Nonreciprocal single-photon state conversion between microwave and optical modes in a hybrid magnonic system

Coherent quantum transduction between microwave and optical signals is of great importance for long-distance quantum communication. Here we propose a novel scheme for the implementation of nonreciprocal single-photon state conversion between microwave and optical modes based on a hybrid magnonic system. A yttrium–iron–garnet (YIG) sphere with both the optomechanical and the optomagnetic properties is exploited to couple with a three-dimensional superconducting microwave resonator. The magnetostatic mode of the YIG sphere is treated as an intermediate to interact with the microwave and optical modes simultaneously. By manipulating the amplitudes and phase differences between the couplings via external driving fields, we show that the nonreciprocal microwave-light single-photon state conversion can be realized via the quantum interference effect.


Introduction
Hybrid quantum systems have been extensively studied to harness the advantages of distinct physical systems for implementing various kinds of quantum information protocols [1][2][3]. The crucial requirement on this subject is to achieve the strong coupling between different physical subsystems [3][4][5]. Recently, the ferromagnetic crystal [6][7][8], especially the yttrium-iron-garnet (YIG) coupled to superconducting microwave cavity has received great attention as an alternative approach to realize strong light-matter interactions [9][10][11][12][13].
On the other hand, the YIG sphere itself can also be shaped into an optical whispering gallery mode (WGM) cavity with high quality factor and relatively small mode volume [67][68][69][70]. When the triple-resonance condition is satisfied, the optical WGMs can couple to magnon modes by means of the Brillouin scattering process [71][72][73][74]; that is, the input and output optical modes are resonant with the frequency difference given by a magnon mode [75]. The triple resonance can be readily achieved by tuning

Model and Hamiltonian
As schematically depicted in figure 1, we investigate a hybrid quantum system of a YIG sphere with both the optomechanical and optomagnetic properties coupled to a three-dimensional microwave cavity through magnetic dipole interaction [15,16]. The YIG sphere is placed at the antinode of microwave magnetic field in the three-dimensional microwave cavity and locally biased with a static magnetic field H. It is well known that many magnetostatic modes can be excited in the YIG sphere, when the magnetic component of the microwave cavity field is perpendicular to the bias magnetic field. Here, we only focus on the simplest magnetostatic mode, i.e. the Kittel mode [108] that has the uniform spin precessions in the whole volume of YIG sphere and the highest magnetic coupling strength with the microwave cavity [93]. Additionally, the YIG itself is shaped into a whispering gallery resonator that supports optical WGMs. The Kittel mode can interact with the optical transverse electric (TE) mode and transverse magnetic (TM) mode through a three-wave process, i.e. a Brillouin light scattering process [71,72]. Owing to the deformation of the geometry structure, we also consider an intrinsic vibrational mode of the YIG sphere, and the associated phonons are coupled to the optical photons via the radiation pressure and to the magnons via the magnetostrictive interaction [40,41]. Thus, the free Hamiltonian of the whole system reads (h = 1 hereafter) where ω β is the resonance frequency, and β and β † are the annihilation and creation operators (β = a, b, c, m, r represents the TM, TE, microwave cavity, magnon and phonon modes, respectively). Furthermore, the interactions between different modes can be described by the following Hamiltonian In the above equation, g m is the single-photon Brillouin coupling rate, G mc represents the coupling strength between the magnon and microwave photon, g mr denotes the single-magnon magnomechanical coupling rate and g br describes the single-photon optomechanical coupling strength between the TE and the phonon mode. Because the vibration direction of the mechanical mode and the field direction of the TM mode is orthogonal [80], there is no direct coupling between the mechanical mode and the TM mode. To enhance and linearize the optomagnonic, magnomechanical as well as the optomechanical interactions, we drive the TM, TE and magnon modes simultaneously. The Hamiltonian that describes coherent driving is given by Figure 1. Schematic diagram of the hybrid quantum system. A YIG sphere is placed inside a three-dimensional microwave cavity near the maximum magnetic field of the cavity mode, and simultaneously biased by a uniform static magnetic field H, enabling a strong coupling between magnons and microwave photons. The magnon mode is directly driven by a microwave source to enhance the magnomechanical coupling. In addition, the TE and TM mode of the YIG microsphere cavity is coupled to the magnon mode through a Brillouin light scattering process.
where Ω a = √ 2κ a P a /ω da (Ω b = √ 2κ b P b /ω db ) denotes the coupling between the TM (TE) mode and the external laser field with phase θ a (θ b ), κ a (κ b ) is the external coupling rates, P a (P b ) is the driving power, and ω da (ω db ) represents the oscillating frequency of the driving field to the TM (TE) mode. In addition, the Rabi frequency Ω m =

Then, the total Hamiltonian yields
Provided that the two conditions ω db − ω da − ω dm = 0 and ω dm ≫ G mc , g m are satisfied, we can safely neglect those fast oscillating terms in equation (4) under the rotating-wave approximation. After that, the total Hamiltonian of the whole system will reduce to the form After a standard linearization procedure under the strong driving conditions, we can obtain the linearized Hamiltonian with ∆ a,c = ∆ ′ a,c , ∆ b = ∆ ′ b + g br (r * s + r s ) and ∆ m = ∆ ′ m + g mr (r * s + r s ). In equation (6), G am = g m b s , G ab = g m m s , G mb = g m a s , G br = g br b s and G mr = g mr m * s describe the effective field-enhanced coupling strengths. Here, a s , b s , c s , m s and r s are the steady state classical mean values of the a, b, c, m and r modes, respectively. Details of the derivations of the above Hamiltonian are given in appendix.
To go a further step, we perform another unitary transformation U 2 (t) = e −iH ′ ′ 0 t to equation (6) with we can use the rotating-wave approximation to neglect those fast oscillating terms; that is, the TM mode can be safely eliminated. Then, the associated interaction Hamiltonian can be written as As depicted in figure 2, the effective couplings of this hybrid quantum system is revealed. Notably, both the magnitude and phase of the couplings can be well controlled by changing the external driving fields [103]. The effective interaction of the hybrid quantum system. The magnon mode as a mediator is used to couple to both the microwave mode and TE mode. Besides, an auxiliary phonon mode interacts with both the magnon mode and TE mode, and a coupling loop is formed.
Without loss of generality, we assume that the coupling constants G mb and G br are positive real numbers, and G mr = |G mr | e iϕ is a complex number with a nontrivial phase ϕ. In such a circumstance, the existence of the nontrivial phase ϕ is related to an artificial synthetic magnetism penetrating the closed-loop [109][110][111][112], which is formed by the couplings G mb , G mr and G br . As a result, the time-reversal symmetry is broken, and we can realize a nonreciprocal quantum transduction between the microwave and optical signals.

Nonreciprocal single-photon state conversion
In this section, we discuss in detail how to realize the nonreciprocal microwave-light single-photon state conversion based on the above mentioned hybrid quantum system, which is an important step towards the long-distance quantum communication. To obtain this goal, we need to investigate the quantum dynamics of our system by solving the quantum Langevin equations (QLEs). To this end, Here, these sources of noise obey the following correlation relations where n α,th = 1/(eh ωα/kBT − 1) (α = c, m, b) and n th = 1/(eh ωr/kBT − 1) are the thermal occupations with the environmental temperature T and the Boltzmann constant k B . To effectively suppress these thermal excitations, we consider the experimental working temperature T = 10 mK. In this situation, the thermal phonon number is about n th = 1.69 for the mechanical frequency ω r /2π = 100 MHz. Additionally, the optical, microwave, and magnon modes have a relative large resonance frequency, whose thermal occupations are negligible. Hence, we neglect the thermal excitations of the optical, microwave, and magnon modes in the following discussions. According to the Heisenberg equations of motion and together with the corresponding damping and noise terms, we can derive the QLE of the vector v as with the coefficient matrix and the diagonal matrix . Here, κ b and κ c are the external coupling rates, and we have neglected the internal decay rates of the TE and microwave modes in our scheme. There are no external input signals for the mechanical and magnetic modes. So, κ r and κ m can be regarded as their total damping rates. Since the time evolution only involves the beam-splitter interactions, our system is always stable for arbitrary parameters. By introducing the Fourier transformation o(t) =´dωe −iωt o(ω)/2π for an operator o, we proceed to convert equation (9) into the frequency domain where I represents the 4 × 4 identity matrix. To achieve the output fields, we further define the output is a unitary matrix. Here, the matrix element T ij (ω) describes the transmission amplitude from the mode j to the mode i (i, j = 1, 2, 3, 4 is referred to c, m, b, r, respectively). In the absence of thermal phonon noise, T(ω) characterizes the transmission probability from the input single-photon state to the output one [113]. To well illustrate the mechanism of nonreciprocal quantum transduction, we start our discussion by neglecting the thermal phonon noise, whose effect will be discussed in the next section.
