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Quantum Optics Quantum Information Quantum Optics & Quantum Information: Decoherence, Measurement & State Preparation University of Innsbruck Peter Zoller Institute for Theoretical Physics , University of Innsbruck, and Austrian Academy Institute for Quantum Optics and Quantum Information of of Sciences the Austrian Academy of Sciences SFB Coherent Control of in collaboration with Quantum Systems C. W. Gardiner (Wellington -> Dunedin, NZ) €U TMR & IP Institute for Quantum Information Leiden 3 motivation slide in lectures 1+2 Quantum Network • Nodes: local quantum computing - store quantum information - local quantum processing channel • Channels: quantum communication - transmit quantum information node • Lecture 1: Quantum computing with trapped ions – entanglement engineering: atoms as qmemory, gates etc. • Lecture 2: Quantum communication – quantum repeater: nested purification protocol – implementation: deterministic / probabilistic; photons & atoms • Lecture 3: Quantum optical systems as open quantum systems – Measurement, state preparation & decoherence Leiden 3 Lecture 1: Quantum computing with trapped ions • trapped ions QC model: 9 qubits: longlived atomic states 9 single qubit gates: laser 9 two qubit gates: via phonon bus 9 read out: quantum jumps requirements: 9 state preparation: phonon cooling 9 [small decoherence] Leiden 3 Trapped ion: the system • system = internal + external degrees of freedom e r dipole-allowed dipole-forbidden transition transition ⊗ g internal: external: electronic levels motion • strong dissipation • small dissipation 9 laser cooling / state preparation 9 Hamiltonian: quantum state engineering 9 qubit / state measurement Leiden 3 System + Reservoir system laser = control ion: internal ion: external = electronic = motion reservoir spontaneous phonon emission heating Development of the theory: • system: Hamiltonian (control) • reservoir: master equation + continuous measurement theory Leiden 3 Lecture 2: Atom –light interfaces & transmission of qubits • deterministic transmission of qubits Node A Node B fiber Laser Laser | transmit • memory: • databus: • memory: atoms photons atoms • system: 9 single atoms 9 high-Q cavities 9 transmission line / fiber (continuum of modes) Leiden 3 Our approach ... Quantum Optics Quantum Information • Open quantum system • Quantum operations system environ- system ment U E environ- |e0 ment harmonic oscillators † ∑ Ek E 9 master equation E k k • Continuous observation counts k time out U “k” system |e0 in 9 Stochastic Schrödinger Equation “Quantum Markov processes“ Leiden 3 1. Quantum Operations Ref.: Nielsen & Chuang, Quantum Information and Quantum Computation Leiden 3 Quantum operations Evolution of a quantum system coupled to an environment: open quantum system system tr † U E envU ⊗ B U |e E environment 0 quantum operation Operator sum representation: † tr U ⊗ |e0〈e 0|U E env † ∑〈ek |U ⊗ |e0〈e 0|U |ek k † ∑ EkE k with Ek 〈ek|U|e 0 operation elements k † Properties: ∑k Ek Ek 1 Leiden 3 Quantum operations Measurement of the environment: Pk ≡ |ek〈e k| state † system k k trenv|e k〈ek |U ⊗ |e0〈e 0|U U “k” † EkE environment |e0 k † † k EkEk /trsysEkEk (normalized) probability † pk trsysenv|ek〈ek|U ⊗ |e0〈e0|U † trsysEkEk Remark: if we do not read out the measurement p E ∑ k k † ∑ EkE k k Leiden 3 2. Quantum Noise Models in Quantum Optics • system + environment model • formulation – operator / c-number stochastic Schrödinger equation – [(operator) Langevin equation] Leiden 3 System + environment model Hamiltonian: environment Htot Hsys HB Hint system “bath” Hsys unspecified 0 ′ ′ harmonic HB db b with b,b − 0− oscillators 0 Hint i dcb − c b 0− system operator Assumptions: • rotating wave approximation Leiden 3 Simplest possible ... Example: spontaneous emission • driven two-level system undergoing spontaneous emission c − |g〈e| e 1 −iLt | i Hsys eg |e〈e| − 2 e h.c. laser Γ photon detector Hint −eg E 0 h.c. g eg | i i db h.c. eg− • ... including the recoil from spontaneous emission ion Hint −eg E x h.c. laser trap 3 ikx ∑ d k… bke h.c. spontaneous emission recoil Leiden 3 System + environment model Hamiltonian: environment Htot Hsys HB Hint system “bath” Hsys unspecified 0 ′ ′ harmonic HB db b with b,b − 0− oscillators 0 Hint i dcb − c b 0− κ(ω) system operator Assumptions: • rotating wave approximation ω0 ω • flat spectrum: → /2 system frequency ω0 ϑω0 + ϑ − flat over bandwidth reservoir bandwidth B Leiden 3 Schrödinger Equation system environ- ment • Schrödinger equation d | i H H H | | |vac dt t − sys B int t ⊗ initial condition • convenient to transform … −iH t interaction picture −it te B |t bbe with respect to bath "rotating frame“ H H̃ c ce−i0t sys sys (transform optical frequencies away) 0 d ̃ ̃ ı− 0t ̃ dt |t −iHsys db e c − h.c. |t 0− 0 → /2 bt : 1 d be−ı−0t 2 0− flat over bandwidth “noise operators” Leiden 3 system environ- • Schrödinger Equation ment d dt |t −iHsys bt c − c bt |t 0 bt : 1 d be−ı−0t 2 0− “noise operators” White noise limit ϑ →∞ bt,b s t − s sys 1/ opt 〈btb s t − s white noise transformed away vacuum limit ϑ after RWA →∞ Remarks: • [We can give precise meaning as a “Quantum Stochastic Schrödinger Equation“ within a stochastic Stratonovich calculus] • We can integrate this equation exactly – counting statistics quantum – master equation operations Leiden 3 3. Integrating the “Quantum Stochastic Schrödinger Equation“ −iH t | |0|t e tot |0 U(t) |t |vac Schrödinger equation: system + environment Leiden 3 What we want to calculate ... • We do not observe the environment: reduced density operator master equation: | t trB|t〈t | U(t) 9decoherence |vac 9preparation of the system (e.g. laser cooling to ground state) • We measure the environment: continuous measurement conditional wave function: | |ct U(t) counts 9counting statistics |vac 9effect of observation on system time evolution (e.g. preparation of the (single quantum) system) Leiden 3 Remark: simple man‘s version of conversion Integration in small timesteps from Stratonovich to Ito • We integrate the Schrödinger equation in small time steps Δt 0 time t |t tf UΔtf…UΔt 1UΔt0|0 • Remark: choice of time step sys Δt ≳ 1/ opt transformed away after RWA allows us to use white noise limit ϑ perturbation theory →∞ in Δt Leiden 3 Δt 1st time step 0 time t • First time step: first order in Δt Δt