# Refereed Publications

• Weak Value Amplification Can Outperform Conventional Measurement in the Presence of Detector Saturation, J. Harris, R.W. Boyd, J.S. Lundeen (Submitted 2016).
• Direct measurement of the photon's spatial wave function, M. Mirhosseini, J.S. Lundeen, R.W. Boyd, Chapter in Advances in Quantum Photonics, Springer (Accepted 2016).
• Weak-value measurements can outperform conventional measurements, O. Magaña, J. Harris, J.S. Lundeen, R.W. Boyd, Physica Scripta (Accepted 2016).
• Measurement of the transverse electric field profile of light by a self-referencing method with direct phase determination, C. Bamber, B. Sutherland, A. Patel, C. Stewart, and J. S. Lundeen, Optics Express, 20, 2034 (2012). Abstract

# Presentations and Proceedings

• Quantum Radiometry at NRC, CORM, Ottawa (2012).
• Seeing is Believing: Direct Observation of the Wavefunction, MIT, Boston (2012).
• Seeing is Believing: Direct Observation of the Wavefunction, Q+ Google Hangout Series (2012). Video is HERE.
• Quantum Metrology with Light, Frontiers in Optics, San Jose, (2011)
• The Search for Fock, Physics Dept.,  University of Delaware Newark, Delaware (2011).
• Single Photons, from Generation to Detection, Physics Dept.,  University of Ottawa (2011).
• Observation of the Quantum Wavefunction, Physics Dept.,  University of Toronto (2011).
• Production and Characterization of Single-Photon States of Light, Cross-Border Workshop on Laser Science, Waterloo  (2011).
• The Search for Fock, Physics Dept., University of Waterloo (2010).
• Production and Characterization of Single-Photon States of Light, Cross-Border Workshop on Laser Science, Waterloo (2010).
• Bridging Particle and Wave Responsivity in a Phase-Sensitive Photon-Number Detector, Single Photon Workshop, Boulder, Colorado (2009).
• Photon Pair Generation in Birefringent Fiber: A Route to Better Photons, Frontiers in Optics, San Jose (2009)
• Designed photons from birefrigent waveguides, Conference on Quantum Information and Quantum Control III, Toronto (2009).
• The search for the perfect photon, Institute for Quantum Computing, University of Waterloo (2009).
• Quantum Metrology with light, National Research Council, Institute for National Measurement Standards, Ottawa (2008).
• J. S. Lundeen et al. The Generation of uncorrelated photon-pairs in an optical fibre, CLEO San Jose (2008).
• K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Conditional Coherence via phase-sensitive post-selection”, CQO8, Rochester (2001). (Poster) The summary of the poster (to be published in the proceedings) is here .

# Abstracts

### Observing Dirac's Classical Phase Space Analog to the Quantum State

C. Bamber and J.S. Lundeen
In 1945, Dirac attempted to develop a "formal probability" distribution to describe quantum operators in terms of two non-commuting variables, such as position x and momentum p [Rev. Mod. Phys. 17, 195 (1945)]. The resulting quasi-probability distribution is a complete representation of the quantum state and can be observed directly in experiments. We measure Dirac's distribution for the quantum state of the transverse degree of freedom of a photon by weakly measuring transverse x so as to not randomize the subsequent p measurement. Further, we show that the distribution has the classical-like feature that it transforms (e.g., propagates) according to Bayes' law.

### We experimentally characterize the performance of the electron multiplying charge coupled device (EMCCD) camera for the detection of single photons. The tests are done with the photon pairs generated from parametric downconversion (PDC). The gain, time response and noise performance of the EMCCD are characterized. In addition we attempt to use the camera to measure the spatial correlations of PDC. The results reveal the capabilities and limits of the EMCCD as a single-photon-detector array for the applications of quantum optics, astronomy and microscopy.

Experimental joint weak measurement on a photon pair as a probe of Hardy's Paradox

J. S. Lundeen, A. M. Steinberg

It has been proposed that the ability to perform joint weak measurements on post-selected systems would allow us to study quantum paradoxes. These measurements can investigate the history of those particles that contribute to the paradoxical outcome. Here, we experimentally perform weak measurements of joint (i.e. nonlocal) observables. In an implementation of Hardy's Paradox, we weakly measure the locations of two photons, the subject of the conflicting statements behind the Paradox. Remarkably, the resulting weak probabilities verify all these statements but, at the same time, resolve the Paradox.

