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Abstract
We propose a scheme to generate squeezed states of magnon and phonon modes and verify squeezing transfer between different modes of distinct frequencies in a cavity magnomechanical system which is composed of a microwave cavity and a yttrium iron garnet sphere. We present that by activating the magnetostrictive force in the ferrimagnet, realized by driving the magnon mode with red-detuned and blue-detuned microwave fields, the driven magnon mode can be prepared in a squeezed state. Moreover, the squeezing can be transferred to the cavity mode via the cavity-magnon beamsplitter interaction with strong magnomechanical coupling. We show that under the weak coupling regime, large mechanical squeezing of phonon mode can be achieved, which verifies that our scheme can find the existence of quantum effects at macroscopic scales. Furthermore, distinct parameter regimes for obtaining large squeezing of the magnons and phonons are given, which is the principal feature of our scheme. The considered scheme can be extended to hybrid optical systems, and can facilitate the advancement for realization of strong mechanical squeezing in cavity magnomechanical systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Cavity magnomechanical systems [1,2], including cavity-magnon system of yttrium iron garnet (YIG), have attracted extensive attention and witnessed magnificent achievement in last decades years. The main reason is that YIGs possess rather high spin density and low damping rate, which is prerequisite for creating strong coupling [3–6] between the magnon mode of YIG and microwave cavity photon mode and makes it possible to accomplish quantum information transfer. Further, beneficial from hybrid systems constructed by the magnon mode of YIG microwave photons [7], some intriguing phenomena, such as magnon-induced nonreciprocal transmission [8], the theoretical study [9] and experimental demonstration [10] of Kerr effect, magnon-induced optical high-order sideband generation [11], and enhanced sideband responses by tuning magnon-photon coupling strength [12] have been widely investigated, leading to novel and potential applications in quantum information processing and quantum communication [13–15]. Besides, some interesting physical problems have also been investigated in the system of cavity-magnon polaritons, such as enhancement of magnon-magnon entanglement inside a cavity [16], steady Bell state generation via magnon-photon coupling [17], nonreciprocal entanglement [18], ground state cooling of magnomechanical resonator [19], and magnon-blockade in a parity-time symmetric cavity-magnon system [20], etc.
Squeezed state, an important quantum state, serves as fundamental and vital resource for continuous variable information processing [21], owing to particular utility for improving measurement sensitivity [22–24], exploring the quantum and classical borderline [25,26], and studying decoherence theories at large scales [27]. Inspired by these superior characteristics, many schemes have been proposed to generate the squeezed state based on a number of platforms or systems, in which the cavity magnomechanical system is a strikingly hot investigated platform. Particularly, the authors in Ref. [28] have proposed the engineering of magnon mode squeezing and the squeezing transfer to the phonon mode through a squeezed vacuum cavity field and a red-detuned laser pulse driving the magnon mode. Cavity optomechanical (COM) system, which drastically enhances the coupling between microwave cavity photons and mechanical motion by radiation-pressure interaction, has made tremendous progress, for example, the quantum squeezing of mechanical motion [29–38], macroscopic entanglement of two massive harmonic oscillators [39,40], the analysis of high-order sideband signals of optomechanical system [41], enhanced photon blockade with parametric amplification [42], the manipulation of multi-transparency windows [43], and the ground-state cooling of the mechanical oscillator [44–50], etc. Similar to the radiation-pressure interaction in the COM system, the magnetostrictive interaction [1] in the cavity magnomechanical system, which is responsible for the coupling between the magnon mode and the phonon mode caused by the deformation of geometry structure of the sphere, can account for many abundent and interesting phenomena occurring in the cavity magnomechanical system.
