]. These findings indicate that the field-localization effect can enable a dynamical enhancement in the refrigeration rate of the mechanical vibration. Our results on the EP cooling differ from what is known in the literatures [77,78], mainly because we are interested not in a single mechanical motion but in the vibrational networks.

To further elucidate this EP-intensive cooling, we plot in Figs. 2(c) and 2(d) the final mean phonon numbers $n_j^f$ versus the environmental temperature ${T_j}$ for both standard cooling (blue curves) and EP cooling (red curves). It is clearly shown that, with increasing ${T_j}$, the motional degree of freedom in the EP-cooling case can be effectively cooled and its cooling performance can be improved up to three orders of magnitude compared with the standard-cooling performance. Remarkably, the threshold reservoir temperature for preserving quantum ground-state cooling in the EP cooling, is nearly three orders of magnitude higher than that in the standard cooling. The underlying physical mechanism behind this counterintuitive phenomenon can be explained as follows: the introduced EP leads to a giant enhancement of the net cooling rate, and this breaks the law that the cooling rate is commonly suppressed by the rugged high-temperature environment. These results provide the means to protect fragile quantum mechanical setups from environmental thermal noises, and pave a way towards noise-tolerant quantum vibrational networks.

To show the immunity of the EP-cooling mechanism against resonator mass, we make a detailed comparison between the standard- and EP-cooling performances versus the mass, using both numerical (curves) and analytical (symbols) results, as shown in Figs. 3(a) and 3(b). Clearly, the numerical and analytical predictions match well with each other. The EP-cooling technique allows us to reach the motional quantum ground state in the large-mass regime, which is very challenging for the standard-cooling schemes [16–31,86–92]. We also reveal that near the EP, the maximum refrigeration rate is insensitive to the resonator mass. Specifically, in the EP-cooling case, the threshold mass for preserving quantum ground-state cooling has been predicted to be more than three orders of magnitude larger than that in the standard-cooling case. Physically, the net cooling rate is reduced due to the decrease in the light-motion coupling strength with increasing the resonator mass, while it can be tremendously compensated for or even amplified by employing the EP-cooling mechanism. Our findings mean that the EP-cooling mechanism paves a route towards exploiting quantum ground-state preparation immune to the device mass.

 

Fig. 3. Final average thermal occupation numbers (a) $n_1^f$ and (b) $n_2^f$ versus the resonator mass ${m_j}$ in both standard cooling (blue curves and symbols, $J/{\kappa _c} = 0$) and EP cooling (red curves and symbols, $J/{\kappa _c} = 0.999$). The curves show the numerical predictions [see Eq. (3)], while the symbols correspond to analytical results [see Eq. (4)]. Clearly, an excellent agreement between the numerical and analytical results is seen. Note that the masses of the two oscillators are in general different; however, for convenience, we consider the case where both masses are equal in our simulations. Here ${T_j} = 1\,\text{K}$, ${\omega _c}/(2\pi) = 2.817 \times {10^{14}}\,\text{Hz}$, $L = 1\,{\text{mm}}$, $\lambda = 1064\,{\text{nm}}$, ${P_L} = 9\,{\text{mW}}$, and other parameters are the same as those in Fig. 1.

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D. Cooling $N$-vibration Optomechanical Networks

The EP-cooling mechanism can be generalized to the exceptional refrigeration of $N$-vibration optomechanical networks, which are coupled to a common passive cavity-field mode, as shown in Fig. 1(a). We see from Fig. 4 that quantum ground-state cooling of the motion cannot be achieved (see green rectangles) in the standard cooling. However, in stark contrast to this, it becomes feasible (see lower red disks) in the EP cooling. Physically, the introduced EP leads to amplifying the net cooling rate of the vibration, and produces a giant enhancement for the refrigeration performance. These results not only open the possibility of further largely suppressing thermal noise in vibrational networks, but also pave a route towards the preparation of nonclassical states of the motion, e.g., large spatial superpositions or non-Gaussian states (e.g., Schrödinger-cat-like states).

 

Fig. 4. Residual motional quanta numbers $n_{j \in [1,N]}^f$ versus the $j$th motion. The horizontal axis denotes the frequency dispersion relation of the vibrations with nine independent vibrational frequencies $({\omega _{1, \ldots ,9}})/{\omega _m} = 0.6, \ldots ,1.4$. The purple disks are the initial thermal occupations, while the green rectangles and red disks denote the final mean phonon numbers in the standard- and EP-cooling cases, respectively. Here ${T_j} = 1\,\text{K}$, ${G_j}/{\kappa _c} = 0.1$, and other parameters are the same as those in Fig. 1.

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3. DISCUSSION AND CONCLUSIONS

In contrast to the previously established demonstrations investigating the exceptional cooling in single-vibration optomechanical platforms [77,78], we here focus on the EP-intensive cooling using multi-vibration optomechanical networks [36–45] showing much richer and more general properties. In particular, our intrinsic motivations are not limited to studying the exceptional refrigeration of the vibration networks by both fully analytical and numerical treatments, but also to overcome a long-standing challenge that quantum ground-state cooling of the motion in the regimes of both large mass and high temperature is hard to achieve. This means that unlike the previous schemes [77,78], where the aim is to cool and isolate a single motion, the aim here is not only to quickly and exceptionally extract thermal phonons from vibrational networks but also to completely overcome the serious cooling obstacles originating from the resonator-mass and environmental-temperature limitations. Our study shows that by simply harnessing the power of an EP, one can achieve an ultra-high-efficiency cooling, which is immune to both resonator mass and environmental noise, without the need of using any high-cost low-loss materials and noise filters [96,97]. These cooling performances are otherwise unachievable in the previous studies [77,78].

In conclusion, we proposed a simple EP-cooling mechanism to amplify the net cooling rate of the vibration, and to effectively cool the motional degree of freedom down to the quantum ground state. We showed that the final mean thermal occupation numbers in the EP-cooling case are three orders of magnitude smaller than those for the standard cooling. In particular, we revealed that in the standard-cooling case, optomechanical cooling becomes much worse with increasing either environmental temperature or resonator mass, while in the EP-cooling case, it is almost independent of these parameters. This study enables the exploration of quantum physics in large-mass and high-temperature quantum mechanical systems, and paves a way towards realizing pure quantum cavity optomechanics, with immunity against both resonator mass and environmental noise.