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Abstract
Phonon lasers, coherent oscillations of phonons, have gradually become one of the emerging frontiers in the last decades, and have promising applications in quantum sensing, information processing, and precise measurement. Recently, phonon lasers based on dissipative coupling have been realized in an active levitated optomechanical (LOM) system for the first time. Here, we further investigated the characteristics of the phonon laser in the system above regarding the oscillator amplitude and the phonon laser linewidth. We established both the experimental system and a physical model of the phonon laser. On the basis of simulations and experiments, the influences of pumping power, numerical aperture, the microsphere’s diameter and refractive index on the performance of the phonon lasers are sufficiently discussed. Our work is of great significance for the high-quality phonon lasers generated by the appropriate parameters, which is the basis for the in-depth research and practical application.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Phonon lasers, with shorter wavelength than lasers, allow measurement and imaging with higher accuracy, and have considerable potential in information processing and quantum physics. For decades, theoretical investigations of phonon lasers had been carried out in various systems, such as ions [1], double-barrier structures [2], semiconductor superlattices [3], nanomachines [4], and nanomagnets [5]. Until 2009, with the remarkable advances of cavity optomechanics, Vahala et al. experimentally generated phonon lasers for the first time by using a cooled Mg+ ion in the Paul trap and applying the blue-detuned laser to induce the ion coherent oscillations [6]. Moreover, they explored the injection locking of phonon lasers based on this system and achieved the detection of ultraweak oscillating forces with a sensitivity of 5(1) × 10−24N [7]. Later on, phonon lasers were gradually achieved through many other systems, such as semiconductor superlattices [8], whispering-gallery microcavities [9,10], F-P cavities [11,12], quantum dot systems [13], and electromechanical systems [14].
In 2019, Pettit et al. realized phonon lasers by levitating nanospheres using optical tweezers [15]. This kind of mesoscopic phonon lasers is of importance for the quantum mechanics and the precision measurement. In our recent work, Kuang et al. generated nonlinear multi-frequency phonon lasers with levitated micro-scale sphere for the first time in an active LOM system [16]. This system is based on the dissipative coupling between the mechanical oscillator and the laser, in contrast to the dispersive coupling of other previous systems. The unique advantage of the active LOM system is that the high optical gain can compensate for the optical loss caused by dissipative coupling. As a result, the fundamental-mode phonon laser has a three-order-of-magnitude enhancement in the power spectrum and a 40-fold improvement in linewidth narrowing, compared with the case without any optical gain. This work provides a pathway for the phonon lasers in active LOM system, which can be applied to nonlinear regime, such as acoustic frequency combs, high-precision metrology, and multi-frequency mechanical sensors.
In this paper, we further analyze the characteristics of phonon lasers in active LOM system. The motion equation of the trapped microsphere in the active LOM system is established and employed to study its characteristics. In addition, the optical damping exerting on the microsphere is analyzed simulatively to figure out the generation of phonon lasers. Furthermore, the oscillator amplitude and phonon laser linewidth are important for the applications of the phonon laser, such as precise measurement [17,18] and acoustic frequency combs [19–21]. Therefore, the influences of the microsphere’s diameter, pumping power, numerical aperture and microsphere’s refractive index on the oscillator amplitude and phonon laser linewidth are fully discussed by simulation and experiments.
2. Experimental setup and principle
2.1 Experimental setup
The experimental setup of the phonon laser is shown in Fig. 1. It consists of an active levitated optomechanical (LOM) system and a set of dual-beam optical tweezers. The LOM system is an active optical cavity using the single-mode Yb3+-doped fiber as the gain medium. A single-mode semiconductor laser with a wavelength of 976 nm is served as the pumping source, which pumps the gain medium through a wavelength division multiplexer (WDM) to produce the laser with a central wavelength of 1030 nm inside the ring cavity. The clockwise (CW) and counterclockwise (CCW) beams are allowed to travel simultaneously in the cavity. The collimators C1 and C2 expand the two lasers with opposite directions into the free space respectively. Through the convergence of lenses L1 and L2 (NA = 0.25), CW and CCW beams are coupled into the fiber by the collimator on the other side separately, which forms an active ring cavity, i.e., intracavity optical tweezers. Meanwhile, the dual-beam optical tweezers is deployed between L1 and L2 along the x-axis direction, vertical to the free-space optical path of the ring cavity, in order to capture and hold the microsphere in the vacuum chamber. Due to the dissipative coupling between the particle position and the intracavity laser, the ring cavity forms an active LOM system. Other details also see in Ref. [16].
