High Capacity Quantum Imaging

University of Rochester: Robert Boyd, Joseph Eberly, John Howell MIT: Vivek Goyal, Jeffrey Shapiro, Franco Wong Boston University: Alexander Sergienko Jet Propulsion Laboratory: Baris Erkmen

Outline

•  Introduction •  Program overview

–  Team structure and roles –  Coordination –  Summary of technical approach –  Program goals

•  Quantum entanglement characterization •  Channel impairment amelioration •  Compressive/Adaptive ghost imaging

Active Imaging for Standoff Sensing

InPho’s goal >1 bit per photon for image formation

Team Structure High Capacity Quantum Imaging

John Howell University of Rochester

Topic 1 Entanglement Characterization

Eberly – Rochester Erkmen – JPL

Howell – Rochester Shapiro – MIT

Topic 2 Channel Impairment Amelioration

Boyd – Rochester Howell – Rochester

Sergienko – Boston Shapiro – MIT

Topic 3 Compressive Adaptive Ghost Imaging

Boyd – Rochester Erkmen – JPL Goyal – MIT

Howell – Rochester Shapiro – MIT

Wong – MIT

Organization and Reporting

•  Team organization – Semiannual team Skype conferences – Semiannual team meeting – Subteam reports sent to John Howell – Each paper written by team will be provided to

John Howell and ARO/DARPA

Team Membership and Roles •  Robert Boyd (Rochester – Experiment)

–  Ghost imaging (GI) through turbulence, compressive sensing •  Joseph Eberly (Rochester – Theory)

–  Entanglement characterization, turbulence modeling •  Baris Erkmen (JPL – Theory)

–  GI theory and compressive sensing •  Vivek Goyal (MIT – Theory)

–  Compressive and adaptive sensing, information theory •  John Howell (Rochester – Experiment and Theory)

–  GI experiment through turbulence, compressive sensing, entanglement theory and experiment

•  Alexander Sergienko (Boston University – Experiment) –  GI through aberrations and atmospheric turbulence

•  Jeffrey Shapiro (MIT – Theory) –  GI theory, atmospheric propagation, compressive adaptive sensing

•  Franco Wong (MIT – Experiment) –  GI experiments in spatial light modulator (SLM) and computational setups

•  Develop high-ebit entanglement measures for pure and mixed states

•  Model entanglement fragility in the presence of atmospheric turbulence

•  Experimentally study entanglement propagation through thin and thick turbulence

Topic 1: Entanglement Characterization

Topic 2: Channel Impairment Amelioration

•  Study theoretically and demonstrate experimentally ghost imaging through turbulence or aberrations: –  via entanglement –  with Bessel beams –  with classical beams

Topic 3: Compressive Adaptive Ghost Imaging

•  Determine fundamental performance limits of ghost imaging for standoff sensing

•  Establish theory for ghost imaging configurations, propagation effects and signal processing

•  Perform proof-of-principle experiments in laboratory settings

Program Goals and Milestones Phase 1 Phase 2 Phase 3

Task 1 Entanglement Characterization

T: Pure and mixed state measures

T: Propagation thin turbulence E: Schmidt, pixelated entanglement

T:Thick turbulence E:Thin and thick turbulence

Task 2 Impairment Amelioration

T&E: GI Entanglement

T&E: Classical Gaussian beams

T&E: Bessel and other exotic beams

Task 3 Compressive Adaptive GI

T: GI turbulence propagation, CCGI E: Entangled photon GI through turbulence

T&E: Adaptive CCGI T&E: Optimize bpp for adaptive CCGI

T=theory E=experiment CCGI=computational compressive ghost imaging

Topic 1: Entanglement Characterization

•  Theory: Transverse entanglement source measures for pure and mixed states –  Attractive approaches

•  Schmidt decomposition •  Entropy as a monotone •  Fedorov ratio •  Joint, conditional entropy •  Mutual information

Entanglement Characterization: Schmidt Decomposition

Gaussian  correlated  State  in  2D  

Examples  of  transverse  eigenmodes  of  the  biphoton  entangled  state    

Schmidt  or  Singular  Value  Decomposi<on  

Schmidt  Coefficients  

Schmidt  Eigenmodes  

Law and Eberly, PRL. 92, 127903 (2004).

Entanglement Characterization

•  Quantum entropy as a monotone – works for pure states – Multiple ebits of information (d-dimensional

state plus quantum communication) – Trace over single particle use reduced density

matrix yielding quantum entropy

Howell, Bennink, Bentley, and Boyd PRL 92, 210403 (2004)

Bennink, Bentley, Boyd, and Howell PRL 92, 033601 (2004)

Position-Position Correlations Pixel size d >> correlation diameter a

Momentum-Momentum Correlation Pixel size d >> correlation diameter a

Measured Joint Probabilities

Pixel entanglement for 3 or 6 dimensions

O'Sullivan-Hale, Ali Khan, Boyd, and Howell PRL 94, 220501 (2006).

