Vibrational resonance via a single-ion phonon laser

Abstract

Vibrational resonances are ubiquitous in various nonlinear systems and play crucial roles in detecting weak low-frequency signals and developing highly sensitive sensors. Here we demonstrate vibrational resonance, for the first time, utilizing a single-ion phonon laser system exhibiting Van der Pol-type nonlinearity. To enhance the response of the phonon laser system to weak signals, we experimentally realize continuously tunable symmetry of the bistability in the phonon laser system via optical modulation, and achieve the maximum vibrational resonance amplification of 23 dB. In particular, our single-ion phonon laser system relaxes the frequency separation condition and exhibits the potential of multi-frequency signal amplification using the vibrational resonance. Our study employs the phonon laser to study and optimize the vibrational resonance with simple and well-controllable optical technology, which holds potential applications in developing precision metrology and single-ion sensors with on-chip ion traps.

Graphical abstract

Keywords

single-ion phonon laser / vibrational resonance / nonlinear system

1 Introduction

Stochastic resonance and vibrational resonance (VR) have been proposed and studied extensively to recover and enhance weak signals [1–3]. For the same nonlinear bistable system, VR can provide more significant amplification than stochastic resonance and thus becomes a promising candidate for various applications [4]. VR shows up as the resonance-like response at a weak low-frequency (LF) signal depending on the amplitude of an additional high-frequency (HF) force. Diving into it also encourages the exploration of novel phenomena, making it a more versatile subject [5]. The basic characteristics of VR have been intensively studied both theoretically and experimentally in a huge variety of nonlinear systems, such as nonlinear oscillators [6–9], analog circuits [10], and biological systems [11]. Recent pertinent advancements in VR, including logical VR [12], entropic VR [13, 14], anti-VR [15], and ghost VR [16, 17], have opened up possibilities for a broad spectrum of applications, including neuralbiology [18, 19], signal processing [20, 21], and precision sensing [8, 22].

Pursuing the weak signal enhancement of VR is a long-standing challenge, particularly for physical systems at nano- and micro-scales. The nonlinear bistability, characterized by tunable bistability, plays a significant role in VR amplification. The tunable bistability symmetry takes the capability to suppress undesired resonances, enhance desired ones, adjust frequencies of VR, and optimize VR results [5, 23]. Nevertheless, thus far, exploration of continuous and precise regulation of bistability symmetry has been limited due to intrinsic physical properties of the oscillators (e.g., the spring coefficients change with temperature in nanomechanical oscillators [3]). These limitations result in insufficient experimental studies on the mechanisms underlying VR, hindering further improvements in amplification [23–26].

In this work, we report modification of the bistability symmetry and enhancement of the VR amplification using a single-ion phonon laser. The single-ion phonon laser, interacting with two laser beams, generates coherent amplification of atomic motion through stimulated emissions of an ion [27, 28]. The phonon laser has been widely utilized to explore nonlinear dynamical behaviors [29–32] and enhance precision measurements [33, 34]. Its strong nonlinear effects and amplified coherent motion give rise to ultrahigh sensitivity to optical and electrical fields. The single-ion phonon laser exhibits self-sustained oscillations with nonlinear damping, characteristic of a Van der Pol (VDP) oscillator rather than the force-driven Duffing oscillators with linear damping [7–10]. This fundamental difference leads to distinct physical mechanisms and VR between these two kinds of oscillators. Compared to the Duffing-type counterparts, the VDP nature of the single-ion phonon laser allows for control nonlinearity via optical fields instead of electrical potential fields. This advantage offers a potential for precise modulation and enhanced coherent control. However, to our knowledge, VR in the VDP oscillator remains unexplored experimentally [35].

Here we investigate the VR phenomenon, for the first time [35], by the Van der Pol oscillator using a phonon laser in a single-ion system. Experimentally, we achieve continuously tunable symmetry of the bistability in the phonon laser system via optical modulation. By adjusting the bistability symmetry, we find the nonlinear VR along with the even-order harmonics of the weak signal is substantially suppressed in our system compared to the others [36–38]. Then we enhance the VR amplification to 23 dB with improved bistability symmetry due to optimizing the system response to the weak signal. Besides, we have observed the VR with HF driving of the frequency close to the weak LF one, which relaxes the commonly required frequency separation condition in theoretical analysis [2, 7]. By injecting signals at various frequencies, we also exhibit the potential for simultaneously amplifying multi-frequency weak signals using a single-ion phonon laser. As clarified later, all the experimental observations qualitatively agree with our numerical simulations.

