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     envU ⊗ 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 †  ∑ EkE 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|

probability

† pk  trsysenv|ek〈ek|U ⊗ |e0〈e0|U  †  trsysEkEk 

Remark: if we do not read out the measurement

    p  E ∑ k k †  ∑ EkE 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   db b with b,b     −   0− oscillators 0   Hint  i  dcb  − 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 −iLt | i Hsys  eg |e〈e| − 2 e   h.c.

laser Γ photon   detector Hint  −eg  E 0  h.c. g eg | i   i  db   h.c. eg−

• ... including the recoil from spontaneous emission ion   Hint  −eg  E x  h.c. laser trap 3 ikx  ∑ d k… bke   h.c. spontaneous  emission recoil

Leiden 3 System + environment model

Hamiltonian: environment Htot  Hsys  HB  Hint system “bath”

Hsys unspecified

0   ′ ′ harmonic HB   db b with b,b     −   0− oscillators 0   Hint  i  dcb  − 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 environment

• Schrödinger equation

d | i H H H | | |vac dt t  − sys  B  int t   ⊗  initial condition • convenient to transform …

−iH t interaction picture −it te B |t bbe with respect to bath

"rotating frame“ H  H̃ c ce−i0t sys sys (transform optical  frequencies away)

0 d ̃ ̃  ı− 0t ̃ dt |t  −iHsys   db e c − h.c. |t 0−

0  → /2 bt : 1  d be−ı−0t 2 0− flat over bandwidth “noise operators” Leiden 3 system environ- • Schrödinger Equation ment

d   dt |t  −iHsys   bt c −  c bt |t

0 bt : 1  d be−ı−0t 2 0− “noise operators” White noise limit ϑ →∞ bt,b s  t − s sys  1/  opt 〈btb 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: | |ct 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   UΔtf…UΔt 1UΔt0|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 UΔt|0  1̂ − iH Δt   c btdt −  c btdt sys 0 0 … |0

| ⊗ |vac

Leiden 3 Δt 1st time step

0 time t

• First time step: first order in Δt

Δt Δt UΔt|0  1̂ − iH Δt   c btdt −  c btdt sys 0 0 Δt t −i2cc dt 2 dt′ btbt′ … |0 0 0

| ⊗ |vac

t t2 t t2    dt2  dt1 bt2 b t1|vac   dt2  dt1 bt2 ,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  ÛΔt|0 ̂   1 − iHeff Δt   cΔB0 |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 : tΔt ΔBt :  bsds t

Leiden 3 Δt 1st time step

0 time t

• First time step: to first order in Δt

|Δt  ÛΔt|0 interpretation: superposition of  1̂ − iH Δt   cΔB0 |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 : tΔt t ΔBt :  bsds Δ t

Remarks and properties: time

 commutation relations:

Δtt t ′ overlapping intervals ΔBt,ΔB† t′  0 t ≠ t ′ nonoverlapping intervals

 one-photon wave packet in time slot Δt ΔB† t |vac ≡ |1t (normalized) Δt

 number operator of photon in time slot t: ΔB†t ΔBt Nt  Δt Δt

′  Nt as set up commuting operators, Nt,Nt   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  |1t ⊗ 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 00E0  E10E 1 †   1 − iHeff Δt 0 1 − iHeff Δt  c0c 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  2c0c − c c0 − 0c c Δt sys 2  

† |     trenvU ⊗ |e0〈e 0|U  | E U † |e0  ∑k Ek Ek Leiden 3 1st time step Discussion 2:

• We read the detector:

| |ct U(Δt) |vac “click” quantum jump  Click: resulting state operator

click E1|0 ≡ | Δt  Δt c|0 (quantum jump)

with probability click 2 p  trsysE10E1   Δt ‖c0‖

Rem.: density matrix 10  E10E 1/tr…

| |k   Ek||ek/||…|| U |e0 2 “k” pk ‖Ek‖ Leiden 3 1st time step Discussion 2:

• We read the detector:

| |ct 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  trsysE00E0   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 − 1Δt |n − 1Δt stroboscopic eff integration

† ≡ 1 − iHeff Δt   c ΔB n − 1Δt 

† … 1 − iHeff Δt   c ΔB 0 |0

9 Note: remember … commute in different time slots Δtt t′ overlapping intervals ΔBt,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 −iHefft−t 1 −iHefft 1  Δt ∑|1t 1  ⊗ e ce |0 t 1 t1 …

n/2 −iHefft−t n −iHefft 1  Δt ∑ |1t 1 1t 2 …1tn  ⊗ e c…ce |0 tn…t 1 …

t1 t2 … tn t

−iHefft no click: |0 → |t  e |0 Leiden 3 • Tracing over the environment we obtain the master equation

| U(Δt) |vac

d t i H , t 1 2c t c c c t t c c dt    − sys    2    −   −   

master equation 9 Lindblad form 9 coarse grained time derivative

Leiden 3 For theorists … Ito-Quantum Stochastic Schrödinger Equation

• taking the limit … Δt  dt ΔBtdBt Ito operator noise ΔB† tdBt† increments

• Quantum Stochastic Schrödinger Equation

i † † d|t  − Hsys dt   cdB t −  c dBt |t 

• Properties of Ito increments: – point to the future: dBt|t  0

– Ito rules: dBt2  dBt2  0, dBtdBt  dt, dBtdBt  0.

Leiden 3 Examples:

• Two-level atom undergoing spontaneous emission Δ e • Driven two-level atom: Optical Bloch Equations | i

laser Γ

g | i

• laser cooling and reservoir engineering of single trapped ion – ground state cooling – squeezed state generation by reservoir engineering

Leiden 3 Example 1: two-level atom undergoing spontaneous decay

|e photodetector  Δt t1 Γ

atom time |g

initial state |c0  cg|g  ce|e

while no photon is our knowledge increases detected that the atom is not in the −iHefft/ −Γt/2 excited state | t e |c0 cg|gcee |e c   ‖…‖  ‖…‖

Γ −|̃ ct a photon is detected | t t |g c Δ  ‖…‖  

t,tΔt 2 −Γt probability that a photon is detected in (t,t+Δt] Γ|ce| e Δt P1 Leiden 3 Example 2: driven two-level atom Δ e | i + spontaneous emission laser Γ

g | i

Fig.: typical quantum trajectory: upper state population

Rabi oscillations

quantum jump: electron returns to the ground state

(prepares the system) Leiden 3 Example 3: laser cooling of a trapped ion

H P̂ 2 1 m 2X̂ 2 |e e| 1 eikX̂ h.c. sys  2m  2  − Δ 〈 − 2  − 

 Master equation (1D): 1 d 1 ikX̂ u ikX̂ u   −i Hsys,  Γ 2  du Nu e −  e − − −  − dt 2 −1

 quantum jump operator: ks recoil from spontaneous emission kL x momentum ~ksu transfer Leiden 3 • Lamb-Dicke limit: adiabatic elimination of internal dynamics

̇  A aa† − 1 a†a −  1 a† a cooling term  2 2 heating term  A a†a − 1 aa† −  1 aa† − 2 2 • processes contributing at low intensity e,n +1 e,n +1 e,n +1 e,n e,n e,n ν e,n −1 e,n −1 e,n −1

2 Γ η Γ ηΩ ηΩ η 2Γ Ω Γ g,n +1 g,n +1 g,n +1 ν g,n ν g,n g,n g,n −1 g,n −1 g,n −1

cooling 2ReS( ν) diffusion D heating 2ReS(+ν) − A =2Re[S( ν)+D] ± ∓

Leiden 3 sideband cooling

• ... as optical pumping to the ground state ...

Γ ν

g, 2 | i g,1 | i g,0 | i • master equation

A a a† 1 a† a 1 a†a A A ̇    − 2  −  2    − "dark state" of the jump operator a: • final state a|0  0 osc  |0〈0| (Γ  , sideband cooling)

Leiden 3 Example 4: reservoir engineering / trapped ion

• Consider the master equation (in interaction picture) 1 2c c  c c cc ̇  2   −  −   • stationary state = dark state of the jump operator assume 1-dim c|D  0    |D〈D| subspace

• reservoir engineering prepare a "pure state"

c|D  0

engineer prepare interesting jump operator quantum state

Leiden 3 • example: squeezed state of the harmonic oscillator

X 2 r Δx1  e i/2 −i/2  c ≡ coshre a  sinh re a −r Δx2  e |D  |r, squeezed vacuum X 1

 ̇  −ia a, X 1  coshrei/2ae −it  sinh re−i/2 ae it … † −…

jump operator Leiden 3 Example 5: State measurement & quantum jumps in 3- level systems

• three level atom e r

dipole-allowed dipole-forbidden transition transition g

• single atom photon counting

photon counting on the photon counting on strong line: strong transition single trajectory

atom

Leiden 3 photon counting on strong transition

P.Z et al. '88

e r

strong line

R. Blatt 9 atomic density matrix conditional to observing an emission window here: with a weak driving field g - r ρ (t) r r preparation in c −→ | ih | metastable state 9 state measurement with 100% efficiency α 2 ... probability NO window ψ = α g + β r | | | i | i β 2 ... probability window | | Leiden 3 4. Cascaded Quantum Systems

• formal theory • example – optical interconnects

Leiden 3 Motivation: Theory of Optical Interconnects

J.I. Cirac, P.Z. H.J. Kimble and H. Mabuchi PRL '97

• A cavity QED implementation

Optical cavities connected by a quantum channel Node A Node B

fiber Laser Laser

• memory: • databus: • memory: atoms photons atoms

• We call this protocol photonic channel

Leiden 3 Cascaded Quantum Systems

• cascaded quantum system = first quantum system drives a second quantum system: unidirectional coupling

in 1 out 1 in 2 system 2: out 2 system 1: ≡ "driven "source" unidirectional system" coupling

Leiden 3 Cascaded Quantum Systems

• example of a cascaded quantum system

counts

in 1 out 2 photon counting time out 1 in 2 ≡

unidirectional coupling

system 1: system 2: "source" "driven system"

Leiden 3 in 1 out 1 in 2 system 2: out 2 system 1: ≡ "driven "source" unidirectional system" coupling

Hamiltonian

H  Hsys1  Hsys2  HB  Hint

0  HB   d b b 0− with b the annihilation operator † ′ ′ b, b     −   position of first system interaction part position of   −i/cx1  i/cx1 Hint t  i d1b e c1 − c1be  second system

 −i/cx 2  i/cx2 i d2b e c2 − c2be x2  x1

Leiden 3 in 1 out 1 in 2 system 2: out 2 system 1: ≡ "driven "source" unidirectional system" coupling

time delay interaction picture   † − −  Hintt  ı 1 b tc1 − btc1   ı 2 b t c2 − bt c2 with t−  t −  where  → 0  1 −ı− t bt ≡ bin t :  dbe 0 2 −

Leiden 3 in 1 out 1 in 2 system 2: out 2 system 1: ≡ "driven "source" unidirectional system" coupling

Stratonovich SSE d i time delay t  − Hsys1  Hsys2  dt    † − −   1 b tc1 − btc1   2 b t c2 − bt c2  t Initial condition: |  | ⊗ |vac

Notation:

1 c1 → c1, 2 c2 → c2, 1

Leiden 3 First time step

Δt

0 time t

Δt ̂ † UΔt|0  1 − iH1  H2Δt   c2  c1   dt b t 0

Δt t2 2  −   † − −i  dt1  dt2−bt1c1 − bt1 c2b t2c1  b t2c2  … |0 0 0 destruction creation

Δt t 2 † †   dt1  dt2−t1 − t2c1c1  t1 − t2  c1c2 0 0 † † time delay! − t1 −  − t2c2c1 − t1 − t2c2c2|vac 1 † † 1 †  − c c1  0 − c c1 − c c2 |vacΔt 2 1 2 2 2 reabsorption

Leiden 3 in 1 out 2 First time step

Δt

0 time t

† UΔt|0  1̂ − iHeffΔt   c2  c1 ΔB 0 |0 one photon

time no photon

• effective Hamiltonian time 1  1   Heff  H1  H2 − i c c1 − i c c2 − ic c1 reabsorption 2 1 2 2 2 1 † † 1 †  H1  H2  i c c2 − c c1  − i c c with c  c2  c1 2 1 2 2 hermitian decay

 we identify  c2  c1  with the "jump operator"

Leiden 3 Summary of results:

• Ito-type stochastic Schrödinger equation:

d|t  |t  dt − |t †  1̂ − iHeffdt   c1  c2 dB t |0

1   1 † Heff  Hsys  i 2 c1c2 − c2c1  − i 2 c c

• master equation for source + system: Version 1:

d † 1 † † † dt   −iHeff − Heff   2 2cc − c c − c c Lindblad form

Version 2: d   −iHsys, dt 1    1     2c1c − c c1 − c c1  2c2c − c c2 − c c2 2 1 1 1 2 2 2 2   − c2,c1  c1,c2.

unidirectional coupling of source to system Leiden 3 Example: Optical Interconnects

J.I. Cirac, P.Z. H.J. Kimble and H. Mabuchi PRL '97

• A cavity QED implementation

Optical cavities connected by a quantum channel Node A Node B

fiber Laser Laser

• memory: • databus: • memory: atoms photons atoms

• We call this protocol photonic channel

Leiden 3 System

• System

Node 1 Node 2

• Hamiltonian: eliminate the excited state adiabatically

Hamiltonian H = H1 + H2

node i Hˆi = δaˆ†aˆi igi(t)[1 i 0 a h.c.](i =1, 2) − i − | i h | − Raman detuning δ = ωL ωc — − gΩi(t) Rabi frequency gi(t)= 2∆

Leiden 3 Ideal transmission

• sending the qubit in state 0

Node 1 Node 2

0 0 0 0 | i| i → | i | i • sending the qubit in state 1

Node 1 Node 2

1 0 0 1 • superpositions | i| i → | i | i

[α 0 + β 1 ] 0 0 [α 0 + β 1 ] | i | i | i → | i | i | i

Leiden 3 Physical picture as guideline for solution

• Ideal transmission = no reflection from the second cavity • Physical picture as guideline for solution: "time reversing cavity decay" – consider one cavity alone

decay

– run the movie backwards

restore

inverse laser pulse

Leiden 3 – two cavities

– design laser pulses to make the outgoing wavepacket symmetric

– we try a solution where the laser pulses are the time reverse of each other

Leiden 3 Description ... as a cascaded quantum systems

• cascaded quantum system

Node 1 in out Node 2

source driven system unidirectional coupling

• a theory of cascaded quantum systems H. Carmichael and C. Gardiner, PRL '94

Leiden 3 ... quantum trajectories

• Quantum trajectory picture: evolution conditional to detector clicks

time detector

Node 1 in out Node 2

• We want no reflection: this is equivalent to requiring that the detector never clicks (= dark state of the cascaded quantum system)

Leiden 3 detector

Node 1 in out Node 2

• system wave function Ψc(t) | i • between the quantum jumps the wave function evolves with time

Hˆeﬀ (t)=Hˆ1(t) + Hˆ2(t) iκ aˆ†aˆ1 + aˆ†aˆ2 +2aˆ† aˆ1 − 1 2 2 ³ ´ • quantum jump

ψ (t + dt) cˆ ψ (t) (with cˆ = aˆ1 + aˆ2) | c i ∝ | c i

• Wave function for quantum trajectories: ansatz

atoms cavity modes

Ψc(t) = 00 00 | i | i| i + α1(t) 10 00 + α2(t) 01 00 | i| i | i| i h +β (t) 00 10 + β (t) 00 01 . 1 | i| i 2 | i| i i ONE excitation in system

• we derive equations of motion ... and impose the dark state conditions • we find exact analytical solutions for pulse shapes leading to "no reflection" ...

Leiden 3 Results

pulse shape

ideal transmission

wave packet

Leiden 3 similar theory developed for … Homodyne Detection

• homodyne detection

current out

local oscillator β time

• conditional system wave function

1  d|Xt  −iH − 2 c cdt   cdXt |Xt

with dXt   〈xtcdt  dWt and dWt a Wiener increment

c+c+ homodyne shot noise current Leiden 3