# Presentation: 484282

## 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**, 153601 (2007)

- Macke et al., _[Phys. Rev. A ](http://link.aip.org/link/?&l_creator=getabs-search&l_dir=FWD&l_rel=CITES&from_key=PRLTAO000098000015153601000001&from_keyType=CVIPS&from_loc=AIP&to_j=PLRAAN&to_v=73&to_p=043802&to_loc=APS&to_url=http%3A%2F%2Flink.aps.org%2Fabstract%2FPRA%2Fv73%2Fe043802)__[73](http://link.aip.org/link/?&l_creator=getabs-search&l_dir=FWD&l_rel=CITES&from_key=PRLTAO000098000015153601000001&from_keyType=CVIPS&from_loc=AIP&to_j=PLRAAN&to_v=73&to_p=043802&to_loc=APS&to_url=http%3A%2F%2Flink.aps.org%2Fabstract%2FPRA%2Fv73%2Fe043802)__[, 043802 (2006)](http://link.aip.org/link/?&l_creator=getabs-search&l_dir=FWD&l_rel=CITES&from_key=PRLTAO000098000015153601000001&from_keyType=CVIPS&from_loc=AIP&to_j=PLRAAN&to_v=73&to_p=043802&to_loc=APS&to_url=http%3A%2F%2Flink.aps.org%2Fabstract%2FPRA%2Fv73%2Fe043802)_.

- H. Tanaka et al., _[Phys. Rev. A ](http://link.aip.org/link/?&l_creator=getabs-search&l_dir=FWD&l_rel=CITES&from_key=PRLTAO000098000015153601000001&from_keyType=CVIPS&from_loc=AIP&to_j=PLRAAN&to_v=68&to_p=053801&to_loc=APS&to_url=http%3A%2F%2Flink.aps.org%2Fabstract%2FPRA%2Fv68%2Fe053801)__[68](http://link.aip.org/link/?&l_creator=getabs-search&l_dir=FWD&l_rel=CITES&from_key=PRLTAO000098000015153601000001&from_keyType=CVIPS&from_loc=AIP&to_j=PLRAAN&to_v=68&to_p=053801&to_loc=APS&to_url=http%3A%2F%2Flink.aps.org%2Fabstract%2FPRA%2Fv68%2Fe053801)__[, 053801 (2003)](http://link.aip.org/link/?&l_creator=getabs-search&l_dir=FWD&l_rel=CITES&from_key=PRLTAO000098000015153601000001&from_keyType=CVIPS&from_loc=AIP&to_j=PLRAAN&to_v=68&to_p=053801&to_loc=APS&to_url=http%3A%2F%2Flink.aps.org%2Fabstract%2FPRA%2Fv68%2Fe053801)__._

- 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**, 043902 (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.

## Slide 11

## 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, 4385 - 4388 (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 (x1-x2)

- Interaction

- Momentum (k1+k2)

- 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 1

- Particle 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, 120401 (2002)
- Discrete Entanglement (violation of separability bounds)
- Hofman and Takeuchi PRA 68 032103
- 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**, 3600-3603 (1995)
- Classical Ghost imaging and Ghost Diffraction
- Bennink et al, PRL 89, 113601 (2002)
- Bennink et al PRL **92, **033601 (2004)]
- Noncommuting observables
- Gatti et al, PRL 90, 133603 (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

## 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

- kp

- ks

- ki

- Collinearly Phase matched type-II in forward direction: Perfect phase matching

- Imperfect Phase matching

- kz

- kz L=/2 gives an approximate size to the birth place.

- 

- ks

- ki

- kp

## 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

- 2f

- 2f

- Correlated distances from optic axis

## Experiments

- Imaging Layout

- Fourier Imaging Layout

- Bennink et al, PRL **92**, 033601 (2004)

- Howell et al, PRL. **92**, 210403 (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**, 220501 (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. Law](http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=PRLTAO&possible1=Law%2C+C.+K.&possible1zone=author&maxdisp=25&smode=strresults&aqs=true)_ and _[J. H. ](http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=PRLTAO&possible1=Eberly%2C+J.+H.&possible1zone=author&maxdisp=25&smode=strresults&aqs=true)__[Eberly](http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=PRLTAO&possible1=Eberly%2C+J.+H.&possible1zone=author&maxdisp=25&smode=strresults&aqs=true)_ Phys. Rev. Lett. **92**, 127903 (2004)

## Vernam Cipher and QKD

- 11010

- 11010

- Plaintext Message: 01101

-  Ciphertext:

- 10111

- 01101

- 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**, 060503 (2007)

## Alice

- PBS

- 

- A1

- A2

- Time Stamping

- Sync Generator

- RS

- PBS

- 

- B1

- B2

- Bob

- Time Stamping

- PPS

- 50/50 BS

- 50/50 BS

- 50/50 Fiber BS

- PPS – Passive Power Splitter
- RS – Raw Sync

**Energy-Time **
**Quantum Cryptography**
**Simplified View**

- Type-I
- BBO

- 50mW cw
- @390nm

- 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

- 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 l1~l2

- 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!