Mode Invisibility and Single Photon Detection

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1 Mode Invisibility and Single Photon Detection Mode Invisibility and Single Photon Detection Marvellous Onuma-Kalu 1 Robert B. Mann 1 Eduardo Martin-Martinez Department of Physics and Astronomy University of Waterloo 2 Institute for Quantum Computing University of Waterloo June 16, 2014

2 Quantum Measurement Problem Photodetection Process Photon is destroyed upon detection

3 Quantum Measurement Problem Photodetection Process Photon is destroyed upon detection

4 Quantum Non-Demolition (QND) Measurement Measurement does not change the state of the measured system Successive measurements yield the same result

5 Quantum Non-Demolition (QND) Measurement Measurement does not change the state of the measured system Successive measurements yield the same result

6 Seeing a single photon without destroying it Haroche et al, Nature 400 (1999) How to measure single photon states in optical cavities Use highly off-resonant atomic transition Off-resonant weak coupling between atom-field mode Non-negligible global phase: Ramsey interferometery How can we improve on the measurement precision? Use a resonant atomic transition But wouldn t this make the measurement destructive again? Mode invisibility

7 Seeing a single photon without destroying it Haroche et al, Nature 400 (1999) How to measure single photon states in optical cavities Use highly off-resonant atomic transition Off-resonant weak coupling between atom-field mode Non-negligible global phase: Ramsey interferometery How can we improve on the measurement precision? Use a resonant atomic transition But wouldn t this make the measurement destructive again? Mode invisibility

8 Enhancing the optical QND scheme Onuma-Kalu et al, Phys. Rev. A 88 (2013) Aim Can the relative phase difference non-destructively provide information about the unknown state of light?

9 Weak adiabatic approximation The probability that the whole system remains in the same state is approximately unity, i.e, ψ(0) U(0, T ) ψ(0) 2 1. (1) Light state stays the same but for a global dynamical phase ψ(t ) = U(0, T ) ψ(0) e iγ ψ(0), (2) where γ is the phase factor to be determined

10 Single photon detection Method Setup a resonant interaction between single atom and target cavity modes using the Unruh DeWitt model H I = κ λ kκ L( σ + e iωt +σ e iωt)( a κe iωκt +a κ e iωκt) sin(k κ x(t)) Observation Strong atom-field interaction evident from the excitation transition probability [ P e (T )=λ 2 I,κ 2 n + I +,κ 2 (n + 1)+ ] I +,β 2 β κ I ±,β = 1 T dte i(ωβ±ω)t sin[k β x(t)] kβ L 0 violates QND technique

11 Single photon detection Method Setup a resonant interaction between single atom and target cavity modes using the Unruh DeWitt model H I = κ λ kκ L( σ + e iωt +σ e iωt)( a κe iωκt +a κ e iωκt) sin(k κ x(t)) Observation Strong atom-field interaction evident from the excitation transition probability [ P e (T )=λ 2 I,κ 2 n + I +,κ 2 (n + 1)+ ] I +,β 2 β κ I ±,β = 1 T dte i(ωβ±ω)t sin[k β x(t)] kβ L 0 violates QND technique

12 Mode invisibility technique largest contributio

13 Mode invisibility continued Excitation transition probability [ P e (T )=λ 2 I,α 2 n + I +,α 2 (n + 1)+ ] I +,β 2 β α I,β = [( 1)β 1]L (βπ) 3/2 v = 0, β = 2, 4, [ P e (T )=λ 2 I +,α 2 (n + 1)+ ] I +,β β α

14 Phase shift on atomic state Leading order correction to phase factor [ ψ (2) T = λ2 n C,α k α L + C ] +,β k β L +(n+1)c +,α ψ(0) + ψ(t ) k α L β α [ η = i ln (1 λ 2 n C,α k α L + C ]) +,β k β L + (n + 1)C +,α, k α L β α C ±,β = T t dt dt e i(ω β±ω)(t t ) sin[k β x(t)] sin[k β x(t )]. 0 0 Then phase on atomic state is γ = Re(η)

15 γ n Figure : Phase plot as a function of photon number

16 Interest is to measure difference between phases for different states containing n and n + m photons. m γ(n) = γ(m + n) γ(n) D m ghnl m=10000 m= m=100 m=10 m= n Figure : Phase resolution required to distinguish between n photons and n + m photons.

17 Interest is to measure difference between phases for different states containing n and n + m photons. m γ(n) = γ(m + n) γ(n) D m ghnl m=10000 m= m=100 m=10 m= n Figure : Phase resolution required to distinguish between n photons and n + m photons.

18 Visibility factor = exp[ Im(η) ] Visibility factor n Figure : Visibility factor as a function of photon number n

19 Current work QND measurement of general states of light ρ 0 = g g ζ, α κ ζ, α κ Figure : Measurement setup

20 Squeezed coherent state α, ζ =S(ζ)D(α) 0 ( ) D(α) = exp αâ α â, α = α e iθ ( 1 S(ζ) = exp 2 ζ ââ 1 2 ζâ â ), ζ = re iφ α and θ are the amplitude and phase of the coherent operator r and φ are the amplitude and phase of the squeezed state

21 Excitation transition probability [ P α,r e = λ 2 k κ L ( I,κ 2 + I +,κ 2 ) ( cosh 2 (r) + sinh 2 (r) ) α 2 2λ2 k κ L ( I,κ 2 + I +,κ 2 ) sinh(r) cosh(r) Re [ α 2 e i(2θ φ)] +( I,κ 2 + I +,κ 2 ) λ2 sinh 2 (r) k κ L + γ ] λ 2 I +,γ k γ L Phase shift on atomic state is [ [ γ = Re i ln 1 λ 2 ( Cκ,κ k κ L sinh2 (r) + γ c +,γ k γ L + C κ,κ k κ L (cosh2 (r) + sinh 2 (r)) α 2 2 k κ L C κ,κ sinh(r) cosh(r) Re [ α 2 e i(2θ φ)] ) ]]

22 g r HaL æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ììììììììììììììììììììì à à à à à à à à à à à à à à à à à à à à à à à à ì ì ò ì ìì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò à æ 1.0 ìà æ 0.8 ò à æ 0.6 ò òòò ì ììììì ì 0.4 ììì æ ò à ìæ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ à à à à à à à à à à à à à à à à ò òòò ììììììì òòòòò òòò à æ ì ì æà ò æà ò a g a HrL æ æ æ æ æ æ æ æ æ æ æ æ æ æ ì ì à ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò àò æ à ì ò 0.06 æ à æà æà æà æà æà æà æà æ à à à à ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò æ æ æ ì ò æ ìæ àò ìæ àò æà æ r

23 æ æ æ æ æ æ æ à à à à à à à ì ì ì ì ì ì ì ò ò ò ò ò ò ò Y ghf,ql Ψ = 2θ φ

24 Conclusion The mode invisibility technique allows for on-resonance QND measurements of single photon states. Being on resonance allows us to amplify the phase-shift. Not limited to single photon states (squeezed light, QND determination of Wigner function,...)

25 Future work Weak measurement of the Wigner function of states of light

26 Thank you

Mode Invisibility Technique A Quantum Non-Demolition Measurement for Light. Marvellous Onuma-Kalu

Mode Invisibility Technique A Quantum Non-Demolition Measurement for Light by Marvellous Onuma-Kalu A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree