Entropy production rate and correlations of cavity magnomechanical system

We present the irreversibility generated by a stationary cavity magnomechanical system composed of a yttrium iron garnet (YIG) sphere with a diameter of a few hundred micrometers inside a microwave cavity. In this system, the magnons, i.e., collective spin excitations in the sphere, are coupled to the cavity photon mode via magnetic dipole interaction and to the phonon mode via magnetostrictive force (optomechanical-like). We employ the quantum phase space formulation of the entropy change to evaluate the steady-state entropy production rate and associated quantum correlation in the system. We find that the behavior of the entropy flow between the cavity photon mode and the phonon mode is determined by the magnon-photon coupling and the cavity photon dissipation rate. Interestingly, the entropy production rate can increase/decrease depending on the strength of the magnon-photon coupling and the detuning parameters. We further show that the amount of correlations between the magnon and phonon modes is linked to the irreversibility generated in the system for small magnon-photon coupling. Our results demonstrate the possibility of exploring irreversibility in driven magnon-based hybrid quantum systems and open a promising route for quantum thermal applications.

Quantum technological and nanofabrication advancements have motivated the design of microscopic and coherent thermodynamic machines -quantum thermal machines [10], as well as investigating the interplay between quantum information and thermodynamic processes [30].Optomechanical thermal machines have been proposed in different configurations [31][32][33][34][35][36][37].The irreversibility (such as, friction, disorder) that influences the machine thermodynamic processes/performance can be quantified by entropy production [38,39].Thus, quantifying the degree of irreversible entropy generated in a dynamic process is useful for the distinctive description of the non-equilibrium processes, and decreasing it, enhances a thermal machine efficiency [40].Based on quantum phase-space method, the measure of quantifying the irreversible entropy production of quantum systems that interact with nonequilibrium reservoirs has been formulated in [41][42][43] and experimentally verified in two distinct setup -an optomechanical system and a driven Bose-Einstein condensate coupled to a high finesse cavity [44].The effect of self-correlation on irreversible entropy production rate in a parametrically driven dissipative system has been investigated [45].It has been recently shown that the presence of nonlinearity via an optical parametric oscillator placed inside the cavity optomechanical system influences the stationary state entropy production rate [46].Moreover, it has been established that the entropy produced in a bipartite quantum system is related to the amount of correlations shared by its subsystems [41].
Here, we investigate the generation of irreversibility in a hybrid cavity magnomechanical setup comprising a microwave cavity and a YIG sphere.In this system, magnons are simultaneously coupled to the phonons of the vibrational sphere via magnetostrictive interaction and to the cavity photons via magnetic-dipole interaction, while there is no direct interaction between the cavity mode and the mechanical mode.We find that the magnon-photon coupling and cavity photon dissipation rate influence the entropy production rate.Furthermore, we demonstrate that the amount of correlations in the cavity magnomechanical system deviates from the steady-state entropy production rate for large magnon-photon coupling.
The rest of this paper is organized as follows.In Section II, arXiv:2401.16857v1[quant-ph] 30 Jan 2024 FIG. 1. Diagrammatic representation of a cavity magnomechanical system consisting of photon, magnon, and phonon modes.The magnon-photon interaction strength and the magnon-phonon interaction of coupling strength are denoted by gam and g mb respectively.
we present the physical model of the cavity magnomechanical system.We derive the linearized Hamiltonian of the system via quantum Langevin equations of motion and standard linearization techniques.In Section III, the stationary entropy production rate and quantum correlation quantified by mutual information are presented using the experimentally feasible parameters.Finally, the conclusions are summarized in Section IV.

II. CAVITY MAGNOMECHANICAL MODEL
We consider a hybrid cavity magnomechanical system, which consists of a microwave cavity and a small sphere (a one mm-diameter, highly-polished YIG sphere is considered in Ref. [47,48]).The YIG sphere is positioned close to the maximal microwave magnetic field of the cavity mode, and a variable external magnetic field H in the z-axis is added to establish the magnon-photon coupling [47,49].The coupling rate can be tailored by adjusting the position of the sphere.The magnons couple to phonons via the magnetostrictive effect.The vibrational modes (phonons) result from the geometric deformation of the YIG sphere because the magnon excitation inside the YIG sphere induces a varying magnetization.The magnomechanical coupling can be enhanced by directly driving the magnon mode with a microwave source [12].In addition, the size of the sphere considered is much smaller than the wavelength of the microwave field, such that the interaction between cavity microwave photons and phonons can be neglected (i.e. the radiation pressure effect is negligible).Thus, the system has three modes: cavity photon, magnon, and phonon modes, which can be schematically depicted by the equivalent coupled harmonic oscillator model as shown in Fig. 1.