A nonclassical light source is used to demonstrate experimentally the absolute efficiency calibration of a photon-number-resolving detector. The photon-pair detector calibration method developed by Klyshko for single-photon detectors is generalized to take advantage of the higher dynamic range and additional information provided by photon-numberresolving detectors. This enables the use of brighter twin-beam sources including amplified pulse pumped sources, which increases the relevant signal and provides measurement redundancy, making the calibration more robust.

Bridging particle and wave sensitivity in a detector of configurable positive operator-valued measures

Graciana Puentes, Jeff S. Lundeen, Matthijs P. A. Branderhorst, Hendrik B. Coldenstrodt-Ronge, Brian J. Smith, Ian A. Walmsley

We report an optical detector with tunable positive operator-valued measures (POVMs). The device is based on a combination of weak-field homodyne techniques and photon-number-resolving detection. The resulting POVMs can be continuously tuned from Fock-state projectors to a variety of phase-dependent quantum-state measurements by adjusting different system parameters such as local oscillator coupling, amplitude and phase, allowing thus not only detection but also preparation of exotic quantum states. Experimental tomographic reconstructions of classical benchmark states are presented as a demonstration of the detector capabilities.

Tailored photon-pair generation in optical fibers

Offir Cohen, Jeff S. Lundeen, Brian J. Smith, Graciana Puentes, Peter J. Mosley, Ian A. Walmsley

We experimentally control the spectral structure of photon pairs created via spontaneous four-wave mixing in microstructured fibers. By fabricating fibers with designed dispersion, one can manipulate the photons' wavelengths, joint spectrum, and, thus, entanglement. As an example, we produce photon-pairs with no spectral correlations, allowing direct heralding of single photons in pure-state wave packets without filtering. We achieve an experimental purity of $85.9\pm1.6%$, while theoretical analysis and preliminary tests suggest 94.5% purity is possible with a much longer fiber.

Optimal Quantum Phase Estimation

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, I. A. Walmsley

By using a systematic optimization approach we determine quantum states of light with definite photon number leading to the best possible precision in optical two mode interferometry. Our treatment takes into account the experimentally relevant situation of photon losses. Our results thus reveal the benchmark for precision in optical interferometry. Although this boundary is generally worse than the Heisenberg limit, we show that the obtained precision beats the standard quantum limit thus leading to a significant improvement compared to classical interferometers. We furthermore discuss alternative states and strategies to the optimized states which are easier to generate at the cost of only slightly lower precision.

### Tomography of Quantum Detectors

J.S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K.L. Pregnell, Ch. Silberhorn, T.C. Ralph, J. Eisert, M.B. Plenio, and I.A. Walmsley

Measurement connects the world of quantum phenomena to the world of classical events. It has both a passive role—in observing quantum systems—and an active one, in preparing quantum states and controlling them. In view of the central status of measurement in quantum mechanics, it is surprising that there is no general recipe for designing a detector that measures a given observable. Compounding this, the characterization of existing detectors is typically based on partial calibrations or elaborate models. Thus, experimental specification (that is, tomography) of a detector is of fundamental and practical importance. Here, we present the realization of quantum detector tomography. We identify the positive-operator-valued measure describing the detector, with no ancillary assumptions. This result completes the triad, state, process and detector tomography, required to fully specify an experiment. We characterize an avalanche photodiode and a photon-number-resolving detector capable of detecting up to eight photons. This creates a new set of tools for accurately detecting and preparing non-classical light.

Conditional preparation of single photons using parametric downconversion: A recipe for purity

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, I. A. Walmsley

In an experiment reported recently [Phys. Rev. Lett. 100, 133601 (2008)], we demonstrated that, through group velocity matched parametric downconversion, heralded single photons can be generated in pure quantum states without spectral filtering. The technique relies on factorable photon pair production, initially developed theoretically in the strict collinear regime; focusing - required in any experimental implementation - can ruin this factorability. Here we present the numerical model used to design our single photon sources and minimize spectral correlations in the light of such experimental considerations. Furthermore, we show that the results of our model are in good agreement with measurements made on the photon pairs and give a detailed description of the exact requirements for constructing this type of source.

