Slow Images and Entangled Photons Ryan Camacho, Curtis Broadbent, Michael Pack, Praveen Vudya Setu, and John Howell University of Rochester Collaborator: Bob Boyd in Institute of Optics $Thanks to$: ARO, PECASE, NSF, Research Corporation, DARPA, DOD MURI, DURIP, University of Rochester

Group

Recent Work

Slow/Fast/Backward Light Frequency Dependent Index of refraction –Accompanies frequency dependent absorption or gain (e.g., absorption resonances, gain bands, spectral holes) –Leads to fast, backward and slow group velocities Hau et al, Nature 397, 594 (1999) Gehring et al, 312, 895 (2006)

Quantum Buffering Use steep linear dispersion between two absorption resonances to slow down Quantum and Classical Images Preserve Images (not just binary signals) for large delays Ultra-low noise (preserve quantum fields in low loss regime) Doppler broadened vapors Have delayed 275 ps pulses in excess of 10 ns and 740 ps pulses in excess of 60 ns. Small broadening: Dispersive broadening dominates absorptive broadening. Cesium Resonances and Delays Camacho et al., Phys Rev Lett 98, (2007) Macke et al., Phys. Rev. A 73, (2006).Phys. Rev. A 73, (2006) H. Tanaka et al., Phys. Rev. A 68, (2003).Phys. Rev. A 68, (2003) Linear Circuits

Group Delay Theory Optical Depth Lorentzian FWHM: Homogeneous Linewidth Real part of index imaginary

Broadening Absorptive Broadening Single Lorentzian Two Lorentzians  

Image Buffering Experiment Camacho et al, Phys. Rev. Lett. 98, (2007)

Classical Image Interference Matched image pulses (high fringe visibility) Unmatched image pulses (low Local oscillator pulse and slow light pulse arrive at same time at the beam splitter (interference) Arrive at different times

Weak Coherent State 2-D Imaging 3ns delay of 2-d image, 0.8 photons per pulse and (right) 2d image propagating through air, 0.8 photons per pulse.

Weak Coherent State 1-D Imaging 0.5 photons, on average, per pulse hit a 2 bar test pattern. The image pulses are delayed by 9 ns. A multimode fiber is used to scan the image.

Potential Digital signal processing Holography Remote imaging Quantum buffers Scenedow? Etc.

Transverse Images What is necessary to preserve images in slow light medium? –It is sufficient to have isotropic medium –Is it necessary? Phys. Rev. Lett. 75, (1995) Steve Harris’ Group

Current Efforts in Slow Light Preservation of entanglement in slow light medium. Biphoton slow light. Double absorption resonances using electromagnetically induced absorption. Images on demand. Matched filtering and single photon image discrimination. Delay of a transverse quantum state using slow light. Fourier transform interferometry using slow light. Measurement of gravitational red shift in the lab using slow light.

Conclusions Delayed transverse image in slow light medium– preserved amplitude and phase Low noise, linear device useful for quantum information preservation. Demonstrated multiple pulse delays with little pulse distortion.

EIA delays Transmission Profile Delay

Single Particle Continuous Variable Uncertainty Relations Continuous observables position and momentum (or e.g., field quadratures) obey 1. Heiserberg’s uncertainty relation. 2. Closely related to the space-bandwidth product in imaging. 3. Continuous quantum cryptography

EPR: Continuous Entanglement Einstein, Podolsky and Rosen questioned the completeness of wavefunction description of Quantum Mechanics in their gedanken experiment [Phys Rev 47, 777 (1935)]. Suppose we have two quantum particles 1 and 2 with their positions governed by

EPR entanglement Position  (x 1 -x 2 ) Interaction Momentum  (k 1 +k 2 ) EPR: no interaction at distant locations. particle 2 must be in both a position and momentum eigenstate, which violates Heisenberg’s uncertainty principle  x  k<1/2. Particle 1Particle 2

Separability Bounds Assume two separable particles Separable Continuous Most general statement of nonentangled two particle system

Entangled statistics Uncertainty product vanish for perfect maximal entanglement.

EPR Entanglement: Previous Work Squeezed light fields (quadrature squeezed correlations) Reid and Drummond, PRL 60, 2731 (1988) Ou et al, PRL 68, 3663 (1992) Collective atomic spin variables (spin observables) Julsgaard, Nature 413, 400 (2001) Modern rephrasing of continuous entanglement Duan et al, PRL 84, 2722 (2000) Simon, PRL 84, 2726 (2000) Mancini et al, PRL 88, (2002) Discrete Entanglement (violation of separability bounds) Hofman and Takeuchi PRA Ali Khan and Howell (to be published)

Transverse Momentum-Position Entanglement Ghost Imaging and Ghost Diffraction –Pittman et al, PRA 52, R3429 (1995) –D. V. Strekalov et al, PRL 74, (1995) Classical Ghost imaging and Ghost Diffraction –Bennink et al, PRL 89, (2002) –Bennink et al PRL 92, (2004)] Noncommuting observables –Gatti et al, PRL 90, (2003) –Equivalent to demonstrating Rotational Invariance, but for continuous variables.