Joint Entropy • Joint entropy as a measure of information for errors and mixedness • Two events, x and y with m and n possible measurement outcomes. The joint entropy is

And single event entropy

Transversely Entangled Photons

• Analog to rotational invariance “correlated plane invariance” • Conditional probabilities computed from

NLC

Joint and Conditional Entropies and Mutual Information

•  Joint entropy preserved for entangled photons is invariant in correlated planes

•  Joint entropy is not preserved for mixed states in the same correlated planes

•  Several avenues of study –  Pixels versus Schmidt modes –  High dimensional mixed state characterization –  Reconciling mode-counting measures –  Measured number of pixels far below Fedorov ratio

Topic 1: Entanglement Characterization Phase 2

•  Theory: –  Model atmospheric turbulence –  Model pure/mixed state through turbulence

•  Experiment: –  Measure eigenmodes of downconversion using

spatial light modulators –  Compare to Fedorov ratio

–  Measure joint and conditional entropies in mutually unbiased bases

•  Correlation area biphoton PSF

•  Model Strehl ratio for correlation area or Fedorov ratio as a function of strength of turbulence

Turbulence Measurements

Topic 1: Entanglement Characterization Phase 3

•  Theory: –  Characterize entanglement after amplitude masks –  Model mixed state joint entropy

•  Experiment: –  Measure dynamic evolution of conditional probabilities

and mutual information in mutually unbiased bases –  Kolmogorov thin and thick turbulence

Turbulence Cell Experiments

Employed in tasks for topics 1, 2 and 3

Entanglement Characterization Milestones

•  Phase I milestones –  Characterize high-ebit entanglement for pure and mixed states

•  Phase II milestones –  Model entanglement through thin atmospheric turbulence –  Perform entanglement measures via SLM in eigenbasis and for

pixelated joint probabilities

•  Phase III milestones –  Model entanglement through thick atmospheric turbulence and

other dynamic media –  Measure entanglement through thin and thick atmospheric

turbulence

Topic 2 (Summary): Image Capacity Enhancement Through Channel Impairment Amelioration

General principle: •  Ghost imaging with correlated or entangled photon pairs

•  One photon samples object, is collected by bucket detector

•  Images formed directly by this photon would be distorted by disruptive effects in transmission channel. Information is lost

•  Second (reference) photon preserves the lost information, allowing undistorted image to be reconstructed

Image Distortion may be introduced by optical system or by propagating medium:

• Aberration

• Dispersion

•  Diffraction

•  Turbulence

Topic 2: Goals

1.  Reduce the number of photons required for high-fidelity identification of each object pixel by ameliorating channel impairments

2.  Develop a theoretical understanding of spatial information gain in correlated ghost imaging due to diffraction and aberration cancellation, as well as turbulence mitigation

3.  Boost prospects for field deployment of correlated ghost imaging with diffraction, aberration, and turbulence noise reduction by using intense correlated Gaussian and Bessel beams. Increase covert capability by extending this technique into IR

Localizing photon’s point of origin by ameliorating channel impairments

Image of point is enlarged and distorted:

Some position information is lost

Image tightly localized around emission point:

Photon carries more information about position

Regular Imaging

Information carried by photon •  Want to estimate location of point in object where photon originated, based on point where it was detected.

•  Conditional probability density for detection position given object position, , is essentially the point spread function centered at

•  Fisher information:

(E = Average over π distribution, T=transpose )

•  measures information gained about object position due to detection of photon.

•  increases when is tightly localized, due to large gradient, .

Example: Diffraction Amelioration

•  Interactions with object or aperture cause bending of light, i.e. photon momentum is changed

•  Result: in far field we see diffraction pattern from object, not actual object shape

•  But second photon is not diffracted by object: retains original momentum information

•  Combining information from two photons via coincidence circuit reconstructs original object without diffraction

Bessel Beams

•  The previous apparatus cancels diffraction due to passage through aperture or around object

•  There is still diffraction during reference beam propagation, causing beam to expand

•  This can be removed by using Bessel beam or other non-diffracting beam

•  Bessel beam formed by placing axicon (conical lens) in Gaussian beam

Topic 2: Milestones

•  Phase I Milestones: –  Demonstrate diffraction and aberration cancellation for GI in reflection via SPDC –  Compute information capacity gain due to diffraction and aberration amelioration

in ghost imaging. Evaluate performance characteristics of GI through atmospheric turbulence

•  Phase II Milestones: –  Demonstrate information capacity increase in ghost imaging by additionally

reducing diffractive spreading with correlated Bessel beams –  Increase robustness and field deployment capability by exploiting intense

correlated beams in ghost imaging –  Provide rigorous Gaussian-state analysis of aberration and diffraction

cancellation in ghost imaging •  Phase III Milestones:

–  Demonstrate non-task-specific imaging with average photon efficiencies ≥ 1 bit/photon using diffraction-cancelled ghost imaging using correlated Bessel beams.

