Past Research

Wavefunction Measurement        Sources of Single Photons        Detector Tomography        Single-Photon nonlinearities        Weak Measurement        Interaction-Free Measurement        Interaction-Free Measurement contd.        Hardy's Paradox        Super-resolving measurements        Young's Double-Slit

Direct Measurement of the Wavefunction

Chosen as 2nd most important Physics Breakthrough of 2011 by Physics World!

Central to quantum theory, the wavefunction is a complex distribution associated with a Artistic depiction of the wavefunction and apparatusquantum system. Despite its fundamental role, it is typically introduced as an abstract element of the theory with no explicit definition. Rather, physicists come to a working understanding of it through its use to calculate measurement outcome probabilities through the Born Rule. Tomographic methods can reconstruct the wavefunction from measured probabilities. In contrast, we demonstrated a method to directly measure the wavefunction so that its real and imaginary components appear straight on our measurement apparatus. At the heart of the method is a joint measurement of position and momentum that is made possible by weak measurement (see below for what that is). As an example of the method we experimentally directly measured the transverse spatial wavefunction of a single photon. This new measurement  gives the wavefunction a plain and general meaning in terms of a specific set of operations in the lab.
For the non-physicists here are a Semi-technical explanation and Idea for direct measurement of the wavefunctionNon-technical explanation of the idea. For physicists, here is video of presentation I gave along with the slides.

Briefly, the idea is as follows: The average result of a weak measurement of A on state psi and which is then strongly measured to be in state phi is called the weak value. It is given by A_w on the right. Weakly measuring the projector |x><x| followed by a strong measurement with result p=0 results in a weak value proportional to the wavefunction.
How the experiment works:Apparatus for measuring the wavefunction
1. Produce a collection of photons possessing identical spatial wavefunctions by passing photons through an optical fiber.
2. Weakly measure the transverse position by inducing a small polarization rotation at a particular position, x.
3. Strongly measure the transverse momentum by using a Fourier Transform lens and selecting only those photons with momentum p=0.
4. Measure the average polarization rotation of these selected photons. This is proportional to the real part of the wavefunction at x.
5. Measure the average rotation of the polarization in the circular basis. (i.e. difference in the number of photons that have left-hand circular polarization and right-hand circular polarization). This is proportional to the imaginary part of the wavefunction at x.
6. Repeat for all x to scan through the wavefunction.

Engineering Sources of Single Photons

Quantum dot sources of single-photons and entangled photon pairs have theQuantum Dots on Pyramidal Bases potential to produce light on demand (i.e. triggered) and a high repetition rate (i.e. they can be bright). Our InAs dots are particularly nice because they are grown (at NRC) on the top of InP pyramids.  Quantum DotsArrays of these pyramids are fabricated giving us quantum dots grown in array of positions, known to within 10 nm or so.  This means the dots can be placed precisely in the antinode of an optical cavity. The optical cavities vary in type but we use 2-d photon crystal cavities. These are an array of holes, drilled into InP. This structure is made to inhibit optical emission of the quantum dot. A defect in the structure then acts as a cavity, capturing the emitted single-photons and, with careful design, channeling them towards an optical fiber. From that point we can use these photons as carries of information and, in the future, use them to build quantum computers or in quantum cryptography. The picture shows a) an array of dots on a  ridge, b) a single dot on top of pyramid, c) electrical gates isolating a single dot on the ridge, d) a pyramid with gates on it. The gates allow us to manipulate the levels and, thus, the emission of the dot.

Another, more common single photon source is spontaneous parametric downconversion. Joint spectrum and Hong-Ou-Mandel dip of photon pairThis process produces photons in pairs. Thus, the detection of on photon heralds its twin. At Oxford University, we developed and demonstrated a heralded single-photon source that is unique in producing photons in a pure spectral-temporal state (i.e. without time or frequency jitter).  The key to this was eliminating spectral correlations between the two photons in the pair. a) and b) show the theoretical and experimental joint spectra, which exhibit no correlations. Eliminating spectral filtering allow us to achieve one of the best photon generation rates in the field, we also observed the highest quality multi-photon interference yet described in the literature (see Figure c)). Photon quantum logic gates function via this interference and it is currently the limiting factor in their functioning. In addition, the ability to build tailored multi-photon states from single photons relies on the photons being in pure quantum states.

The crucial design element in the source above is the control of momentum The generation of pure spectral state photons from microstructured fibersconservation. There are exciting opportunities to apply the same techniques in other systems, particularly microstructured waveguide systems, where dispersion (and thus the photon momentum) is controlled via the structure, or atomic sources where dispersion is controlled via ancillary lasers (e.g. slow light). These new systems hold the promise of quantum light sources with flexible spectral, spatial and entanglement properties. On the right is a) Energy-momentum conservation curves for photon-pair (signal, s, and idler, i) generation in the photonic crystal fiber shown in b) for positive, zero, and negative birefringence (red, green, blue) and pump frequency ω_p. Points A to F bound regions along the curves in which spectrally pure photons can be generated. c) The theoretical joint spectrum of the photon pair at point D.

Measuring Measurement: Tomography of detectors

Measurement plays an important role in quantum physics, causing the collapse of the wavefunction. Yet, given a particular device, how do we know what measurement it actually does? Using the technique of tomography, for the first time we reconstruct a full “image” of the measurement a detector performs. Tomography is used, for example, in CAT scans, reconstructing full three-dimensional images of the body by taking a series of two-dimensional pictures from different angles. A generalization of this idea allows us to reconstruct the measurement performed by a new type of quantum detector that can count photons.  

detector tomography
Despite our belief that quantum physics should describe everything, the measurement process itself cannot be rigorously modeled by quantum dynamics alone. We approach this problem from the opposite direction with a general procedure for characterizing a detector by the information it gives us. Remarkably, this finite set of information is enough to predict the detector’s response to any possible input. We successfully characterized a photon number counting detector, an exciting new device at the ultimate limit of light detection. The subjects of our tomography
The tomography procedure:
1. Create a series of coherent states with different intensities.
2. Measure the intensities of the coherent states with the power meter in a monitor arm.
3.  For each intensity measure the rates of each detector outcome.
4.  These rates allow us to reconstuct the set of matrices describing the detector action (The detector POVM).
The Wigner functions on the right depict the POVM element for the one click outcome of an Avalanche Photodiode and a Photon Number Resolving Detector.
Detector Wigner functions

Single-photon nonlinearities

Entanglement and interference are often identified as two of the defining phenomena of quantum physics.  Quantum computation algorithms rely on both to perform classically impossible tasks such as factoring large numbers.  In these algorithms entanglement is created in two-particle quantum logic gates, but in optics we don't have a strong enough interaction (nonlinearity) between photons to create these. Can quantum interference be used as the nonlinear interaction in a quantum gate?  Usually, the answer appears to be no. In a series of experiments, we show that interference can be used to enhance a pre-existing nonlinearity by eleven orders of magnitude up to the level where single photons can scatter from each other.

A switch that allows a single photon to control the transmission of another photon
In two experiments, spontaneous parametric down-conversion is suppressed or enhanced by interference with two photons from a classical local-oscillator field. When there is destructive interference, any photon pairs from the laser fields undergo sum-frequency generation and are removed from the beams. In this sense, there is an effective two-photon nonlinearity that is significant even when only two photons are present at a time.  While the underlying physics is that of incipient two-mode squeezing, it comprises qualitatively new effects when studied at the single-photon level.  For instance, it functions essentially as an absorptive two-photon switch.  If the phase of the pump is shifted relative to the incoming photons, then it functions similar to a conditional-phase quantum logic gate; if and only if both photons are present will the output state be shifted in phase.  We have found uses for these effects in a universal Bell-state analyzer and in Hardy's Paradox (see below).

In a third experiment, we show that exchange effects, responsible for Pauli exclusion among other things, can modify photon-photon interactions from what would be predicted for distinguishable particles.  In collaboration with Geoff Lapaire and John Sipe, we demonstrate an exchange effect with two single-photon states that involves real transitions in media.  Interference occurs between the two paths in which both photons can be absorbed.  When the photons are completely overlapped or completely separated in time, the absorption is that of two distinguishable particles.  But for intermediate times, the absorption is unexpectedly suppressed as if there was an effective interaction between the two photons.
An effective nonlinearity in two-photon absorption

Measurement in quantum mechanics

Weak Measurement

Weak measurement was originally proposed by Aharanov, Albert and Vaidman as an extension to the standard von Neumann ("strong") model of measurement.   A weak measurement can be performed by sufficiently reducing the coupling between the measuring device and the measured system.   In this case, the pointer of the measuring device begins in a state with enough position uncertainty that any shift induced by the weak coupling is insufficient to distinguish between the eigenvalues of the observable in a single trial.   While at first glance it may seem strange to desire a measurement technique that gives less information than the standard one, recall that the entanglement generated between the quantum system and measurement pointer is responsible for collapse of the wavefunction.  Furthermore, if multiple trials are performed on an identically-prepared ensemble of systems, one can measure the average shift of the pointer to any precision -- this average shift is called the weak value.  A surprising characteristic of weak values is that they need not lie within the eigenvalue spectrum of the observable and can even be complex.  On the other hand, an advantage of weak measurements is that they do not disturb the measured system nor any other simultaneous weak measurements or subsequent strong measurements, even in the case of non-commuting observables.  This makes weak measurements ideal for examining the properties and evolution of systems before post-selection and might enable the study of new types of observables.  Weak measurements have been used to simplify the calculation of optical networks in the presence of polarization-mode dispersion by Gisin, applied to slow- and fast-light effects in birefringent photonic crystals by Chiao, and bring a new, unifying perspective to the tunneling-time controversy as Steinberg showed.  Wiseman used weak values to physically explain the results of a cavity QED experiment.  Weak measurement can be considered the best estimate of an observable in a pre and post-selected system.
Weak measurement

Interaction-Free Measurement

This is a type of measurement proposed by Elitzur and Vaidman in 1991 in which the presence of an object can be discerned without it being disturbed.  The simplest example of this type of measurement is a Mach-Zehnder interferometer (shown to the right) that is aligned so that all the photons entering the interferometer leave through the "bright" output port C (and none through D, the "dark" output port).  Both the first and last beam splitters (BS1 and BS2) reflect and transmit with 50% probability.  Consider what would happen if an object, say a highly light-sensitive bomb, were positioned in the right-hand path (I).  If we let only one photon enter the interferometer, it has a 50% chance of taking the left path (O), thus avoiding triggering the bomb.  It then has a 50% chance of exiting through port C, which conveys no information about the presence of the bomb, and a 50% chance of exiting through the previously dark port, D.  The latter then indicates the presence of the bomb even though the photon did not interact with it.  In total, one has a 25% chance of success (detecting the bomb without detonating it), 25% chance of receiving no knowledge and a 50% chance of failure (an explosion). We setup two Interaction-Free Measurements of this type in our lab to show that sometimes their results can be in contradiction to what any reasonable person would logically expect (see Hardy’s Paradox below).

A logical paradox in quantum mechanics

Interaction-Free Measurement continued


The above analysis of Interaction-Free Measurement (IFM) naturally leads one to the question of whether the photon was in the bomb arm in trials which were successful.  If not, how could the bomb's presence influence which output port the photon took.  If so, why did the bomb not explode and how did the photon get to the detector? As the next step towards Hardy's Paradox, consider what would happen if the bomb were in an equal superposition of two position states, one in the arm I and one outside of the interferometer?  In this scenario, if a photon is detected in the dark port D, the state of the bomb collapses to a state in the arm I.  On the other hand, if the photon exits from the bright port C, no collapse occurs.  Evidently, IFM's can also detect objects which exhibit wave-like properties such as a superposition of states.


An interaction-free measurement still works when acting on an object in a superposition


Hardy's Paradox

Hardy's Paradox takes this scenario one step further.  Lucien Hardy envisioned using an electron as the test particle in the IFM and a positron as the bomb (see Figure 2).   The positron is placed into a superposition by a 50/50 beamsplitter.  The extra step is that after the electron and positron would have annihilated each other in region W, the positron superposition is recombined at another 50/50 beamsplitter.  We now have a symmetric experiment where both the electron and positron enter Mach-Zehnder interferometers (distinguished by the subscripts - and + respectively).  Each interferometer performs an IFM on the particle in the other interferometer. 
The original form of Hardy's Paradox is two nested IFM's. The electron is the object for the positron interferometer and vice versa. A coincident detection at D₊ and D₋ implies both particles were in region W and should have annihilated each other.
If one detects the electron at D this implies that the positron was in I and conversely, a positron at D implies the electron was in I. Surprisingly, it is possible for both the electron and positron to exit their respective D port in the same trial, implying they were each in their respective I arm.  Consequently, this appears to answer the question in the last section of which arm the electron was in before a successful measurement:  The arm with the object in it.  But then how did the electron and positron avoid annihilation and yet arrive at our detectors?.  These two contradictory conclusions arise from the false classical idea that a particle has determinate properties, such as position, before observation.  This is the essence of Hardy's Paradox and the form of it that experimentally implement.  Instead of electron, and positron we use H and V polarized photons and we use an absorptive two-photon switch in region W (see above). Experimental setup for Hardy's Paradox

Weak Measurement in Hardy's Paradox

The IFM dark port detections only allow us to make counterfactual claims about which MZI arms the photons were in.  Any strong measurements designed to verify these claims would disturb at least one of the MZIs, so that it would no longer act as an IFM.  Thus the paradox would be destroyed.  On the other hand, because they do not disturb the system, weak measurements can test these claims.  Aharonov et al. used weak measurement to find which Mach-Zehnder arms the photons were in, individually and as a pair, in the subensemble of systems for which the IFMs give their paradoxical result.  The resulting weak values for the single-photon occupation numbers (eg. |O+>< O+|) were found to be NO+ = 0, NO- = 0, NI+ = 1, NI- = 1, where I is the inner arm and O is the outer arm.  These simply agree with the results of the IFMs:  Individually, the photons were in the inner arms.  However, the impossibility of detecting two photons which were both in the inner arms (they would have “annihilated”) means that NI,I =0.  From this we find that NO,I  and NI,O equal one since we already know 1= NI+ = NI,I + NI,O for instance.  But since we know neither photon was in the outer arms, 0= NO+ = NO,I + NO,O, implying the strange result NO,O = –1.  We experimentally implemented the proposed weak measurements using a small polarization rotation &theta; in the relevant arm(s). In particular, we found that when weakly measuring NO,O the photons emerged at the dark ports with a polarization rotated in the opposite direction from what the rotation &theta;, we imposed.  The exact measurement came out to roughly NO,O = –0.75.

Super-resolving measurements

Resolution is a limiting factor for many scientific endeavors.  For example, electron microscopes can see smaller objects than light microscopes because they have higher resolution.  This paper describes a way to break through some resolution limits using quantum-mechanical entanglement.  By entangling three photons in the state |30>+|03>, we were able to increase their resolution threefold over the single-photon limit.  This concept in general entails creating a quantum state containing N photons, where the N photons are always found together in one interferometer arm or the other, sometimes called a "high-noon" state |N0>+|0N>.  Essentially, the grouped N photons of wavelength lambda behave like one photon of wavelength lambda/N.   In comparison to a laser beam containing on average N photons, this concept offers an improvement in resolution of square root N.
In comparison to a laser beam containing on average N photons, this concept offers an improvement in resolution of square root N.  Typically to create this state from unentangled photons one requires strong nonlinearities at the single-photon level.  We create our three-photon state from unentangled photons without strong nonlinearities by using a generalizable technique that combines linear optics transformations with post-selection. For this reason, this could be extended to arbitrary resolution improvement with some future advances in single-photon production.  In general this work has possible ramifications for a variety of resolution- and/or sensitivity-limited measurements.  Specific applications of this state have been suggested for spectroscopy, precise metrology, and optical lithography.
Data from our three-noon experimenta

Does a which-way measurement kick the particle? 

Young's double-slit revisited.

The double-slit was one of the first experiments with which scientists realized that quantum mechanics has some very odd properties. In particular, when a particle has a chance of travelling through two slits, the particles always arrive at a screen on the other side in a peculiar pattern. This pattern is set of parallel lines, like the teeth of comb. However, if there is any way of knowing which slit the particle travelled through these lines disappear, replaced with the pattern one gets with a single slit (containing no lines). Bohr said this is because whatever device you used to measure which slit the particle went through also gives the particle a little kick, blurring out the lines. This became to be known as the Heisenberg uncertainty principle. Einstein came up with many ingenious attempts to try and avoid this fact but in the end failed. In the 1990s, Marlan Scully and coworkers seemed to finally come up with measurement device that got around Bohr's rule. It was controversial and nobody could directly test whether it kicked the particles or not. This is where our experiment comes in. See the amusing video to the left for more information about the famous Young's double-slit experiment.

This paper addresses the longstanding debate over how measurement may destroy interference, which dates back to Bohr and Einstein  in the 1920s. It is known that any which-way measurement destroys interference, but it has remained controversial what mechanism enforces this complementarity. In particular, this was debated in the pages of Nature through the 1990s. One camp (Storey et al.) asserted that the mechanism is always momentum disturbance, in accordance with Heisenberg’s uncertainty principle, as Bohr had argued. The other camp (Scully et al.) argued that no such disturbance is necessary, so that complementarity is not enforced by the uncertainty principle. In this paper we present the first measurement of the momentum transfer caused by a which-path measurement.  With a tilted glass sliver we weakly measure the momentum p1 of a photon after the double slit. We then do a which path measurement with a half-wave plate (HWP). After, we measure the final momentum p2 of the photons with a CCD camera after a lens.
The difference p2-p1=q the momentum transfer. Repeating this many times, we measure the probability distribution P(q) of momentumtransfers q, but find that this distribution is negative for some values of q because of the quantum nature of the momentum transfer. This allows the distribution we measure (a weak-valued distribution) to satisfy theoretical claims made by both sides of the debate: it has a width of order that expected by the uncertainty principle, but has a variance consistent with zero (see figure). Thus our experiment has the potential to reconciles the two camps, by fully demonstrating the connections between three of the most fundamental issues in quantum mechanics: complementarity, measurement, and the uncertainty principle. Moreover, this was done using the archetype of quantum mechanics experiments — Youngs double-slit — which is familiar to all physicists. Data from our three-noon experimenta