Transverse Momentum-Position Entanglement Created? –Used first order (two-photon) spontaneous parametric down conversion. –One photon downconverts into two photons. Momentum conserved (momentum correlation) Photons emitted from a small birth place region (position correlation) Thin crystal, paraxial and narrow filter approximation Angular Spectrum of pump Phase matching condition

Momentum Correlation kpkp kpkp

Momentum Correlation Measurement The momentum correlation: detect photons in far field (focal or Fourier plane of lens). The k- vectors mapped to position. f f Optic axis Anti correlated distance from optic axis

Position Correlation BBO crystal Pump Laser Beam 1mm Pair birth place 10’s  m

Position Correlation kpkp ksks kiki Collinearly Phase matched type-II in forward direction: Perfect phase matching Imperfect Phase matching kzkz  k z L=  /2 gives an approximate size to the birth place.  ksks kiki kpkp

Position Correlation Both Photons created inside birthplace region. Photons measured in near field (image planes). Strong entanglement in limit birth place diameter r much smaller pump beam diameter d (EPR bound). 2f Optic axis 2f Correlated distances from optic axis

Experiments Imaging Layout Fourier Imaging Layout Bennink et al, PRL 92, (2004) Howell et al, PRL. 92, (2004)

EPR Result Inferred uncertainty product for particle 2 is approximately Single-Particle variance product Conditional Variance product

Pixel Entanglement: Discretizing continuous entanglement Same Basis: correlated or anticorrelated measurements. (3 possible coincidence measurements ) Different basis: uncorrelated measurements (9 possible coincidence measurements). Generalization of Ekert cryptographic protocol to qudits of arbitrary dimension d (d=3) O’Sullivan-Hale Phys. Rev. Lett. 94, (2005)

Pixel Entanglement Results Position- Position Momentum- Momentum

Pixel Entanglement

6 pixel array

Generalization to large state spaces Generalization to arbitrarily large APD arrays. Current limit to dimensionality is due to detectors. Reminder: APD arrays inside single photon emission cones.

Time-Energy Entanglement Time-Energy experiment –Violated separability bound by more than 3 orders of magnitude. –Fiber transmission –High bandwidth quantum information

Time-Energy: Why? Quantum Communication –Transverse entanglement requires wavefront preservation: multimode –Time-Energy: Single Mode (fiber transportable) –Very High Bandwidth (qubits vs. large d qudits)

Time-Energy Correlations Time Correlations (100’s of fs) –Need ultra fast detectors –HOM dip is local measurement –Use Franson Interferometer to measure space-like fourth order correlations Energy Correlations (MHz set by pump) –Grating spectral decomposition Large Potential Information Content –Bandwidth of Down Conversion divided by the Bandwidth of the Pump Laser Information Eigenmmodes C. K. LawC. K. Law and J. H. Eberly Phys. Rev. Lett. 92, (2004)J. H. Eberly

Vernam Cipher and QKD Plaintext Message: Ciphertext: Public Domain One-time pad

Quantum Key Distribution Use Quantum Mechanics to make secure keys –Typical Binary random strings Photon Linear Polarization: Alice sends Bob 0,90 basis 45/135 Bob measures in one of two bases Bennett Brassard 84 Protocol Eavesdropper intercepts in wrong basis and retransmits errors 50% of time

Information capacity No need to limit photon information to binary outcomes Photons can, in theory, carry infinite information (only bounded by the dimensions of the universe) Demonstrated QKD with over 1000 measurable states per pair. –Ali Khan et al, Phys. Rev. Lett. 98, (2007)

PBS  A1 A2 Alice Time Stamping Sync Generator RS PBS  B1B2 Bob Time Stamping PPS 50/50 BS 50/50 Fiber BS PPS – Passive Power Splitter RS – Raw Sync Energy-Time Quantum Cryptography Simplified View Type-I BBO 50mW Internet Computer B Computer A

Time Scales 4 Relevant times (7 orders of magnitude) –Pump Coherence Length 1 microsecond –Time difference short-long paths 10 ns –Detector Response time 50 ps –Two photon correlation 100 fs Usable information content per pair) –Ratio of Long-Short difference and Detector response time (~100)

Time-Time Correlations Multistate state cryptosystem. Time bin Alice and Bob both measure their Photon at the same point in the time bin 0123 Alice Bob 0,34,25,92,45 Large Bin Synch Pulses ever 64 ns

Sifted Key Bit Rates Qudit to Qubit conversion Example: 1024 alphabet qudit converted to 10 bits

Franson Interferometer: Measure Energy Correlations Output ports of Michelson with postselection of short-short and long long Using temporal commutator relation, normal ordering and assuming l 1 ~ l 2 QND Measurement

Franson Visibility as a function of Eve’s measurement in time for Gaussian POVM Fringe Visibility Eve’s Temporal Uncertainty for QND 3 meters 10 meters

Franson visibility Measured greater than 95% fringe visibility –Eve cannot get timing information better than approximately 20 ns before revealing herself Used 3 meter interferometer arms With faster detectors can shorten arms

Thank you for coming

Conclusion Demonstrated large continuous variable entangled states in transverse and longitudinal momentum-position. Used CV for large alphabet QKD Thank you!