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Cold Atoms As “Driven Open” UNIVERSITY of INNSBRUCK Quantum Optical Systems Quantum Noise in Correlated Systems, Weizmann Institute Jan 6-11 2008 Cold atoms as “driven open” UNIVERSITY OF INNSBRUCK quantum optical systems quantum noise quantum optics cold atoms IQOQI AUSTRIAN ACADEMY OF SCIENCES SFB Coherent Control of Quantum Peter Zoller Systems €U networks A. Micheli, S. Diehl, A. Kantian, HP. Büchler, and PZ, in preparation Quantum State Engineering (in Many Body Systems) • thermodynamic equilibrium - standard scenario of cond mat & cold atom physics T 0 ✓interesting ground states H/kB T → H Eg = Eg Eg ρ e− Eg Eg quantum phases | ! | ! ∼ −−−→ | $ % | ✓ ✓excitations Hamiltonian (many body) cooling to ground state driven / dissipative dynamical equilibrium • drive - quantum optics system bath dρ t = i [H, ρ] + ρ ρ(t) →∞ ρ mixed state dt − L −−−→ ss !? competing dynamics = D D pure state (“dark states”) | # $ | master equation steady state ✓many body pure states / driven quantum phases ✓mixed states ~ “finite temperature” Outline • Quantum noise & quantum optics introduction & review: - master equation, continuous measurement etc. “photon bath” • Dark states in quantum optics - examples: quantum state engineering / cooling • Cold atom mixtures as open dissipative quantum system “Bogoliubov bath” • Dissipatively driven BECs atoms in an optical lattice: - “BEC” 1d/2d/3d dissipative coupling to a “current” - η-condensate local dissipation A. Micheli, S. Diehl, A. Kantian, HP. Büchler, and PZ Quantum Noise & Quantum Optics • open quantum systems • dark states Quantum Optics: Open Quantum Systems • open quantum system drive system environment / bath role of the environment: • noise / dissipation (decoherence) • quantum optics … tool: state preparation - e.g. laser cooling, optical pumping bath / reservoir: harmonic oscillators • quantum optics - radiation field, [spin bath] - here: bath of Bogoliubov excitations Quantum Optics: Open Quantum Systems • open quantum system time drive out system in environment click ~ quantum jump role of the environment: • continuous observation quantum optical tools and techniques: • Quantum Markov processes • master equation • stochastic Schrödinger equation Quantum Optics: Open Quantum Systems H = Hsys + HB + Hint environment / system HB = dωωb† bω bath of oscillators bath ω ! bω,b† = δ(ω ω!) ω! − ! " Example: spontaneous emission H = i dωκ(ω) b† c b c† int ω − ω from TLS ! " # e | ! Rotating wave approximation ! ! photon Markov / white noise g | ! photodetector c = g e σ− | !" | ≡ Quantum Optics: Open Quantum Systems H = Hsys + HB + Hint environment / system HB = dωωb† bω bath of oscillators bath ω ! bω,b† = δ(ω ω!) ω! − ! " H = i dωκ(ω) b† c b c† int ω − ω ! " # Rotating wave approximation system frequency Markov / white noise reservoir bandwidth Time evolution of system + environment e | ! ! ! photon … time g | ! photodetector Time evolution of system + environment e | ! ! ! photon time g | ! photodetector system + reservoir Time evolution of system + environment e | ! ! ! photon time g | ! photodetector one photon nonhermitian system time Hamiltonian no photon time photon emission … … system wave function for count click: trajectory t , t , etc. 1 2 “quantum jump” = effect of gives photon count statistics detecting a photon on system can be simulated as a stochastic