The Hamiltonian of the hybrid quantum system under rotating-wave approximation in a frame rotating with the frequency ω d of the driving field can be expressed as (ℏ = 1) [23,48]: 4 γ g √ N t B 0 (assuming low-lying excitations) denotes the coupling strength of the drive magnetic field [23].The amplitude and frequency of the drive field are B 0 and ω d , respectively, and the total number of spins N t = ρ V , where V is the volume of the sphere and ρ = 4.22 × 10 27 m −3 is the spin density of the YIG.Each of the modes is coupled to an independent noise reservoir, their energy decay rates are γ a , γ m , and γ b , for photon, magnon and phonon respectively.Experimentally, g mb is extremely weak [12], but the magnomechanical interaction can be enhanced by driving the magnon mode with a strong microwave field [47,49].The magnon-photon coupling rate g am can be larger than the dissipation rates of the cavity and magnon modes, γ a and γ m , entering into the strong coupling regime, g am > max{γ a , γ m } [13,14].
As a result of strong driving, the Hamiltonian in Eq. ( 1) can be linearized around the coherent steady-state amplitude: where O s and the operators Ô, represent the steady-state amplitudes and quantum fluctuations of the corresponding mode.We have the steady-state amplitudes Then, the linearized Hamiltonian can be derived as where the enhanced magnon-phonon coupling G mb = g mb m s , and ∆m = ∆ m − g mb (b s + b * s ) is the effective magnon detuning incorporating the magnetostriction.For the considered parameters, g mb (b s + b * s ) ≪ ∆ m , so we can have, ∆m ≈ ∆ m .Since the driving field affects m s , we can improve G mb by adjusting the external driving field Ω d .
From the Hamiltonian in Eq. ( 3), we obtain the quantum Langevin equations as; where fin ∈ {â in , min , bin } are input noise operators for the cavity, magnon and mechanical modes, respectively, which are zero mean and characterized by the following correlation functions [51]: , where N k = 1/ e ℏω k /kBT − 1 (k ∈ {a, m}), are the equilibrium mean thermal photon, and magnon number, respectively, while N b = 1/ e ℏΩ b /kBT − 1 is the equilibrium mean thermal phonon number, T is the environmental temperature and k B is the Boltzmann constant .Eq. ( 4) represents the evolution of the fluctuation incorporating the interplay with the environment via the noise operators [25].The photon number and magnon occupation number are approximately zero, i.e., N a,m ≈ 0, due to the high frequencies of their modes.
Since the nature of the noise is Gaussian, all the information is contained in the first and second-order moments of the operators.In particular, it is convenient to introduce the quadratures x and ŷ of the photon, magnon, and phonon modes by using the relation b}, and elements of the corresponding covariance matrix are defined as The system must be stable for a steady state to exist, to ascertain this, the Routh-Hurwitz criterion [52] is employed to characterize the stability of the system.To achieve this, the real part of the spectrum of the drift matrix A , Eq. ( 5), must be negative, this means that all the eigenvalues of the drift matrix A have non-positive real parts.

III. ENTROPY PRODUCTION RATE
The basic thermodynamics principle asserts the entropy of an open system, (classical or quantum) evolves as: where Π ⩾ 0 is the irreversible entropy production rate and Φ is the entropy flow from the system to the reservoir.In thermal equilibrium, the steady states are characterized by dS/dt = Π = Φ = 0.However, when the system is connected to multiple reservoirs or being externally driven, it may instead reach a nonequilibrium steady states where dS/dt = 0 but Π = Φ ⩾ 0. In this nonequilibrium steady state case, the system is characterized by the continuous production of entropy, all of which flows to the reservoir.
We now move to study the entropy production rate from a multipartite system.In analogy with the bipartite case [41], we combine quantum phase-space methods and the Fokker-Planck equation to characterise the irreversible entropy production of quantum systems interacting with reservoirs.In general, the entropy production rate of the quantum system described in Section II is given by (see, Appendix A): where Here, we focus on characterizing the entropy production rates in a cavity magnomechanical system.Without loss of generality, we consider the effective entropy flow between the magnon mode and the mechanical resonator.Thus, the rate of entropy production Π s at a steady-state reads We remark, when the system is in the equilibrium state, we have V 11 + V 22 = 1, V 33 + V 44 = 2N b + 1, and hence, Π s ≡ 0.
To proceed, we study the entropy production rate in a magnon-phonon-photon system at a steady state in the resolved sideband, where the magnon dissipation rate is comparable to or well below the mechanical resonance frequency (i.e., γ m < Ω b ).We assume the following parameters close to those employed in the experimental realizations [12], as the phonon frequency Ω b /2π = 10 MHz, the cavity dissipation rate γ a /2π = 3 MHz, the magnon dissipation rate γ m /2π = 1 MHz, the phonon damping rate γ b = 300 Hz, the phonon-magnon coupling g mb /2π ≃ 1 Hz, and the temperature T = 10 − 100 mK (i.e., N b ≃ 200).In the following analysis, we will utilize dimensionless quantities; that is, the quantities will be expressed in units of the phonon frequency, Ω b .
In Fig. 2, we present the entropy production rate Π s is plotted as a function of the normalized magnon detuning ∆ m /Ω b for various values of the photon-magnon coupling g am .In Fig. 2(a) and (b), we consider the dissipation rate of the cavity γ a = 10 −1 Ω b for distinct occupation number N b = 10 and N b = 100, respectively.It can be seen that in the absence of magnon-photon interaction g am = 0, the entropy production rate Π s peaks at ∆ m /Ω b = ±1.The two peaks in the rate of entropy production for positive/negative detuning imply the cooling/heating processes behave differently in two distinct regimes of the system.For non-zero g am , the smaller peak smears out with reduced maximum Π s .We observe that for g am = Ω b (g am = 2 Ω b ), the maximum Π s is in the red (blue) sideband region.The maximum entropy flow between the phonon mode and the effective magnon mode occurs when g am = 2Ω b , at ∆ m = ω b .Fig 2 (c) and (d) show more clearly the effect and interplay between the cavity photon dissipation rate and the magnon-photon coupling.For the case of γ b = Ω b , Fig. 2(c) and (d), the value of Π s further decreases with a broaden peak as magnon-phonon coupling g am increases.In the blue-sideband, for g am ̸ = 0, the value of Π s is always reduced compare to the g am = 0 scenario.It can be seen that Π s could be enhanced close to the negative detuning region ∆ m < 0. The effects of thermal fluctuations of the environment on the entropy production rate is shown in Fig. 2. The Fig. 2 shows that increasing the phonon thermal excitation (occupation number) N b , increases Π s in magnitude (see the magnitude of Π s in Fig. 2(a) and (c) compare to (b) and (d) respectively).The non-uniform behaviour when varying g am for different γ a is a direct implication of the complex nature of the interaction between the dissipation processes in the cavity magnomechanical system.In Fig. 3, the effects of the cavity photon dissipation rate of the environment on the entropy production rate is explored.We present in Fig. 3 (a), the plot of the entropy production rate as a function of magnon-photon coupling with various values of the cavity decay rate, while Fig. 3 (b) shows the entropy production rate as function of cavity decay rate with different values of the magnon-photon coupling.For large thermal excitations, N b = 100, and ∆ m = Ω b , the entropy production rates decrease in oscillatory form as the magnon-photon coupling increases.For γ a = 10 −1 Ω b , the maximum entropy production rate corresponds to the g am = 2Ω b .It can be seen that the entropy production rate can be increase/decrease by tuning the cavity decay value γ a and the magnon-photon coupling, see g am ≃ 0.2 − 3 Ω b .The entropy production rate Π s linearly decreases with respect to the magnon-photon coupling when g am > 3/Ω b but the value slightly increase when we increase γ a .This non-monotonic behaviour (increase/decrease) of the entropy production with respect to the magnon-photon coupling can be attributed to the imbalance in populations between the interacting modes induced by the environment.Furthermore, Fig. 3 (b) shows that the entropy production rate can increase/decrease when γ a < g am but increases slightly after γ a = g am .In the region, γ a ≫ g am , the entropy production rate saturates to a quasi-constant value with decreasing value as the magnon-photon g am increases.This is due to the fact that the individual modes are far from resonance and they are effectively decoupled, in such a way that the individually thermalize their own bath.