First of all, we consider the situation that an input single-photon is in resonance with the cavity frequency, i.e. ω = 0. The nonreciprocal conversion of the single-photon state with a finite bandwidth will be later analyzed. To realize a perfect one-way quantum transduction from the microwave mode c to the optical mode b, we must guarantee that the transmission matrix elements satisfy T 13 (0) = 0 and T 31 (0) = 1. According to equation (12), we can derive out the ratio of the transmission matrix element T 13 (0) to T 31 (0); that is Particularly, if the parameters are chosen to be |G mr | = G mb κ r /2G br and ϕ = π/2, we have T 13 (0) = 0 and T 31 (0) ̸ = 0. This is the obvious nonreciprocal conditions, under which the input quantum signal of microwave mode c can be converted to the output one of the optical mode b, but not vice versa. Physically, this nonreciprocity originates from the artificial synthetic magnetism. As seen in figures 3(a) and (b), the phase difference ϕ among couplings G mb , G mr and G br is equivalent to creating a synthetic magnetic flux threading the closed loop. When ϕ ̸ = nπ is satisfied (n is an integer), the time-reversal symmetry is broken, leading to the desired optical nonreciprocity. For the specifical magnetic flux of ϕ = π/2, the three modes m, b and r play the role of a three-port circulator; that is, it allows for the quantum signal transmission along the counterclockwise direction due to the quantum interference effect (m → b → r → m) [114,115]. As shown in figure 3(a), the input single-photon state of microwave cavity c can be transferred to the output of optical cavity b along two possible paths, where one path is along c → m → b and the other one is along c → m → r → b. Under the nonreciprocal conditions, the constructive interference occurs between the two paths, which results in the enhancement of the matrix element T 31 (0). On the contrary, as displayed in figure 3(b), the transmission from the mode b to c is completely suppressed due to the destructive interference between the two paths (b → m → c and b → r → m → c). This is the mechanism for the realization of nonreciprocal quantum transduction. Besides, it is also noteworthy here that the optical nonreciprocity can be reversed if the relative phase is switched to ϕ = −π/2. Then, we can acquire an unidirectional single-photon state conversion from the optical to microwave fields, i.e. T 13 (0) ̸ = 0 and T 31 (0) = 0. To simplify our discussion, we only concentrate on the situation of ϕ = π/2, i.e. the one-way quantum conversion from the microwave to optical photons.
As discussed above, T 13 (0) = 0 can be achieved in terms of the nonreciprocal conditions |G mr | = G mb κ r /2G br and ϕ = π/2. However, to ensure an ideal nonreciprocal optical component, the condition T 31 (0) = 1 should be met at the same time. Consequently, it also requires |T i1 (0)/T 31 (0)| ≪ 1 (i = 2, 4), which can prevent the loss of input photons to other modes. By setting G br = √ κ b κ r /2, we can deduce T 41 (0) = 0 from the equation (12), indicating that the transmission from the microwave mode to mechanical mode is totally inhibited. We now can give the full transmission matrix on resonance (i.e. ω = 0) with Γ b = 4G 2 mb /κ b and Γ c = 4G 2 mc /κ c . It is now clear that the matrix elements T 21 (0) and T 31 (0) take the form  To highly suppress the state conversion from the microwave mode c to the magnon mode m, we should control the parameters for κ m ≪ Γ b , such that the condition |T 21 (0)/T 31 (0)| ≪ 1 can be satisfied. If we further set Γ b ≈ Γ c (i.e. G mb ≈ G mc √ κ b /κ c ), the transmission amplitude T 31 (0) ≈ 1 can be achieved. Therefore, by fulfilling the nonreciprocal conditions |G mr | = G mb κ r /2G br , G br = √ κ b κ r /2 and ϕ = ±π/2, as well as κ m ≪ Γ b ≈ Γ c , we can implement a high-performance nonreciprocal quantum transduction between the optical and microwave modes. To confirm the above discussion, we plot the transmission matrix element T 31 (0) = 2 √ Γ b Γ c / (κ m + Γ b + Γ c ) as a function of the coupling constant G mb in figure 4. Under the nonreciprocal conditions, it can be observed that the optimal nonreciprocal single-photon state conversion from the microwave to optical mode (i.e. T 31 (0) = 1 and T 13 (0) = 0) can be realized when Γ b = Γ c and κ m = 0. Hence, it proves the validity of our scheme. In a real situation, the dissipation of magnon mode is inevitable to weaken the optical nonreciprocity. However, for the most relevant value κ m /2π ≈ 1 MHz in experiments, the condition κ m ≪ Γ b , Γ c is well satisfied, which can ensure a high-performance nonreciprocal single-photon state conversion.