2 Experimental setup and the model

2.1 Experimental setup

We employ a surface-electrode trap (SET) of 500 μm size, whose measured secular frequency along z-axis is ${\omega }_z/(2\pi )$ = 183.41 $\pm$ 0.01 kHz. The side view of the structure consisting of the endcap electrodes (EEs), the RF electrodes, and the axial electrode (AE) is drawn in Fig.1(a). The single ${}^{40}$Ca${}^{+}$ ion, located at 800 μm above the center of the SET, behaves as a phonon laser in the trapping potential.

Fig.1 Experimental setup and principle of tunable bistability. (a) Sketch of SET and single-ion phonon laser. (b) Schematic of experimental setup and the corresponding energy level scheme. After frequency and power adjustment by the AOMs, the three laser beams are then combined and introduced into the SET, where the solid arrows represent the laser beams and the corresponding transitions they couple (blue, red, and grey denote blue-tuned 397-nm laser, red-tuned 397-nm laser and 866-nm laser, respectively), the yellow wavy arrow represents the spontaneous emission. The amplitudes of the signals are transformed to frequency unit, with respect to the modulation depth of 3.6 MHz/Vpp. (c) The amplitude of the single ion’s oscillation versus the detuning $\mathrm{\Delta }$ between the 397-nm laser and energy gap (for two energy-levels $4^2{\mathrm{S}}_{1/2}$ and $4^2{\mathrm{P}}_{1/2}$) in a sweep-up (red solid line) and sweep-down (blue dashed line) experiment. The ion exhibits thermal motion $|t⟩$ and coherent motion $|c⟩$ corresponding to the lower and upper branches of the bistability, respectively. The yellow (blue) area represents the thermal (coherent) oscillation regime. (d) Distribution of photon counts of the ion collected by the PMT and relevant to the two stable states, which correspond to ${\mathrm{\Delta }}_a$, with a driving set with frequency ${\mathrm{\Omega }}_{\mathrm{d}}/(2\pi )$ = 1 Hz and amplitude ${\omega }_{\mathrm{d}}/(2\pi )$ = 2.16 MHz. With ${\mathrm{\Delta }}_a/(2\pi )$ detuned to −0.75, 0, and 0.75 MHz, the ratios of frequency counts of the ion regarding the two states are 561:369, 468:462, 346:589, respectively.

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As illustrated in Fig.1(a) and (b), the three laser beams (red-, blue-detuned 397-nm and 866-nm laser beams) are combined together and applied to the ion at an angle of 10.8° to the z-axis. This angle is limited by the structure of the vacuum chamber. The phonon laser is generated by amplifying the ion’s oscillation in the direction of ${\omega }_z$ under irradiation of two 397-nm laser beams. The two beams own different detunings with one being red-detuned ${\mathrm{\Delta }}_r$ = $\mathrm{\Delta }+{\mathrm{\Delta }}_r^0$ and the other blue-detuned ${\mathrm{\Delta }}_b$ = $\mathrm{\Delta }+{\mathrm{\Delta }}_b^0$. Then, we can adjust their frequencies and power via acousto-optic modulators (AOMs) after beam splitting, as drawn in Fig.1(b). Here we employ the frequency detunings of ${\mathrm{\Delta }}_r^0/(2\pi )$ = −90 MHz, ${\mathrm{\Delta }}_b^0/(2\pi )$ = 56 MHz, and $\mathrm{\Delta }$ is tunable detuning. We modify the power to achieve the desired saturation ratio $r=s_b/s_r$ = 0.13, where $s_{r(b)}$ represents the saturation of the red (blue)-detuned laser beam. During the whole experimental period, the 866-nm laser beam keeps acting on the ion at an over-saturated power of over 400 μW, for repumping of the 6% leakage of spontaneous emission onto the metastable state 3${}^2{D}_{3/2}$ back to the state 4${}^2{P}_{1/2}$. This setup ensures the generation and oscillation of a stable phonon laser system with an oscillation amplitude of 13 μm.

2.2 The model

In our scheme, the ion oscillates with the secular frequency ${\omega }_z/(2\pi )$ = 183.41 $\pm$ 0.01 kHz along the $z$-axis under additional modulation. The motion functions as a Van der Pol oscillator governed by the following equation:

(1)$m\ddot{x}(t)=-m{\omega }_z^{2}x(t)+F_c+F_g+m\chi (t),$

where $x(t)$ and $m$ are the displacement and mass of the ion, and the terms on the right side of the equator correspond to the force of the trapping potential $-m{\omega }_z^{2}x(t)$, the cooling (gain) force $F_c$($F_g$) induced by the detuned laser beam, and a stochastic noise induced force $m\chi (t)$. More details of our theoretical model are presented in Appendix A.

We introduce a weak LF signal with frequency (amplitude) of ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}$(${\omega }_{\mathrm{s}\mathrm{i}\mathrm{g}}$) and an HF driving with frequency (amplitude) of ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$(${\omega }_{\mathrm{H}\mathrm{F}}$). They are generated and combined from an arbitrary waveform generator (AWG) and introduced as frequency modulation (FM) of the driving signal to the AOM1, performing and adjusting both red- and blue-tuned laser beams, as sketched in Fig.1(b). The responses of the phonon laser bistable system are investigated when the mixture of the LF signal and HF driving is applied with different variables. The amplitude ${\omega }_{\mathrm{s}\mathrm{i}\mathrm{g}}$ of the weak signal is set to be far smaller than the switching threshold for the inter-well jumps between the two motions at any semiperiod when the ion is initially prepared in the thermal motion with small photon counts.

3 The bistability of single-ion phonon laser system

To achieve VR, we must first generate the bistability of the phonon laser as describe by Eq. (1). As the hysteresis of the bistability illustrated in Fig.1(c), the thermal motion $|t⟩$ and coherent motion $|c⟩$ correspond to the lower and upper branches of the bistability of the ion’s motion, respectively. We obtain the hysteresis curve of bistability between thermal and coherent oscillations by slowly increasing and decreasing the detuning $\mathrm{\Delta }$ of the 397-nm laser. Then we can identify the threshold for the generation (annihilation) of the phonon laser between the two motions, positioned at $\mathrm{\Delta }={\mathrm{\Delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ ($\mathrm{\Delta }={\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$) for the rising (falling) edge. We define the bistable region of our system ranging from ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ to ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$. In this optically controlled bistability, the width $W_{bi}$ of the bistable region depends on the intensity ratio $r=I_r/I_b$ of the laser beams in red- and blue-detunings. In the following, we set $r$ = 0.47 and $W_{bi}$ = 3.0 MHz, with thresholds at ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}/(2\pi )$ = −24 MHz and ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}/(2\pi )$ = −21 MHz.

Moreover, to illustrate the continuously tunable symmetry of bistability from asymmetry to symmetry in our system, we modify the center offset of the bistability region at the detuning ${\mathrm{\Delta }}_a=\mathrm{\Delta }+2\pi \times 22.50$ MHz. For instance, we configure a driving with frequency ${\mathrm{\Omega }}_{\mathrm{d}}/(2\pi )$ = 1 Hz and amplitude ${\omega }_{\mathrm{d}}/(2\pi )$ = 2.16 MHz (in units of frequency, corresponding to the modulation depth). We inject the driving into an optical modulator via FM of the AOM1, which can be described as

(2)$V(t)=V_0\cos \{{\omega }_0+{\omega }_d\mathrm{s}\mathrm{g}\mathrm{n}[\cos ({\mathrm{\Omega }}_{d}t)]\}t,$

where $V_0$ and ${\omega }_0$ are the modulation amplitude and frequency of the AOM1, respectively, and $\mathrm{s}\mathrm{g}\mathrm{n}(K)$ = 1 in the case of $K\ge 0$ and $\mathrm{s}\mathrm{g}\mathrm{n}(K)$ = −1 otherwise.