Focusing on factorability: space-time coupling in the generation of pure heralded single photons

Peter J. Mosley,  Jeff S. Lundeen,  Brian J. Smith, and Ian A. Walmsley

The interference of single heralded photons from multiple parametric downconversion sources requires photon pairs in factorable states. Typically, these are selected from an ensemble of pairs by narrow filters that remove any exhibiting correlations. In order to eliminate these lossy filters, factorable photon pairs free from any spatio-temporal correlations must be created directly at each source. This requires careful engineering of the group velocity dispersion of the nonlinear crystal in which pair generation takes place. Several schemes have been proposed to do this in the plane-wave regime, but in a realistic experiment one must also take into account the effects of focusing on the two-photon state. Focusing leads to space-time coupling between the pump structure and the downconverted pairs that has the potential to reduce their factorability, but if carefully managed can actually increase it. In this paper, we consider some of the effects of focusing and their consequences for pure single photon generation.

A proposed testbed for detector tomography

H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith,  W. Mauerer, C. Silberhorn, J. Eisert, M. B. Plenio, I. A. Walmsley

Measurement is the only part of a general quantum system that has yet to be characterized experimentally in a complete manner. Detector tomography provides a procedure for doing just this; an arbitrary measurement device can be fully characterized, and thus calibrated, in a systematic way without access to its components or its design. The result is a reconstructed POVM containing the measurement operators associated with each measurement outcome. We consider two detectors, a single-photon detector and a photon-number counter, and propose an easily realized experimental apparatus to perform detector tomography on them. We also present a method of visualizing the resulting measurement operators.

### Heralded Generation of Ultrafast Single-Photons in Pure Quantum States

Peter J. Mosley, Jeff S. Lundeen, Brian J. Smith, Piotr Wasylczyk, Alfred B. U'Ren, Christine Silberhorn, and Ian A. Walmsley

We present an experimental demonstration of heralded single photons prepared in pure quantum states from a parametric down-conversion source. It is shown that, through controlling the modal structure of the photon pair emission, one can generate pairs in factorable states and thence eliminate the need for spectral filters in multiple-source interference schemes. Indistinguishable heralded photons were generated in two independent spectrally engineered sources and Hong-Ou-Mandel interference observed between them without spectral filters. The measured visibility of 94.4% sets a minimum bound on the mean photon purity.

### A short perspective on long crystals: broadband wave mixing and its application to ultrafast quantum optics

P. Wasylczyk, A. B. U’Ren, P. Mosley, J. Lundeen, M. P. A. Branderhorst, S.-P. Gorza, A. Monmayrant, A. Radunsky and I. A. Walmsley

We present an overview of recently developed ideas in ultrafast nonlinear optics, and describe three applications where these ideas have had an impact. A closer look at three wave mixing of broadband electromagnetic fields in birefringent nonlinear crystals shows that not only phase matching, but also group velocity matching is important for understanding the process of up- and down-conversion with ultrashort laser pulses. In fact the higher-order dispersion of nonlinear crystalline materials provides an underused degree of freedom that allows tailoring the interaction so that it is suitable for a number of different applications. We analyse the processes of parametric downconversion for the production of pure single photon states, and upconversion for ultrashort pulse characterization and for quantum state and process tomography in molecules.

### Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber

K. Garay-Palmett, H. J. McGuinness, Offir Cohen, J. S. Lundeen, R. Rangel-Rojo, M. G. Raymer, C. J. McKinstrie, S. Radic, A. B. U’Ren and I. A.Walmsley

We study theoretically the generation of photon pairs by spontaneous four-wave mixing (SFWM) in photonic crystal optical fiber. We show that it is possible to engineer two-photon states with specific spectral correlation (entanglement'') properties suitable for quantum information processing applications. We focus on the case exhibiting no spectral correlations in the two-photon component of the state, which we call factorability, and which allows heralding of single-photon pure-state wave packets without the need for spectral post filtering. We show that spontaneous four wave mixing exhibits a remarkable flexibility, permitting a wider class of two-photon states, including ultra-broadband, highly-anticorrelated states.

### Classical dispersion-cancellation interferometry

K. J. Resch, P. Puvanathasan, J. S. Lundeen, M. W. Mitchell, K. Bizheva

Even-order dispersion cancellation, an effect previously identified with frequency-entangled photons, is demonstrated experimentally for the first time with a linear, classical interferometer. A combination of a broad bandwidth laser and a high resolution spectrometer was used to measure the intensity correlations between anti-correlated optical frequencies. Only 14% broadening of the correlation signal is observed when significant material dispersion, enough to broaden the regular interferogram by 4250%, is introduced into one arm of the interferometer.

### A double-slit which-way' experiment addressing the complementarity-uncertainty debate

R. Mir, J.S. Lundeen, M.W. Mitchell, A.M. Steinberg, H. M. Wiseman, and J. L. Garretson

Since the Einstein-Bohr debates, it has been known that any ‘which-path’ measurement destroys interference, but it has remained controversial what mechanism enforces this ‘complementarity.’ One camp asserts that the mechanism is momentum disturbance, in accordance with the uncertainty principle, while another argues that no such disturbance is necessary. We present the first direct measurement of the momentum transfer caused by a ‘which-path’ measurement, and reconcile the differing points of view.

### Comment on “Linear optics implementation of weak values in Hardy's paradox”

J.S. Lundeen, K.J. Resch, and A.M. Steinberg

A recent experimental proposal [S.E. Ahnert and M.C. Payne, Phys. Rev. A 70, 042102 (2004)] outlines a method to measure the weak value predictions of Aharonov in Hardy's paradox. This proposal contains flaws such as the state preparation method and the procedure for carrying out the requisite weak measurements. We identify previously published solutions to some of the flaws.

### Practical measurement of joint weak values and their connection to the annihilation operator

J.S. Lundeen and K.J. Resch

Weak measurements are a new tool for characterizing post-selected quantum systems during their evolution. Weak measurement was originally formulated in terms of von Neumann interactions which are practically available for only the simplest single-particle observables. In the present work, we extend and greatly simplify a recent, experimentally feasible, reformulation of weak measurement for multiparticle observables [Phys. Rev. Lett. 92 (2004) 130402]. We also show that the resulting “joint weak values” take on a particularly elegant form when expressed in terms of annihilation operators.

### Super-resolving phase measurements with a multi-photon entangled state

M.W. Mitchell, J.S. Lundeen, and A.M. Steinberg

Interference phenomena are ubiquitous in physics, often forming the basis of demanding measurements. Examples include Ramsey interferometry in atomic spectroscopy, X-ray diffraction in crystallography and optical interferometry in gravitational-wave studies. It has been known for some time that the quantum property of entanglement can be exploited to perform super-sensitive measurements, for example in optical interferometry or atomic spectroscopy. The idea has been demonstrated for an entangled state of two photons, but for larger numbers of particles it is difficult to create the necessary multiparticle entangled states. Here we demonstrate experimentally a technique for producing a maximally entangled three-photon state from initially non-entangled photons. The method can in principle be applied to generate states of arbitrary photon number, giving arbitrarily large improvement in measurement resolution. The method of state construction requires non-unitary operations, which we perform using post-selected linear-optics techniques similar to those used for linear-optics quantum computing.

### Photon-exchange effects on photon-pair transmission

K.J. Resch, G.G. Lapaire, J.S. Lundeen, J.E. Sipe, A.M. Steinberg

It has been proposed that photon-exchange effects associated with virtual atomic absorption could have widespread application in quantum information processing. Here we investigate simpler exchange effects associated with real absorption as modeled by an equivalent linear optical filter. Using nonclassical pairs of photons with variable time separation, we observe a maximum suppression of pair transmission by at least 5% with respect to the result for independent photons.

### Experimental Realization of the Quantum Box Problem

K.J. Resch, J.S. Lundeen, A.M. Steinberg

The three-box problem is a gedankenexperiment designed to elucidate some interesting features of quantum measurement and locality. A particle is prepared in a particular superposition of three boxes, and later found in a different (but nonorthogonal) superposition. It was predicted that appropriate “weak” measurements of particle position in the interval between preparation and post-selection would find the particle in two different places, each with certainty. We verify these predictions in an optical experiment and address the issues of locality and of negative probability.

### Experimental application of decoherence-free subspaces in a optical quantum computing algorithm

M. Mohseni, J.S. Lundeen, K.J. Resch, and A.M. Steinberg

For a practical quantum computer to operate, it is essential to properly manage decoherence. One important technique for doing this is the use of "decoherence-free subspaces" (DFSs), which have recently been demonstrated. Here we present the first use of DFSs to improve the performance of a quantum algorithm. An optical implementation of the Deutsch-Jozsa algorithm can be made insensitive to a particular class of phase noise by encoding information in the appropriate subspaces; we observe a reduction of the error rate from 35% to 7%, essentially its value in the absence of noise.

### Practical creation and detection of polarization Bell states using parametric down-conversion

K.J. Resch, J.S. Lundeen, and A.M. Steinberg

The generation and detection of maximally-entangled two-particle states, 'Bell states,' are crucial tasks in many quantum information protocols such as cryptography and teleportation.  Unfortunately, they require strong inter-particle interactions lacking in optics.  For this reason, it has not been possible to perform complete Bell state determination in optical systems.  In this work, we show how a recently developed quantum interference technique for enhancing optical nonlinearities can make Bell state measurement possible.  We also discuss weaknesses of the scheme including why it cannot be used for unconditional teleportation.

### A conditional-phase switch at the single-photon level

K.J. Resch, J.S. Lundeen, and A.M. Steinberg

We present an experimental realization of a two-photon conditional-phase switch, related to the `c-phi'' gate of quantum computation. This gate relies on quantum interference between photon pairs, generating entanglement between two optical modes through the process of spontaneous parametric down-conversion (SPDC). The interference effect serves to enhance the effective nonlinearity by many orders of magnitude, so it is significant at the quantum (single-photon) level. By adjusting the relative optical phase between the classical pump for SPDC and the pair of input modes, one can impress a large phase shift on one beam which depends on the presence or absence of a single photon in a control mode.

### Quantum state preparation and conditional coherence

K.J. Resch, J.S. Lundeen, and A.M. Steinberg

It is well known that spontaneous parametric down-conversion can be used to probabilistically prepare single-photon states.  We have performed an experiment in which arbitrary superpositions of zero- and one-photon states can be prepared by appropriate post-selection.  The optical phase, which is meaningful only for superpositions of photon number, is related to the relative phase between the zero- and one-photon states.  Whereas the light from spontaneous parametric down-conversion has an undefined phase, we show that this technique collapses one beam to a state of well-defined optical phase when a measurement succeeds on the other beam.

### Electromagnetically induced opacity for photon pairs

K.J. Resch, J.S. Lundeen, and A.M. Steinberg

We show that quantum interference with classical beams may be used to suppress or enhance the rate of spontaneous photon-pair production from a nonlinear crystal.  Sum-frequency generation of the classical beams is simultaneously enhanced or suppressed via interference with a classical pump.  In the extreme case, a crystal which is transparent to individual photons may block all photon pairs, converting them to 2\omega.  This constitutes a coherent nonlinear response at the single-photon level, enhanced by a factor of approximately 10^10.  We present experimental data and a theoretical description, and attempt to delineate the classical and quantum aspects of these effects.

### Nonlinear optics with less than one photon

K.J. Resch, J.S. Lundeen, and A.M. Steinberg

We demonstrate suppression and enhancement of spontaneous parametric down-conversion via quantum interference with two weak fields from a local oscillator (LO).  Effectively, pairs of LO photons upconvert with high efficiency for appropriate phase settings, exhibiting an effective nonlinearity enhanced by at least 10 orders of magnitude.  This constitutes a two-photon switch, and promises to be applicable to a wide variety of quantum nonlinear optical phenomena.

### Total reflection cannot occur with a negative time delay

K.J. Resch, J.S. Lundeen, and A.M. Steinberg

It was recently predicted [1] that the frustrated Gires-Tournois interferometer exhibits a negative delay time for reflection.  Given its 100% reflectivity, this appears to contradict causality.  We demonstrate that an additional, positive, contribution comes from consideration of the transverse dimension.  We prove that this contribution is always large enough to enforce a positive total delay.

### Experimental observation of nonclassical effects on single-photon detection rates

K.J. Resch, J.S. Lundeen, and A.M. Steinberg

It is often asserted that quantum effects can be observed in coincidence detection rates or other correlations, but never in the rate of single-photon detection. We observe nonclassical interference in a singles rate, thanks to the intrinsic nonlinearity of photon counters. This is due to a dependence of the effective detection efficiency on the quantum statistics of the light beam. Such measurements of detector response to photon pairs promise to shed light on the microscopic aspects of silicon photodetectors, and on general issues of quantum measurement and decoherence.