2.2 Motion equation of the trapped microsphere
For the microsphere trapped in the ring cavity, the forces on it mainly consist of the intracavity optical force, the optical force exerted by the dual-beam optical tweezers, the mechanical damping force caused by the gas molecules, and the Brownian motion force. Thus, the motion equation of the microsphere can be described as
In the ring cavity, the optomechanical dissipative coupling is formed between the microsphere and the laser, which can be described as [22]
Since there is a response time between the intracavity power change and the particle displacement, namely the intracavity photon lifetime τ, the intracavity power is a time-dependent function, i.e., ${F_{opt}}({x,t} )= Q(x )P({x({t - \tau } )} )$. By exploiting the Taylor expansion, the intracavity optical force can be written as ${F_{opt}}({x,t} )= Q(x )\left[ {P(x )- \frac{{dP}}{{dx}}{\tau_c}(x )\frac{{dx}}{{dt}}} \right]$. To sum up, the motion equation of the microsphere can be simplified as
In this system, optical damping can amplify the vibration of the microsphere, which is the reason for the microsphere switching from the thermal regime into the phonon lasing state. Therefore, it is necessary to analyze the role of the optical damping and its influence mechanism, to study the generation of the phonon laser of the active LOM system in depth.
For the microsphere oscillating radially in the cavity, the optical damping rate is always less than or equal to 0 by calculating Eq. (4), as illustrated in Fig. 2. Thus, the optical damping is a negative damping, which is able to accelerate the microsphere nonlinearly [16]. When analyzing the optical damping, the optical force exerted by the dual-beam optical tweezers can be not taken into account. It is because that the dual-beam optical tweezers only provides the microsphere with a fixed vibrational frequency Ω0 and that the optical damping is independent of the dual-beam optical tweezers according to Eq. (4). Therefore, the optical damping can be considered to be radial symmetric and the optical damping of the negative radical displacement is the same as that of the positive one. The microsphere vibration region can be divided into three parts, according to the value of optical damping. In Part I, the optical damping is zero due to the large scattering loss. When the microsphere is close to the center of the beam, the scattering loss will prevent the ring cavity from producing laser. As the microsphere gradually moves away from the optical axis of the intracavity optical tweezers, the ring cavity starts to emit laser and thus optical damping is induced, as shown in Part II. When the microsphere moves out of the light spot, the microsphere is no longer subjected to the optical force. Therefore, the optical damping returns to zero, i.e., Part III. The microsphere is accelerated by the optical damping every time it passes through Part II, with the condition that the optical damping counteracts the mechanical damping. It can be indicated that the generation of the phonon laser and its performance are significantly influenced by the range of the Part II and the strength of the optical damping to some extent.
3. Results and discussions
The displacement data of the microsphere were collected for three seconds at a sampling rate of 200kHz. The power spectrum is calculated as shown in Fig. 3. The microsphere experiences the coherent oscillation and that the phonon laser is generated. Based on the experimental parameters, the relationships of the intracavity power and the optical damping versus the microsphere displacement by calculating the Eq. (2) and Eq. (4). Then, the motion equation of the microsphere can be solved, according to Eq. (3). The simulated power spectrum of the phonon laser is illustrated by the blue curve. The simulation results are similar to the experiments, indicating that our model can well simulate the phonon lasing dynamic. In addition, other noises such as laser noise, circuit noise, and low-frequency mechanical vibration are still existing in the experiment, although the LOM system can dramatically reduce the mechanical noise of the experimental setup. These noises make the experimental linewidth broader, compared with the simulated one.
Amplitude and linewidth are two key parameters of phonon laser to describe its performance. We further analyze the phonon laser characteristics of this active cavity LOM system as below.
The size of the microsphere substantially influences the dissipative coupling between the oscillator and the laser. The relationship between the oscillator amplitude and the microsphere’s diameter is presented in Fig. 4(a). The oscillator amplitude is increased with the diameter increasing, which results from the greater scattering loss caused by the larger microsphere, making the dissipative coupling stronger. In the meanwhile, the increase of the microsphere’s diameter will cause the higher power variation per unit displacement, which consequently causes the increase of the optical damping. Both the enhanced optomechanical coupling effect and the higher optical damping allow more phonons to be stimulated from the thermal state to the coherent state and increases the intensity of phonon laser. It is also the reason why the phonon laser linewidth drops notably at first, as shown in Fig. 4(b).
Fig. 4. Characteristics of (a) the oscillator amplitude and (b) the phonon laser linewidth versus the microsphere’s diameter.
However, when the diameter is larger than about 2 µm, the vibration amplitude of the microsphere is weakened and the phonon laser linewidth is broadened. It is because the scattering loss further increases, leading to a decrease of the intracavity power and the narrower range of the non-zero region(Part II) of the optical damping. Microspheres with different diameter are selected for experiments, as shown by the gray triangles in Fig. 4. The experiment has the same trend as the simulation, which further proves the validity of the simulation results.
