JQI summer school. Aug 12, 2013 Mohammad Hafezi

Transcription

1 JQI summer school Aug 12, 2013 Mohammad Hafezi Electromagnetically induced transparency (EIT) (classical and quantum picture) Optomechanics: Optomechanically induced transparency (OMIT)

2 Ask questions! Others try to answer them The speed and level will be dynamically tuned

3 Plan: 1. Introduction to EIT, a simple picture 2. Semi-classical description 3. Quantum description 4. Optomechanics (?)

4 What is EIT? medium Transparency Slow light

5 Simple picture: Dark state Light-matter interaction can be described by a Hamiltonian: H = a b + E c b b B a + E c H = B b D E a c Control field a (t) E c Probe (Quantum field)

6 Maxwell equation in the medium: E = B t B = 1 c 2 t ( ( E + 1 ) P ɛ 0 2 E 1 c 2 2 E t 2 = 1 ɛ 0 c 2 2 P t 2. Slowly varying envelope approximation: E(r, t) =E(r, t)e i kz i νt + c.c. P(r, t) =P(r, t)e i kz i νt + c.c. 1 2i k 2 E + E z + 1 c E t = i k relation to microscopic properties: 2ɛ 0 P in the rotating frame. P =(N/V )µ ρ 12 e i kz

7 Assuming the response of the system is linear and the solution is a plane wave: 1 2i k 2 E + E z + 1 c E t = i k 2ɛ 0 P E z = iδν c E + i k 2 χ(δν)e, }{{} }{{} P Solving the equation in the Fourier domain: E(δν,z)=E(δν, 0)e iz(δν/c+ kχ(δν)/2. E(t, z) = d(δν)e iδνt E(δν, 0)e iz(δν/c+ kχ(δν)/2. Group velocity: χ(δν) χ(0) + dχ δν +... dν E(t, z) = d(δν)e(δν, 0)e iδνt dχ iz(δν/c+ k(χ(0)+ e dν δν)/2 = e iz k(χ(0)/2 d(δν)e(δν, 0)e iδν(t z/v g ) = e iz k(χ(0)/2 E(t z/v g,z =0), v g = c 1+ ν 2 dχ dν Two-level atom susceptibility: χ(δν) =i N V µ 2 ( ρ 0 ɛ 11 ρ 0 ) γ 12 iδν

8 Three-level atom: 3 Δ 2 Δ 1 Ω 2 Ω 1 rotating frame: Ĥ = δ 2 2 (Ω Ω h.c.) 2 1 Using Stochastic wavefunction approach, instead of master equation ignoring jump form the excited state: ψ = c 1 (t) 1 + c 2 (t) 2 + c 3 (t) 3 atomic equations of motion ċ 1 = iω 1c 3 ċ 2 = ( γ 2 2 iδ) c 2 + iω 2c 3 }{{} Γ 12 ċ 3 = ( γ 3 2 i 1) c 3 + iω 1 c 1 + iω 2 c 2 }{{} Γ 13 a weak probe: c 1 (t) 1.

9 ρ 13 = c 1 c 3 = iω 1 Γ 12 Γ 12 Γ 13 + Ω 2 2. Limit of large detuning: caresinc γ 13 ρ 13 Ω 1 i Γ 13 i Ω 2 2 / 2 γ 12 + γ 13 ( Ω 2 2 / 2 ) }{{} γ eff i (δ Ω 2 2 / ) }{{} δ On resonance: 2 =0. Im[ ρ 13 Ω 1 ]= γ 12 γ 13 γ 12 + Ω 2 2 0asγ Transparency Probe field susceptibility as 0 Absorption ~ Im[χ] Single photon resonance ν = ω 1 13 Two-photon resonance ν = ν + ω ω large = 0 Probe frequency ν 1 Dressed state picture and splitting with (use board)

10 Slow light: Re[χ] Re[ ρ 13 Ω 1 ] δ Ω γ 12 γ 13 divergence when γ 12 0andΩ 2 0 v g c 1+ γ 13 ν3π(n/v )(λ/2π) 3 Ω 2 2 +γ 13 γ 12 the bandwidth shrinks: w Ω 2 2 γ what are the conditions under which an entire pulse can be trapped?

11 Quantum description: what happens to the excitations? can the stored pulse be retrieved? what happens to the quantum properties of light?

12 keeping both photonic and atomic operators introducing bright and dark polaritons: (explain Fock state of excitations) z,t = cos E p z,t sin ϱ 21 z,t e i kz z,t = sin E p z,t + cos ϱ 21 z,t e i kz controllable mixing angle (dynamical): nce, obeys the simple shor under EIT conditions: t + c cos 2 z z,t =0, hich describes a form-stable p

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14 Tempting: Increase the propagation (interaction) time! Can we use this to improve nonlinearity and reach few photon nonlinearity, single photon switch, quantum gates, etc.? control photons input photons output photons

15 Photon Number classical nonlinear fiber, selffocusing, solitons, parametric down conversion in crystal Quantum Many-body effects: Tonks gas, Mott insulator photon blockade, single-photon transistor, quantum phase gate Weak (classical) Strong (Quantum)

16 Few photon nonlinearity Free Space: Rb, dye molecule, QD Wrigge et al Nat. Phys.4 60 (2008) Tey et al. Nat. Phys. 4, 924 (2008) 1D waveguide: optical, plasmons Ghosh et al. PRL 94, (2005) LeKien et al. PRA 77, (2005) Akimov, Mukherjee, et al. Nat Phys (2008) Evanescent coupling Rauschenbautel, JQI λ2/α Cavity QED Kimble, Haroche, Rempe,... ƒ λ2/α Quantum Dot+ PC nanocavity Vuckovic, Waks, Englund,... ƒʹλ2/α2 Vahala, Painter

17 Estimation of nonlinear effect: photonic excitation are converted to polaronic excitations, one-to-one map interaction time will increase by the slowing factor while the effective Kerr nonlinearity decreases by the same amount Therefore, the net effect is zero.

18 EIT, slow light... is due to interference between three modes: optical field - atomic polarization - ground state excitation which can be dynamically controlled via an external field. Most of the physics can be explain by wave mechanics. 3 Ω 2 Δ 2 Δ 1 Ω Can we generalize such idea to other systems? yes, we can!

19 Optomechanics (on board)

20 (b) cal guide ex ar (z+d) ar (z) al (z+d) al (z) optical waveguide ex optical cavities a2 in ar (z+d) al (z+d) active optical cavity a in m(t) mechanical cavity mechanical cavity b m m (d) 1 (b) 0.8 Ωm = 0 Ω = κ/4 nmm, n ex -0.1 in κ bj m(t)3 ω (π c / a) a 0 j ar (z,t) nm, n 1 > > Ωm = κ/2 Ωm = κ m(t) nm + 1, n 1 > 2 4Ωm ~ κ (a) Illustration of a double0.2optical cavity system forming the unit optical 0 0.5waveguide is 1 coupled to the optomechanical array. A two-way m K (π/d) of optical cavity modes a1 and a2, whose resonance frequencies differ 0.60 frequency of the mechanical mode b. Both optical modes (d) (e) (f ) leak energy

21 References: Lukin RMP (2003) Fleischhauer et al RMP (2005) Scully quantum optics book and Lukin lecture notes Chang et al. NJP (2011) Weis et al, Science (2010), Safavi-Naeini et al, Nature (2011)