Then, we can realize hoppings between thermal and coherent motions, offering the opportunity to observe the tunable symmetry of the bistability. Finally, we record the distribution of fluorescence counts using a photo-multiplier tube (PMT), calculate the ion’s occupation in thermal and coherent motions, and derive the tunable symmetry of the bistability from the ion’s motion in thermal motion $|t⟩$ and coherent motion $|c⟩$, as drawn in Fig.1(d). See more details of data sampling and processing in Appendix B.

In Fig.1(d), we present examples of the continuously tunable bistability symmetry in the single-ion phonon laser system by adjusting the detuning ${\mathrm{\Delta }}_a$. We first consider the symmetric case (${\mathrm{\Delta }}_a$ = 0). In this case, the ratio of the occupation probabilities in the two motions is 468: 462, indicating almost equal probabilities of occupying them. Thus we obtain a phonon laser system exhibiting symmetric bistability, and the switching threshold to drive inter-well jumps between the two states is $W_{bi}/2=$1.5 MHz. Next, we consider the asymmetric cases, in which the system is not centered in its bistability region (i.e., −1.5 MHz $<{\mathrm{\Delta }}_a/(2\pi )<0$ MHz and 0 MHz $<{\mathrm{\Delta }}_a/(2\pi )<$ 1.5 MHz). Then, we observe the evolution of the asymmetric distribution of fluorescence counts at two motions, as displayed in Fig.1(d). When 0 MHz < ${\mathrm{\Delta }}_a/(2\pi )$ < 1.5 MHz, the ion has a larger probability to stay in $|c⟩$; when −1.5 MHz < ${\mathrm{\Delta }}_a/(2\pi )$<0, the occupation probability in $|t⟩$ is higher. For instance, the probability ratios at ${\mathrm{\Delta }}_a/(2\pi )$ = −0.75 MHz and ${\mathrm{\Delta }}_a/(2\pi )$ = 0.75 MHz are, respectively, 561: 369 and 346: 589, which are nearly inverse. With $|{\mathrm{\Delta }}_a|$ increasing, the asymmetry becomes stronger until the ion stays in one of the states and hardly makes a leap.

4 Results

4.1 VR in a Van der Pol oscillator

For a bistable system, a periodic HF driving with sufficiently high strength combined with a weak signal with extremely low frequency (much lower than ${\omega }_z$) can drive the system to overcome the barrier height and induce a series of synchronized inter-well jumps between the two stable states, exhibiting oscillations. The weak LF signal used in our experiment is a square waveform signal with frequency ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}/(2\pi )$= 1 Hz $\ll {\omega }_z$, and amplitude ${\omega }_{\mathrm{s}\mathrm{i}\mathrm{g}}/(2\pi )$ = 0.72 MHz, far lower than the switching threshold. The HF driving is fixed with frequency ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}/(2\pi )=$ 100 Hz and variable amplitude ${\omega }_{\mathrm{H}\mathrm{F}}$. We rewrite $V(t)$ in Eq. (2), describing the weak LF signal and the HF driving of the FM of AOM1, as $V(t)=V_0\cos \{{\omega }_0+{\omega }_{\mathrm{s}\mathrm{i}\mathrm{g}}\mathrm{s}\mathrm{g}\mathrm{n}[\cos ({\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}t)]+{\omega }_{\mathrm{H}\mathrm{F}}\sin ({\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}t)\}t$.

4.1.1 Symmetric case

To achieve the VR amplification, we set our experimental parameters satisfying the symmetric bistability. The left column of Fig.2(a) displays the temporal behavior of the photon counts acquired by the PMT for different ${\omega }_{\mathrm{H}\mathrm{F}}$, increasing from top to bottom. Counts falling in the yellow and blue areas represent the ion in the thermal motion and the coherent motion, respectively. For only the weak square-wave signal, the ion stays in the initial motion and no prominent spectral component is observed in its fast Fourier transform (FFT) spectrum. When the strength of the additional HF driving is not large enough [${\omega }_{\mathrm{H}\mathrm{F}}/(2\pi )\le$1.62 MHz], rare inter-well jumps occur, and the signal peak begins to rise from the background in the corresponding FFT spectrum, indicating the occurrence of VR amplification [see Fig.2(a) for ${\omega }_{\mathrm{H}\mathrm{F}}/(2\pi )$ = 0.72 MHz]. Then, increasing ${\omega }_{\mathrm{H}\mathrm{F}}$ leads to more regular switchings between the motions and enhances the signal peak, and we find the optimal amplification of the weak signal occurring at ${\omega }_{\mathrm{H}\mathrm{F}}={\omega }_{\mathrm{H}\mathrm{F}}^{\mathrm{m}\mathrm{a}\mathrm{x}}=2\pi \times 2.88$ MHz. With further increase of ${\omega }_{\mathrm{H}\mathrm{F}}$, HF driving gradually turns to be dominant, accompanied by a decline in ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}$ in the spectrum until it is completely drowned out by the over-strong HF driving. Because our sampling rate is 20 Hz $<{\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$, the photon counts oscillate in the middle region (100, 200) between the yellow and blue areas. Thus the amplification of the weak signal is highly relevant to ${\omega }_{\mathrm{H}\mathrm{F}}$.

Fig.2 VR in a Van der Pol oscillator. The response of the phonon laser with respect to the strength of the HF driving modulation, where the weak signal with the strength ${\omega }_{\mathrm{s}\mathrm{i}\mathrm{g}}/(2\pi )=$ 0.72 MHz is of the frequency ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}/(2\pi )=$ 1 Hz and the HF modulation frequency is ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}/(2\pi )=$ 100 Hz. (a) The time trace of the photon counts acquired by the PMT and the corresponding FFT spectrum under the condition of a symmetric bistable system with ${\mathrm{\Delta }}_a=$ 0, where the panels from top to bottom correspond to the HF driving strength of ${\omega }_{\mathrm{H}\mathrm{F}}/(2\pi )=$ 0.72 MHz, 2.16 MHz, 2.88 MHz, 5.76 MHz, and 14.40 MHz. The blue (yellow) semitransparent area in the left panel represents the mean photon counts of the system in the phonon-laser (thermal) motion. The vertical dashed lines in the right panel corresponds to the frequency of the weak signal ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}/(2\pi )=$ 1 Hz and its odd-order harmonics of 3 Hz and 5 Hz for guiding the eye. (b) The FFT spectra of the time trace results with the HF driving modulation strength ${\omega }_{\mathrm{H}\mathrm{F}}/(2\pi )=$ 2.16 MHz, 2.52 MHz, 2.88 MHz, and 3.24 MHz under the condition of asymmetric bistable system, where ${\mathrm{\Delta }}_a/(2\pi )=$ −0.3 MHz and the vertical dashed lines mean the frequencies of the two even harmonics at 2 Hz and 4 Hz. (c) Experimental SNR as the function of the HF driving for different symmetry of the bistable system with ${\mathrm{\Delta }}_a/(2\pi )=$ 0, −0.3 MHz, −0.6 MHz, −0.9 MHz, and −1.2 MHz. The stars correspond to the SNR for the FFT spectrum with the same color in Fig.2(b). The dashed line marks the corresponding ${\omega }_{\mathrm{H}\mathrm{F}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$. The colored areas under the threshold ${\mathrm{S}\mathrm{N}\mathrm{R}}_{\mathrm{T}}$ = 3 dB indicate the weak LF signal indistinguishable from the noises. (d) Experimental observation and numerical simulation of SNR${}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ as the function of ${\mathrm{\Delta }}_a$. The shape of the experimental data matches the curves in Fig.2(c). Due to the symmetry of the effective bistable potential, we present only the results of ${\mathrm{\Delta }}_a<0$ for simplicity.

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The corresponding FFT spectra are presented in the right column of Fig.2(a), in which we also notice the damping of the signal peaks at 3 Hz and 5 Hz, corresponding to odd-order harmonics of ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}$. Since we choose square wave as the weak signal to be detected, resonances do not occur in the even-order harmonics due to their time symmetry. For a quantitative comprehension of the phenomenon described above, we employ the output SNR to quantize the amplification of the weak signal induced by VR, where the SNR is defined as the intensity ratio of the signal to the background noise in the close vicinity in the spectrum [9]. Then we only observe the LF weak signal in the presence of an HF driving field. Experimentally, we consider the weak LF signal to be distinguishable from the noises when the SNR exceeds a threshold of ${\mathrm{S}\mathrm{N}\mathrm{R}}_{\mathrm{T}}$ = 3 dB, and thus we define the VR amplification as $\mathrm{S}\mathrm{N}\mathrm{R}-{\mathrm{S}\mathrm{N}\mathrm{R}}_{\mathrm{T}}$ [9]. From the experimental results in Fig.2(d), we assess the maximum VR amplification to be 23 dB and the maximum ${\mathrm{S}\mathrm{N}\mathrm{R}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ = 26 dB at ${\mathrm{\Delta }}_a$ = 0. This indicates that the maximum VR amplification corresponds to the symmetric bistability.

