Optomechanically induced transparency of x-rays via optical control: Supplementary Information Wen-Te Liao 1, and Adriana Pálffy 1 1 Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Department of Physics, National Central University, 3001 Taoyuan City, Taiwan In the following we present in detail the analytic derivations. TABLE S1: Symbol table. Symbol Explanation â â b b x x ω c ω n Ω k x ω x ω l c P h m L δl cavity optical photon annihilation operator cavity optical photon creation operator phonon annihilation operator for the movable microlever phonon creation operator for the movable microlever x-ray photon annihilation operator x-ray photon creation operator resonant angular frequency of the optical cavity nuclear transition angular frequency Rabi frequency coupling between the nuclear transition currents and the x-ray x-ray wave vector x-ray photon angular frequency optical laser angular frequency = ω l ω c optical laser detuning to the cavity angular frequency = ω x ω n the x-ray detuning to the nuclear transition angular frequency optical laser power reduced Planck constant mass of the movable microlever cavity length = hω c n cav /(Lmω0 ) averaged cavity length shift 1
ω 0 inherent oscillation (phonon) angular frequency of the movable microlever ( ) δω 0 = 4G ω0 microlever s optomechanically induced oscillation angular frequency shift κ +16ω0 ω m = ω 0 + δω 0 optomechanically modified oscillation angular frequency of the microlever κ spontaneous decay rate of nuclear excited state optical cavity photon decay rate γ 0 inherent mechanical damping rate of the microlever ( ) δγ 0 = 4G 1 κ κ optomechanically induced mechanical damping rate shift κ +16ω0 γ m s = γ 0 + δγ 0 optomechanically modified mechanical damping rate of the microlever = + κ + γ m total decoherence rate of the system κp n cav = averaged cavity optical photon number hω l[(ω l ω c ) +(κ/) ] Y ZPF = h/(mω m ) zero-point fluctuation of the movable microlever G 0 G η F m n = ω c Y ZPF /L optomechanical coupling constant = G 0 ncav optomechanical coupling constant in the perturbation region = k x Y ZPF Lamb-Dicke parameter = m e iη( b + b) n Franck-Condon coefficient HAMILTONIAN The full Hamiltonian of the system sketched in Fig. 1a is a combination of the optomechanical Hamiltonian Ĥ opto and nuclear interaction with x-ray photons Ĥ nx Ĥ = Ĥ opto + Ĥ nx. (S1) Here Ĥ opto = hω 0 b b + hωc â â hg 0 â â( b + b), (S) Ĥ nx = hω n e e + hω ( ) e iω xt+ik x Y ZPF( b + b) x e g + e iω x t ik x Y ZPF( b + b) x g e, (S3) which are standard expressions and can be found in Refs. [1] and [33,36], respectively. Derivation of Equation (3) In order to derive the expression of the Hamiltonian in Eq. (S1) in the interaction picture, an unitary transformation to the rotating frame [1] can be done by using Ĥ in = ÛĤ old Û i hû Û t and
( Û = exp iω l b bt + iωx x xt ). Moreover, as the quantum state of x-rays used in nuclear scattering experiments can be approximated by a coherent state, we invoke the semiclassical treatment for coupling between the x-ray and nuclei, namely, Ω x Ω. Both derivations give Ĥ = hω 0 b b h c â â hg 0 â â( b + b) + h e e hω ] [ e g e ik xy ZPF( b + b) + H.c..(S4) Derivation of Equation (1) Interesting physics happens at the fluctuation about the averaged cavity photon number n cav [1]. In this region, one can apply the replacement â n cav + â to Eq. (S4). The operator â on the right hand side now denotes the new annihilation operator for photon number fluctuations around the average value n cav, namely, in the perturbation region. Also, in this region, the microlever s oscillation frequency, the mechanical damping rate and the optomechanical coupling constant are optomechanically modified as ω m, γ 0 + δγ 0 and G, respectively [1], ( Ĥ = hω m b b h c â â hg â b ) + â b + h e e hω [ e g e ik xy ZPF( b + b) + H.c. ].(S5) MASTER EQUATION To calculate the Hamiltonian matrix elements, for η n < 1, we can approximate e g e ik xy ZPF( b + b) e g [1 ( b )] + ik x Y ZPF + b, [ ( b )] = e g 1 + iη + b. (S6) Equation (S6) shows there are three paths for the system to reach the nuclear excited state e via absorbing an x-ray photon. The x-ray absorption can be accompanied by either deexciting a phonon, a zero phonon line or by exciting a phonon, as illustrated in Fig. S1. The corresponding Franck-Condon coefficients F m n = m e iη( b + b) n are F m n n F m<n n = (iη) m n m! m n! = (iη) m n m n! n!, (S7) n! m!. (S8) 3
FIG. S1: Level scheme of the effective nuclear harmonic oscillator, adapted from Fig. 1b in the manuscript. Lower (upper) three states correspond to the ground (excited) state g (e) while v (n) denotes the number of cavity photons (number of phonons). Vertical green arrows depict the x-ray absorption by nuclei (with x-ray detuning ), and red diagonal arrows illustrate the beam splitter interaction between cavity photons and the microlever s mechanical motion. The full yellow ellipse indicates the initial state of the system. In derivations, for simplicity, we use state vectors labeled with numbers, i.e., 1,, 3, 4, 5, and 6. Using the state notation introduced in Fig. S1, the explicit forms of each matrix in the master ] equation t ρ = i h [Ĥ, 1 ρ + ρ dec are given by Ĥ = (1 v) c + (1 + n)ω m G (1 + n)v 0 1 ΩFm+1 n+1 0 0 G (1 + n)v v c + nω m G n(1 + v) 0 1 ΩFm n 0 0 G n(1 + v) (1 + v) c + (n 1)ω m 0 0 1 ΩFm 1 n 1 1 ΩFm+1 n+1 0 0 + (1 v) c + (1 + m)ω m G (1 + m)v 0 0 1 ΩFm n 0 G (1 + m)v v c + mω m G m(1 + v) 0 0 1 ΩFm 1 n 1 0 G m(1 + v) (1 + v) c + (m 1)ω m, (S9) ρ dec = (s )ρ 11 + ρ 44 + κρ (s )ρ 1 (s )ρ 13 sρ 14 sρ 15 sρ 16 (s )ρ 1 (s )ρ + ρ 55 + κρ 33 (s )ρ 3 sρ 4 sρ 5 sρ 6 (s )ρ 31 (s )ρ 3 (s )ρ 33 + ρ 66 sρ 34 sρ 35 sρ 36 sρ 41 sρ 4 sρ 43 (s + )ρ 44 + κρ 55 (s + )ρ 45 (s + )ρ 46 sρ 51 sρ 5 sρ 53 (s + )ρ 54 (s + )ρ 55 + κρ 66 (s + )ρ 56 sρ 61 sρ 6 sρ 63 (s + )ρ 64 (s + )ρ 65 (s + )ρ 66, (S10) 4
ρ 11 ρ 1 ρ 13 ρ 14 ρ 15 ρ 16 ρ 1 ρ ρ 3 ρ 4 ρ 5 ρ 6 ρ 31 ρ 3 ρ 33 ρ 34 ρ 35 ρ 36 ρ =. (S11) ρ 41 ρ 4 ρ 43 ρ 44 ρ 45 ρ 46 ρ 51 ρ 5 ρ 53 ρ 54 ρ 55 ρ 56 ρ 61 ρ 6 ρ 63 ρ 64 ρ 65 ρ 66 Typically, only low nuclear excitation is achieved in nuclear scattering with x-rays, such that the master equation in the perturbation region / + κ + γ m > G Ω can be used. We assume all population ρ ii (t) = 0 but ρ (t) = 1 due to the prepared initial condition. Also, we focus on the red-detuned regime, namely, cavity detuning c = ω m. In the perturbation regime, we neglect all the dynamics of populations and consider only coherences related to state : t ρ 1 = (γ m + κ)ρ 1 i [ G ] (1 + n)v, (S1) t ρ 3 = (γ m + κ)ρ 3 + i [ G ] (1 + v)n, (S13) ( ) t ρ 4 = + γ m + κ ρ 4 Fn+1 m+1 Ωρ 1 + G (1 + m)vρ 5 + [ + (n m)ω m ]ρ 4 }, (S14) ( ) t ρ 5 = + γ m + κ ρ 5 [ Fn m Ω + [ + (n m)ω m ]ρ 5 + G (1 + m)vρ4 + ]} (1 + v)mρ 6,(S15) ( ) t ρ 6 = + γ m + κ ρ 6 F m 1 n 1 Ωρ 3 + G (1 + v)mρ 5 + [ + (n m)ω m ]ρ 6 }. (S16) 5
Derivation of Equation (6) The standard method to calculate the dispersion relation for x-rays is solving the steady state solution of the master equation, i.e., using t ρ i j = 0 to get 0 = (γ m + κ)ρ 1 i [ G ] (1 + n)v, (S17) 0 = (γ m + κ)ρ 3 + i [ G ] (1 + v)n, (S18) ( ) 0 = + γ m + κ ρ 4 Fn+1 m+1 Ωρ 1 + G (1 + m)vρ 5 + [ + (n m)ω m ]ρ 4 }, (S19) ( ) 0 = + γ m + κ ρ 5 [ Fn m Ω + [ + (n m)ω m ]ρ 5 + G (1 + m)vρ4 + ]} (1 + v)mρ 6, (S0) ( ) 0 = + γ m + κ ρ 6 Fn 1 m 1 Ωρ 3 + G (1 + v)mρ 5 + [ + (n m)ω m ]ρ 6 }. (S1) We thus obtain the most relevant coherence term ρ 5, since most of the population remains in state during the period of interest. The real and imaginary parts of ρ 5 are associated with dispersion and absorption, respectively [36], { [ ]} Ω Fn m (s )[is + (m n)ω m ] ig Fn+1 m+1 v (1 + m)(1 + n) + Fn 1 m 1 (1 + v) mn ρ 5 ( ) = (s ) {G (m + v + mv) + [s i( (m n)ω m )] }. (S) When steady state arrives, v = 0 should be used and the coherence reads ρ 5 ( ) = Ω{ Fn m (s )[is + (m n)ω m ] ig Fn 1 m 1 } mn. (S3) (s ) {G m + [s i( (m n)ω m )] } Here the asymmetric missing of F m+1 n+1 term is the consequence of the relation â 0 = 0. Equation (S3) demonstrates that the x-ray spectra can be controlled by changing the optomechanical coupling constant via altering the averaged cavity photon number. Eigensystem With s Ω, the eigensystem for the setup is governed by the following simplified Hamiltonian 6
Ĥ = (n + v)ω m G (1 + n)v 0 0 0 0 G (1 + n)v (n + v)ω m G n(1 + v) 0 0 0 0 G n(1 + v) (n + v)ω m 0 0 0 0 0 0 + (m + v)ω m G (1 + m)v 0 0 0 0 G (1 + m)v + (m + v)ω m G m(1 + v). (S4) 0 0 0 0 G m(1 + v) + (m + v)ω m The corresponding eigenenergies and eigenstate vectors are presented in Table S. TABLE S: Eigensystem of Hamiltonian Eq. (S11). Eigenenergy Eigenstate vector ( ) (n + v)ω (1+v)n m (1+n)v,0,1,0,0,0 (1+v)m ) + (m + v)ω m (0,0,0, (1+m)v,0,1 G ) m + v + mv + (m + v)ω m (0,0,0, (1+m)v (1+v)m, m+v+mv (1+v)m,1 G ) m + v + mv + (m + v)ω m (0,0,0, (1+m)v (1+v)m, m+v+mv (1+v)m,1 G ( ) n + v + nv + (n + v)ω (1+n)v m (1+v)n, n+v+nv (1+v)n,1,0,0,0 G ( ) n + v + nv + (n + v)ω (1+n)v m (1+v)n, n+v+nv (1+v)n,1,0,0,0 OPTICAL THICKNESS ESTIMATE The realistic microlever parameters considered in the manuscript were adopted from Ref. [38]. In particular, a microlever mass of M = 0.14 µg was considered in the numerical calculations. Here we determine the corresponding optical thickness values for the x-ray absorption. We consider the cases of ZnO, Ge and 10 18 cm 3 Th nuclei doped in CaF VUV-transparent crystals. The densities of these materials are 5.61 g/cm 3, 5.3 g/cm 3 and 3.18 g/cm 3, respectively. Depending on the exact sample geometry, the mass value M = 0.14 µg could correspond for example to a sample size of approx. 30x30x30 µm 3. The optical thickness is given by ξ = nσl, where n is the concentration, σ the resonant nuclear absorption cross section and l the thickness of a sample. The resonant nuclear cross section values are σ ZnO = 5 10 0 cm [39], σ Ge = 10 0 cm [39] and σ Th = 10 10 cm [40]. For ZnO and Ge we consider a number density of nuclei of 10 cm 3. The corresponding optical thickness ξ 7
is of order unity for ZnO and Ge and approx. 10 5 for 10 18 cm 3 Th nuclei doped in CaF crystals samples. By varying the ratio between sample concentration and sample thickness, larger optical thickness parameters can be achieved. 8