Photoelectron counting in quantum optics

Photoelectron counting in quantum opticsTobias BrandesManchester7th January 2006

1 Introduction2 Photoelectric counting: classical fieldMandel formula3 Photo-count formula in quantum opticsMandel formula generalisation: discussion4 Some quantum optics techniquesMaster equations and quantum dissipationApplication: microscopic field-detector theoryQuantum optics techniques: P-representation

Overview: photons, photon-counting, fluctuationsCounting photonsCounting photons, but......‘the eternal question: what is a photon’.‘What is light ?’

Overview: photons, photon-counting, fluctuationsCounting photonsCounting photons, but......‘the eternal question: what is a photon’.‘What is light ?’Einstein 1951: ‘...these days everyfool pretends to know what a photonis. I have been thinking about thisfor the whole of my life, and Ihaven’t found the answer‘.

Overview: photons, photon-counting, fluctuationsCounting photonsCounting photons, but......‘the eternal question: what is a photon’.‘What is light ?’Einstein 1951: ‘...these days everyfool pretends to know what a photonis. I have been thinking about thisfor the whole of my life, and Ihaven’t found the answer‘....cavity mode H = ωa † a, n-photon eigenstate |n〉.

Overview: photons, photon-counting, fluctuationsCounting photonsCounting photons, but......‘the eternal question: what is a photon’.‘What is light ?’Einstein 1951: ‘...these days everyfool pretends to know what a photonis. I have been thinking about thisfor the whole of my life, and Ihaven’t found the answer‘....cavity mode H = ωa † a, n-photon eigenstate |n〉....photon as gauge-boson of QED .

Overview: photons, photon-counting, fluctuationsIrony of history: quantum mechanics‘No photons’ for thephotoelectric effect.Quantum mechanics wasdiscovered in its own classicallimit.

Overview: photons, photon-counting, fluctuationsPhoton counting: some issuesCount photo-electrons instead of photons.Counting statistics: correct theory forp n (t, t + T ) probability for n photo-electrons in [t, t + T ).Detector back-action. System-bath problem ‘with two baths’.... no entirely trivial!

Semiclassical theory for p n (t, t + T ): Mandel formulaPhotodetector model: ionize single atomClassical electromagnetic field,vector potentialA(r)e −iωt + A ∗ (r)e iωt .Probability p 1 (t, t + ∆t) of one count: Fermi’s Golden Rulep 1 (t, t + ∆t) =∫ ∞0∣dEν(E) ∣〈E| e m pA(r)|E 0〉 ∣ 2 D ∆t (E − E 0 − ω)= ηI (r)∆t, I (r) = |A(r)| 2 (intensity). (1)D ∆t (ε) ≡ ( [sin 1 2 ε∆t]/[ 1 2 ε]) 2 , ∆t → 0. Polarisation A(r) = ⃗εA(r).

Mandel formula: many countsHow to obtain probability of n transitions p n (t, t + T )Short-time probability p 1 (t, t + ∆t) = ηI (r)∆t for single electrontransition (ηI (r) transition rate).Long-time probability of n transitions p n (t, t + T ) ↔ n electrons.Individual transitions are statistically independent... Poisson distribution.Characterized by average ¯n only p n (t, t + T ) = ¯nnn! e−¯n , ¯n = ηI (r)T . (2)

Mandel formula: many countsHow to obtain probability of n transitions p n (t, t + T )Short-time probability p 1 (t, t + ∆t) = ηI (r)∆t for single electrontransition (ηI (r) transition rate).Long-time probability of n transitions p n (t, t + T ) ↔ n electrons.Markovian master equation for probabilities. p n (t) ≡ p n (0, t),p n (t + dt) = p n (t) × [1 − ηI (r)dt] + p n−1 (t) × ηI (r)dt (2)ddt p n(t) = ηI (r)[p n−1 (t) − p n (t)]. (3)Generating function G(s, t) ≡ ∑ ∞n=0 sn p n (t),∂ t G(s, t) = ηI (r)(s − 1)G(s, t).Solve with p 0 (0) = 1, p n (0) = 0, n > 0, G(s, 0) = 1.Thus G(s, t) = exp[ηI (r)t(s − 1)] = ∑ ∞ ¯nnn=0snn! e−¯n , ¯n = ηI (r)t.

Mandel formula: many countsHow to obtain probability of n transitions p n (t, t + T )Short-time probability p 1 (t, t + ∆t) = ηI (r)∆t for single electrontransition (ηI (r) transition rate).Long-time probability of n transitions p n (t, t + T ) ↔ n electrons.SUMMARY so far:Classical photo-electron counting formula (Mandel formula)Poisson process.p n (t, t + T ) = ¯nnn! e−¯n , ¯n = ηI (r)T .Generating function G(s, t) ≡ ∑ ∞n=0 sn p n (t) = exp[ηI (r)t(s − 1)].Nothing said here about PHOTONS! This is a DETECTOR theory.

‘Quantum Mandel formulas’Kelley-Kleiner, Carmichael, etc. versionp n (t, t + T ) = 〈: ˆΩ nn! e−ˆΩ :〉 with ˆΩ ≡ ξ ∫ t+Ttdt ′ Ê − (t ′ )Ê + (t ′ ).No backaction of detector on field.‘Non-absorbed photons escape, open system.’Typically many field degrees of freedom, field is a ‘BIG QUANTUMSYSTEM’.

‘Quantum Mandel formulas’Kelley-Kleiner, Carmichael, etc. versionp n (t, t + T ) = 〈: ˆΩ nn! e−ˆΩ :〉 with ˆΩ ≡ ξ ∫ t+Ttdt ′ Ê − (t ′ )Ê + (t ′ ).No backaction of detector on field.‘Non-absorbed photons escape, open system.’Typically many field degrees of freedom, field is a ‘BIG QUANTUMSYSTEM’.Mollow; Scully/Lamb; Srivinas/Davies; Ueda etc. versionBackaction of detector leads to damping (continuous measurement)of the field.‘Eventually all photons absorbed, closed system.’Typically few field degrees of freedom, field is a ‘SMALL QUANTUMSYSTEM’

Scully-Lamb photodetectorM. Scully, W. Lamb Jr., Phys. Rev. 179, 368 (1969)‘Photon statistics’ means (reduced) density operator ρ(t) of a lightfield (more generally: boson field).‘Photon statistics’ is inferred by photoelectric counting techniques.

