Analogue of dynamic Hall effect in cavity magnon polariton system and coherently controlled logic device

Cavity magnon polaritons are mixed quasiparticles that arise from the strong coupling between cavity photons and quantized magnons. Combining high-speed photons with long-coherence-time magnons, such polaritons promise to be a potential candidate for quantum information processing. For harnessing coherent information contained in spatially distributed polariton states, it is highly desirable to manipulate cavity magnon polaritons in a two-dimensional system. Here, we demonstrate that tunable cavity magnon polariton transport can be achieved by strongly coupling magnons to microwave photons in a cross-cavity. An analog to the dynamic Hall effect has been demonstrated in a planar cavity spintronic device, where the propagation of cavity-magnon-polaritons is deflected transversally due to hybrid magnon-photon dynamics. Implementing this device as a Michelson-type interferometer using the coherent nature of the dynamic Hall and longitudinal signals, we have developed a proof-of-principle logic device to control the amplitude of cavity-magnon-polaritons by encoding the input microwave phase.


Supplementary Note 1. Theoretical description of magnon-photon coupling in x-cavity
Hamiltonian of X-CMP dynamics The whole system can be viewed as a coupled cavity magnon system in a photon bath. The Hamiltonian of the whole system has the following form: Here, H sys is the Hamiltonian of the coupled cavity magnon system isolated with surroundings, and H bath is the Hamiltonian of the photon bath. As for the H int , it represents the interaction effect between the coupled cavity magnon system and the photon bath. The detailed expressions of these three parts are shown as following: , a y (a † y ) represent the annihilation (creation) operators of X-cavity modes in x-and y-directions. m(m † ) is the annihilation (creation) operator of the magnon. p k,r (p † k,r ) is the annihilation (creation) operator of the photon with wave vector k, while the subscript index r =A,B,C,D corresponds to the port A,B,C,D respectively.

The equation motion for the extra-cavity photon
The equation of motion for the extra-cavity photon operator in Heisenberg representation reads The solution of this dynamic equation can be formally written as Analogously, we can also get the expressions of the photon operators for the other three ports: The equation motion for the intra-cavity photon Similar to Supplementary equation (5), we can write the equations of motion for the two orthogonal X-cavity modes in the Heisenberg representation.
By substituting H sys and Supplementary equation (6)-(9) into Supplementary equation (10) and (11), we obtain a new form of equations of motion for x-and y-directions, respectively.
where κ p is the coupling strength between feedlines and X-cavity, which follows the relation √ κ p = √ πλ.

Supplementary Note 2. Input-output relations
The extra-cavity asymptotic output operators at t = +∞ can be related to the input operators at t 0 = −∞ and the cavity photon ones through a linear relationship 1,2 . The standard definitions of the input and output photons are p in k,r = p k,r (t 0 )e iω k (t0) and p out k,r = p k,r (t)e iω k (t) . By substituting them into Supplementary equation (6)- (9), we can get the response formula of the photon with wave vector k at each port, shown below Here, the a(ω k ) is the Fourier Transformation of a x (t ′ ) or a y (t ′ ) depending on the port number, r.
The input and output signals are the sum of photons with different wave vectors i.e. the wave packets. Therefore, we define the input and output wave packets as Substituting these two definitions into the Supplementary equation (14) and transferring a r (ω k ) from the frequency domain back to the time domain a(t), we can obtain the input-output relation as

Supplementary Note 3. Transmission of X-CMP dynamics
Start from the dynamic equation of magnon in Heisenberg representation If we assume the solution of this equation follows the general form as m = |m|e −iωt , then Supplementary equation (19) should be: Now, we put the intrinsic damping of the magnon into this equation i.e. ω m → ω m − iαω m , then Supplementary equation (20) becomes In experiment, for general assumption, the port A and port C are the input ports, and the port B and port D are output ports. Under this condition, we can set p in C and p in D as zero and obtain the equations of motion of the coupled cavity magnon system in the cross cavity as following: Here, β in ω c is the intrinsic damping of the cavity. We can also use a lumped damping factor β to represent the loaded damping of the cavities, i.e. βω c = β in ω c + 2κ p .

Dynamics hall effect in X-CMP
Firstly, if we only input microwave power at port A. As a result, input condition can be simplified as p in A = V in x and p in B = 0, using Supplementary equation (22), we get the expressions of a x and a y in the frequency domain as: Substituting these two expressions into Supplementary equation (18), we get the output signals from the port C and port D, respectively, i.e. p out Because of the geometric symmetry of our device, it's straightforward to draw a conclusion that if signal was input from port B i.e. p in A = 0 and p in B = V in y , we can also observe the normal transmission signal and the dynamic Hall signal from the port C and port D, respectively. By means of this symmetric property, we have transformed supplementary equation (24) into a matrix equation which is shown as: To better describe the dynamic Hall effect, this matrix equation has been transformed into a much more straightforward form: Here,T is the dynamic Hall tensor in X-CMP system, which plays an important role in controlling the direction of polariton flow. The detail ofT is shown below aŝ with the denominator det(T ) as, Because of the non-zero off-diagonal terms of the transfer matrix, the Hall signal is generated in the experiment. And since these off-diagonal terms exhibit a resonance response of the external magnetic field, the dynamic Hall signal can be modulated by the external field. We have shown a detailed description of the dynamic Hall signal in the manuscript.

Two-port input experiment
The second step of our experiment is the Two-port input experiment. The input signal is V in , and we split it into two branches with same amplitude. A mechanical phase shifter was added into one branch to induce a phase difference between the two coherent signals (Φ). Then, these two signals were injected into the cross cavity from port A and port B, simultaneously. In theory, they can be described as: By substituting these two input signals into Supplementary equation (26), it's straightforward to obtain the output signal from port C: And the spectra of the output signal in frequency domain is: