SUPPLEMENTARY INFORMATION

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1 Realizing effective magnetic field for photons by controlling the phase of dynamic modulation: Supplementary information Kejie Fang Department of Physics, Stanford University, Stanford, California 94305, USA Zongfu Yu and Shanhui Fan Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA NATURE PHOTONICS 1

2 I. TWO OPTICAL RESONATORS DYANAMICALLY COUPLED THROUGH AN INTERMEDIATE OPTICAL RESONATOR For the structure shown in Fig. 4a, the coupled mode equations for the four states are da m dt = iω A a m + iv 1 a px, (1) da px dt = iω A a px + iv 1 a m + iv d cos(ωt + φ)a py, (2) da py = iω B a py + iv 2 a q + iv d cos(ωt + φ)a px, dt (3) da q dt = iω Ba q + iv 2 a py, (4) where a m, a px, a py, a q are the amplitudes of the monopole, dipole and quadrupole modes respectively. Using rotating wave approximation and eliminating p x and p y from the equations, the coupled mode equations reduce to d 2 a m = i V dv 1 e iφ da q dt 2 2V 2 dt V 1 2 a m, (5) d 2 a q dt = iv dv 2 e iφ da m V a q. (6) 2V 1 dt If we do a further transformation ã m = e iv 1t a m, ã q = e iv 2t a q, (7) and assuming (5)-(6) reduces to V d, 1 a m da m dt, 1 a q da q dt V 1 V 2, (8) dã m = V d dt 4 eiφ ã q, (9) dã q dt = V d 4 e iφ ã m. (10) 2 NATURE PHOTONICS

3 SUPPLEMENTARY INFORMATION We see the coupling between the monopole and quadrupole reduces to an effective dynamic coupling of the form V d /2cos(Ωt + φ), and the phase of the modulation controls the phase of effective coupling between the monopole and quadrupole-xy modes. Another situation in which the coupled mode equations can be reduced is V 1,V 2 V d. In this case, (5)-(6) can be reduced to da m = i V 1V 2 e iφ a q, dt V d (11) da q dt = iv 1V 2 e iφ a m. V d (12) In this case the strength of the coupling between the monopole and quadrupole is V = 2V 1 V 2 /V d. II. COUPLED MODE THEORY FOR MICROWAVE RESONATORS A. Coupling between a microwave resonator and a transmission line For the structure in Fig. 5a, we first consider the coupling of one of the resonators to the transmission line. Suppose the impedance of a microwave resonator is Z, and the characteristic impedance of the transmission line is Z c. The outgoing field and incoming field are related by = Z Z c Z + Z c E in. (13) Expand Eq. (13) around the complex resonant frequency ω 0 + iγ (γ is the loss) of the resonator (Z(ω 0 + iγ) = 0), we have = (ω ω 0 iγ) Z c /Z (ω 0 + iγ) (ω ω 0 iγ)+z c /Z (ω 0 + iγ) Ein. (14) NATURE PHOTONICS 3

4 Phenomenologically, we can describe the evolution of the mode amplitude a of the resonator ( a 2 is the energy of the mode), near the resonant frequency, using input-output theory, da dt =(iω 0 γ κ)a + i 2κE in, (15) = E in + i 2κa. (16) The transmission property of the coupled mode equation is Comparing Eq. (13) with Eq. (17), we obtain = (ω ω 0 iγ)+iκ (ω ω 0 iγ) iκ Ein. (17) κ = iz c /Z (ω 0 + iγ). (18) In particular, for the serial RLC circuit as shown in Fig. 5, Z = iωl +1/iωC + R, and κ = Z c /2L, (19) where we have assumed R << L/C. B. Effective coupling between microwave resonators connected by mixers We provide a detailed description of the dynamic coupling for the structure shown in Fig. 5. The mixer is described by a scattering matrix E1 in E2 in = 0 αe i(φ+φs) αe i( φ+φs) 0 1 2, (20) where the E s are the incoming and outgoing field amplitudes at the ends of the transmission line, φ is phase of local oscillator in the mixer, φ s represents the free propagation phase due to propagation through the transmission line, and α is the conversion efficiency of the mixer. 4 NATURE PHOTONICS

5 SUPPLEMENTARY INFORMATION We omit the e ±i(ω A ω B )t factors in the off-diagonal terms, by assuming that the field on the left(right) of the mixers only has frequency ω A (ω B ) component. The input-output equations for the two resonators are da 1 dt =(iω A γ 1 κ 1 )a 1 + i 2κ 1 E in 1, (21) 1 = E in 1 + i 2κ 1 a 1, (22) da 2 dt =(iω B γ 2 κ 2 )a 2 + i 2κ 2 E in 2, (23) 2 = E in 2 + i 2κ 2 a 2, (24) where a n 2 is energy in the n th resonator, γ n is the radiation loss of the resonator and κ n represents the coupling between the waveguide and resonators. Using Eq. (20), (22) and (24), we can eliminate the waveguide components in Eq. (21), (23), and obtain the coupled-mode equations for the two-resonator system, where da 1 dt =(iω A γ 1 κ 12 )a 1 iv 12 e i(ω B ω A )t a 2 (25) da 2 dt =(iω B γ 2 κ 21 )a 2 iv 21 e i(ω B ω A )t a 1, (26) is κ 12 = κ 1 + 2α2 κ 1 e 2iφs, (27) 1 α 2 2iφs e κ 21 = κ 2 + 2α2 κ 2 e 2iφs, (28) 1 α 2 2iφs e V 12 = i 2α κ 1 κ 2 e i(φ+φs), (29) 1 α 2 e 2iφs V 21 = i 2α κ 1 κ 2 e i( φ+φs) 1 α 2 e 2iφs. (30) If we have φ s =(n+ 1 )π, then the dynamic coupling coefficient between the two resonators 2 V = 4α κ 1 κ 2 1+α 2 cos(ωt + φ), (31) NATURE PHOTONICS 5

6 where Ω = ω A ω B, and the total loss γ tn of the resonators is γ t1 = γ α2 1+α 2 κ 1, (32) γ t2 = γ α2 1+α 2 κ 2. (33) From (31) we see the phase of effective coupling between two microwave resonators can be implemented by the phase of local oscillator in the mixer. Suppose κ 1 = κ 2 = κ, and γ 1 = γ 2 = γ, then strong coupling condition V >γ tn requires α 2 +4α 1 κ > γ. (34) α 2 +1 Typical mixer has a conversion efficiency of 4 7 db. As an illustration, we take α =0.63, corresponding to conversion efficiency of 4 db. Under these conditions, Eq. (34) requires 1.37Z c > R, (35) where R is the resistance in the resonator, and we have used Eq. (19). 6 NATURE PHOTONICS