no click: c-number Schrödinger equation Time evolution of system + environment e | ! ! ! photon g | ! photodetector Fig.: typical quantum trajectory: upper state population Rabi oscillations quantum jump: electron returns to the ground state (prepares the system) Quantum Optics: Open Quantum Systems • open quantum system drive out we do not read system the measurements in master equation: quantum Markov processes dρ = i [H, ρ] + ρ quantum jump operators dt − L Nc ρ κ 2c ρc† c† c ρ ρc† c L ≡ α α α − α α − α α α=1 ! " # Lindblad Dark States as Pure Steady States • master equation dρ = i [H, ρ] + ρ quantum jump operators dt − L Nc Lindblad ρ κ 2c ρc† c† c ρ ρc† c L ≡ α α α − α α − α α α=1 ! " # • dark states as pure steady states: eigenstates of the quantum jump operators t sufficient α cα D = 0 ρ(t) →∞ ρss D D ✓ ∀ | " −−−→ ≡ | $ % | ✓conditions for uniqueness ... (H = 0, or D eigenstate of H) | ! Example 1: Dark States & Λ-Systems • 3-level systems dark state bright state internal states Aspects: D Ω2 g 1 Ω1 g+1 | ! ∼ | − ! − | ! ✓coherent: STIRAP B Ω1 g 1 + Ω2 g+1 ✓incoherent: optical pumping into dark | ! ∼ | − ! | ! state t ρ(t) →∞ D D Applications: −−−→ | # $ | ✓EIT / slow light / quantum memory... ✓sub recoil laser cooling steady state as pure “dark state” ✓... Application: sub-recoil laser cooling • Laser cooling: VSCPT (Cohen-Tannoudji et al., see also: Kasevich & Chu) dark state D, p g 1, p !k + g+1, p + !k | ! ∼ | − − ! | ! bright state B, p g 1, p !k g+1, p + !k | ! ∼ | − − ! − | ! momentum distribution t ρ(t) →∞ D, p = 0 D, p = 0 −−−→ | # $ | Levy flights single p particle Example 2: Squeezed State of Motion as a Dark State • squeezed states of the harmonic oscillator (µa + νa†) squeezed state = 0 | ! c ~ quantum jump operator ! ≡ quadrature operators • example: cooling of trapped ions / cantilevers … we "cool" to a squeezed state of motion … I. Cirac & PZ 2000 P. Rabl, A. Shnirman & PZ PRB 2006 • master equation: in rotating frame with respect to H = νa†a dρ t = κ 2cρc† c†cρ ρc†c ρ(t) →∞ squeezed squeezed dt − − −−−→ | # $ | ! " Cold Atoms & Bath of Bogoliubov Excitations • dissipative Hubbard models • … as a quantum optical system A. Griessner, A. Daley et al. PRL 2006; NJP 2007 (noninteracting atom) Driven Dissipative Hubbard Dynamics (Many Body) • BEC as a “phonon reservoir” - quantum reservoir engineering BEC • master equation - reduced system dynamics - Quantum Markov process ‣ validity (as in quantum optics) dρ ✓ interband transitions = i [Hsys, ρ] + ρ ✓ RWA + Born + Markov dt − L ✓ when is a reservoir a reservoir? 2 band Hubbard model BEC phonons “think quantum optics” • driven two-level atom + spontaneous • trapped atom in a BEC reservoir emission e energy scale! | ! optical 1 ! ! photon | ! 0 g BEC | ! | ! “phonon” (cycling transition) laser atom photon laser assisted atom + BEC collision • reservoir: vacuum modes of the • reservoir: Bogoliubov excitations of the radiation field (T=0) BEC (@ temperature T) ? cold atoms / many body / Quantum optics ideas / techniques non-equilibrium situation i ρ˙ = [H,ρ] + L ρ −h¯ † † † i L ρ !