4.1.2 Asymmetric case

Similar to the above symmetric case, we investigate the responses of our system with asymmetric bistability to illustrate the VR amplification of the single-ion phonon laser. Fig.2(b) presents unique characteristics of the nonlinear VR [36–38] that besides the occurrence of the peaks of ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}$ and the associated odd-order harmonics, the even-order harmonics at 2 Hz and 4 Hz are also observed in the case of ${\mathrm{\Delta }}_a/(2\pi )$ = −0.3 MHz. This is in much contrast to the previous observations in other nonlinear systems, which reported the emergence of additional resonances of even-order harmonics significantly degraded the SNR for the weak LF signal, and was only suppressed for certain ${\omega }_{\mathrm{H}\mathrm{F}}$ [23–26]. In our case here, the Van der Pol oscillator still exhibits amplification effects dominated by the frequency of the LF signal. The even-order harmonics occur only as weak and narrow peaks close to ${\omega }_{\mathrm{H}\mathrm{F}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ in the spectrum (see the panel with ${\omega }_{\mathrm{H}\mathrm{F}}/(2\pi )=2.88$ MHz), and then disappears quickly. Besides, we only find their occurrence when ${\mathrm{\Delta }}_a/(2\pi )$ = −0.3 MHz; With other values of ${\mathrm{\Delta }}_a$ we obtain similar spectra to the symmetric case. The above diversities indicate the effective suppression of even-order harmonics of our system.

We have also observed the dependencies of SNR as a function of ${\omega }_{\mathrm{H}\mathrm{F}}$ for different asymmetry of the bistable system, as presented in Fig.2(c). In each curve, the response initially decreases and then rapidly increases with increasing ${\omega }_{\mathrm{H}\mathrm{F}}$. It soon reaches a maximum value SNR${}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, but further increase of ${\omega }_{\mathrm{H}\mathrm{F}}$ results in degradation of SNR. Besides the resonance behaviors observed under the symmetric condition ($|{\mathrm{\Delta }}_a|=$0), more nonlinearity involved, i.e., increasing $|{\mathrm{\Delta }}_a|$, leads to enhancement of the effective potential so that ${\omega }_{\mathrm{H}\mathrm{F}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ is shifted to higher values. As presented in Fig.2(d), we can obtain SNR${}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ comparable to the symmetric case within a considerable range of ${\mathrm{\Delta }}_a$, indicating the excellent robustness of the phonon laser system. Due to the effective potential of our bistable system approaching a symmetric double-well potential, both positive and negative values of ${\mathrm{\Delta }}_a$ share similar amplification results [23]. Hence, we present only the experimental demonstration of ${\mathrm{\Delta }}_a<0$ for simplicity. Moreover, the quantititative discrepancy between numerical simulations and experiments arises from thermal noise and experimental limitations.

4.2 The relaxed frequency separation condition

To fully illustrate the VR amplification and unique nonlinearity-generating machanism by the single-ion phonon laser, we explore the impacts of the frequency separation between the weak LF signal and the HF driving. Since symmetric and asymmetric bistabilities lead to similar results, below we only focus on the symmetric case to achieve the optimized outcomes.

