Authors: Stefanie Barz*, W. Steve Kolthammer, Ian Walmsley
Affiliation:
Clarendon Laboratory, Department of Physics, University of Oxford, UK.
*Current address: Institute for Functional Matter and Quantum Technologies and Center for Integrated Quantum Science and Technology IQST, University of Stuttgart, Germany.
The exchange symmetry of identical particles is a foundational principle in quantum physics. If two particles are identical, then the outcomes of any measurements of the system should be unaffected if the particles are swapped. A theory of such particles should therefore give the same predictions if all of the degrees of freedom of the two particles are exchanged. Indeed, this exchange symmetry provides a useful classification for quantum particles: Bosons are described by many-particle wavefunctions that are symmetric under such exchange, whereas fermions have anti-symmetric wavefunctions.
A simple example of invariance under exchange is illustrated in figure 1. Two particles are characterized by their position (p) and shape (s). We can label their joint state as (p, s; p', s'), as shown in Fig. 1a. If the particles are exchanged, then the new joint state is (p', s'; p, s), as in Fig. 1b. It is obvious that there is no difference between these two states. What happens, though, if we were to exchange only some of the labels? Say, the positions of the particles, but not their shapes. In that case the state would be (p, s'; p', s), as in Fig.1c. It’s clear that this partially-exchanged state is different than the original state. Now, let’s imagine the case when the two particles have the same shape, so s=s'. In that case we might say that the particles were indistinguishable, since exchanging the other degree of freedom would leave the state unchanged.
Fig 1. Full and partial exchange of the degrees of freedom of two particles. The single-particle states are denoted by position, left (l) or right (r), and shape, triangle (t) or star (s).
In fact, most particles have more complicated specifications of their states, and this allows a third option: partially distinguishable particles. For instance, imagine two photons in non-overlapping positions (left and right) with identical colour, or spectra: they are both red, say. These photons are would be indistinguishable, since exchange of their positions results in no change to the two-photon state. In contrast, let’s say the photons have non-overlapping spectra: the photon on the left is blue, while the other is red. If we now exchange their positions, we obtain a state that has no overlap with the initial configuration. Therefore, a measurement can completely distinguish whether or not the exchange occurred. The two photons are said to be distinguishable with respect to exchange of position. If two photons had partly overlapping spectra, they would be partially distinguishable. In this case, the nature of the particles can be roughly understood as an interpolation of completely indistinguishable and distinguishable cases, governed by the degree of their spectral overlap.
Remarkably, however, this perspective does not work for more than two particles. In this case the fact that we are dealing with wavefunctions characterized by complex probability amplitudes is critical. In our recent study of the properties of particle exchange among several photons, we have discovered some intriguing new features, which indicate how three particles are very different from two particles and why more particles can be decomposed into properties of three photons – or triads, as we’ll call them. interpolation of completely indistinguishable and distinguishable cases, governed by the degree of their spectral overlap.
How do we investigate the nature of distinguishability for several particles? It turns out light provides an excellent opportunity to do so. Photons are massless bosons with no charge. They do not interact easily, and effectively cannot do so at all in free space. Thus it is only their bosonic character that links the evolution of multi-particle states. Further, photons can be engineered to have a wide variety of wavefunctions, specified by their external degrees of freedom that result in properties such as time, spectrum, space, direction and polarization. This makes it possible to make them distinguishable, indistinguishable or partly distinguishable. Moreover, they can be generated, manipulated and detected in ambient conditions, revealing quantum phenomena without complicated apparatus.
What we have done is build photon networks that allow the particles to co-propagate through a series of beam splitters. Because these networks consist entirely of linear optical elements the only means by which the photons can react is through their intrinsic “boson-ness”. When two similar photons enter the two input ports of a beam splitter, for instance, they tend to “bunch” together, leaving through the same output port. This is the well-known Hong-Ou-Mandel effect, first demonstrated in 1987.
It turns out that this effect is in fact a direct measurement of the symmetry of the wavefunction under exchange of the particles’ positions. The reason for this is straightforward to see, and is illustrated in the first movies, Video 1a and 1b. One may consider the beam splitter as a 2X2 network: there are two input beams and two output. Either photon could pass directly through the network: input 1 to output 1, say. Or the two could exchange: input 1 to output 2. These two options offer two paths for one photon to emerge in each output port. The degree to which the quantum amplitudes for these paths interfere is governed by the distinguishability of the “unexchanged” and “exchanged” states.
Video 1a by Adrian Menssen:
Video 1b by Adrian Menssen:
(Caption for videos above) Beam splitter: In the case of a beam-splitter there are two possible paths the photons can take, when considering coincident detection of one photon in each output mode of the interferometer: One where both photons remain in the waveguide they started in, or where they exchange positions. Interference from paths where two photons are exchanged leads to non-classical Hong-Ou-Mandel interference.
There is a further reason why this is important. Optical quantum information processing schemes rely on the interference of multiple photons for their enhanced performance. A key goal is to demonstrate a quantum machine that can outperform a conventional computer at a well-posed computational task, and thus show quantum “supremacy” or quantum advantage.