γα 2cα ρcα cα cα ρ ρcα cα ρ˙ = [H,ρ] + L ρ ≡ α − − −h¯ ! " 2 2 † h¯ " † † ˆ 3 3† † † HBEC = ψˆ b(r) ψˆ b(r)d r + gbb ψˆ b(r)Lψˆ bρ(r)ψˆ b(γr)ψˆ 2b(cr)ρdcr c c ρ ρc c − 2mb ! α α α α α α α # $ % # ≡ α − − † ! " E0 + ε(k)bˆ bˆk 2 2 ! k † h¯ " 3 † † 3 → k=Hˆ0BEC = ψˆ (r) ψˆ (r)d r + gbb ψˆ (r)ψˆ (r)ψˆ (r)ψˆ (r)d r $ b − 2m b b b b b # $ b % # Hˆ = E + (k)bˆ†bˆ (1) BEC 0 ! ε k k † k=0 E0 + ε(k)bˆ bˆk $ ! k → k=0 where $ ˆ† ˆ ˆ† ˆ bk HBEC = E0 + ! ε(k)bkbk (1) k=0 and $ wherebˆ k ˆ† bk h¯k and ε(k) bˆk ˆ † † 3 hk Hint = gab ψˆ a(r)ψˆ a(r)ψˆ b(r)ψˆ b(r)d r ¯ (2) # 2 ε(k) gab = 4πh¯ aab/(2mr) Hˆ = g ˆ †(r) ˆ (r) ˆ †(r) ˆ (r)d3r (2) mr = (mamb)/(ma + mb) int ab ψa ψa ψb ψb # ˆ ˆ 2 ψb = √ρ0 + δψb gab = 4πh¯ aab/(2mr) 1 ˆ ik.r ˆ† ik.r mr = (mamb)/(ma + mb) δψˆ b = ! ukbke + vkbke− , (3) √V k & ' ψˆ b = √ρ0 + δψˆ b L 1 u = k , v = 1 (4) k k ˆ ˆ ik.r ˆ† ik.r 1 L2 1 L2 δψb = ! ukbke + vkbke− , (3) k k √V k − − & ' ( 2 ( 2 εk (hk¯ ) /(2m) mc Lk 1 Lk = − − . uk = , vk =(5) (4) mc2 2 2 1 Lk 1 Lk 2 2 4 2 1/2 − − εk = [c (hk¯ ) + (hk¯ ) /(2mb) ] ( (6)(spectrum of ε (hk¯ )2/(2m) mc2 k Bogoliubov excitations “Spontaneous Emission” Lk = − 2 − . (5) c = gbbρ0/mb mc ( ε = [c2(hk¯ )2 + (hk¯ )4/(2m )2]1/2 (6) gbb = 4πha¯ bb/mb 1 k b | ! ρ0 ˆ ik.r ˆ† ik.r c = gbbρ0/mb δρˆ = ! (uk + vk)bke + (uk + vk)bke− . V k ( ) & BEC 0 ' gbb = 4πha¯ bb/mb | !2 S(k,ω) = (uk + vk) ρ0 ik.r † ik.r δρˆ = (u + v )bˆ eE =+ (u c+2kv2 +)bˆke4−/(2m .)2 1/2 V ! k k k k k k k b S(k,ω) ) k & ! ' ˆ 1/2 ik.r ˆ 2 Hint gab BEC!S( kreservoir,ω) w1 e w0 bk 1 0 + h.c. Sinteraction:(k,ω) = (uk inter+ vkband) 0 - 1 ∼ • k ' | | ( | (' | • 1/2 ˆ ˆ ˆ S(k,ω) 1/2 Hb = Ekbk† bk Hˆ g S(k, ω) 1 0 ˆb + h.c. spontaneous emission rate collision ∼ ab | " # | k k Hˆ g S(k, )1/2 w eik.r wk bˆ 1 0 + h.c. a!s = 100a0 int ab ! ω 1 !0 k Bogoliubov ∼ k '“spontaneous| | ( emission”| (' | 14 3 n = 5 10 cm− × “spontaneousspontaneous emissionemission rate” rate • ω = 2π 100 KHz × as = 100a0 scattering length 14 3 # = 2π 1.1 KHz ρ = 5 10 cm− density × 0 × weak coupling2 tunable ! # ρ0as √ω ω = 2π 100 KHz trap frequency ∼ × # = 2π 1.1 KHz [tayloring the reservoir, e.g. phononic band gap]× • # ρ a2√ω ∼ 0 s A. Griessner, A. Daley et al. PRL 2006; NJP 2007 Application: Raman Cooling within a Blochband step 2: (dissipative) step 1: (coherent) • • decay to ground band quasimomentum selective excitation ~ subrecoil Raman cooling by Kasevich and Chu (also VSCPT: Cohen et al.) A. Griessner, A. Daley et al. PRL 2006; NJP 2007 Raman Cooling within a Blochband Temperature: k T=2J0(Δ q)2 ✓ no heating / cooling due to B intraband transitions Levy flights ✓ We can cool to temperatures lower than the BEC “think quantum optics” • Λ-system three electronic levels dark state bright state • N atom on M sites t ρ(t) →∞ BEC BEC −−−→ | # $ | driven dissipative BEC cond mat: ~ spatial array of dissipative Josephson junctions quantum optics: spatial chain of Λ-systems “think quantum optics” • Λ-system dark state bright state • 1 atom on 2 sites 1 2 (a† + a†) vac (a† a†) vac 1 2 | ! 1 − 2 | " symmetric anti-symmetric J “in-phase” “out-of-phase” ~ dissipative Josephson junction pumping into symmetric state “phase locking” 2 2 N (non-) interacting atoms on 2 sites H = J a†a + U a† a • − i j i i <i,j> i ! ! U 1 2 N BEC (a† + a†) vac | ! ∼ 1 2 | ! J - every “spontaneous emission event” associated with quantum jump ψ (a† + a†)(a a ) ψ c ψ quantum jump operator | c! → 1 2 1 − 2 | ! ≡ 12 | c! - master equation dρ = i [H, ρ] + κ 2c ρc† c† c ρ ρc† c Lindblad form dt − 12 12 − 12 12 − 12 12 ! " - steady state c12 BEC = 0 | ! ✓BEC as a “dark state” t ρ(t) →∞ BEC BEC ✓compatible with H for U=0 −−−→ | # $ | ✓unique A.
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