PART 2 : BALANCED HOMODYNE DETECTION Michael G. Raymer Oregon Center for Optics, University of Oregon raymer@uoregon.edu 1 of 31
OUTLINE PART 1 1. Noise Properties of Photodetectors 2. Quantization of Light 3. Direct Photodetection and Photon Counting PART 2 4. Balanced Homodyne Detection 5. Ultrafast Photon Number Sampling PART 3 6. Quantum State Tomography 2 of 31
DC-BALANCED HOMODYNE DETECTION I Goal -- measure quadrature amplitudes with high Q.E. and temporal-mode selectivity E S = signal field (ω O ), 1-1 photons E L = laser reference field (local oscillator) (ω O ), 1 6 photons E S (t) E 1 = E S + E L PD BS PD dt dt n 1 n 2 N D E L (t) τ d delay θ E 2 = E S - E L N D E ( ) 1 (t τ d ) E ( ) 2 (t τ d ) E 1 ( +) (t) dt E 2 ( +) (t) dt 3 of 31
DC-BALANCED HOMODYNE DETECTION II integrator circuit PD dt n 1 dt n 2 N D PD θ 4 of 31
DC-BALANCED HOMODYNE DETECTION III Φ S = signal amplitude; Φ L = laser reference amplitude EΦ S (t) S EΦ L (t) L τ d delay θ BS dt dt n 1 n 2 N D ˆ N D = T dt Det d 2 x ( ) Φ ˆ L (x,,t τ d ) Φ ˆ ( +) S (x,,t) + h.c. overlap integral (+) Φ ˆ S (r,t) = i c a ˆ k k v k (r,t) v k (r,t) = C k j u j (r) exp( iω j t) c T dt j wave-packet d 2 x v * k (x,,t) v m (x,,t) = δ Det k m modes 5 of 31
DC-BALANCED HOMODYNE DETECTION IV ˆ N D T dt Det d 2 x ˆ Φ L ( ) (x,,t τ d ) a ˆ k v k (x,,t) + h.c. k wave-packet modes Assume that the LO pulse is a strong coherent state of a particular localized wave packet mode: LO phase (+) Φ ˆ L (r,t) α L exp(iθ) v L (r,t) + vacuum ˆ N D (θ) = ˆ a = a ˆ k c k α L ( ˆ a e iθ + T dt ˆ a e iθ ) d 2 x v * L (x,,t τ d ) v k (x,,t) = ˆ Det a k= L The signal field is spatially and temporally gated by the LO field, which has a controlled shape. Where the LO is zero, that portion of the signal is rejected. Only a single temporal-spatial wavepacket mode of the signal is detected. 6 of 31
DC-BALANCED HOMODYNE DETECTION V signal : (+) Φ ˆ S (r,t) a ˆ v L (r,t) + a ˆ k k v k (r,t) wave-packet modes quadrature operators: q ˆ = ( a ˆ + a ˆ ) / 2 1/2 p ˆ = ( a ˆ a ˆ ) / i2 1/2 detected quantity: ˆ q θ ˆ N D (θ) α L 2 = ˆ a e iθ + 2 a ˆ e iθ LO phase ˆ q θ ˆ N D (θ) α L 2 = ˆ q cosθ + ˆ p sinθ ˆ ˆ q θ p θ cosθ sinθ q ˆ = sinθ cosθ p ˆ 7 of 31
ULTRAFAST OPTICAL SAMPLING Conventional Approach: Ultrafast Time Gating of Light Intensity by NON-LINEAR OPTICAL SAMPLING strong short pump (ω p ) delay sum-frequency (ω p + ω s ) weak signal(ω s ) second-order NL crystal 8 of 31
LINEAR OPTICAL SAMPLING I BHD for Ultrafast Time Gating of Quadrature Amplitudes detected quantity: ˆ q θ ˆ N D (θ) α L 2 = ˆ q cosθ + ˆ p sinθ LO phase q ˆ = ( a ˆ + a ˆ ) / 2 1/2 p ˆ = ( a ˆ a ˆ ) / i2 1/2 ˆ a = a ˆ k c k T dt d 2 x v * L (x,,t τ d ) v k (x,,t) = ˆ Det LO signal a k= L t θ 9 of 31
LINEAR OPTICAL SAMPLING II Ultrafast Time Gating of Quadrature Amplitudes LO mode: ˆ N D (τ d ) = i c α L * v L (x,,t) α L v L (x) f L (t τ d ) T dt f * L (t τ d ) φ S (t) + h.c. φ S (t) = Det d 2 x v L *(x) ˆ Φ S ( +) (x,,t) if signal is band-limited and LO covers the band, e.g. f L (t) (1/ t)sin(b t / 2) ˆ N D (τ d ) α L* α L * signal LO ν Β/2 ν+β/2 ω f * dω L (ν) ν +B /2 exp( iω τ ν B /2 d ) φ S (ω) + h.c. 2π f * L (ν) φ S (τ d ) + h.c. exact sampling 1 of 31