–  Use rigorous Gaussian-state analysis to evaluate the information capacity of ghost imaging through turbulence with aberration and diffraction cancellation

Topic 3: Compressive/Adaptive Ghost Imaging

•  Determine fundamental performance limits of ghost imaging for standoff sensing

•  Establish theory for ghost imaging configurations, propagation effects and signal processing

•  Perform proof-of-principle experiments in laboratory settings

Four Types of Active Ghost Imaging

SPDC

correlator

cw laser

correlator

cw laser

correlator

cw laser correlator

Biphoton ghost imaging

Pseudothermal ghost imaging

Spatial light modulator ghost imaging

Computational ghost imaging

Erkmen & Shapiro, Phys Rev A 2008, 2009; Shapiro, Phys Rev A 2008 Erkmen & Shapiro, Adv Opt Photon 2010

Compressed Sensing Paradigm

•  Typical info acquisition paradigm (esp. science): –  collect raw data with simple system model

(e.g., acquire lowpass portion up to bandwidth B) –  possibly compress, using sophisticated signal model

(e.g., signal is approximately sparse in DWT domain) •  Compressed sensing:

–  collect raw data with sophisticated signal model –  often, better quality at low numbers of measurements

(i.e., more information per photon) –  models of signal and system become more important –  computation becomes much more important –  desirable properties of acquisition may not be obvious

Compressed Sensing for Imaging •  Signal modeling:

–  Sparsity/compressibility is suitably universal (DCT, DWT)

•  Good sensing: isometric on signals of interest –  Leads to restricted

isometry property •  Computation:

–  ℓ1-regularization popular –  Will develop belief

propagation methods for MMSE estimation

Toward Ghost Imaging for Standoff Sensing

•  Compressive ghost imaging

Katz et al. Appl Phys Lett 2009

•  Ghost imaging in reflection

Meyers et al. Phys Rev A 2008

•  Ghost imaging through turbulence

Cheng, Opt Express 2009

Compressive/Adaptive Ghost Imaging Research Program

•  Compressive ghost imaging [Katz et al. (2009)] –  impairment modeling lacking (equality-constr recon) –  signal modeling limited (ℓ1-regularized DCT)

•  Tasks: –  relaxed belief-propagation

algorithm to reflect Poisson signal and noise character

–  Realistic acquisition model including turbulence and reflection

–  study better signal models

Compressive/Adaptive Ghost Imaging Research Program

•  Bayesian analysis of compressed sensing much more predictive than restricted isometry property or coherence-based analyses [Rangan, Fletcher, Goyal (2009)]

•  Adaptive sensing (active learning) techniques are mostly non-Bayesian

•  Tasks: –  Develop average posterior information measure to

evaluate value of a “potential measurement” –  Develop method for (approximate) informativeness

maximization

Ghost Imaging Theory Research Program

•  Theory for key performance metrics –  spatial resolution, image contrast, signal-to-noise ratio

•  Theory for principal ghost imaging configurations –  biphoton, pseudothermal, SLM, computational

•  Theory for vacuum and turbulence propagation –  operation in transmission and in reflection

•  Theory for optimizing photon efficiency –  compressive sensing and adaptive illumination

Adaptive/Compressive SLM Ghost Imaging Research Program

•  single-spatial mode classical light or high-flux SPDC pulses •  speedup and resolution improvement by compressed sensing •  additional enhancement with adaptive algorithms •  SLM implementation of phase patterns

Achieve more efficient GI by compressed sensing and by feedback

Compressive Computational Ghost Imaging Research Program

•  single-arm CGI with computationally derived reference •  3D CGI with different imaging planes without physical

adjustment •  compressed sensing techniques applied directly to CGI with

SLM implementation of phase patterns

Compressed sensing applied to computational ghost imaging (CGI)

Compressive/Adaptive Ghost Imaging •  Phase I milestones

–  ghost imaging theory for vacuum and turbulence propagation –  compressive sensing algorithms for ghost imaging –  SLM and compressive computational ghost imaging experiments –  biphoton ghost imaging experiments through turbulence

•  Phase II milestones –  algorithms for adaptive illumination in compressive SLM and

compressive computational ghost imaging –  SLM and compressive computational ghost imaging experiments

with adaptive illumination

•  Phase III milestones –  theoretical limits on photon efficiency in compressive/adaptive

ghost imaging in SLM and computational configurations –  experiments in compressive/adaptive ghost imaging that

approach fundamental performance limits

Entanglement Characterization Channel Impairment Amelioration

Compressive Adaptive Ghost Imaging

Thank You