In this paper, we propose a scheme to generate the squeezed states of magnons and phonons and demonstrate the squeezing transfer in a cavity magnomechanical system consisting of cavity microwave photons, magnons, and phonons. Different from the method in Ref. [28], we apply two-tone microwave fields to drive the YIG, which leads to the generation of squeezed state of the driven magnon mode [34], instead of the prerequisite with a squeezed vacuum cavity field. Moreover, making use of equal resonance frequencies, we find that the reversible squeezing transfer process occurs between the magnon mode and the cavity mode via the magnetic dipole interaction (cavity-magnon beamsplitter interaction) [2]. Under the rotating-wave approximation, large mechanical squeezing of phonon mode can be obtained with weak effective coupling strengths between the magnon mode and the phonon mode, namely, the amplitudes of two-tone microwave fields. The squeezing is in the stationary state in the long time limit and is robust against environment thermal noise. We investigate the system through the standard Langevin formalism and the linearized Hamiltonian, and give the ratio of blue-detuned driven to red-detuned driven magnomechanical coupling for the maximal squeezing of phonon mode. Also, our scheme could inspire other experimental application of parametrically coupled bosonic modes, e.g., superconducting circuits.
The paper is organized as follows: In Sec. 2, we describe the model, give the quantum Langevin equations and the steady-state mean values of the magnon mode, and obtain the final effective Hamiltonian. In Sec. 3, by using the effective Hamiltonian derived in Sec. 2, we calculate the quantum Langevin equations for fluctuation operators, and further give the dynamical equation for covariance matrix, i.e., the solutions of quantum Langevin equations. In Sec. 4, we first show that the driven magnon mode can be squeezed due to the simultaneous driving of two microwave fields, and then we show that the squeezing can be transferred to the cavity mode through equal resonance frequencies. We also demonstrate the squeezing swapping between the phonon mode and the cavity or magnon mode, via altering red-detuned and blue-detuned magnomechanical couplings. Finally, a conclusion is given in Sec. 5.
2. Model and Hamiltonian
As schematically shown in Fig. 1, the proposed system is a hybrid cavity magnomechanical system, where magnons couple to microwave cavity photons and phonons via magnetic dipole interaction and magnetostrictive interaction, respectively. The magnon mode is represented by collective motion of a large number of spins in the ferrimagnet. The phonon mode originates from the deformation of the geometry structure caused by the magnetostrictive interaction. The YIG is adjusted in the suitable direction of bias magnetic field ($z$ direction) such that the magnetostrictive interaction is activated, which can be significantly enhanced by driving the magnon mode with a microwave field. Here, we use two microwave fields (not shown in the system sketch) with distinct amplitudes $E_{\pm }$ applied to the magnon mode. The total Hamiltonian of the syetem is written as ($\hbar =1$)
It is not difficult to calculate the solution of $m_{s}$ from the dynamical equations of steady-state mean value in Eq. (3). By dropping the small term $\eta (b_{s}^{\ast }+b_{s})$, we obtain
It is worth emphasizing that, $\omega _{+}$ is the blue-detuned driven frequency, $\omega _{+}=\omega _{m}+\omega _{b}$, and $\omega _{-}$ is the red-detuned driven frequency, $\omega _{-}=\omega _{m}-\omega _{b}$. Now we linearize the Hamiltonian in Eq. (1) by dropping the small term $\eta (b_{s}^{\ast }+b_{s})$, the linearized Hamiltonian is thus given by
It is convenient to work in the interaction picture. Transforming the Hamiltonian of Eq. (5) into the interaction picture through the unitary time evolution operator $U(t)=\exp [-it(\omega _{a}\delta a^{\dagger} \delta a + \omega _{m} \delta m^{\dagger} \delta m +\omega _{b}\delta b^{\dagger}\delta b)]$, assuming $\omega _{a}=\omega _{m}$, we obtain the final effective Hamiltonian, followed by
Fig. 1. Sketch of the system. One macroscopic YIG is placed inside the cavity near the maximum magnetic field of the cavity mode, and simultaneously in a uniform bias magnetic field which mediates the coupling between the cavity mode and the magnon mode. The bias magnetic field ($z$ direction), the drive magnetic field ($y$ direction), and the magnetic field ($x$ direction) of the cavity mode are mutually perpendicular at the site of the sphere.
3. Quantum Langevin equations and solution
In this section, we first utilize the effective Hamiltonian of the previous section to gain the QLEs, and then derive the dynamical equations of quadrature fluctuations via the covariance matrix approach. Considering that the dynamics of system evolves as dictated by effective Hamiltonian of Eq. (6), we obtain the QLEs for quantum fluctuations, given by
Combining Eq. (10) and Eq. (11), we can obtain
whereFurther, by substituting Eq. (15) into Eq. (12), the derivative of the covariance matrix is given by
Therefore, the time evolution of covariance matrix can completely govern the dynamics of the present system. Next we concentrate on our interested squeezing. The degree of squeezing can be expressed in the dB unit, which can be evaluated by
4. Squeezing dynamics with and without rotating-wave approximation
4.1 With rotating-wave approximation
At the beginning of our study, we pay attention to the case of the weak coupling regime, namely, $G_{-}, G_{+}<< \omega _{b}$, which implies high frequency oscillating terms can be discarded in Eq. (9). We adopt experimentally realistic parameters [1]: $\gamma _{a}=0.1\omega _{b}$ (units of $\omega _{b}$), $\gamma _{m}=\gamma _{a}$, and $\gamma _{b}=10^{-5}\omega _{b}$. The equal resonance frequencies are employed, $\Delta _{am}=0$, which plays a vital role in achieving squeezing transfer process. In Fig. 2, we show the mean square fluctuations (variances) as a function of time $t$ for the phonon mode, the magnon mode, and the cavity mode under two conditions of RWA and without RWA, which demonstrates that the numerical simulation under the RWA is consistent with the consequence without the RWA. It is clear that at the beginning parts of time increasing, the mean square fluctuations of the magnon mode and the cavity mode both start to decline as a result of the combination of red-detuned and blue-detuned driven microwave fields [34]. However, as the time grows, their evolution curves finally approch to vacuum fluctuation. By contrast, the phonon mode variance is always below vacuum fluctuation value and finally decreases to a stable value. In order to illustrate the squeezing phenomena of the three modes in the so-called phase space, we depict the respective Wigner functions [21] in the steady state in Fig. 3, which agree well with the squeezing behavior of Fig. 2. Figure 3(a) shows a contraction along the amplitude quadrature $\delta X_{b}$ axis and a stretch along the phase quadrature $\delta Y_{b}$ axis. In comparison, no squeezing occurs for the magnon mode and the cavity mode because of circular images in Fig. 3(b) and Fig. 3(c). In the following, the role of the blue-detuned microwave field which plays in the squeezing generation is explained.
Fig. 2. The evolution of mean square fluctuations for the phonon mode $\left \langle \delta X_{b}(t)^{2} \right \rangle$ (a), the magnon mode $\left \langle \delta X_{m}(t)^{2} \right \rangle$ (b), and the cavity mode $\left \langle \delta Y_{a}(t)^{2} \right \rangle$ (c) with the RWA (dashed lines) and without the RWA (solid lines). Other parameters are: $G_{-}=0.01\omega _{b}$, $G_{+}=0.009\omega _{b}$, $g=0.01\omega _{b}$, and $n_{th}=0$.
Fig. 3. The Wigner function for the phonon mode (a), the magnon mode (b), and the cavity mode (c) with the RWA. All the figures are drawn in the steady state, and the parameters are the same as in Fig. 2.
In Fig. 4, we plot the phonon mode quadrature squeezing as a function of $G_{+}/G_{-}$ under the condition of RWA. The squeezing of the phonon mode first increases with the enhancement of the ratio $G_{+}/G_{-}$ and then decreases with the ratio $G_{+}/G_{-}$. There exists an optimal ratio for the strongest squeezing of the phonon mode. It is concluded that the squeezing of phonon mode can not surpass 3dB if blue-detuned driven microwave field is not applied or is not large enough, implying that two-tone field pulse is an essential contribution to the squeezing generation. To clearly show the robustness of phonon mode squeezing to the environment, we plot the squeezing degree for phonon mode in the steady state at different mechanical noises $n_{th}=0,10,50$ as function of time $t$ in Fig. 4(a). From the plot, we find even when phonon thermal occupation $n_{th}=50$, the squeezing of the phonon mode still exists, which is robust about environment thermal noise. The squeezing degree for phonon mode as a function of the ratio $G_{+}/G_{-}$ at different $\gamma _{m}$ for the steady state is displayed in Fig. 4(b). By comparing the three lines, we see that for larger magnon decay rate $\gamma _{m}$, smaller $G_{+}/G_{-}$ is needed to obtain mechanical squeezing. From the Fig. 4(c), we see that the mechanical squeezing is immune to the cavity decay rate $\gamma _{a}$, which implies that our proposal can be effectively simplified as a protocal of two coupled bosonic modes. For larger $\gamma _{a}$, the largest squeezing requires larger ratio $G_{+}/G_{-}$. In Fig. 4(d), we show that it is only in the region $G_{-}>G_{+}$ which ensures stability that the mechanical squeezing occurs. In addition, it can also intuitively show the optimal proportion region of maximum squeezing value.
Fig. 4. The squeezing degree of the phonon mode as function of $G_{+}/G_{-}$ at (a) different $n_{th}$, the red dashed, blue dot, and pink dot-dashed curves correspond to $n_{th}=0, 10$, and $50$, respectively, (b) different magnon decay rate $\gamma _{m}$, the red dashed, blue dot, and pink dot-dashed lines correspond to $\gamma _{m}=0.1\omega _{b}, \omega _{b}, 10\omega _{b}$, and (c) different cavity decay rate $\gamma _{a}$, the red dashed, green solid, and black dot curves correspond to $\gamma _{a}=0.1\omega _{b}, 10\omega _{b}, 100\omega _{b}$. (d) The squeezing degree of the phonon mode versus $G_{+}$ and $G_{-}$. $\gamma _{m}=0.1\omega _{b}$ in (a) and (c), $n_{th}=0$ in (b) and (c), and $\gamma _{a}=0.1\omega _{b}$ in (a) and (b). $G_{-}=0.01\omega _{b}$ is used in (a), (b), and (c). In (d), $n_{th}=0$, $\gamma _{m}=0.1\omega _{b}$, and $\gamma _{a}=0.1\omega _{b}$. The other parameters are the same as in Fig. 2.
The physical mechanism for the influence of the blue-detuned driven microwave filed can be explained in detail as follows. Under the RWA, the direct magnetostrictive force between the magnon mode and the phonon mode of YIG is denoted by $\delta m^{\dagger}(G_{+}\delta b^{\dagger}+G_{-}\delta b)$+H.c. in Eq. (6). Making use of the standard squeezing transformation, we can rewrite the term as $G_{eff}(\delta m^{\dagger}\delta B)$+H.c., with $G_{eff}=\sqrt {G_{-}^{2}-G_{+}^{2}}$, $\delta B = \textrm{cosh} (r) \delta b + \textrm{sinh} (r) \delta b^{\dagger}$ being the Bogoliubov mode, and $r$=ln[$(G_{-}+G_{+}) / (G_{-}-G_{+})$]/2 being the squeezing parameter. When there is no blue-detuned microwave field causing magnetostrictive coupling $G_{-}$, the squeezing parameter $r$ is zero, which accounts for no mechanical squeezing. Indeed, two competing factors make joint decisions for the mechanical squeezing. Apart from squeezing parameter $r$, the effective direct coupling between the phonon mode and the magnon mode, represented by $G_{eff}$, is also a decisive factor. The squeezing parameter $r$ increases with the enhancement of the ratio $G_{+}/G_{-}$, while $G_{eff}$ decreases with the enhancement of the ratio $G_{+}/G_{-}$. As a consequence, a tradeoff between these two competing factors leads to the maximal squeezing.
4.2 Without rotating-wave approximation
In this subsection, we consider the case where the magnon-phonon effective couplings ($G_{-}, G_{+}$) are comparable to the mechanical resonance frequency $\omega _{b}$. We need to numerically solve Eq. (16) without the RWA. From the Fig. 5, it is observed that mean square fluctuation curves for the cavity mode and the magnon mode evolve with time periodically. Here it is necessary to compare the main differences between our scheme and the previous literatures in the generated squeezing of the magnon mode. The authors in Ref. [51] proposed a scheme involved using a nonmagnetic conductor and a ferromagnet, and demonstrated the generated squeezing via dipolar interaction, while two-tone driven microwave fileds and specific magnon-phonon effective couplings ($G_{-}$, $G_{+}$) are essential for our scheme. In Ref. [52], the authors proposed a setup where a ferromagnetic ellipsoid tuned by an external magnetic field is kept inside a microwave cavity, the magnetic Hamiltonian of the reference includes a component which is similar to the coupling between the cavity field and the parametric amplifier in optomechanical system with an optical parametric amplifier [53], while the component $[(E_{+}e^{-i\omega _{+}t}+E_{-}e^{-i\omega _{-}t})m^{\dagger}+\mathrm {H.c.}]$ in Eq. (1) is contribute to the squeezing generation. Therefore, the physical mechanism of the squeezing of the magnon mode in our scheme is obviously different from the two schemes. In fact, under the strong coupling regime, the mean square fluctuation for the phonon mode is beyond the vacuum fluctuation (not shown in the figure). This is because heat swapping emerges when the system is in the strong coupling regime which enables reversible energy exchange between cavity photons and magnons. Meanwhile, quantum backaction heating resulted from the counter-rotating terms are the accompanying effect opposed to radiation pressure-like utilized to cool the mechanical motion. Note that the instant of the biggest squeezing for the magnon mode is the instant of the smallest squeezing for the cavity mode, and vice versa, which verifies that the optimal squeezing appears alternately between the two modes and could be highly prospective for the squeezing transfer between distinct modes in hybrid optical systems.
Fig. 5. The evolution of mean square fluctuations of the magnon mode (green solid line), and the cavity mode (black dot line) with strong magnomechanical couplings without the RWA. The parameters are: $G_{-}=\omega _{b}$, $G_{+}=0.9\omega _{b}$, and $g=0.32\omega _{b}$, and the other parameters are the same as in Fig. 2. The insert shows the evolution of mean square fluctuations for the two modes in the long time limit.
5. Conclusions
In conclusion, we have proposed a scheme to acheive the squeezing generation of magnons and phonons in a hybrid cavity-magnon system, by driving the magnon mode with two-tone microwave fields, without any explicit measurement or feedback. We show that strong squeezing of the phonon mode can be achieved under the weak coupling regime. Furthermore, we also show that the driven magnon mode is first squeezed and the squeezing transfers to the cavity mode by stimulating cavity-magnon beamsplitter (state-swap) interaction. We demonstrate that distinct parametric regimes for obtaining ideal squeezing for phonon mode or cavity mode and magnon mode are required, namely, different red-detuned and blue-detuned driven magnomechanical couplings. Besides, when only the red-detuned laser is applied or the applied blue-detuned laser is not strong enough, the squeezing can not be generated. We realize the squeezed states of magnons and phonons in a large object, which represent genuinely macroscopic quantum states [28]. Thus our scheme is enlightening in the field of quantum-to-classical borderline, tests of decoherence theories, as well as for improvement of measurement precision.
Funding
National Natural Science Foundation of China (61822114, 12074330, 62071412).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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