LINEAR OPTICAL SAMPLING III M. E. Anderson, M. Munroe, U. Leonhardt, D. Boggavarapu, D. F. McAlister and M. G. Raymer, Proceedings of Generation, Amplification, and Measurment of Ultrafast Laser Pulses III, pg 142-151 (OE/LASE, San Jose, Jan. 1996) (SPIE, Vol. 271, 1996). Ultrafast Laser (optical or elect. synch.) Signal Source Spectral Filter Time Delay τ d Phase Adjustment θ Signal Reference (LO) LO Signal Balanced Homodyne Detector n n 1 2 Computer mean quadrature amplitude in sampling window at time t ˆ q θ (t) ψ 11 of 31
LINEAR OPTICAL SAMPLING IV 84 nm, 17 fs Sample: Microcavity exciton polariton LO coherent signal θ scan LO delay τ d Balanced Homodyne detector ˆ q θ (t) ψ 12 of 31
LINEAR OPTICAL SAMPLING V Mean Quadrature Measurement - sub ps Time Resolution 1 q ˆ θ (t) ψ Sample: Microcavity exciton polariton 5 1 4 mean quadrature amplitude <q> at time t < n(t) > 1 1 1 3 2 1 g (2) (t,t).1.1-1 2 4 6 8 Time (ps) LO delay τ d (ps) 1 12 coherent field --> ˆ q θ +π /2 (t) ψ = p ˆ θ (t) ψ 13 of 31
LINEAR OPTICAL SAMPLING VI Phase Sweeping for Indirect Sampling of Mean Photon Number and Photon Number Fluctuations detected quantity: ˆ q θ ˆ N D (θ) α L 2 = ˆ q cosθ + ˆ p sinθ Relation with photon-number operator: ˆn = â â = 1 ( 2 ˆq i ˆp )( ˆq + i ˆp ) = ˆq 2 + ˆp 2 + 1 2 Phase-averaged quadrature-squared: 2 q ˆ θ θ = 1 π π q ˆ 2 θ dθ = 1 π π (θ = LO phase) ( q ˆ cosθ + p ˆ sinθ ) 2 dθ = 1 q 2 ˆ 2 + ˆ ( p 2 ) ˆ n = 2 q ˆ θ θ 1 2 ensemble average ˆ n (t) ψ = ˆ q θ 2 (t) θ ψ 1 2 works also for incoherent field (no fixed phase) 14 of 31
LINEAR OPTICAL SAMPLING VII Phase Sweeping --> Photon Number Fluctuations detected quantity: n (r ) ψ ˆ q θ ˆ N D (θ) α L 2 = ˆ q cosθ + Richter s formula for Factorial Moments: ˆ p sinθ = [n(n 1)...(n r +1)] p(n) = ( a ˆ ) r ( a ˆ ) r ψ n = = (r!)2 2 r (2r)! 2π Hermite Polynomials: n (1) = ˆ a ˆ a = 1 4 n (2) = ˆ a 2 ˆ a 2 = 2π 2π dθ 2π dθ 2π dθ 2π H 2r ( ˆ q θ ) ψ H (x) =1, H 1 (x) = 2x, H 3 (x) = 4x 2 2 4 ˆ q θ 2 2 ψ 2 3 ˆ q θ 4 2 ˆ q θ 2 + 1 2 ψ ˆ n (t) ψ = ˆ q θ 2 (t) θ ψ 1 2 15 of 31
LINEAR OPTICAL SAMPLING VIII Phase Sweeping --> Photon Number Fluctuations Variance of Photon Number in Sampling Time Window: var(n)=< n 2 > - < n > 2 var(n) = 2π dθ 2π 2 q 3 ˆ 4 θ 2 q ˆ θ 2 2 1 q ˆ θ + 4 Second-Order Coherence of Photon Number in Sampling Time Window: g (2) (t,t )=[< n 2 > - < n >]/< n > 2 g (2) (t,t) = 2 corresponds to thermal light, i.e. light produced primarily by spontaneous emission. g (2) (t,t) =1 corresponds to light with Poisson statistics, i.e., light produced by stimulated emission in the presence of gain saturation. 16 of 31
LINEAR OPTICAL SAMPLING IX Photon Number Fluctuations if the signal is incoherent, no phase sweeping is required 8MHz 1-5kHz Ti:Sapphire Regen. Amplifier λ/2 Electronic Delay Trigger Pulse Sample LO λ/2 Signal Alt. Source PBS1 Voltage Pulser Computer AD/DA GPIB controller n 1 n 2 Stretcher Charge-Sensitive Pre-Amps Shaper Shaper Photodiodes Balanced Homodyne Detector λ/2 PBS2 M. Munroe 17 of 31
LINEAR OPTICAL SAMPLING X Superluminescent Diode (SLD) Optical Amplifier metal cap 6 o 6 µm 3 µm (AR) SiO 2 p-clad layer quantum wells n-clad layer ~ n-gaas substrate ~ p-contact layer undoped, graded confining layers (Sarnoff Labs) Superluminescent Emission M. Munroe 18 of 31
LINEAR OPTICAL SAMPLING XI (no cavity) (a) Intensity (a.u.) 1..8.6.4.2 (a) Output Power (mw) 25 2 15 1 5 1 2 Drive Current (ma) (b) Intensity (a.u.). 81 1..5. 82 83 84 Wavelength (nm) (b) 85 76 8 84 Wavelength (nm) 88 M. Munroe 19 of 31
LINEAR OPTICAL SAMPLING XII SLD in the single-pass configuration 3. <n(t,t)> g (2) (t,t) 2.4 2.5 2.2 2. <n(t)> 2. 1.5 1..5 5 1 time (ns) 15 2 1.8 1.6 1.4 1.2 1. g (2) (t,t) Photon Fluctuation is Thermal-like, within a single time window (15 fs) M. Munroe 2 of 31
LINEAR OPTICAL SAMPLING XIII SLD in the double-pass with grating configuration 14 12 1 <n(t)> g (2) (t,t) 4. 3.5 3. <n(t)> 8 6 4 2 5 1 time (ns) 15 2 2.5 2. 1.5 1..5 g (2) (t,t) Photon Fluctuation is Laser-like, within a single time window (15 fs) M. Munroe 21 of 31
Single-Shot Linear Optical Sampling I -- Does not require phase sweeping. Measure both quadratures simultaneously. Dual- DC-Balanced Homodyne Detection LO1 signal 5/5 BHD q q 2 + p 2 = n BHD p π/2 phase shifter LO2 22 of 31
Fiber Implementation of Single-shot Linear Optical Sampling Of Photon Number MFL: mode-locked Erbium-doped fiber laser. OF: spectral filter. PC: polarization controller. BD: balanced detector. 23 of 31
Measured quadratures (continuous and dashed line) on a 1-Gb/s pulse train. Waveform obtained by postdetection squaring and summing of the two quadratures. 24 of 31
Two-Mode DC-HOMODYNE DETECTION I LO is in a Superposition of two wave-packet modes, 1 and 2 signal ˆ Φ L (+) (r,t) = i c α L exp(iθ) v 1 (r,t)cosα + v 2 (r,t)exp( iζ )sinα [ ] Dual temporal modes: Dual LO 1 2 (temporal, spatial, or polarization) BHD Q β = θ ζ Q ˆ = cos(α) [ q ˆ 1 cosθ + p ˆ 1 sinθ ] + sin(α) q ˆ 2 cosβ + [ p ˆ 2 sinβ ] q ˆ 1θ q ˆ 2β quadrature of mode 1 quadrature of mode 2 25 of 31
SLD Two-Mode DC-HOMODYNE DETECTION II ultrafast two-time number correlation measurements using dual- LO BHD; super luminescent laser diode (SLD) 1 2 Dual LO signal BHD t 1 t 2 Q two-time secondorder coherence g (2) (t 1,t 2 ) = : n ˆ (t ) n ˆ (t ): 1 2 n ˆ (t 1 ) n ˆ (t 2 ) D. McAlister 26 of 31
Two-Mode DC-HOMODYNE DETECTION III Alternative Method using a Single LO. Signal is split and delayed by different times. Polarization rotations can be introduced. source signal LO BHD polarization rotator Q two-pol., two-time second-order coherence (2) (t 1,t 2 ) = : n ˆ (t ) n ˆ (t ): i 1 j 2 n ˆ i (t 1 ) n ˆ j (t 2 ) g i, j A. Funk 27 of 31
Two-Mode DC-HOMODYNE DETECTION IV Single-time, two-polarization correlation measurements on emission from a VCSEL -2π phase sweeping and time delay -2π relative phase sweeping E. Blansett 28 of 31
Two-Mode DC-HOMODYNE DETECTION V Single-time, twopolarization correlation measurements on emission from a VCSEL at low temp. (1K) (2) (t 1,t 2 ) = : n ˆ i(t 1 ) n ˆ i (t 2 ): n ˆ i (t 1 ) n ˆ i (t 2 ) g i,i (2) (t 1,t 2 ) = : n ˆ (t ) n ˆ (t ): i 1 j 2 n ˆ i (t 1 ) n ˆ j (t 2 ) g i, j uncorrelated E. Blansett 29 of 31
Two-Mode DC-HOMODYNE DETECTION VI Single-time, twopolarization correlation measurements on emission from a VCSEL at room temp. (2) (t 1,t 2 ) = : n ˆ i(t 1 ) n ˆ i (t 2 ): n ˆ i (t 1 ) n ˆ i (t 2 ) g i,i (2) (t 1,t 2 ) = : n ˆ (t ) n ˆ (t ): i 1 j 2 n ˆ i (t 1 ) n ˆ j (t 2 ) g i, j anticorrelated Spin-flip --> gain competition 3 of 31
SUMMARY: DC-Balanced Homodyne Detection 1. BHD can take advantage of: high QE and ultrafast time gating. 2. BHD can provide measurements of photon mean numbers, as well as fluctuation information (variance, second-order coherence). 3. BHD can selectively detect unique spatial-temporal modes, including polarization states. 31 of 31