As depicted in Fig.3(a), we investigate VR with ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}/(2\pi )$ = 1 Hz, ${\omega }_{\mathrm{s}\mathrm{i}\mathrm{g}}/(2\pi )$ = 0.72 MHz, and ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$ varied from 1.1 Hz to 500 Hz, covering the frequency range from ${\omega }_{\mathrm{s}\mathrm{i}\mathrm{g}}$ to far exceeding it. In previous publications, VR was investigated only under the condition of ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}\ll {\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$. However, here we can observe the phenomenon of VR in every curve, highlighting when tuning ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$ to be only marginally higher than ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}$. This is beyond the interpretation of previous research on VR, whose motion equations had analytical solutions under some approximations [2, 6–10]. But the dynamical equation of our Van der Pol oscillator only has numerical solutions and they are in agreement with our experimental results (see more details in Appendix D). It is easier to trigger the resonance with smaller ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$. When ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$ increases, we observe the elevation and broadening of the curves.

Fig.3 The relaxed frequency separation condition. (a) The SNR of the output signal as a function of ${\omega }_{\mathrm{H}\mathrm{F}}$ in variation with ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$ in symmetric bistable system, where we set ${\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}/(2\pi )=$ 1 Hz, ${\omega }_{\mathrm{s}\mathrm{i}\mathrm{g}}/(2\pi )=$ 0.72 MHz and the waveform of the HF driving as sinusoid. The dashed line marks the corresponding ${\omega }_{\mathrm{H}\mathrm{F}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$; The colored areas under the threshold ${\mathrm{S}\mathrm{N}\mathrm{R}}_{\mathrm{T}}=$ 3 dB indicate the weak LF signal indistinguishable from the noises. (b) The SNR${}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ as the function of ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$.

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Fig.3(b) indicates that higher ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$ yields more improved amplification, until SNR${}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ reaches its saturation around 100 Hz, which exactly corresponds to the lower bound of the previous theoretical range of ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$. Fortunately, since the small systematic noise and excellent coherence of the phonon laser system allow for high frequency resolution, we are able to observe these novel results and obtain qualitative consistency with theoretical work on the Duffing oscillator that, ${\omega }_{\mathrm{H}\mathrm{F}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ is positively correlated with ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}$ [24].

Moreover, the optimal amplification occurs at the HF driving of ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}/(2\pi )$ = 100 Hz, ${\omega }_{\mathrm{H}\mathrm{F}}/(2\pi )$= 2.88 MHz, with the SNR of 26 dB and the amplification of 23 dB. Compared to the recent result of 20 fold ($\approx$ 13 dB) in single trapped ${}^{40}$Ca${}^{+}$ as well [8], our enhancement of the optimized VR amplification arises from the amplified coherent motion. The oscillation amplitude of the ion was less than 3 μm in Ref. [8], while the phonon laser provides a saturable amplification of the motion to an amplitude of 13 μm, accompanied with the broadening of the width of the bistable region $W_{bi}$. In previous studies, $W_{bi}$ determines the maximum amplification and was constrained by factors such as sideband effects and cooling transition saturation [7–9]. In the single-ion phonon laser system, the $W_{bi}$ can be further expanded via enlarging the intensity of the blue-detuned laser beam. Nevertheless, stronger blue-detuned field also induces extra heating and reduces the enhancement. As a result, in our experiments, the increased $W_{bi}$ does not improve VR amplification further (see Appendix E). It implies that we should suppress experimental noise or improve fluorescence collection efficiency to improve the SNR and then enhance the VR amplification in our scheme.

4.3 Responses to various weak signals

Now we consider more complex weak signals by adding appropriate amplitudes of HF driving to the phonon laser system to significantly improve their SNR. In addition to weak LF signals with different amplitudes (see Appendix F), our scheme is also compatible with multi-frequency signals and binary aperiodic signals.

4.3.1 Multi-frequency weak signal

In contrast to the previous work basically focusing on amplifying signals of single frequencies, here we introduce a combination of two square waves of 1 Hz and 2.3 Hz for instance with the FFT spectra of their optimal amplification presented in Fig.4(a). We find plenty of peaks of multi-frequency components in this spectrum, including the two original ones and their odd-order harmonics (e.g., 3 Hz, 5 Hz, 6.9 Hz) as well as the terms of difference frequency and sum frequency (i.e., 1.3 Hz, 3.3 Hz). Since the intriguing extra resonances on the missing frequencies of the input make the phonon laser bistable system functioning like a frequency mixer, this phenomenon may be interpreted as the effects of ghost VR [17]. The input frequencies are still predominant components in the spectrum, whereas the amplification of ghost frequencies leads to dispersion of the system energy and reduction of the effective gain.

Fig.4 Detection of a mixed signal and aperiodic signal using VR. (a) The FFT spectrum of a mixed signal under the condition of ${\omega }_1/(2\pi )={\omega }_2/(2\pi )=$ 0.72 MHz, ${\mathrm{\Omega }}_1/(2\pi )=$ 1 Hz, ${\mathrm{\Omega }}_2/(2\pi )=$ 2.3 Hz, ${\omega }_{\mathrm{H}\mathrm{F}}/(2\pi )=$ 4.32 MHz and ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}/(2\pi )=$ 100 Hz. The vertical dashed lines lie at the frequencies of the peaks marked as from A to G. (b) The input aperiodic signal (I) and the time trace of the photon counts under the condition of ${\omega }_{\mathrm{H}\mathrm{F}}/(2\pi )=$ 0 (II), 1.44 MHz (III), 2.16 MHz (IV), 3.60 MHz (V) and 5.76 MHz (VI), where we set ${\omega }_{ap}/(2\pi )=$ 0.72 MHz and ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}/(2\pi )=$ 100 Hz. In (I) the corresponding part of the input pattern of the weak binary aperiodic signal is shown with an arbitrary scale.

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4.3.2 Binary weak signal

Here we consider the scenario of using a binary aperiodic signal as the input weak signal [5]. In Fig.4(b)(I), we present a part of time series of the aperiodic binary signal employed in our experiment. Since the binary signal is with an amplitude of ${\omega }_{ap}/(2\pi )$= 0.72 MHz, much smaller than the switching threshold, no inter-well jump would happen in the absence of the additional HF driving, see Fig.4(b)(II). Similar to the periodic signal scenario, we find in the present case that, with the increase of ${\omega }_{\mathrm{H}\mathrm{F}}$, rare inter-well jumps between two states, induced by the joint effect of the input aperiodic binary signal and HF driving, are replaced by more frequent switching events. The optimal case occurs corresponding to the complete synchronization between the aperiodic binary signal and the output of the phonon laser system, as shown in the panel (IV). Then the further increase of ${\omega }_{\mathrm{H}\mathrm{F}}$ worsens the input−output synchronization, as represented in panels (V) and (VI), indicating that more frequent jumps distort the resolution of the output signal. These results hold potential applications in detecting and recovering weak aperiodic signals, such as pattern recognition and digital data transmission.

5 Conclusion

We have experimentally investigated the VR amplification with a single-ion phonon laser, working as a Van der Pol oscillator. We have realized the precisely continuous tuning of the bistability symmetry in the phonon laser. The improved symmetric bistability can optimize the VR amplification to 23 dB, accompanied by the suppression of even-order weak signals, increased robustness of VR effects, and reduction of experimental frequency difference (from ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}\gg {\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}$ [6–10] to ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{F}}>{\mathrm{\Omega }}_{\mathrm{s}\mathrm{i}\mathrm{g}}$). Additionally, we have exhibited the application of enhanced VR amplification for multi-frequency weak signals [39]. Notably, our approach allows for enhancement of system response due to the amplified coherent motion of the phonon laser rather than the decrease of noises [40, 41], providing the sensitive detection and significant amplification of the weak signal in considerably noisy environment.

The significant weak signal enhancement results from amplified coherent motions and unique nonlinearity-generating mechanism. The idea in this work can be generalized to various platforms, ranging from single-atom [27] to mechanical resonators [42], highlighting the interplay between nonlinear damping and metrology in open systems. This work opens up new possibilities to optimize weak signal detection and amplification with coherent motion, providing potential applications in digital communication and signal processing at the single-atom level.

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Declarations

The authors declare no competing interests and no conflicts.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. U21A20434, 12074390 and 92265107, the Key Lab of Guangzhou for Quantum Precision Measurement under Grant No. 202201000010, the Science and Technology Projects in Guangzhou under Grant Nos. 202201011727 and 2023A04J0050, Guangdong Provincial Quantum Science Strategic Initiative under Grant No. GDZX2305004, and Nansha Senior Leading Talent Team Technology Project under Grant No. 2021CXTD02.

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