In the simulation above, the effects of the microsphere size on many other parameters besides the optomechanical coupling are also considered, such as the mechanical damping caused by the residual gas in the vacuum chamber and Brownian motion force. The mechanical damping can be described as [23]
In addition, the pumping power of the ring cavity also influences the quality of the phonon laser. Below the threshold, the optical damping provided by the intracavity tweezers is not adequate to generate phonon laser, as indicated in Fig. 5. The intracavity power is risen with the pumping power increasing, which can enhance optical damping and broaden the non-zero region(Part II) of the optical damping, as described in Eq. (4). By surpassing the lasing threshold, the oscillator amplitude increases while the phonon laser linewidth is narrowed remarkably. Furthermore, the dissipative coupling is strengthened, resulting in the significant enhancement of the coherent oscillation.
Fig. 5. Characteristics of (a) the oscillator amplitude and (b) the phonon laser linewidth versus the pumping power.
Experimental results are illustrated by the gray triangles, which are similar to the simulations. As the pumping power increases, the phonon laser linewidth in the simulation has a significant decrease, while the linewidth change in the experiment is not so dramatic, by comparison. This is because the simulation model is more idealized than the actual experiment and the threshold behavior of the phonon laser can be observed more clearly. Although the experimental and simulation results do not overlap completely, their trends are basically the same and the difference is not considerable. It can be believed that the simulation can guide experiments to some extent.
Furthermore, the characteristics of the phonon laser can be influenced by the numerical aperture NA, as illustrated in Fig. 6. The oscillator amplitude gradually rises with the NA increasing initially, as the Fig. 6(a) indicates. Increasing NA can result in the amplification of the scattering by the microsphere and an increase in the optical gradient force. Therefore, the increase of the NA will cause the higher power variation per unit displacement, which causes the increase of the optical damping. It is the enhanced optical damping that makes the dissipative coupling stronger and the vibration more coherent. Meanwhile, the phonon laser linewidth fluctuates, yet it is not significant, shown as Fig. 6(b). With the further increase of NA, however, excessive scattering loss leads to the apparent decrease in intracavity power. The larger NA can result in an extension in the Part I range of the optical damping and a decrease in the Part II range. In this way, the amplitude starts to decrease while the phonon laser linewidth broadens.
Fig. 6. Characteristics of (a) the oscillator amplitude and (b) the phonon laser linewidth versus the NA.
The relationship between the microsphere refractive index and phonon laser characteristics is shown in Fig. 7. The refractive index of the microsphere mainly affects the optical force exerted on the microsphere, which can be explained by the theory of geometric optics [24]. However, the refractive index mainly affects the scattering force, and have a little effect on the gradient force which is the intracavity optical force exerted on the microsphere vibrating radically. From Fig. 7(a), it can be observed that the oscillator amplitude decreases gradually, which is the consequence of the influence of the refractive index on the gradient optical force, with the refractive index increasing. In addition, the microsphere’s refractive index is almost independent of the phonon linewidth, as shown in Fig. 7(b). This means the refractive index has a relatively weak effect on the coherence of the vibration. By comparing the influence of the microsphere size and the refractive index on the phonon laser, it can be seen that the dissipative coupling is associated with microsphere size, to a large extent. Therefore, we can adopt many approaches, such as the core-shell structure [25], to change the refractive index without affecting the microsphere size and the dissipative coupling. In this way, the microsphere with more interesting optical properties can be used to further explore the phonon laser in the active LOM system in the future.
Fig. 7. Characteristics of (a) the oscillator amplitude and (b) the phonon laser linewidth versus the microsphere’s refractive index.
4. Conclusions
The nonlinear multi-frequency phonon lasers with a micro-size sphere have been realized based on the dissipative coupling between the sphere and the intracavity laser in the active cavity LOM system [16]. In this work, we further investigated the characteristics of the phonon laser in active LOM system regarding the oscillator amplitude and the phonon laser linewidth.
We point that the parameters of the active cavity LOM system affect the characteristics of the phonon laser through impacting the dissipative coupling strength and the intracavity power. The numerical simulation and experimental result show that the microsphere’s diameter and the pumping power have more pronounced influence on the phonon laser. An appropriate size of trapped particle, which is about 1µm for our experimental system, is needed to ensure the largest oscillator amplitude and the narrowest phonon laser linewidth. While the higher the pumping power is applied, the larger the oscillator amplitude and the narrower phonon laser linewidth can be achieved. In addition, the optimal NA of the lens can enhance oscillator amplitude a little since there is an extremal point, i.e., NA = 0.35 for our experimental system. Although its impact on linewidth is not significant. Meanwhile the oscillator amplitude will slightly rise with the reduction of the refractive index of the microsphere.
Our work will pave the way towards the realization of the phonon laser with high performance in the active cavity LOM system. In the meanwhile, the universal approaches to studying the phonon laser characteristics are presented, which is instructive and significant for applications in the future, such as the quantum precision measurement and the phonon laser frequency combs.
Funding
National Natural Science Foundation of China (6197523); Scientific Research Project of the National University of Defense Technology (ZK20-14); Natural Science Foundation of Hunan (2021JJ40679).
Acknowledgments
The authors wish to thank the anonymous reviewers for their valuable suggestions.
Disclosures
The authors declare no competing financial interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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