Cavity optomechanics in new regimes and with new devices Andreas Nunnenkamp University of Basel 1. What? Why? How? 2. Single-photon regime 3. Dissipative coupling
Introduction
Cavity optomechanics radiation pressure force mechanical cavity mode oscillator
Cavity optomechanics: the mechanical oscillator ˆx = x ZPF (ˆb + ˆb ) Ĥ = ω M ˆb ˆb + 1 phonons 2 thermal state with n th = k BT ω M
Cavity optomechanics: the cavity mode photons L
Cavity optomechanics: the optomechanical coupling L
Standard model of optomechanics Ĥ = ω C (ˆx)â â + ω Mˆb ˆb with Standard model of optomechanics radiation-pressure force ˆx = x ZPF (ˆb + ˆb ) with N.B. We neglect the dynamical Casimir effect. Marquardt & Girvin, Physics 2, 40 (2009) Kippenberg & Vahala, Science 321, 1172 (2008)
Standard model of optomechanics!"#$%*,-.*#) '(%)+$%*,-.*#)!"#$%!"#$%&!"#!'(%)*+$%*, $+#(/*%#$!+ '(%)*+$%& with the dimensionless parameters Ω/κ /ω m drive strength & detuning ω m /κ good/bad-cavity limit ω m /γ mechanical quality factor g/κ cavity shift per phonon in units of line width g/ω m oscillator displacement per photon in units of its ZPF
Single-photon regime
The single-photon strong-coupling regime with b D. Stamper-Kurn (Berkeley) g/ω M 2 ω M /κ 0.1 Brooks et al., Nature 488, 476 (2012) O. Painter (Caltech) g/ω M 2.5 10 4 ω M /κ 8 Chan et al., Nature 478, 89 (2011) g/κ 0.007 0.1 ω M /κ 30 Chan et al., arxiv:1206.2099 2 µm J. Teufel / K. Lehnert (Boulder) g/ω M 5 10 6 ω M /κ 50 Teufel et al., Nature 475, 359 (2011) What happens when g/ω M 1 and ω M /κ 1?
Single-photon regime Completing the square photon-photon interaction new equilibrium position of mechanical oscillator polaron transformation Interaction is diagonal, but drive and damping are not!
Single-photon regime Photon number conserved â â, Ĥ =0 2ω R 4g 2 ω M At fixed photon number n, the mechanical oscillator is displaced by ω R g 2 ω M nx 0 /x ZPF = 2ng/ω M oscillator displacement per photon in units of its ZPF The energy spectrum is E nm = ω R n g 2 n 2 /ω M +ω M m n and m. The anharmonicity is d E 0 ω M x/x ZPF 2g ω M The nonlinear spectrum leads to many interesting effects! 2g ω M Mahan, Many-Particle Physics (Springer) Nunnenkamp et al., PRL 107, 063602 (2011)
Single-photon regime: the cavity response 1 0.8 g =0 ce, i.e. n â â /n0 0.6 0.4 ω R g 2 ω M 0.2 g = ω M 4 2 0 2 4 /ω M E 0 ω M x/x ZPF 2g 2g ω M ω M Franck-Condon factors In the single-photon regime multiple resonances appear. ω M /κ =2, ω M /γ = 20,n 0 =4Ω 2 /κ 2 solution of quantum Langevin equations Nunnenkamp et al., PRL 107, 063602 (2011) also: Rabl, PRL 107, 063601 (2011)
Single-photon regime: the cavity output spectrum S(ω) = S(ω)/n0 2 1.5 1 0.5 dt e iωt â (t)â(0) g = ω M n th =0 0 6 4 2 0 2 ω/ω M =0. E ω R 0 ω M x/x ZPF 2g 2g ω M ω M g 2 ω M nonlinear photon-phonon coupling In the single-photon regime multiple sidebands appear. ω M /κ =2, ω M /γ = 20,n 0 =4Ω 2 /κ 2 solution of quantum Langevin equations Nunnenkamp et al., PRL 107, 063602 (2011) also: Liao et al., arxiv:1201.1696
Dissipative coupling
Dispersive vs. dissipative optomechanics Dispersive coupling Dissipative coupling C(x) C C out L C out (x) L κ(ˆx) =κ + κ x ˆx Elste et al., PRL 102, 207209 (2009) Xuereb et al., PRL 107, 213604 (2011)
Sideband cooling: the quantum noise approach At weak coupling all you need to know is the force spectrum s arising from S FF (ω) = dt e iωt ˆF (t) ˆF (0) Ĥ int = ˆF ˆx Obtain the rates with Fermi s Golden is real-valued Rule and non-negativ = x2 ZPF 2 S FF( ω M ). enter And calculate the complete the steady-state maste equation for the density m phonon number Γ opt = x2 ZPF 2 S FF(ω M ) Γ opt n M = Γ M n th + Γ opt n O M Γ M + Γ opt erage of the thermal and t Clerk et al., RMP 82, 1155 (2010) Γ opt = Γ opt n O M = Γopt Γ opt Γ opt optical damping minimal phonon number Marquardt et al., PRL 99, 093902 (2007) Wilson-Rae et al., PRL 99, 093901 (2007)
Sideband cooling: the quantum noise approach Calculate the force (i.e. shot-noise) spectrum S FF (ω) = is real-valued ωr and 2 non-negativ κ S FF (ω) = n p L (ω + ) 2 +(κ/2) 2 minimum phonon number 100 10 1 0.1 0.01 0.001 0.0001 sidebandresolved regime 0.001 0.01 0.1 1 10 100 mechanical frequency vs. ring-down rate figure by F. Marquardt (b) Clerk et al., RMP 82, 1155 (2010) dt e iωt ˆF (t) ˆF (0) with photon number circulating inside the cav ˆF = ω R L â â figure by F. Marquardt Red-sideband cooling at ω L = ω R ω M ω out = ω in + ω M = ω M Ground-state cooling is possible in the sidebandresolved regime ω M κ Marquardt et al., PRL 99, 093902 (2007) Wilson-Rae et al., PRL 99, 093901 (2007)
Fano resonance in the force spectrum 1 0.8 =+κ/2 SFF(ω) 0.6 0.4 with Linear theory with two noise sources! ˆF 1 ˆd in + ˆd in ˆF = ˆF 1 + ˆF 2 ˆF 2 ˆd + ˆd 0.2 0 4 2 0 2 4 ω/κ Zero temperature bath! Γ opt S FF ( ω M ) They are random, but not independent! with S FF (ω) 1 κ 2 + i χ R (ω) =[κ/2 i(ω + )] 1 ˆd(ω) χr (ω) ˆd in χ R (ω) 2 Fano line shape This enables ground state cooling outside the resolved-sideband limit! Elste et al., PRL 102, 207209 (2009)
Fano resonance for dissipative coupling unstable unstable Γ opt κ nosc n M 100 10 1 0.1 0.01 Ω m opt 0.001 2 1 0 1 2 Ω m b For dissipative coupling there are two cooling regions cooling and two instability regions. Weiss, Bruder, and Nunnenkamp, arxiv:1211.7029 also: Tarabrin et al., arxiv:1212.6242
Normal-mode splitting in strong-coupling regime S cc (ω) = dt e iωt ĉ (t)ĉ(0) = ω M Fano interference Regions of instability Normal-mode splitting Weiss, Bruder, and Nunnenkamp, arxiv:1211.7029