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We demonstrate the distance sensitivity of thermal light second-order interference beyond spatial coherence. This kind of interference, emerging from the measurement of the correlation between intensity fluctuations on two detectors, is sensitive to the distances separating a remote mask from the source and the detector, even when such information cannot be retrieved by first-order intensity measurements. We show how the sensitivity to such distances is intimately connected to the degree of correlation of the measured interference pattern in different experimental scenarios and independently of the spectral properties of light. Remarkably, in specific configurations, sensitivity to the distances of remote objects can be preserved even in the presence of turbulence. Unlike in previous schemes, such a distance sensitivity is reflected in the fundamental emergence of new critical parameters which benchmark the degree of second-order correlation, describing the counterintuitive emergence of spatial second-order interference not only in the absence of (first-order) coherence at both detectors but also when first-order interference is observed at one of the two detectors.

We present feasible Gaussian schemes for the estimation with Heisenberg-scaling sensitivity of an unknown parameter distributed within an arbitrary optical net-work. We discuss how these schemes overcome some typical experimental drawbacks of quantum-enhanced metrology.

We demonstrate the ultimate sensitivity allowed by quantum physics in the estimation of the time delay between two photons by measuring their interference at a beam-splitter through frequency-resolving sampling measurements. This sensitivity can be increased quadratically by decreasing the photonic temporal bandwidth even at values smaller than the time delay when standard two-photon interferometers become inoperable and without adapting the path of the reference photon, nor the need of time-resolving detectors with an unfeasible high resolution. Applications can range from more feasible imaging of nanostructures, including biological samples, and nanomaterial surfaces to quantum enhanced estimation based on frequency-resolved boson sampling in optical networks., 15 pages, 9 figures

We show a protocol achieving the ultimate Heisenberg-scaling sensitivity in the estimation of a parameter encoded in a generic linear network, without employing any auxiliary networks, and without the need of any prior information on the parameter nor on the network structure. As a result, this protocol does not require a prior coarse estimation of the parameter, nor an adaptation of the network. The scheme we analyse consists of a single-mode squeezed state and homodyne detectors in each of the $M$ output channels of the network encoding the parameter, making it feasible for experimental applications., Comment: 11 pages, 1 figure

We propose a measurement setup reaching Heisenberg scaling precision for the estimation of any distributed parameter $\varphi$ (not necessarily a phase) encoded into a generic $M$-port linear network composed only of passive elements. The scheme proposed can be easily implemented from an experimental point of view since it employs only Gaussian states and Gaussian measurements. Due to the complete generality of the estimation problem considered, it was predicted that one would need to carry out an adaptive procedure which involves both the input states employed and the measurement performed at the output; we show that this is not necessary: Heisenberg scaling precision is still achievable by only adapting a single stage. The non-adapted stage only affects the value of a pre-factor multiplying the Heisenberg scaling precision: we show that, for large values of $M$ and a random (unbiased) choice of the non-adapted stage, this pre-factor takes a typical value which can be controlled through the encoding of the parameter $\varphi$ into the linear network., Comment: 14 pages, 3 figures

We demonstrate quantum interference of three photons that are distinguishable in time, by resolving them in the conjugate parameter, frequency. We show that the multi-photon interference pattern in our setup can be manipulated by tuning the relative delays between the photons, without the need for reconfiguring the optical network. Furthermore, we observe that the symmetries of our optical network and the spectral amplitude of the input photons are manifested in the interference pattern. We also demonstrate time-reversed Hong-Ou-Mandel like interference in the spectral correlations using time-bin entangled photon pairs. By adding a time-varying dispersion using a phase modulator, our setup can be used to realize dynamically reconfigurable and scalable boson sampling in the time-domain as well as frequency-resolved multi-boson correlation sampling.

We propose an interferometric scheme for the estimation of a linear combination with non-negative weights of an arbitrary number M > 1 of unknown phase delays, distributed across an M-channel linear optical network, with Heisenberg-limited sensitivity. This is achieved without the need of any sources of photon-number or entangled states, photon-number resolving detectors or auxiliary interferometric channels. Indeed, the proposed protocol remarkably relys upon a single squeezed state source, an antisqueezing operation at the interferometer output, and on-off photodectors. Office of Naval Research Global (N62909-418-1-2153) Istituto Nazionale di Fisica Nucleare (INFN) through the project “QUANTUM” University of Portsmouth Italian National Group of Mathematical Physics (GNFM-INdAM)

We report a detailed analysis of the optical realization [1, 3, 2, 4] of the analogue algorithm described in the first paper of this series [5] for the simultaneous factorization of an exponential number of integers. Such an analogue procedure, which scales exponentially in the context of first order interference, opens up the horizon to polynomial scaling by exploiting multi-particle quantum interference., to be published in Quantum Information Processing (QIP)

We address the problem of distributed quantum metrology with a single squeezed-vacuum source by using the formalism of quantum mechanics in phase space. In particular, we demonstrate Heisenberg-limited sensitivity in the measurement of the average of two arbitrary phase shifts in the arms of a Mach–Zehnder interferometer. We obtain exact results for the measurement probability at the interferometer output for any value of the phases, which give us insight into the emergence of Heisenberg-limited sensitivity for periodical values of the phases.

Interference of multiple photons via a linear-optical network has profound applications for quantum foundation, quantum metrology, and quantum computation. Particularly, a boson sampling experiment with a moderate number of photons becomes intractable even for the most powerful classical computers. Scaling up from small-scale experiments requires highly indistinguishable single photons, which may be prohibited for many physical systems. Here we report a time-resolved multiphoton interference experiment by using photons not overlapping in their frequency spectra from three atomic-ensemble quantum memories. Time-resolved measurement enables us to observe nonclassical multiphoton correlation landscapes, which agree well with theoretical calculations. Symmetries in the landscapes are identified to reflect symmetries of the optical network. Our experiment can be further extended to realize boson sampling with many photons and plenty of modes, which thus may provide a route towards quantum supremacy with nonidentical photons.

Cavity-enhanced single photon sources exhibit mode-locked biphoton states with comb-like correlation functions. Our ultrabright source additionally emits either single photon pairs or two-photon NOON states, dividing the output into an even and an odd comb respectively. With even-comb photons we demonstrate revivals of the typical non-classical Hong-Ou-Mandel interference up to the 84th dip, corresponding to a path length difference exceeding 100 m. With odd-comb photons we observe single photon interference fringes modulated over twice the displacement range of the Hong-Ou-Mandel interference., 10 pages, 7 figures

Recently, quantum Fourier transform interferometers have been demonstrated to allow a quantum metrological enhancement in phase sensitivity for a small number n of identical input single photons [1-3]. However, multiphoton distinguishability at the detectors can play an important role from an experimental point of view [4]. This raises a fundamental question: How is the phase sensitivity affected when the photons are completely distinguishable at the detectors and therefore do not interfere? In other words, which role does multiphoton interference play in these schemes? Here, we show that for small phase values the phase sensitivity achievable in the proposed schemes with indistinguishable photons is enhanced only by a constant factor with respect to the case of completely distinguishable photons at the detectors. Interestingly, this enhancement arises from the interference of only a polynomial number (in n) of the total n! multiphoton path amplitudes in the n-port interferometer. These results are independent of the number n of single photons and of the phase weight factors employed at each interferometer channel., accepted for publication in International Journal of Quantum Information on Nov 27, 2017

The boson-sampling problem has triggered a lot of interest in the scientific community because of its potential of demonstrating the computational power of quantum interference without the need of non-linear processes. However, the intractability of such a problem with any classical device relies on the realization of single photons approximately identical in their spectra. In this paper, we discuss the physics of boson sampling with non-identical single-photon sources, which is strongly relevant in view of scalable experimental realizations and triggers fascinating questions in the complexity theory.

We report observation of a counter-intuitive phenomenon in multi-path correlation interferometry with thermal light. The intensity correlation between the outputs of two unbalanced Mach-Zehnder interferometers (UMZI) with two classically correlated beams of thermal light at the input exhibits genuine second-order interference with the visibility of 1/3. Surprisingly, the second-order inter- ference does not degrade at all no matter how much the path length difference in each UMZI is increased beyond the coherence length of the thermal light. Moreover, the second-order interference is dependent on the difference of the UMZI phases. These results differ substantially from those of the entangled-photon Franson interferometer which exhibits two-photon interference dependent on the sum of the UMZI phases and the interference vanishes as the path length difference in each UMZI exceeds the coherence length of the pump laser. Our work offers deeper insight into the interplay between interference and coherence in multi-photon interferometry.