System-bath theoryDivide ‘total universe’ into system Sand bath B,H = H S + H B + H SB≡ H 0 + V , V ≡ H SB . (2)Total density matrix χ(t) obeys the Liouville-von-Neumann equationdχ(t) = −i[H, χ(t)]. (3)dt

Master equationEffective density matrix of the system ρ(t) ≡ Tr B [χ(t)].Interaction picture with respect to H 0 ,∫dtdt ˜ρ(t) = −iTr B[Ṽ (t), χ(t = 0)] −0dt ′ Tr B [Ṽ (t), [Ṽ (t′ ), ˜χ(t ′ )]].Born approximation, ˜χ(t ′ ) ≈ R 0 ⊗ ˜ρ(t ′ ), R 0 bath density matrix.System-bath interaction as V = ∑ k S k ⊗ B k ,Bath correlation functions C kl (t, t ′ ) ≡ Tr B[˜Bk (t)˜B l (t ′ )R 0],Tr B ˜B k (t)R 0 = 0.

Master equationEffective density matrix of the system ρ(t) ≡ Tr B [χ(t)].Interaction picture with respect to H 0 ,∫dtdt ˜ρ(t) = −iTr B[Ṽ (t), χ(t = 0)] −0dt ′ Tr B [Ṽ (t), [Ṽ (t′ ), ˜χ(t ′ )]].Born approximation, ˜χ(t ′ ) ≈ R 0 ⊗ ˜ρ(t ′ ), R 0 bath density matrix.System-bath interaction as V = ∑ k S k ⊗ B k ,Bath correlation functions C kl (t, t ′ ) ≡ Tr B[˜Bk (t)˜B l (t ′ )R 0],Tr B ˜B k (t)R 0 = 0.∫dtdt ˜ρ(t) = −0dt ′ ∑ kl+ C lk (t ′ − t)[}C kl (t − t ′ ){˜S k (t)˜S l (t ′ )˜ρ(t ′ ) − ˜S l (t ′ )˜ρ(t ′ )˜S k (t){˜ρ(t ′ )˜S l (t ′ )˜S k (t) − ˜S} ]k (t)˜ρ(t ′ )˜S l (t ′ ) . (4)

Scully-Lamb PhotodetectorDetector modelSystem: single photon mode a and N detector single level ‘quantumdots’ j with one (|1〉 j ) or zero (|0〉 j ) electrons.Photon absorption empties dots into bath: leads j, c † αj |vac〉.H SB = ∑ αj(V j αc † αj |0〉 j〈1|a + ¯V j αc αj |1〉 j 〈0|a †) ≡ ∑ kS k ⊗ B k . (5)

NScully-Lamb PhotodetectorDetector modelSystem: single photon mode a and N detector single level ‘quantumdots’ j with one (|1〉 j ) or zero (|0〉 j ) electrons.Photon absorption empties dots into bath: leads j, c † αj |vac〉.H SB = ∑ αj(V j αc † αj |0〉 j〈1|a + ¯V j αc αj |1〉 j 〈0|a †) ≡ ∑ kS k ⊗ B k . (5)12....

Scully-Lamb PhotodetectorDetector modelSystem: single photon mode a and N detector single level ‘quantumdots’ j with one (|1〉 j ) or zero (|0〉 j ) electrons.Photon absorption empties dots into bath: leads j, c † αj |vac〉.H SB = ∑ αj(V j αc † αj |0〉 j〈1|a + ¯V j αc αj |1〉 j 〈0|a †) ≡ ∑ kS k ⊗ B k . (5)Master equation: trace out the leadsTerms C kl (t − t ′ )˜S k (t)˜S l (t ′ )˜ρ(t ′ ); C kl (t − t ′ ) = 〈˜B k (t)˜B l (t ′ )〉.‘Broadband detection’ at all energies, ∑ α |V j α| 2 δ(ε − ε αj ) = ν.ddt ˜ρ t = −πν ∑ j{}|1〉 j 〈1|a † a˜ρ t + ˜ρ t a † a|1〉 j 〈1| − 2|0〉 j 〈1|a˜ρ t a † |1〉 j 〈0| .

Scully-Lamb PhotodetectorState with m excitationsDetector states |m; λ〉 ≡ ˆΠ λ |0〉 1 ...|0〉 m |1〉 m+1 ...|1〉 N . Permutationsm-resolved field ‘pseudo’ density matrix ˜ρ (m)t≡ ∑ λ 〈m; λ|˜ρ t|m; λ〉.

Scully-Lamb PhotodetectorState with m excitationsDetector states |m; λ〉 ≡ ˆΠ λ |0〉 1 ...|0〉 m |1〉 m+1 ...|1〉 N . Permutationsm-resolved field ‘pseudo’ density matrix ˜ρ (m)t≡ ∑ λ 〈m; λ|˜ρ t|m; λ〉.ddt ˜ρ t = −πν ∑ {}|1〉 j 〈1|a † a˜ρ t + ˜ρ t a † a|1〉 j 〈1| − 2|0〉 j 〈1|a˜ρ t a † |1〉 j 〈0|j∑〈m; λ|˜ρ t |1〉 j 〈1|m; λ〉 = (N − m)〈m; λ|˜ρ t |m; λ〉ddt ˜ρ(m) t = −πνj∑〈m; λ|0〉 j 〈1|˜ρ t |1〉 j 〈0|m; λ〉 =j{ [(N − m) a † a˜ρ (m)t+ ˜ρ (m)t a † amterms ∑〈m − 1; λ ′ |˜ρ t |m − 1; λ ′ 〉λ ′ ]− 2(N − m + 1)a˜ρ (m−1)t a †}