Δt UΔt|0 1̂ − iH Δt c btdt − c btdt sys 0 0 … |0 | ⊗ |vac Leiden 3 Δt 1st time step 0 time t • First time step: first order in Δt Δt Δt UΔt|0 1̂ − iH Δt c btdt − c btdt sys 0 0 Δt t −i2cc dt 2 dt′ btbt′ … |0 0 0 | ⊗ |vac t t2 t t2 dt2 dt1 bt2 b t1|vac dt2 dt1 bt2 ,b t1|vac 0 0 0 0 t t2 dt2 dt1 t2 − t1|vac 0 0 1 Δt|vac first order in Δt 2 Leiden 3 Δt 1st time step 0 time t • First time step: to first order in Δt |Δt ÛΔt|0 ̂ 1 − iHeff Δt cΔB0 |0 We define: effective (non-hermitian) system Hamiltonian H : H − i c †c eff sys 2 annihilation / creation operator for a photon in the time slot Δt : tΔt ΔBt : bsds t Leiden 3 Δt 1st time step 0 time t • First time step: to first order in Δt |Δt ÛΔt|0 interpretation: superposition of 1̂ − iH Δt cΔB0 |0 eff vacuum and one-photon state one photon time no photon time Leiden 3 Discussion: 1st time step annihilation / creation operator for a photon in the time slot Δt : tΔt t ΔBt : bsds Δ t Remarks and properties: time commutation relations: Δtt t ′ overlapping intervals ΔBt,ΔB† t′ 0 t ≠ t ′ nonoverlapping intervals one-photon wave packet in time slot Δt ΔB† t |vac ≡ |1t (normalized) Δt number operator of photon in time slot t: ΔB†t ΔBt Nt Δt Δt ′ Nt as set up commuting operators, Nt,Nt 0, which can be measured "simultaneously" Leiden 3 Δt 1st time step: quantum operations 0 time t • Summary of first time step: to first order in Δt † |Δt 1 − iHeff Δt c ΔB 0 |0 |vac ⊗ 1 − iHeff Δt |0 |1 t ⊗ Δt c|0 ≡ |vac ⊗ E0|0 |1t ⊗ E 1|0 operation elements where we read off the operation elements E0 1 − iHeff Δt (no photon) time E1 Δt c (1 photon) time | ||e0| U||e 0 U | |e0 ∑|e k〈ek |U|e0| ≡ ∑|e k Ek | k Leiden 3 1st time step Discussion 1: • We do not read the detector: reduced density operator | Δt trB|Δt〈Δt| U(Δt) † † |vac E 00E0 E10E 1 † 1 − iHeff Δt 0 1 − iHeff Δt c0c Δt no photon one photon master equation: t 0 i H 0 0 H† t c 0 c t Δ − − eff − eff Δ Δ ≡−i H ,0 Δt 1 2c0c − c c0 − 0c c Δt sys 2 † | trenvU ⊗ |e0〈e 0|U | E U † |e0 ∑k Ek Ek Leiden 3 1st time step Discussion 2: • We read the detector: | |ct U(Δt) |vac “click” quantum jump Click: resulting state operator click E1|0 ≡ | Δt Δt c|0 (quantum jump) with probability click 2 p trsysE10E1 Δt ‖c0‖ Rem.: density matrix 10 E10E 1/tr… | |k Ek||ek/||…|| U |e0 2 “k” pk ‖Ek‖ Leiden 3 1st time step Discussion 2: • We read the detector: | |ct U(Δt) |vac “no click” No click: resulting state decaying norm no click −iHeffΔt E0|0 ≡ | Δt 1 − iHeff Δt |0 ≈ e |0 with probability 2 no click −iHeffΔt p trsysE00E0 e 0 | |k Ek||ek/||…|| U |e0 2 “k” pk ‖Ek‖ Leiden 3 Δt 2nd time step etc. … 0 time t • Second and more time steps: |nΔt 1 − iH Δt c ΔB†n − 1Δt |n − 1Δt stroboscopic eff integration † ≡ 1 − iHeff Δt c ΔB n − 1Δt † … 1 − iHeff Δt c ΔB 0 |0 9 Note: remember … commute in different time slots Δtt t′ overlapping intervals ΔBt,ΔB †t′ 0 t ≠ t′ nonoverlapping intervals Leiden 3 Final result for solution of SSE • Wave function of the system + environment: entangled state iH t |t |vac ⊗ e− eff |0 1/2 −iHefft−t 1 −iHefft 1 Δt ∑|1t 1 ⊗ e ce |0 t 1 t1 … n/2 −iHefft−t n −iHefft 1 Δt ∑ |1t 1 1t 2 …1tn ⊗ e c…ce |0 tn…t 1 … t1 t2 … tn t 1.
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