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 ! !

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. Micheli, H.P. Büchler, Driven Dissipative BEC: lattice with many sites S. Diehl

• Setup local dissipation

2 2 H = J a†a + U a† a − i j i i i ! ! • quantum jumps

pairs i, j ψ (a† + a†)(a a ) ψ c ψ ∀ " # | c# → i j i − j | # ≡ ij | c#

spatial correlation function momentum correlation function

a†a a†aq ! i j" ! q ! "

off-diagonal long range order macroscopic occupation of q=0 What happens for ﬁnite interactions U?

• master equation

d ρ = i [H, ρ]+κ 2c ρc† c† c ρ ρc† c dt − ij ij − ij ij − ij ij ij !! " " #

interaction wants to squeeze: dissipation wants to pump into ~ Bogoliubov “BEC”

mixed (“ﬁnite temperature”) squeezed state

• 3D: condensate = long range order @ ﬁnite temperature - Bogoliubov

linearized • 1D/2D: phase ﬂuctuations theory - phase / amplitude First, a simpliﬁed problem ... Two Sites: Dissipation vs. Interactions

dρ = i [H, ρ] + κ 2c ρc† c† c ρ ρc† c dt − 12 12 − 12 12 − 12 12 ! " interaction vs. dissipation

two sites Bloch sphere

1 2

Hamiltonian 2 1 2 2 H = J(a†a + h.c.) + U a† a − 1 2 2 i i i=1 quantum jump operator !

c = a† + a† a a 1 2 1 − 2 ! " ! " Two Sites: Dissipation vs. Interactions

dρ = i [H, ρ] + κ 2c ρc† c† c ρ ρc† c dt − 12 12 − 12 12 − 12 12 ! " interaction vs. dissipation

two sites Bloch sphere

1 2

Schwinger representation 1 S = a†a + a†a angular momentum x 2 1 2 1 2 1 ! " [Sx, Sy] = iSz etc. Sy = a†a2 a†a1 2 2 − 1 1 i ! " ∆Sx∆Sy Sz S = a†a a†a ≥ 2 |" #| z 2 1 2 − 2 1 ! " Two Sites: Dissipation vs. Interactions

dρ = i [H, ρ] + κ 2c ρc† c† c ρ ρc† c dt − 12 12 − 12 12 − 12 12 ! " interaction vs. dissipation

two sites Bloch sphere

1 2

Hamiltonian

H = US2 2JS z − x quantum jump operator raising operator c = [S ] S + iS + x ≡ y z • steady state: no interaction U=0 - dark state (BEC): P N Q-distribution [S+]x S, S = 0 with S = | !x 2 max angular momentum eigenstate X

S N ∆Sy = ∆Sz √N (SQL) ! x" ∼ ∼ 1 P ∆X = ∆P = √ - for states near the BEC: linearize 2

angular momentum [S , S ] = iS N/2 y z x → X

harmonic oscillator [X, P ] = i dissipation wants to pump into pure

BEC vac | ! ∼ | ! • Interaction vs. Dissipation

dρ = i [H, ρ] + κ 2c ρc† c† c ρ ρc† c dt − 12 12 − 12 12 − 12 12 ! "

interaction (Hamiltonian) dissipation P 2 H = US2 “free particle” 1 z → 2M ∆X = ∆P = P P P √2

evolution

X X X

Hamiltonian wants to squeeze dissipation wants to pump into pure ~ Bogoliubov BEC vac | ! ∼ | !

Teﬀ U/κ ρ exp ( H /k T ) ∼ ss ∼ − Bog B eﬀ • steady state: numerical results no interaction interaction: squeezing

» 4 entropy squeezing N=2 S 1 1.2 N=26 1.1 N=28 1 SQL 0.5 0.9

0.8 (J =0)

0 0.7 “effective −2 −1 0 1 2 0.010.1 110 100 10 10 10 10 10 U/κ U/· temperature” What happens for ﬁnite interactions U?

• master equation

d ρ = i [H, ρ]+κ 2c ρc† c† c ρ ρc† c dt − ij ij − ij ij − ij ij ij !! " " #

interaction wants to squeeze: dissipation wants to pump into ~ Bogoliubov “BEC”

mixed (“ﬁnite temperature”) squeezed state

• 3D: condensate = long range order @ ﬁnite temperature - Bogoliubov

linearized • 1D/2D: phase ﬂuctuations theory - phase / amplitude H.P. Büchler, S. Diehl 3D: Bogoliubov !e iθ 2 • We replace the condensate by a c-number a0 = √N0e Hamiltonian (momentum space) • !e1

1 2iθ 2iθ H = !q aq† aq + a† qa q + µ e aq† a† q + e− aqa q 2 − − − − q=0 !! " # " # 2 qeλ !q = 4J sin µ = UN/V Un] 2 ≡ !λ " # • quantum jump operators (momentum space) qe iθ iqeλ 2 λ c = √Ne− 2 1 e a q=0 is “dark” κq = κ 16n sin q,λ − q 2 λ ! " ! " # • master equation d ρ = i [H, ρ]+ κ 2a ρa† a† a ρ ρa† a dt − q q q − q q − q q q=0 !! " # • Exact timedependent solution as Gaussian Density operator ☺ H.P. Büchler, S. Diehl 1D/2D: phase ﬂuctuations

• phase amplitude modes

iφi ai = √n + Πie− with [φi, Πi] = i

Hamiltonian φq =i U/2Eq(dq d† q) • − − U ! H = (n!qφ qφq + Π qΠq) Πq = Eq/2U(dq + d† q) − 2 − − q ! ! Eq = 2Un!q • quantum jump operators (momentum space) !

iqa c =2√n(1 e− )a q − q • master equation d ρ = i E [d† d , ρ]+ κ (2(a ρa† a† a ρ ρa† a ) dt − q q q q q q − q q − q q q q ! !