Scully-Lamb PhotodetectorState with m excitationsDetector states |m; λ〉 ≡ ˆΠ λ |0〉 1 ...|0〉 m |1〉 m+1 ...|1〉 N . Permutationsm-resolved field ‘pseudo’ density matrix ˜ρ (m)tN ≫ m, γ ≡ 2πNν ddt ρ(m) t = −i[H F , ρ (m)t ] − γ (a † aρ (m)t2≡ ∑ λ 〈m; λ|˜ρ t|m; λ〉.+ ρ (m)t a † a − 2aρ (m−1)t a †) . (5)

Scully-Lamb PhotodetectorState with m excitationsDetector states |m; λ〉 ≡ ˆΠ λ |0〉 1 ...|0〉 m |1〉 m+1 ...|1〉 N . Permutationsm-resolved field ‘pseudo’ density matrix ˜ρ (m)tN ≫ m, γ ≡ 2πNν ddt ρ(m) t = −i[H F , ρ (m)t ] − γ (a † aρ (m)t2≡ ∑ λ 〈m; λ|˜ρ t|m; λ〉.+ ρ (m)t a † a − 2aρ (m−1)t a †) . (5)Now counting statistics as p m (t) ≡ Trρ (m)t !Jump super-operator J, Jρ ≡ γaρa † , time evolution generator L 0Define L 0 ρ ≡ Y ρ + ρY † with Y ≡ −iH F − γ 2 a† a.˙ρ (m)t= L 0 ρ (m)t+ Jρ (m−1)t . (6)

Summary: counting statistics in Scully-Lamb detectormodelm-resolved field density matrix˙ρ (m)t= L 0 ρ (m)t+ Jρ (m−1)t .Counting statistics as p m (t) ≡ Trρ (m)t !

Summary: counting statistics in Scully-Lamb detectormodelm-resolved field density matrix˙ρ (m)t= L 0 ρ (m)t+ Jρ (m−1)t .Counting statistics as p m (t) ≡ Trρ (m)t !Generating operator Ĝ(s, t)Define Ĝ(s, t) ≡ ∑ ∞m=0 sm ρ (m)t, s: counting variable.Usually s complex, e.g. s = e iφ with real φ.Infinite set of master equations now becomes a single equation,∂∂t Ĝ(s, t) = (L 0 + sJ)Ĝ(s, t). (7)

Solve d dt Ĝ = −i[H F, Ĝ] − γ 2(a † aĜ + Ĝa † a − 2saĜa † )

Solve d dt Ĝ = −i[H F, Ĝ] − γ 2(a † aĜ + Ĝa † a − 2saĜa † )

Solve d dt Ĝ = −i[H F, Ĝ] − γ 2(a † aĜ + Ĝa † a − 2saĜa † )

(a † aĜ + Ĝa † a − 2saĜa † )Solve d dt Ĝ = −i[H F, Ĝ] − γ 2P-representation in harmonic oscillator Hilbert spaceGlauber introduced coherent states |z〉, a|z〉 = z|z〉.Glauber-Sudarshan representation of operators such as Ĝ asĜ = ∫ d 2 zP(Ĝ; z, z ∗ )|z〉〈z|.z and z ∗ independent variables. Short form P(z) instead P(Ĝ; z, z ∗ ).Rules aĜa † ↔ zz ∗ P(z), a † aĜ ↔ (z ∗ − ∂ z )P(z),Ĝa † a ↔ (z − ∂ z ∗)P(z).

(a † aĜ + Ĝa † a − 2saĜa † )Solve d dt Ĝ = −i[H F, Ĝ] − γ 2P-representation in harmonic oscillator Hilbert spaceGlauber introduced coherent states |z〉, a|z〉 = z|z〉.Glauber-Sudarshan representation of operators such as Ĝ asĜ = ∫ d 2 zP(Ĝ; z, z ∗ )|z〉〈z|.z and z ∗ independent variables. Short form P(z) instead P(Ĝ; z, z ∗ ).Rules aĜa † ↔ zz ∗ P(z), a † aĜ ↔ (z ∗ − ∂ z )P(z),Ĝa † a ↔ (z − ∂ z ∗)P(z).PDE for P-function of generating operatorField Hamiltonian H F = Ωa † a.∂∂t P s(z, t) = [ −yz∂ z − y ∗ z ∗ ∂ z ∗ + γ(1 + |z| 2 (s − 1)) ] P s (z, t)y ≡ −iΩ − γ 2 . (8)

Solve ∂ ∂t P s = [ −yz∂ z − y ∗ z ∗ ∂ z∗ + γ(1 + |z| 2 (s − 1)) ] P sCase s = 1: simply damped harmonic oscillator1st order PDE’s are solved by method of characteristicsP 1 (z, t) = e γt P (0) ( ze i(Ω−iγ/2)t) (9)Example (G(s, t = 0) ≡ ρ (0) (t = 0) = |z 0 〉〈z 0 |)P 1 (z, t = 0) = δ (2) (z − ( z 0 ) (10)) (P 1 (z, t > 0) = e γt δ (2) ze i(Ω−iγ/2)t − z 0 = δ (2) z − z 0 e −i(Ω−iγ/2)t)(two-dimensional Delta-function!). State spirals into the origin.

Solve ∂ ∂t P s = [ −yz∂ z − y ∗ z ∗ ∂ z∗ + γ(1 + |z| 2 (s − 1)) ] P sArbitrary s:P s (z, t) = e γt P (0) ( ze i(Ω−iγ/2)t) exp{−|z| 2 (s − 1)(1 − e γt )}Now TrĜ(s, t) ≡ ∑ ∞m=0 sm Trρ (m)tt .distribution p m (t) ≡ Trρ (m)TrĜ(s, t) ==, read off photoelectron counting∫∫d 2 zP s (z, t) = d 2 zP (0) (z)e −|z|2 (s−1)(e −γt −1)∞∑∫(|z|s m d 2 zP (0) 2 ) mη t(z) e −|z|2 η t, η t ≡ 1 − e −γt .m!m=0Use normal ordering property of P-representation,p m (t) = Trρ(0) : (a† aη t) mm!e −a† aη t:, η t ≡ 1 − e −γt (11)

Single-mode counting formula: discussion ofp m (t) = Trρ(0) : ( a † aη t) mm!e −a† aη t:, η t ≡ 1 − e −γtCoherent state ρ(0) = |z 0 〉〈z 0 | p m (t) = (〈n〉η t) me −〈n〉ηt .m!◮ Poisson-distribution.◮ Average 〈n〉 ≡ 〈a † a〉 = |z 0 | 2 .◮ Coincides with semiclassical Mandel formula for γt ≪ 1.Fock-state ρ(0) = |n〉〈n| ( np m (t) =m)η m t (1 − η t ) n−m , n ≥ m.◮◮Bernoulli-distribution.m successful events (counts), n − m failures (no counts) regardless oforder.