We present the experimental characterization of two distant double-slit masks illuminated by chaotic light, in the absence of first-order imaging and interference. The scheme exploits second-order interference of light propagating through two indistinguishable pairs of {\it disjoint} optical paths passing through the masks of interest. The proposed technique leads to a deeper understanding of biphoton interference and coherence, and opens the way to the development of novel schemes for retrieving information on the relative position and the spatial structure of distant objects, which is of interest in remote sensing, biomedical imaging, as well as monitoring of laser ablation, when first-order imaging and interference are not feasible., Comment: 9 pages, 3 figures

Multiphoton quantum interference underpins fundamental tests of quantum mechanics and quantum technologies. Consequently, the detrimental effect of photon distinguishability in multiphoton interference experiments can be catastrophic. Here, we employ correlation measurements in the photonic inner modes, time or frequency, to restore quantum interference between photons differing in their colors or injection times in arbitrary linear optical networks, without the need for additional filtering or post selection. Interestingly, we demonstrate how harnessing the multiphoton inner-mode quantum information enables to unravel symmetries of multiphoton networks and states and the generation of an entire class of multipartite entangled states with a fixed interferometer. These results are therefore of profound interest for future applications of universal inner-mode resolved linear optics across fundamental science and quantum technologies with photons with experimentally different spectral properties., Comment: Comments are welcome

A novel thermal light interferometer was recently introduced in V. Tamma and J. Seiler, New J. Phys. 18, 032002 (2016). Here, two classically correlated beams, obtained by beam splitting a thermal light beam, propagate through two unbalanced Mach-Zehnder interferometers. Remarkably, second-order interference between the long and the short paths in the two interferometers was predicted independently of how far, in principle, the length difference between the long and short paths is beyond the coherence length of the source. This phenomenon seems to contradict our common understanding of second order coherence. We provide here a simple description of the physics underlying this effect in terms of two-photon interference., in honour of Professor Wolfgang P. Schleich on the occasion of his 60th birthday

We report a recent experimental simulation of a controlled-NOT gate operation based on polarization correlation measurements of thermal fields in photon-number fluctuations. The interference between pairs of correlated paths at the very heart of these experiments has the potential for the simulation of correlations between a larger number of qubits.

We give a full description of the problem of Multi-Boson Correlation Sampling (MBCS) at the output of a random interferometer for single input photons in arbitrary multimode pure states. The MBCS prob- lem is the task of sampling at the interferometer output from the probability distribution associated with polarization- and time-resolved detections. We discuss the richness of the physics and the complexity of the MBCS problem for nonidentical input photons. We also compare the MBCS problem with the standard bo- son sampling problem, where the input photons are assumed to be identical and the system is "classically" averaging over the detection times and polarizations., appears in Quantum Information Processing (2015)

We take advantage of the interesting connection between truncated exponential sums with continuous arguments (CTES) in number theory and interference in Physics in order to investigate the challenging problem of factoring large numbers. In particular we develop a novel method of factorization based on the use of an optical computer able to reproduce “CTES interferograms” by exploiting polychromatic interference. The scaling properties at the core of such “factoring” interferograms allows, in principle, the prime number decomposition of several large integers.

We introduce a new factorization algorithm based on the optical computation by multi-path interference of the periodicity of a "factoring" function given by exponential sums at continuous arguments. We demonstrate that this algorithm allows, in principle, the prime number decomposition of several large numbers by exploiting a remarking rescaling property of this periodic function. Such a function is recorded by measuring optical interferograms with a multi-path Michelson interferometer, a polychromatic light source and a spectrometer. The information about factors is encoded in the location of the inteferogram maxima.

The phase φ of any wave is determined by the ratio x/λ consisting of the distance x propagated by the wave and its wavelength λ. Hence, the dependence of φ on λ constitutes an analogue system for the mathematical operation of division, that is to obtain the hyperbolic function f(ξ)≡1/ξ. We take advantage of this observation to decompose integers into primes and implement this approach towards factorization of numbers in a multi-path Michelson interferometer. This work is part of a larger program geared towards unraveling the connections between quantum mechanics and number theory. We briefly summarize this aspect.

Path-entangled N-photon systems described by NOON states are the main ingredient of many quantum information and quantum imaging protocols. Our analysis aims to lead the way toward the implementation of both NOON-state sources and their applications. To this end, we study the functionality of "real" NOON-state sources by quantifying the effect real experimental apparatuses have on the actual generation of the desired NOON state. In particular, since the conditional generation of NOON states strongly relies on photon counters, we evaluate the dependence of both the reliability and the signal-to-noise ratio of "real" NOON-state sources on detection losses. We find a surprising result: NOON-state sources relying on nondetection are much more reliable than NOON-state sources relying on single-photon detection. Also the comparison of the resources required to implement these two protocols comes out to be in favor of NOON-state sources based on nondetection. A scheme to improve the performances of "real" NOON-state sources based on single-photon detection is also proposed and analyzed.

We provide a compact full description of multiboson correlation measurements of arbitrary order N in passive linear interferometers with arbitrary input single-photon pure states. This allows us to physically analyze the novel problem of multiboson correlation sampling at the output of random linear interferometers. Our results also describe general multiboson correlation landscapes for an arbitrary number of input single photons and arbitrary interferometers. In particular, we use two different schemes to demonstrate, respectively, arbitrary-order quantum beat interference and 100% visibility entanglement correlations even for input photons distinguishable in their frequencies., Comment: accepted for publication in Physical Review Letters

We construct an analog computer based on light interference to encode the hyperbolic function f({\zeta}) = 1/{\zeta} into a sequence of skewed curlicue functions. The resulting interferogram when scaled appropriately allows us to find the prime number decompositions of integers. We implement this idea exploiting polychromatic optical interference in a multipath interferometer and factor seven-digit numbers. We give an estimate for the largest number that can be factored by this scheme., Comment: 4 pages, 2 figures

The boson sampling problem has brought a lot of attention in the quantum information field because it is not efficiently solvable with a classical computer; nonetheless it can be implemented with linear optical interferometers with single-boson sources. Recently, we introduced a more general problem, the multi-boson correlation sampling problem, which allows us to take advantage of the multi-mode spectral distribution of the bosonic sources together with time-correlated measurements in order to achieve sampling not only over the output ports of the interferometer but also over the joint detection times. This problem was analyzed for both single-photon sources and thermal sources. In this work, we demonstrate that it is possible to use single qubit bosonic sources in order to sample not only over the described "space-time" degree of freedom but also over all the possible exponential number of multi-qubit output states., Comment: 8 pages, 2 figures

We demonstrate how the physics of multiboson correlation interference leads to the computational complexity of linear optical interferometers based on correlation measurements in the degrees of freedom of the input bosons. In particular, we address the task of MultiBoson Correlation Sampling (MBCS) from the probability distribution associated with polarization- and time-resolved detections at the output of random linear optical networks. We show that the MBCS problem is fundamentally hard to solve classically even for nonidentical input photons, regardless of the color of the photons, making it also very appealing from an experimental point of view. These results fully manifest the quantum computational supremacy inherent to the fundamental nature of quantum interference., Comment: v6: fixed missing equation numbers

We develop a general description of multiboson interferometry based on correlated measurements in arbitrary passive linear interferometers for multimode thermal sources with arbitrary spectral distributions. The multiorder correlation functions describing the multiboson detection probability rates can be expressed in terms of permanents of positive semidefinite matrices, depending on the interferometer evolution, the spectral distribution of the sources, and the times when the correlated measurements occur. The permanent structure of these multiorder probability rates is a manifestation of the underlying physics of multiboson interference and yields an interesting connection with the so-called boson sampling problem.

The connection between the boson sampling problem in a linear optical interferometer with single photon sources and the computation of matrix permanents has recently brought a strong interest for quantum computing applications. In this work we explore the physics of N-order interference in terms of N-order correlation functions in a generic linear optical interferometer with both Fock states sources and thermal sources. We will show how also in this case the connection of such a problem with matrix permanents play a fundamental role.

Atom interferometers covering macroscopic domains of space-time are a spectacular manifestation of the wave nature of matter. Due to their unique coherence properties, Bose-Einstein condensates are ideal sources for an atom interferometer in extended free fall. In this paper we report on the realization of an asymmetric Mach-Zehnder interferometer operated with a Bose-Einstein condensate in microgravity. The resulting interference pattern is similar to the one in the far-field of a double-slit and shows a linear scaling with the time the wave packets expand. We employ delta-kick cooling in order to enhance the signal and extend our atom interferometer. Our experiments demonstrate the high potential of interferometers operated with quantum gases for probing the fundamental concepts of quantum mechanics and general relativity., Comment: 8 pages, 3 figures; 8 pages of supporting material

In a recent experiment, we have shown how optical interference can be exploited in order to perform computations and in particular factoring large integers. This has lead us to the developing of a novel analogue algorithm which allows the factorization of several large integers in a single run. This work represents a step stone towards the developing of novel physical networks able to implement quantum digital algorithms other than Shor’s algorithm. We describe the main ideas leading to this new line of research referring to future publications for a detailed description.

We first describe the experimental computation of a novel factoring algorithm based on continuous Gauss sums. Than, interestingly, we show how an exponential speed-up can be achieved by exploiting multi-photon n$^{th}$-order interference.

We present a novel factorization algorithm which can be computed using an analogue computer based on a polychromatic source with a given wavelength bandwidth, a multi-path interferometer and a spectrometer. The core of this algorithm stands on the measurement of the periodicity of a "factoring" function given by an exponential sum at continuous argument by recording a sequence of interferograms associated with suitable units of displacement in the inteferometer. A remarking rescaling property of such interferograms allows, in principle, the prime number decomposition of several large integers. The information about factors is encoded in the location of the inteferogram maxima.