Bogoliubov type operators aq = uqdq vqd† q − − • Exact timedependent solution as Gaussian Density operator ☺ Gaussian model: decoupled modes

Gaussian model for decoupled modes: dq,σ =(dq + σd q) (σ = 1) • − ± d 2 ρ = iE[d†d, ρ] dt − 2 2 +κ(2(u d†ρd + v dρd† σuv(d†ρd† + dρd)) 2 2 − u d†d + v dd† σuv(d†d† + dd), ρ ) −{ − }

ansatz: Gaussian characteristic function χ(η, η∗)=tr(ρ exp(ηd† exp η∗d)) • − moment equation •

d 2 d†d = κ d†d + v κ, dt" # − " # d 1 d†d† = (κ iE) d†d† + σuvκ , dt" # 2 − − " # d 1! " dd = (κ +iE) dd + σuvκ dt" # 2 − " # ! " !q Eq fq κq uq,vq 2 2 iθ iqa 2 1 #q+Un0 1/2 d = 3 4J sin(qeλ/2) 2Un0!q + ! 2√n0e− (1 e− ) 16κn0 sin (qeλ/2) 1 q √2 Eq λ − λ ± " ! 2 iθ iqa ! 2 1 # Un 1 $Eq d =1, 2 Jq 2Un!q 2√ne− (1 e− )4κnq − √2 Eq ± 2 Un % #" " $ Results: steady state

excitations on top of the ground state d G =0 • | # 30 mixed state • no interactions: v2 0, no excitations • → - pure BEC ﬁnite interactions • - thermal state ∼ 1 equilibrium: d† d = (exp βE 1)− T/E & q q# q − ≈ q 2 here: d† d = v Un/(2E ) & q q# q ≈ q

Teﬀ = Un/2

31 H.P. Büchler, S. Diehl Results for the steady state

• ﬁnite interactions as “ﬁnite temperature”

1 thermodynamic equilibrium: d† d = (exp βE 1)− T/E ! q q" q − ≈ q 2 here: d† d = v Un/(2E ) ! q q" q ≈ q

Teﬀ = Un/2 effective temperature • 3D: depletion of the condensate

d3q 2 n0 = n aq† aq q 0 1 (Un) − (2π)3 " # a a → ! q† q 2 2 ! " −−−→ 2 Eq + κq low energy behavior

standard condensate S. Diehl

• 1D/2D dissipation of phase coherence

i(φ φ ) (φ φ )2 /2 a†a e i− 0 = e−" i− 0 # ! i 0" ∼ ! " d 2 d q iqxi (φi φ0) = 1 e φ qφq ! − # (2π)d − ! − # ! " # 2 q 0 U n → φ qφq 2 2 ! − " −−−→ Eq + κq

T eff x i(φ φ ) e− 8Jn (d = 1) e i− 0 Teff /4TKT with ! "∼ ! (x/x0)− (d = 2) Teﬀ = Un/2

TKT = πJn Teﬀ ! 1/2 x0 =2κn(Teﬀ J)− S. Diehl Results: time dependence

• time dependent moments

κ t d† d d† d e− q ! q q"t −! q q"eq ∼ 1 (κ +iE )t d d d d e− 2 q q ! q q"t −! q q"eq ∼ • 3D depletion Un 1 n0,eq n0(t) − ∼ ! 8J 2κnt

• 1D/2D build up of phase correlations

ddq 1 d iφ(x) d φ qφq Ψ(t) d x e− t = e− (2π) " − # ∼ Ld " # R ! 1 U√κn √ 4√π J t Teﬀ = Un/2 = e− (d = 1) Teff /8TKT " (t/t0)− (d = 2) TKT = πJn Teﬀ ! compare Altman et al., Demler et al. Open Questions

• Phase transitions as a function of the “effective temperature” - non-equilibrium distribution

• imperfections - non-ideal realization of master equation / jump operators

• Which (other interesting) many body states can be engineered by (local) dissipation? - quantum info η-condensate A. Kantian “Cooling” to Excited (Metastable) States: η-Condensate

• Preparation of excited (metastable) states of Hubbard Hamiltonians • The η-state is an exact excited eigenstate of the two-species fermi Hubbard Hamiltonian in d-dimensions [Yang ʼ89, high-Tc ...]

† † † H = J cxσcx!σ + U cx cx cx cx − ↑ ↓ ↓ ↑ x,x ,σ x ! !!" !

D d=1 xd η† ( 1) cx† cx† ∼ − P ↑ ↓ !x η-particle

N N H(η†) 0 = NU(η†) 0 | ! | ! exact eigenstate: off-diagonal long range order

• The η-state is an exact and unique dark state of the jump operators

C = (η† η†)(η + η ) x,y x − y x y