Summary part 1Done so farPhoton counting: photo-electron counting.Semiclassical Mandel formula.Photo-detector theory: Scully/Lamb.Some techniques: quantum master equations, P−representation,counting variables and generating functions/operators.

Summary part 1Done so farPhoton counting: photo-electron counting.Semiclassical Mandel formula.Photo-detector theory: Scully/Lamb.Some techniques: quantum master equations, P−representation,counting variables and generating functions/operators.Still to doMore general situations.Sources, fields, and detectors.

Photoelectron counting in quantum opticsTobias BrandesManchester7th January 2006

5 Correlation functions6 Source-field dynamics and countingQuantum optics basicsQuantum sources of lightResonance fluorescence: driven spontaneous emission7 Master equations and quantum jumpsCounting the jumpsQuantum trajectories

Revision: towards a counting formula in quantum opticsMandel (Poissonian)p n (t, t + T ) = ¯nnn! e−¯n , ¯n = ηI (r)T .◮ Classical field with intensity I (r). Golden rule (photo-electric effect).Mollow, Scully-Lamb single mode1( ) np n (0, t) = Trρ(0) : a † aη t exp(−a † aη t ) :, η t ≡ 1 − e −γt .n!◮ Correctly describes detector backaction. ‘Closed system’. Free cavityfields only, no sources.‘Quantum Mandel’, Kelley-Kleiner◮◮p n (t, t + T ) = 〈: ˆΩ nn! e−ˆΩ :〉.∫ t+THeisenberg operators, Ω ≡ ξ dt ′ Ê − (t ′ )Ê + (t ′ ).tNot correct for long times. ‘Open system’. Various generalisations onthe market.

Coherence functionsDefinitionsNotation x = (r, t).G (1) (x, x ′ ) ≡ 〈E (−) (x)E (+) (x ′ )〉 (12)G (2) (x 1 , x 2 , x ′ 2, x ′ 1) ≡ 〈E (−) (x 1 )E (−) (x 2 )E (+) (x ′ 2)E (+) (x ′ 1)〉. (13)Based on photon absorption intensity 〈I (x)〉 = G (1) (x, x).G (1) describes first order coherence: Mach-Zehnder (Young,Michelson) interference.G (2) describes second order coherence: Hanbury Brown, Twiss.

Coherence functionsSpecial cases, normalised versions, single-mode example H = ωa † aG (1) (t, t + τ) ≡ 〈E (−) (t)E (+) (t + τ)〉 (12)G (2) (t, t + τ) ≡ 〈E (−) (t)E (−) (t + τ)E (+) (t + τ)E (+) (t)〉(13)g (2) G (2) (t, t + τ)(t, t + τ) ≡G (1) (t, t)G (1) (14)(t + τ, t + τ)number state ρ(0) = |n〉〈n| g (2) n(n − 1)(τ) =n 2 = 1 − 1 ncoherent state ρ(0) = |z〉〈z| g (2) (τ) = z∗ z ∗ zz|z ∗ = 1. (15)z| 2Definition (bunching, antibunching; sub/super-Poissonian)- Bunching: g (2) (τ) < g (2) (0), anti-bunching g (2) (τ) > g (2) (0).- Super-P. g (2) (0) > 1, sub-P. g (2) (0) < 1: relation to p n (t, t + T ).

Coherence functionsExample for bunching: cavity mode a † in thermal bath (temperature β −1 )Mode angular frequency ω, damping κ.Master equation.Use quantum regression theorem.Long-time limit, t → ∞, n B = [e βω − 1] −1limt→∞ 〈a† (t)a(t + τ)〉 = n B e −(κ+iω)τ (12)limt→∞ 〈a† (t)a † (t + τ)a(t + τ)a(t)〉 = nB2 (1 + e−2κτ ) . (13)Thus, g (2) (τ) = 1 + e −2κτ and g (2) (τ) < g (2) (0): photon bunching.(cf. Carmichael book etc.)

Now from single mode a † to many modes a † Q .

Quantization of Maxwell’s equationsVector potential in Coulomb gauge.Fourier expansion into field modes u Q (r), mode index Q.(∇ 2 + ω 2 Q )u Q(r) = 0.Quantization, annihilation operator a Q , creation operator a † Q .Electric field operatorE(r) = i ∑ Q( ωQ2ε 0) 1/2u Q (r)a Q + H.c. = E (+) (r) + E (−) (r).

The most basic case: two-level atom...

Spontaneous emission from a two-level atomTwo-level atom with states |1〉, |0〉H = ω 02 σ z + ∑ Qγ Q(σ + a Q + σ − a † Q)+ ∑ Qω Q a † Q a Q. (14)Pauli matrices, photon creation operators a † Q .

Spontaneous emission from a two-level atomTwo-level atom with states |1〉, |0〉H = ω 02 σ z + ∑ Qγ Q(σ + a Q + σ − a † Q)+ ∑ Qω Q a † Q a Q. (14)Pauli matrices, photon creation operators a † Q .Algebra of Pauli matrices( 0 1σ x ≡1 0( 0 0σ − ≡1 0) ( 0 −i, σ y ≡i 0), σ + ≡( 0 10 0) ( 1 0, σ z ≡0 −1))σ ± = 1 2 (σ x ± iσ y ), σ x = σ + + σ − , σ y = −i(σ + − σ − )[σ + , σ − ] = σ z , [σ z , σ ± ] = ±2σ ± . (15)

Spontaneous emission from a two-level atomTwo-level atom with states |1〉, |0〉H = ω 02 σ z + ∑ Qγ Q(σ + a Q + σ − a † Q)+ ∑ Qω Q a † Q a Q. (14)Pauli matrices, photon creation operators a † Q .Schrödinger equation for total wave function|Ψ(t)〉 = c(t)|1〉|vac〉 + ∑ Qb Q (t)|0〉a † Q|vac〉, c(0) = 1 (15)Can be solved (Wigner-Weisskopf) within some approximations. Inparticular, c(t) = e −Γt/2−iω 0t .No re-absorption of any emitted photon ↔ single mode model (onlyone Q, Jaynes-Cummings Hamiltonian, revivals).

Spontaneous emission from a two-level atomTwo-level atom with states |1〉, |0〉H = ω 02 σ z + ∑ Qγ Q(σ + a Q + σ − a † Q)+ ∑ Qω Q a † Q a Q. (14)Pauli matrices, photon creation operators a † Q .Electric field E (+) (r, t) = E (+)f(r, t) + E (+)s (r, t), source field in termsof source operatorsHeisenberg EOM ȧ Q (t) = −iω Q a Q (t) − iγ k σ − (t) ∫ ta Q (t) = a Q e −iωQt − iγ Q dt ′ σ − (t ′ )e −iω Q(t−t ′) . (15)0

Spontaneous emission from a two-level atomTwo-level atom with states |1〉, |0〉H = ω 02 σ z + ∑ Qγ Q(σ + a Q + σ − a † Q)+ ∑ Qω Q a † Q a Q. (14)Pauli matrices, photon creation operators a † Q .Field at the detector in terms of atom dipole operator⎡⎤∫ tE (+)s (r, t) = dt ′ ⎣ ∑ f Q (r)e −iω Q(t−t ′ )Q≈0∫ t0⎦ σ − (t ′ ) (15)dt ′ [ E(ˆr)δ(t − t ′ − r/c) ] σ − (t ′ ) = E(ˆr)σ − (t − r/c).- Note dipole form of E(ˆr).

Spontaneous emission from a two-level atomTwo-level atom with states |1〉, |0〉H = ω 02 σ z + ∑ Qγ Q(σ + a Q + σ − a † Q)+ ∑ Qω Q a † Q a Q. (14)Pauli matrices, photon creation operators a † Q .Not too much can be learned here: transient process, exponentiallydecaying probability.We want to describe stationary processes ‘driven spontaneousemission’ (resonance fluorescence).Analogy to tunneling of a single electron from a single level quantumdot.

Resonance fluorescence: analogy to single electrontunneling

Resonance fluorescence model- Spontaneous emission from TLS plus driving with classical fieldE cos(ω L t), Rabi-frequency Ω ≡ dE/, d dipole moment.H t ≡ H SE + Ω 2(e−iω L t σ + + e iωt σ −), (RWA). (15)- Time-dependent unitary trafo leaves Liouville-v.Neumann equationinvariant¯H t ≡ −iU † t∂U t∂t + U† t H t U t , ¯ρ t ≡ U † t ρ t U t . (16)- The form U t = exp(−i ˆN F ω L t)diag(e −iω Lt , 1) leads to (ω 0 = ω L )¯H t ≡ Ω 2 (σ + + σ − ) + ∑ Q γ Q()σ + a Q + σ − a † Q+ ∑ Q (ω Q − ω L )a † Q a Q (17)

Master equation for TLS-‘source’ density operator ρ t˙ρ t = i Ω 2 [σ + + σ − , ρ t ] − β (σ + σ − ρ t + ρ t σ + σ − − 2σ − ρ t σ + )Spontaneous emission rate β = π ∑ Q γ2 Q δ(ω L − ω Q ), effect of drivingin β neglected (↔ ‘intra-collisional field effect).Compare with our previous ( detector equation,)˙ρ (m)t = −i[H F , ρ (m)t ] − γ 2a † aρ (m)t + ρ (m)t a † a − 2aρ (m−1)t a † .

Master equation for TLS-‘source’ density operator ρ t˙ρ t = i Ω 2 [σ + + σ − , ρ t ] − β (σ + σ − ρ t + ρ t σ + σ − − 2σ − ρ t σ + )Spontaneous emission rate β = π ∑ Q γ2 Q δ(ω L − ω Q ), effect of drivingin β neglected (↔ ‘intra-collisional field effect).Compare with our previous ( detector equation,)˙ρ (m)t = −i[H F , ρ (m)t ] − γ 2a † aρ (m)t + ρ (m)t a † a − 2aρ (m−1)t a † .Remember spontaneous emission: field at the detector in terms ofatom dipole operator,E (+)s (r, t) ≈ E(ˆr)σ − (t − r/c).Thus a ∼ E (+)s ∼ σ − . detector photon absorption ∼ electron jumps from up to down, σ − .

Cook’s ‘counting at the source’R. J. Cook PRA 23, 1243 (1981)n-resolved master equation for resonance fluorescence of driven TLS()˙ρ (n)t = i Ω 2 [σ + + σ − , ρ (n)t ] − β σ + σ − ρ (n)t + ρ (n)t σ + σ − − 2σ − ρ (n−1)t σ +Splitting up ρ t = ∑ ∞n=0 ρ(n) t, n photon emissions.

Cook’s ‘counting at the source’R. J. Cook PRA 23, 1243 (1981)n-resolved master equation for resonance fluorescence of driven TLS()˙ρ (n)t = i Ω 2 [σ + + σ − , ρ (n)t ] − β σ + σ − ρ (n)t + ρ (n)t σ + σ − − 2σ − ρ (n−1)t σ +Splitting up ρ t = ∑ ∞n=0 ρ(n) t, n photon emissions.Cook’s original idea: momentumtransfers between atom anddriving field.Count number of discretedisplacements nk.Alternatively, count number ofspontaneous emission events.