A promising computing problem for this challenge is called BosonSampling. It addresses the following question: given N non-interacting bosons at the input of an M-port random network (where M > N2), what is the distribution of the bosons measured at the output ports? It turns out that the probability of each possible measurement outcome is related to the permanent of the complex-valued transfer matrix that describes the relevant network section. Furthermore, estimating such matrix permanents is known to be a hard computational problem for classical computers. However, since photons are boson and multiport optical networks can be composed from simple linear optical elements, it seems that constructing a quantum machine that naturally tackles BosonSampling might be feasible. Remarkably, such a machine would accomplish a task known to be unreachable by classical algorithms.
One of the challenges in building these kinds of quantum information processing technologies is to ensure that all the photons are identical. The details of how distinguishable photons interfere is known in complete detail only for one and two photons. In our work we extended such studies to three photons. That may seem a small step, but it reveals a complex landscape, not all the features of which were known. For example, we identified a new phase – the triad phase – that is not present in two-photon interference, and which appears to be a qualitatively new element that characterizes all higher order states.
We measured the probability that three photons put into the three ports of a “tritter”, labelled, say, 1, 2, and 3, would end up at the three output ports. A tritter is a device that couples each input port symmetrically to each output port. In our experiment we used on constructed out of optical fibers, in which all three modes are connected by allowing the evanescent fields of the guided modes to overlap, thereby coupling all the modes to each other with equal strength. This enables the input photons to scatter into all the output modes.
The probability that there is a photon in each of the output ports can now happen in six ways, as shown in the second set of videos, 2a and 2b. First, none of the photons are scattered out of their input mode and end up at the same mode at the output, represented as 123 -> 123. Second, pairs of photons exchange modes, and one photon remains in its original mode, so 123 - > 213, for example. There are three such terms. Third, pairs of photons exchange twice. There are two such terms: 123 - > 213 -> 231; 123 - > 132 ->312. These pathways represent possible actual events and therefore the probability of their occurence must be determined by adding together their amplitudes coherently, taking the modulus squared of this quantity and tracing over the unmeasured degrees of freedom. The resulting number is a direct measure of the difference between the input three-photon wavefunction and its exchanged versions. The more distinguishable the photons become, the larger this distance and the disappearance of any interference pattern.
Video 2a by Adrian Menssen:
Video 2b by Adrian Menssen:
(Caption for videos above) Tritter: Here we illustrate two possible scenarios how three photons can traverse a tritter. In the first case, they remain in the waveguide they started in, while in the second scenario each photon couples into a different waveguide, leading to a full exchange of all three photons. The triad phase arises precisely from interference between paths, where a full three-photon exchange occurs. We could also imagine paths where only two photons exchange position, while the third one does not participate. Contributions from these paths lead to Hong-Ou-Mandel like “two-photon” interference. Isolating “genuine” three photon interference from two-photon interference is one of the hallmarks of our paper.
In our experiment the photons can be made distinguishable by adjusting the delays between them. But delays alone are insufficient to explore the entire state space. That’s because a there are three complex numbers specifying the three state overlaps and delays provide only two parameters. Therefore we used an additional degree of freedom – the polarization of each of the photons to provide a means to access the necessary parameters.
By setting all three polarizations to be the same, the triad phase is zero (as long as some simple conditions on the photon spectra are satisfied). In this case, the three-photon exchange term dominates, yielding a coincidence rate that is larger than when the photons are completely distinguishable. As the photons are made more distinguishable, the pair-wise exchange terms dominate, leading to a reduction in the coincidences below that of fully distinguishable photons, reminiscent of the Hong-Ou-Mandel dip. But they never completely dominate, so that there is never a complete suppression of coincidences.
The triad phase can be set to a non-zero value by adjusting the relative polarizations of the three input photons. For example, when they are polarized at 120 degrees to each other, the triad phase is π. In this case, the three-photon exchange and the two-photon exchange contributions are of the same sign, and both contribute to a dip. The photons are of course in this case rather distinguishable, so in some ways it is surprising to see such a suppression of coincidences arising from interference.
It is possible to adjust the polarizations as the photons are rendered more distinguishable by delaying them in such a way as to keep all of the two-photon coincidences constant. This is analogous to the one-photon rates in the Hong-Ou-Mandel experiment, which should be constant in order that the two-photon dip can be proven to be non-classical in origin.
We find in our experiment that the three-photon coincidence exhibits an analogous dip, which is the first time that the fully exchanged three-photon wavefunction has been isolated by interference.
Are there more complex structures for more than three photons? Of course the interference patterns are richer, and we can expect that new features appear. But it might be that there are no new classes of parameters that are needed. In other words, by measuring the triad phase of all photon triplets entering the device one can determine whether the photons are identical. If so, then pair-wise two-photon correlations would be insufficient to fully characterize a linear optical interferometer, and three-photon correlations both necessary and sufficient.
The nature of multi-particle wavefunctions, and in particular the degree to which partially distinguishable particles can still exhibit non-classical phenomena (not to say provide quantum enhancement in technological applications) is an area ripe for exploration. New features arise when one goes from one to two particles, and the richness of three particles is startling. What new ones will be found in the wavefunctions of higher particle numbers remains to be seen.
Reference:
[1] Adrian J. Menssen, Alex E. Jones, Benjamin J. Metcalf, Malte C. Tichy, Stefanie Barz, W. Steven Kolthammer, Ian A. Walmsley, "Distinguishability and many-particle interference". Physical Review Letters, 118, 153603 (2017). Abstract. arXiv:1609.09804.
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