In principle, the different frequencies of the input excitations will influence the nonreciprocal conversion. In practice, a microwave (optical) cavity has an internal (intrinsic) decay rate κ in = ω 0 /Q, where ω 0 is the eigenfrequency and Q is the quality factor. It is obvious that the cavity's intrinsic loss will reduce the conversion fidelity. For a fixed quality factor Q, the different frequencies of input states will suffer from different internal decay rates. From this perspective, the conversion fidelity is frequency-dependent. In our scheme, however, we have considered the microwave (optical) cavity with a high-quality factor, and the associated internal decay rate, which is much smaller than the external one, has been neglected. So, the nonreciprocal conversion has not been affected by the frequencies of the input excitations when the input single-photon is in resonance with the cavity frequency.
Up to now, we have only focused on the nonreciprocal quantum transduction when the input field is in resonance with the cavity frequency, i.e. ω = 0. In practice, the single-photon is typically with a finite bandwidth. In order to investigate the frequency dependence of the optical nonreciprocity, we derive the transmission matrix elements T 13 (ω) and T 31 (ω) as figure 5, we display the frequency-dependence matrix elements |T 31 (ω)| and |T 13 (ω)|. Based on the chosen parameters in the caption of figure 5, we can see that transmission amplitudes |T 31 (ω)| > 0.99 and |T 13 (ω)| < 0.1 can be achieved around the central frequency, which can enable a high-performance nonreciprocal single-photon state conversion from the microwave to optical modes. Without loss of generality, we consider an input single-photon microwave state that is described as a Gauss wave packet denoted by |ψ c,in (ω)⟩ =´dωϕ in (ω) |1(ω)⟩ c , where ϕ in (ω) = [2/(π d 2 )] 1/4 e −(ω/d) 2 is the normalized spectra amplitude with pulse width d, and |1(ω)⟩ c represents the microwave single-photon state of frequency component ω. Since the microwave and optical modes are coupled to the other modes and the environment, the input single-photon microwave state cannot be fully transmitted to the output of the optical mode. As a result, the output optical state is described by So, the conversion fidelity of the single-photon state is quantified by In our scheme, |T 31 (ω)| can be larger than 0.99 around the central frequency, such that most of the microwave single-photon component can be transmitted to the output of optical mode. Figure 6 exhibits the fidelity F 31 (d) versus the pulse width d of the single-photon state. For κ m /2π = 0.5 MHz, we can observe that the transmitted fidelity yields F 31 (d) ≈ 0.97 for an input microwave single-photon state with the bandwidth d/2π = 1 MHz. Since the nonreciprocal conversion of a single-photon wave packet with a relatively high fidelity can be achieved between the microwave and optical modes, our scheme may find important applications in long-distance quantum communication and quantum networks. Finally, it is noted here that our scheme is also applicable to an input signal of weak coherent state. For a weak coherent state |α⟩, α can be treated as a classical number, so that the output state is still a coherent state. In addition, we should emphasize that it is not feasible for an input signal of a strong coherent state. This is because the strong coherent state input can be treated as an additional strong driving for the microwave or optical modes. Then the assumptions for deriving the linearized Hamiltonian (7), as well as the non-reciprocal conditions will be modified. Therefore, our scheme is only available to a weak coherent state input, whose amplitude is much smaller than the steady-state values of microwave or optical modes.

Effect of the thermal phonon noise
To investigate the effect of thermal phonon noise to the single-photon state conversion, we introduce the output spectra S b_out (ω) of the cavity mode b, which is defined as . The correlation functions of noise operators in the frequency domain are given by where S c_in (ω) denotes the input spectrum of the microwave mode. Then, the output spectrum of the optical cavity b can be derived as with The presence of thermal phonon noise will degrade the quality of single-photon state conversion. For the experimental working temperature T = 10 mK, n th is about 1.62 for a mechanical frequency ω r /2π = 100 MHz. To quantitatively describe its effect to the quantum transduction, we exhibit the value n th |T 34 (ω)| 2 as a function of the frequency ω under the nonreciprocal conditions (see figure 7). It is observed that the added thermal excitation is smaller than 0.01 around the central frequency, indicating that the implementation of a high-quality nonreciprocal single-photon state conversion is feasible.

Experimental feasibility
Let us now discuss the experimental feasibility of our proposal. To achieve the high-performance of nonreciproccal single-photon state conversion, we have employed the coupling parameters [14,15,40,41,71,72]: G mc /2π = 8 MHz, G br /2π = 1.58 MHz, G mr /2π = 5.66 MHz, G mb /2π = 8.94 MHz, κ b /2π = 5 MHz, κ c /2π = 4 MHz, and κ r /2π = 2 MHz. In practical situation, the single excitation coupling rates are typically weak, where the magnomechanical (g mr ) [40], optomagnonic (g m ) [71] and optomechanical (g br ) [116,117] interactions are proportional to 1/D 2 , 1/ √ D 3 and 1/D, respectively, i.e. D is the diameter of a YIG sphere. For a magnetic sphere with D ∼ 30 µm, the single excitation coupling rates are calculated as g mr = 20 Hz, g m = 100 Hz and g br = 50 Hz. Therefore, to obtain the desired couplings, the strong external driving fields are needed with driving powers P a = 98.8 mW, P b = 34.2 mW, and P m = 4.6 mW. Up to now, most experiments have mainly exploited the submillimeter diameter YIG sphere. And the one with a 30 µm diameter hasn't been reported in experiment. Considering the 36 µm diameter Silica microsphere [116] having already been demonstrated in experiment, the YIG sphere about 30 µm diameter is expected to be experimentally realized with the rapid development of magnonic systems. In addition, the smaller size of YIG sphere (i.e. 1 µm diameter or even smaller) has already been studied theoretically for the generation of magnonic cat states [79,118] and frequency conversion between microwave and optical photons [90]. Therefore, we believe that our work will be instructive to the future experiments.
On the other hand, the single excitation coupling rates (g mr , g m , g br ) can be further enhanced by engineering the optomagnonic cavity structure such as microrings to increase the mode overlap [72,84,90,117,119]. Especially, single-photon coupling rate g m can be increased by purifying and doping YIG [120], and utilizing the epsilon-near-zero medium [88]. With the rapid development in cavity magnomechanics and optomagnonics, we believe our scheme is expected to be realized in the future experiments.

Summary
In summary, we have put forward an efficient scheme to realize the nonreciprocal single-photon state conversion between the microwave and optical domains based on a hybrid magnonic system, in which a YIG sphere with both the optomechanical and optomagnetic properties is coupled to a three-dimensional microwave cavity. The magnetostatic mode as a mediator is coupled to the microwave and optical modes simultaneously. By controlling the magnitude ratios and phase differences of the couplings via external driving fields, we can acquire a relatively high-fidelity nonreciprocal microwave-light single-photon state conversion via the quantum interference effect. Our work provides an appealing way for implementing unidirectional quantum transduction between the microwave and optical photons, which is vital for long-distance quantum networks and distributed quantum information processing.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors. with ∆ a = ∆ ′ a , ∆ b = ∆ ′ b + g br (r * s + r s ), ∆ c = ∆ ′ c and ∆ m = ∆ ′ m + g mr (r * s + r s ). For |α s | ≫ 1 and |r s | ≫ 1, we can discard those nonlinear terms and derive the linearized QLEs of the quantum fluctuation operators with the field-enhanced coupling strengths G am = g m b s , G ab = g m m s , G mb = g m a s , G br = g br b s and G mr = g mr m * s . Then, we can figure out the effective linearized Hamiltonian where we have redefined δα −→ α and δr −→ r. It is just the Hamiltonian in equation (6) of the main text.