Next, we analyze the behaviour of the entropy production rate with respect to the amount of correlations shared between the magnon and phonon modes.For coupled quantum system, it has been demonstrated that the irreversibility generated by the dissipative system at steady state and the total amount of correlations shared between the subsystems are closely related in small coupling limit as [41]; I ≈ Π s /(2 γ tot ), where I is the quantum mutual information between the modes at the stationary state and γ tot is the sum of the dissipation rates to the local baths.Since the quantum noises are Gaussian, the mutual information between the magnon and phonon modes can be computed as [53]: where V a (V b ) is the covariance matrix of magnon (mechanical) mode.In Fig. 4, we show a comparative plot of the entropy production rate Π S to the correlations established by the cavity magnomechanical system, as quantified by the mutual information I. Considering small thermal phonon excitation, N b = 10, Fig. 4 (a) shows a good similarity in both Π s and I curves for |∆ m /Ω b | ⩽ 2. As the magnon-photon coupling increases, both the entropy production rate and mutual information are decreasing as well as an increase in both quantities deviation, see Fig. 4 (b) and (c).The increase in deviation between Π s and I for increasing detuning ∆ m , clearly demonstrates the interplay among magnon-photon coupling and dissipation rates.This shows that degree of irreversibility induced by the stationary process is directly related to the amount of correlations shared in the system.

IV. CONCLUSIONS
We have investigated the entropy production rate in a hybrid magnomechanical system where a microwave cavity mode is coupled to a magnon mode in a YIG sphere, and the latter is simultaneously coupled to a phonon mode via magnetostrictive force.Specifically, within the quantum phase space formulation of the entropy change, we evaluate the steady-state entropy production rate and associated quantum correlation in the hybrid system.We have shown that the entropy flow between the effective magnon mode and the phonon mode is influenced by the magnon-photon coupling and the cavity photon dissipation rate.Our numerical analysis shows nonuniform behavior of irreversibility resulting from the complex and competing nature of the interactions in the cavity magnomechanical system.We find that the entropy production rate can be increased/decreased when the cavity decay rate is less than the magnon-photon coupling.Furthermore, we studied the range of validity of the link between irreversibility and the amount of correlations in a mesoscopic quantum system.These results provide insight into the impact of magnonics on the thermodynamics processes of hybrid quantum systems.Finally, we anticipate that our study will open new perspectives for quantum thermodynamics applications, such as realizing thermal machines in the deep quantum regime and quantum thermal transport.
where the bosonic annihilation (creation) operators â (â † ), b ( b † ) and m ( m † ) denote, respectively, cavity photon, phonon and maganon modes whose resonant frequencies are taken to be ω a , Ω b , and ω m corrospondingly.The detuning parameters ∆ a := ω a − ω d , where ω a /2π = 10 GHz[12], and ∆ m := ω m − ω d .The uniform magnon mode frequency in the YIG sphere is ω m = γ g H, where γ g /2π = 28 GHz/T is the gyromagnetic ratio, and we set ω m at the Kittel mode frequency[50], which can lead to cavity polaritons by strongly coupling magnon and cavity photons.The parameters g am and g mb are the optomagnon (photon-magnon) and magnomechanical (magnon-phonon) interaction coupling strengths respectively.The last term, (Ω d m † − Ω * d m) in the Hamiltonian describes the external driving of the magnon mode.The Rabi frequency Ω d ≡ √ 5 Here, R i (t) := [x a , ŷa , xm , ŷm , xb , ŷb ] ⊤ is the column vector of quadratures and the stationary covariance matrix is obtained by solving the algebraic equation A ⊤ V + A V + D = 0, where D = diag (γ a , γ a , γ m , γ m , γ b (2N b + 1), γ b (2N b + 1)) and the drift matrix A is expressed as: