1. PART 2 : BALANCED HOMODYNEDETECTION Michael G. RaymerOregon Center for Optics, University of Oregon raymer@uoregon.eduM.G.Raymer_TTRL2b_V2_2005 1 of 31

2. OUTLINEPART 11. Noise Properties of Photodetectors2. Quantization of Light3. Direct Photodetection and Photon CountingPART 24. Balanced Homodyne Detection5. Ultrafast Photon Number SamplingPART 36. Quantum State TomographyM.G.Raymer_TTRL2b_V2_2005 2 of 31 3. DC-BALANCED HOMODYNE DETECTION IGoal -- measure quadrature amplitudes with highQ.E. and temporal-mode selectivity ES = signal field (O), 1 - 1000 photons EL = laser reference field (local oscillator) (O), 106 photonsn1 E1 =dtES (t)ES + E L PD NDBSn2 PDdtEL (t) E2 = ES - ELND E1( )(t d ) E1(+) (t) dt ddelay E 2( )(t d ) E 2(+) (t) dt M.G.Raymer_TTRL2b_V2_20053 of 31 4. DC-BALANCED HOMODYNE DETECTION II integrator circuit n1dtPD ND n2 PD dtM.G.Raymer_TTRL2b_V2_2005 4 of 31 5. DC-BALANCED HOMODYNE DETECTION IIIS = signal amplitude; L = laser reference amplitude n1 dt SES (t)ND BS n2 dtEL (t) L d delayoverlap dt Det d x L TND = ( ) (x,0,t d ) (+) (x,0,t) + h.c. 2S 0integral (+ ) (r,t) = i cS ak v k (r,t)k v k (r,t) = Ck j u j (r) exp(i j t) j wave-packet c 0 dt Det d x v *k (x,0,t) v m (x,0,t) = k m T 2 modes M.G.Raymer_TTRL2b_V2_20055 of 31 6. DC-BALANCED HOMODYNE DETECTION IV ( ) (x,0,t d ) ak v k (x,0,t) + h.c. dt Det d x L TND 2 0k wave-packet modesAssume that the LO pulse is a strong coherent state of a particularlocalized wave packet mode: LO phase (+ ) (r,t) | | exp(i ) v L (r,t) + vacuumL LN D ( ) = | L | ( a ei + a e i )a = ak c 0 dt Det d 2 x v *L (x,0,t d ) v k (x,0,t) = ak= L T k 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 wave- packet mode of the signal is detected.M.G.Raymer_TTRL2b_V2_2005 6 of 31 7. DC-BALANCED HOMODYNE DETECTION V wave-packetsignal : S (r,t) a v L (r,t) + ak v k (r,t) (+ ) k modesquadrature operators:q = ( a + a ) / 21/2 p = (a a ) / i21/2 detectedN D ( ) a ei + a e i LO phaseq =quantity:|L | 2 2 N D ( )q = q cos + p sin |L | 2 q cos sin q = p sin cos p M.G.Raymer_TTRL2b_V2_20057 of 31 8. ULTRAFAST OPTICAL SAMPLING Conventional Approach: Ultrafast Time Gating of Light Intensity byNON-LINEAR OPTICAL SAMPLINGstrong shortpump (p )delaysum-frequency (p + s ) weak signal(s ) second-order NL crystalM.G.Raymer_TTRL2b_V2_2005 8 of 31 9. LINEAR OPTICAL SAMPLING IBHD for Ultrafast Time Gating of Quadrature Amplitudesdetected N D ( )q = q cos + p sin quantity:|L | 2LO phase q = ( a + a ) / 21/2 p = (a a ) / i21/2 a = ak c 0 dt Det d 2 x v *L (x,0,t d ) v k (x,0,t) = ak= LT k LO signal tM.G.Raymer_TTRL2b_V2_2005 9 of 31 10. LINEAR OPTICAL SAMPLING II Ultrafast Time Gating of Quadrature Amplitudes LO mode:v L (x,0,t) L v L (x) f L (t d )TN D ( d ) = i c * dt f L* (t d ) S (t) + h.c.L 0 S (t) = Det d x v L * (x) S 2 (+) (x,0,t)if signal is band-limited and signalLO covers the band, e.g.LOf L (t) (1 / t)sin(B t / 2)/2+/2 D ( d ) * fL* ( ) +B /2 d exp(i d ) S ( ) + h.c. NL B /2 2 L fL* ( ) S ( d ) + h.c.*exact sampling M.G.Raymer_TTRL2b_V2_2005 10 of 31 11. LINEAR OPTICAL SAMPLING IIIM. E. Anderson, M. Munroe, U. Leonhardt, D. Boggavarapu, D. F. McAlister and M. G. Raymer, Proceedings ofGeneration, Amplification, and Measurment of Ultrafast Laser Pulses III, pg 142-151 (OE/LASE, San Jose, Jan.1996) (SPIE, Vol. 2701, 1996). UltrafastSignal Laser(optical or Source elect. synch.) Spectral SignalFilterSignalReference (LO) Time PhaseLOBalanced DelayAdjustment HomodyneDetector dn1 n2 Computer mean quadrature amplitude in sampling q (t) window at time t M.G.Raymer_TTRL2b_V2_200511 of 31 12. LINEAR OPTICAL SAMPLING IV LOscan LO840 nm, 170 fs delay dSample: Microcavityexciton polariton coherent signal Balanced Homodyne detectorq (t) M.G.Raymer_TTRL2b_V2_2005 12 of 31 13. LINEAR OPTICAL SAMPLING VMean Quadrature Measurement - sub ps Time Resolution Sample: Microcavity q (t)10000 exciton polariton 5 10004mean1003quadratureg < n(t) > (2) 102(t,t)amplitude at11time t0.10 0.01-1 0 24 6 8 1012Time (ps) LO delay d (ps) coherent field --> q + /2 (t) = p (t) 0 M.G.Raymer_TTRL2b_V2_2005 13 of 31 14. LINEAR OPTICAL SAMPLING VIPhase Sweeping for Indirect Sampling of Mean Photon Number and Photon Number Fluctuationsdetected N D ( ) q = q cos + p sin ( = LO phase)quantity: |L | 2Relation with photon-number operator:1 1 n = a a = ( q i p )( q + i p ) = q + p + 2 22 2 Phase-averaged quadrature-squared:1 2 1 1 2q = 0 q d = 0 (q cos + p sin ) d = (q + p 2 )2 2 2 1 ensemble 1 n = q 2 n (t) = q (t) 2 2 2 averageworks also for incoherent field (no fixed phase)M.G.Raymer_TTRL2b_V2_200514 of 31 15. LINEAR OPTICAL SAMPLING VIIPhase Sweeping --> Photon Number Fluctuations detectedN D ( ) quantity: q | | 2 = q cos + p sin L Richters formula for Factorial Moments: n (r ) = [n(n 1)...(n r + 1)] p(n) = ( a ) r ( a) r n= 0(r!) 2 2 d= r 2 (2r)! 0 2 H 2r (q ) Hermite Polynomials: H 0 (x) = 1, H1 (x) = 2x, H 3 (x) = 4 x 2 2 12d 1 n (1) = a a = 2 2 4q (t) = q 2 (t) n 40 2 22d 2 4 1 n (2) = a a 22= 0 2 3 q 2 q + 2 2 M.G.Raymer_TTRL2b_V2_200515 of 31 16. LINEAR OPTICAL SAMPLING VIII Phase Sweeping --> Photon Number FluctuationsVariance of Photon Number in Sampling TimeWindow: var(n)=< n 2 > - < n >22 d 2 4 1 2var(n) =q q q2 2 + 02 3 4Second-Order Coherence of Photon Number inSampling Time Window:g(2)(t,t )=[< n 2 > - < n >]/< n >2g(2) (t,t) = 2 corresponds to thermal light, i.e. light producedprimarily by spontaneous emission.g(2) (t,t) = 1 corresponds to light with Poisson statistics, i.e., lightproduced by stimulated emission in the presence of gain saturation. M.G.Raymer_TTRL2b_V2_200516 of 31 17. LINEAR OPTICAL SAMPLING IXPhoton Number Fluctuationsif the signal is incoherent, no phase sweeping is required 80MHz1-50kHzTi:Sapphire Regen. Amplifier /2 Electronic Trigger PulseSample LO Delay /2 SignalAlt. SourcePBS1 /2Voltage Charge-SensitivePBS2 Pulser Pre-AmpsComputerPhotodiodes n1 ShaperAD/DA Stretcher n2 Shaper M. GPIB controllerBalanced Homodyne Detector MunroeM.G.Raymer_TTRL2b_V2_2005 17 of 31 18. LINEAR OPTICAL SAMPLING XSuperluminescent Diode (SLD) Optical Amplifier metal capo6 600 m 3 m (AR) SiO 2p-clad layerp-contact layerquantum wells~~undoped, graded~~n-clad layerconfining layersn-GaAs substrateSuperluminescent(Sarnoff Labs)Emission M. Munroe M.G.Raymer_TTRL2b_V2_200518 of 31 19. LINEAR OPTICAL SAMPLING XI (no cavity)1.0(a)(a) 0.8 Intensity (a.u.)0.60.40.2250.0Output Power (mW)810 820830840850 Wavelength (nm)20151051.0 (b)0Intensity (a.u.) 0100 200 0.5 Drive Current (mA)(b) 0.0 760 800840880 Wavelength (nm) M. MunroeM.G.Raymer_TTRL2b_V2_2005 19 of 31 20. LINEAR OPTICAL SAMPLING XII SLD in the single-pass configuration 3.0 2.4(2) g (t,t) 2.5 2.2 2.0 2.0 1.8 g(2)(t,t) 1.6 1.5 1.4Photon Fluctuationis Thermal-like, 1.0 1.2within a single time 1.0window (150 fs) 0.5 0510time (ns)1520M. Munroe M.G.Raymer_TTRL2b_V2_200520 of 31 21. LINEAR OPTICAL SAMPLING XIII SLD in the double-pass with grating configuration4.014 (2)g (t,t)3.5123.0102.5g(2)(t,t) 82.0 6Photon Fluctuation1.5 4is Laser-like, within 21.0 a single time 00.5window (150 fs) 0510 15 20time (ns) M. Munroe M.G.Raymer_TTRL2b_V2_200521 of 31 22. Single-Shot Linear Optical Sampling I -- Does not require phase sweeping. Measure both quadratures simultaneously. Dual- DC-Balanced Homodyne DetectionLO1BHD qsignal 50/50 q2 + p2 = nBHD p/2 phase LO2shifter M.G.Raymer_TTRL2b_V2_200522 of 31 23. Fiber Implementation of Single-shot Linear Optical Sampling Of Photon NumberMFL: mode-locked Erbium-doped fiber laser. OF: spectral filter.PC: polarization controller. BD: balanced detector.M.G.Raymer_TTRL2b_V2_2005 23 of 31 24. Measured quadratures(continuous and dashedline) on a 10-Gb/spulse train.Waveform obtained bypostdetection squaringand summing of the twoquadratures.M.G.Raymer_TTRL2b_V2_2005 24 of 31 25. Two-Mode DC-HOMODYNE DETECTION I LO is in a Superposition of two wave-packet modes, 1 and 2 (+ ) (r,t) = i c | L |exp(i ) [v1 (r,t)cos + v 2 (r,t)exp(i )sin ]LDual temporal modes:1 2 (temporal, Dual LOspatial, orsignalpolarization)BHD Q = Q = cos( )[q1 cos + p1 sin ] + sin( )[q2 cos + p2 sin ] q1 q2 quadrature of mode 1quadrature of mode 2 M.G.Raymer_TTRL2b_V2_200525 of 31 26. Two-Mode DC-HOMODYNE DETECTION II ultrafast two-time number correlation measurements using dual- LO BHD; super luminescent laser diode (SLD) 1 2Dual LO signalt1 t2SLDBHD Qtwo-time second-order coherence: n (t1 ) n (t2 ): g (t1,t2 ) =(2)n (t1 ) n (t2 )D. McAlisterM.G.Raymer_TTRL2b_V2_2005 26 of 31 27. Two-Mode DC-HOMODYNE DETECTION IIIAlternative Method using a Single LO. Signal is split and delayed by different times. Polarization rotations can be introduced.signalLOsourceBHDQ polarization rotatortwo-pol., two-time : n i (t1 ) n j (t2 ): second-order g (t1,t2 ) = (2) i, jcoherencen i (t1 ) n j (t2 ) A. Funk M.G.Raymer_TTRL2b_V2_2005 27 of 31 28. Two-Mode DC-HOMODYNE DETECTION IVSingle-time, two-polarization correlation measurements on emission from a VCSEL0-2 phasesweepingand timedelay 0-2 relative phase sweepingE. BlansettM.G.Raymer_TTRL2b_V2_200528 of 31 29. Two-Mode DC-HOMODYNE DETECTION VSingle-time, two- polarization correlationmeasurements on emission from a VCSELat low temp. (10K): n i (t1 ) n i (t2 ): g (t1,t2 ) = (2) i, in i (t1 ) ni (t2 ): n i (t1 ) n j (t2 ): uncorrelatedg (t1,t2 ) =(2)i, jn i (t1 ) n j (t2 ) E. BlansettM.G.Raymer_TTRL2b_V2_200529 of 31 30. Two-Mode DC-HOMODYNE DETECTION VISingle-time, two- polarization correlationmeasurements on emission from a VCSELat room temp.: n i (t1 ) n i (t2 ): g (t1,t2 ) = (2) i, in i (t1 ) ni (t2 ) : n i (t1 ) n j (t2 ): anticorrelatedg (t1,t2 ) =(2)i, j n i (t1 ) n j (t2 ) Spin-flip --> gain competitionM.G.Raymer_TTRL2b_V2_2005 30 of 31 31. SUMMARY: DC-Balanced Homodyne Detection1. BHD can take advantage of: high QE and ultrafast timegating.2. BHD can provide measurements of photon meannumbers, as well as fluctuation information (variance,second-order coherence).3. BHD can selectively detect unique spatial-temporalmodes, including polarization states. M.G.Raymer_TTRL2b_V2_200531 of 31