Cook’s ‘counting at the source’R. J. Cook PRA 23, 1243 (1981)n-resolved master equation for resonance fluorescence of driven TLS()˙ρ (n)t = i Ω 2 [σ + + σ − , ρ (n)t ] − β σ + σ − ρ (n)t + ρ (n)t σ + σ − − 2σ − ρ (n−1)t σ +Splitting up ρ t = ∑ ∞n=0 ρ(n) t, n photon emissions.Jump super-operator J with Jρ = 2βσ − ρσ + = 2β|−〉〈+|ρ|+〉〈−〉.Generating operator as usual, G(s, t) ≡ ∑ ∞variable s.Counting statistics as p n (0, t) = Trρ (n)t .Photons are integrated out: just 4 by 4 equationn=0 sn ρ (n)t∂ t G = i Ω 2 [σ + + σ − , G] − β (σ + σ − G + Gσ + σ − − 2sσ − Gσ + ).; counting

Cook’s ‘counting at the source’R. J. Cook PRA 23, 1243 (1981)n-resolved master equation for resonance fluorescence of driven TLS()˙ρ (n)t = i Ω 2 [σ + + σ − , ρ (n)t ] − β σ + σ − ρ (n)t + ρ (n)t σ + σ − − 2σ − ρ (n−1)t σ +Splitting up ρ t = ∑ ∞n=0 ρ(n) t, n photon emissions.Solution G = exp{(L 0 + sJ)t}ρ(0), needs diagonalisation.In Laplace space, Ĝ(s, z) = [z − L 0 − sJ] −1 ρ(0), needs Laplaceinversion.Ĝ as vector, resolvent matrix⎛[z − L 0 − sJ] −1 =⎜⎝z + 2β 0 0 −Ω−2βs z 0 Ω0 0 z + β 0Ω2− Ω 20 z + β⎞⎟⎠ .

Cook’s ‘counting at the source’R. J. Cook PRA 23, 1243 (1981)n-resolved master equation for resonance fluorescence of driven TLS()˙ρ (n)t = i Ω 2 [σ + + σ − , ρ (n)t ] − β σ + σ − ρ (n)t + ρ (n)t σ + σ − − 2σ − ρ (n−1)t σ +Splitting up ρ t = ∑ ∞Result in Laplace spacen=0 ρ(n) t, n photon emissions.TrĜ(s, z) = (18)(z + β)(z + 2β) + Ω 2 + (s − 1)2β [ (z + β)ρ ++0+ ΩImρ +− ]0z(z + β)(z + 2β) + Ω 2 .[z + β(1 − s)]

Resonance fluorescence: sub-Poissonian counting statisticsInformation contained inTrĜ(s, z) =f (z)zf (z) + βΩ 2 (1 − s) , f (z) ≡ (z + β)(z + 2β) + Ω2 .

Resonance fluorescence: sub-Poissonian counting statisticsInformation contained inTrĜ(s, z) =f (z)zf (z) + βΩ 2 (1 − s) , f (z) ≡ (z + β)(z + 2β) + Ω2 .Need to transform back into time-domain.p n (0, t) = ∂n∂s n TrG(s, t)| s=0 . (19)〈n〉 t = ∂ ∂s TrG(s, t)| s=11st moment. (20)〈n(n − 1) t 〉 = ∂2∂s 2 TrG(s, t)| s=12nd factorial moment.(21)

Resonance fluorescence: sub-Poissonian counting statisticsInformation contained inTrĜ(s, z) =f (z)zf (z) + βΩ 2 (1 − s) , f (z) ≡ (z + β)(z + 2β) + Ω2 .Large t: pole z 0 closest to z = 0.Expand z 0 = ∑ ∞m=1 c m(s − 1) mβΩ 2 〈n〉 t→∞ =2β 2 + Ω 2 t (19)[ σt 2 ≡ 〈∆n 2 〉 t→∞ = 〈n〉 t→∞ 1 − 6β2 Ω 2 ](2β 2 + Ω 2 ) 2 .Negative Mandel Q-parameter Q ≡ F − 1, Fano factorF ≡ 〈∆n 2 〉/〈n〉 < 1.

Resonance fluorescence: sub-Poissonian counting statisticsInformation contained inTrĜ(s, z) =f (z)zf (z) + βΩ 2 (1 − s) , f (z) ≡ (z + β)(z + 2β) + Ω2 .Large t ≫ β −1 : counting statistics p n (t) becomes a Gaussian!lim p n(t) =t→∞(D. Lenstra, PRA 26, 3369 (1982)).1√2πσ2 te −(n−¯nt)2 /2σ 2 t. (19)

Resonance fluorescence: sub-Poissonian counting statisticsInformation contained inTrĜ(s, z) =f (z)zf (z) + βΩ 2 (1 − s) , f (z) ≡ (z + β)(z + 2β) + Ω2 .

Counting in quantum optics: towards a counting formula...Short revisionDirect ‘counting at the source’: the savest option...◮ n-resolved master equations with ‘jumpers’ J → sJ, generatingoperators. Cook 81 (Lesovik 89, Gurvitz 99, Bagrets/Nazarov 03 ...)Mandel (Poissonian) p n (t, t + T ) = ¯nnn! e−¯n , ¯n = ηI (r)T .◮ Classical field with intensity I (r).◮ Golden rule (photo-electric effect) plus Markov.Mollow, Scully-Lamb single modep n (0, t) = Trρ(0) :n!( 1 a † ) naη t exp(−a † aη t ) :, η t ≡ 1 − e −γt .◮ Correctly describes detector backaction. ‘Closed system’.◮ Free cavity fields only, no sources.‘Quantum Mandel’, Kelley-Kleiner p n (t, t + T ) = 〈: ˆΩ nn! e−ˆΩ :〉.◮ Heisenberg operators, Ω ≡ ξ ∫ t+Tdt ′ Ê − (t ′ )Ê + (t ′ ).t◮ Not correct for long times. ‘Open system’. Various generalisations onthe market.

Three parties (source, field, detector).

Ueda’s photodetector theoryM. Ueda PRA 41, 3875 (1990). (Relatively) consistent attempt to puteverything together ?Source-field interaction.Detector-field backaction.Three parties (source, field, receiver/detector).

Multi-mode photodetectorH = H 0 + H D + H FD , H 0 = H S + H FS + H FH FD = ∑ Qkj()V Q k c† kj |0〉 j〈1|a Q + H.c. , field-detector interaction(19)

Multi-mode photodetectorH = H 0 + H D + H FD , H 0 = H S + H FS + H FH FD = ∑ Qkj()V Q k c† kj |0〉 j〈1|a Q + H.c. , field-detector interaction(19)Neglect H FS in deriving non-unitary part of master equation for χ t(field-source density operator).ddt χ(m) t = −i[H 0 , χ (m)t ] (20)− 1 ∑ ()a †2Q a Q ′χ(m) t + χ (m)t a † Q a Q ′ − 2a Q ′χ(m−1) t a † Q.QQ ′ γ QQ ′Assumes ‘broadband detection’, γ QQ ′ = 2πN ∑ k V Q kN ≫ m detector atoms.Q′ ¯Vkδ(ε − ε kj),

Formal solutionGenerating operator G, ‘damper’ L 0 , ‘jumper’ J.Write ∂ t G = L 0 G + sJG, G(s, t) ≡ ∑ ∞t .L 0 X ≡ YX + XY † , Y ≡ −iH 0 − 1 ∑2 QQ ′ γ QQ ′a† Q a Q ′.JX ≡ ∑ QQ ′ γ QQ ′a Q ′Xa† Q .m=0 sm χ (m)

Formal solutionGenerating operator G, ‘damper’ L 0 , ‘jumper’ J.Write ∂ t G = L 0 G + sJG, G(s, t) ≡ ∑ ∞t .L 0 X ≡ YX + XY † , Y ≡ −iH 0 − 1 ∑2 QQ ′ γ QQ ′a† Q a Q ′.JX ≡ ∑ QQ ′ γ QQ ′a Q ′Xa† Q .m=0 sm χ (m)Interaction picture G(s, t) ≡ S t ˜G(s, t), S t ≡ e L0t .Here, S t X ≡ e L0t X = e Yt Xe Y †t .Counting and jumping in interaction picture,∂ t ˜G(s, t) = se −L0t Je L 0t ˜G(s, t). (21)

Formal solutionGenerating operator G, ‘damper’ L 0 , ‘jumper’ J.Write ∂ t G = L 0 G + sJG, G(s, t) ≡ ∑ ∞t .L 0 X ≡ YX + XY † , Y ≡ −iH 0 − 1 ∑2 QQ ′ γ QQ ′a† Q a Q ′.JX ≡ ∑ QQ ′ γ QQ ′a Q ′Xa† Q .m=0 sm χ (m)Solution of ∂ t ˜G(s, t) = se−L 0 t Je L 0t ˜G(s, t) as formal power series,∫ {t∫ }t ′˜G(s, t) = ˜G(s, 0) + dt ′ se −L 0t ′ Je L 0t ′ ˜G(s, 0) + dt ′′ s...00∞∑∫ t ∫ t2= s m dt m ... dt 1 S −tm JS tm−tm−1 J...JS tm χ(0)G(s, t) =m=00∞∑∫ ts mm=000∫ t2dt m ... dt 1 S t−tm JS tm−tm−1 J...JS tm χ(0). (21)0

Formal solutionGenerating operator G, ‘damper’ L 0 , ‘jumper’ J.Write ∂ t G = L 0 G + sJG, G(s, t) ≡ ∑ ∞t .L 0 X ≡ YX + XY † , Y ≡ −iH 0 − 1 ∑2 QQ ′ γ QQ ′a† Q a Q ′.JX ≡ ∑ QQ ′ γ QQ ′a Q ′Xa† Q .m=0 sm χ (m)Single-mode case first for simplicity (A(t) ≡ e −Yt ae Yt ):˜ρ (m)t = γ m ∫ t0ρ (m)t = γ m ∫ t0∫ t2dt m ...dt m ...0∫ t20dt 1 A(t m )...A(t 1 )χ(0)A † (t 1 )...A † (t m )dt 1 e Yt A(t m )...A(t 1 )χ(0)A † (t 1 )...A † (t m )e Y †t .

Formal solutionGenerating operator G, ‘damper’ L 0 , ‘jumper’ J.Write ∂ t G = L 0 G + sJG, G(s, t) ≡ ∑ ∞t .L 0 X ≡ YX + XY † , Y ≡ −iH 0 − 1 ∑2 QQ ′ γ QQ ′a† Q a Q ′.JX ≡ ∑ QQ ′ γ QQ ′a Q ′Xa† Q .Single mode case, taking traces:Tr˜ρ (m)t = γ m ∫ tTrρ (m)t = γ m ∫ t00∫ t2dt m ...dt m ...0∫ t20m=0 sm χ (m)dt 1 〈A † (t 1 )...A † (t m )A(t m )...A(t 1 )〉dt 1 〈A † (t 1 )...A † (t m )e Y †t e Yt A(t m )...A(t 1 )〉.

Relation with Kelley-Kleiner formulaUeda vs Kelley-Kleiner∫ t ∫ t2pm(t) U = γ m dt m ... dt 1 〈A † (t 1 )...A † (t m )e Y †t e Yt A(t m )...A(t 1 )〉00pm KK (t) = 〈: ˆΩ m∫ te−ˆΩ :〉, ˆΩ ≡ ξ dt ′ a † (t ′ )a(t ′ ) (21)m!0

Relation with Kelley-Kleiner formulaUeda vs Kelley-Kleiner∫ t ∫ t2pm(t) U = γ m dt m ... dt 1 〈A † (t 1 )...A † (t m )e Y †t e Yt A(t m )...A(t 1 )〉00pm KK (t) = 〈: ˆΩ m∫ te−ˆΩ :〉, ˆΩ ≡ ξ dt ′ a † (t ′ )a(t ′ ) (21)m!No detector backaction in KK.Replace damped time-evolution A(t) ≡ e −Yt ae Yt by freetime-evolution a(t) ≡ e iH 0t ae −iH 0t .Remember single mode case (Mollow, Scully-Lamb)p m (t) = Tr { ρ(0) :1m!(a † aη t) m exp(−a † aη t ) : } , η t ≡ 1 − e −γt .KK is short-time limit γt ≪ 1 η t = γt.0

Relation with Kelley-Kleiner formulaUeda vs Kelley-Kleiner∫ t ∫ t2pm(t) U = γ m dt m ... dt 1 〈A † (t 1 )...A † (t m )e Y †t e Yt A(t m )...A(t 1 )〉00pm KK (t) = 〈: ˆΩ m∫ te−ˆΩ :〉, ˆΩ ≡ ξ dt ′ a † (t ′ )a(t ′ ) (21)m!Up to first order in γ(e Y †t e Yt = 1 + γ 2(= 1 + γ(= : exp γ∫ t0∫ t0∫ t00) (dt ′ a † (t ′ )a(t ′ )... 1 + γ 2)∫ t0)dt ′ a † (t ′ )a(t ′ )...dt ′ a † (t ′ )a(t ′ )...)dt ′ a † (t ′ )a(t ′ ) : (22)

Relation with Kelley-Kleiner formulaUeda vs Kelley-Kleiner∫ t ∫ t2pm(t) U = γ m dt m ... dt 1 〈A † (t 1 )...A † (t m )e Y †t e Yt A(t m )...A(t 1 )〉00pm KK (t) = 〈: ˆΩ m∫ te−ˆΩ :〉, ˆΩ ≡ ξ dt ′ a † (t ′ )a(t ′ ) (21)m!Sum-rule ∑ ∞m=0 p m(0, t) = 0 fulfilled for0p m (0, t) ≡ Trρ (m)t = (22)∫ t ∫ t2= γ m dt m ... dt 1 〈: a † (t 1 )a(t 1 )...a † (t m )a(t m )e γ R t0 dt′ a † (t ′ )a(t ′) :〉00[ ∫1 t] m= 〈: γ dt ′ a † (t ′ )a(t ′ ) e γ R t0 dt′ a † (t ′ )a(t ′) :〉.m! 0

Multi-mode form×p m (0, t) ≡ Trρ (m)t =∫ t0∑Q 1 Q ′ 1 ...QmQ′ mγ Q1 Q ′ 1 ...γ Q mQ ′ m ×∫ t1 (dt m ... dt 1 Tr χ 0 a † Q 1(t 1 )...a † Q m(t m )e Y †t )e Yt a Q ′ m(t m )...a Q ′1(t 1 ) .0Somewhat impractical ...Counting-at-source method much simpler.Alternative: integrate out fields in ∂ t G = L 0 G + sJG (?)

Quantum trajectories: this should now be easy...

Quantum jump method, Monte-Carlo for master equationExample: spontaneous emission from TLS (rotating frame)˙ρ t = −β (σ + σ − ρ t + ρ t σ + σ − − 2σ − ρ t σ + )

Quantum jump method, Monte-Carlo for master equationExample: spontaneous emission from TLS (rotating frame)˙ρ t = −β (σ + σ − ρ t + ρ t σ + σ − − 2σ − ρ t σ + )Jump super-operator J with Jρ = 2βσ − ρσ +Solve ∂ t ρ t = (L 0 + J)ρ t .Interaction picture with respect to L 0 : ρ t ≡ S t ˜ρ t , S t ≡ e L 0t .

Quantum jump method, Monte-Carlo for master equationExample: spontaneous emission from TLS (rotating frame)˙ρ t = −β (σ + σ − ρ t + ρ t σ + σ − − 2σ − ρ t σ + )Jump super-operator J with Jρ = 2βσ − ρσ +Solve ∂ t ρ t = (L 0 + J)ρ t .Interaction picture with respect to L 0 : ρ t ≡ S t ˜ρ t , S t ≡ e L0t .Solution of ∂ t ˜ρ(t) = e −L0t Je L 0t ˜ρ(t) as formal power series,ρ(t) =∞∑m=0∫ t0∫ t2dt m ... dt 1 S t−tm JS tm−tm−1 J...JS t1 ρ(0). (23)0m quantum jumps occuring at times t 1 , ..., t m .Sum over all ‘trajectories’ with m = 0, 1, ..., ∞ jumps between ‘free’(but damped) time-evolution.

Quantum jump method, Monte-Carlo for master equationExample: spontaneous emission from TLS (rotating frame)˙ρ t = −β (σ + σ − ρ t + ρ t σ + σ − − 2σ − ρ t σ + )Monte-Carlo procedure. Fixed time step ∆t.Step 1: start with pure wave function |Ψ〉.Step 2: calculate collaps probability, P col = β∆t〈Ψ|σ + σ − |Ψ〉Step 3: compare P col with random number 0 ≤ r ≤ 1.◮ If P col > r, replace |Ψ〉 → σ − |Ψ〉/‖σ − |Ψ〉‖.◮ If Pcol ≤ r, no emission but time-evolution |Ψ〉 → (1 − i∆tH eff |Ψ〉/N ,where H eff ) = −iβσ + σ − .Go back to Step 2.Repeat procedure in order to obtain average.

Quantum jump method, Monte-Carlo for master equationExample: spontaneous emission from TLS (rotating frame)˙ρ t = −β (σ + σ − ρ t + ρ t σ + σ − − 2σ − ρ t σ + )Monte-Carlo procedure. Fixed time step ∆t.Step 1: start with pure wave function |Ψ〉.Step 2: calculate collaps probability, P col = β∆t〈Ψ|σ + σ − |Ψ〉Step 3: compare P col with random number 0 ≤ r ≤ 1.◮ If P col > r, replace |Ψ〉 → σ − |Ψ〉/‖σ − |Ψ〉‖.◮ If Pcol ≤ r, no emission but time-evolution |Ψ〉 → (1 − i∆tH eff |Ψ〉/N ,where H eff ) = −iβσ + σ − .Go back to Step 2.Repeat procedure in order to obtain average.Widely used in quantum optics community.Note: splitting L = L 0 + J is not unique.Literature: Carmichael (book); Plenio,Knight (review).

SummaryMulti-mode quantum optics: field as ‘bath’.Correlation (coherence) functions.Resonance fluorescence: ‘counting at the source’, sub-Poissonian,anti-bunched.Multi-mode photo-detector theory.Quantum trajectories.

SummaryMulti-mode quantum optics: field as ‘bath’.Correlation (coherence) functions.Resonance fluorescence: ‘counting at the source’, sub-Poissonian,anti-bunched.Multi-mode photo-detector theory.Quantum trajectories.Still to doMicroscopic models for source-field-detector.Further understanding of counting statistics p n (t).More complex quantities, e.g. time-resolved probabilitiesP n (t 1 , ..., t n ; [t, t + T ]).