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Journal of Optics

Observation of photonic states dynamics in 3-D integrated Fourier circuits

Fulvio Flamini, Niko Viggianiello, Taira Giordani, Marco Bentivegna, Nicolò Spagnolo, Andrea Crespi, Giacomo Corrielli, Roberto Osellame, Miguel Angel Martin-Delgado, Fabio Sciarrino

Published May 21, 2018• 4 citations
Publisher

Abstract

Abstract Entanglement is a fundamental resource at the basis of quantum-enhanced performances in several applications, such as quantum algorithms and quantum metrology. In these contexts, Fourier interferometers implement a relevant class of unitary evolutions which can be embedded in a large variety of protocols. For instance, in the single-particle regime it can be adopted to implement the quantum Fourier transform, while in the multi-particle scenario it can be employed to generate quantum states possessing useful entanglement for quantum phase estimation purposes, or as a tool to verify genuine multi-photon interference. In this article, we study experimentally the dynamics of single-photon and two-photon input states during the evolution provided by a 8-mode Fourier transformation, implemented by exploiting a three-dimensional architecture enabled by the femtosecond laser micromachining technology. In such a way, we fabricated three devices to study the evolution after each step of the decomposition. We observe that the probability distributions obey a step-by-step majorization relationship, where the quantum state occupies a progressively larger portion of the Hilbert space. Such behaviour can be related to the majorization principle, which has been conjectured as a necessary condition for quantum speedup.

DOI
10.1088/2040-8986/aac68b

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Abstract Entanglement is a fundamental resource at the basis of quantum-enhanced performances in several applications, such as quantum algorithms and quantum metrology. In these contexts, Fourier interferometers implement a relevant class of unitary evolutions which can be embedded in a large variety of protocols. For instance, in the single-particle regime it can be adopted to implement the quantum Fourier transform, while in the multi-particle scenario it can be employed to generate quantum states possessing useful entanglement for quantum phase estimation purposes, or as a tool to verify genuine multi-photon interference. In this article, we study experimentally the dynamics of single-photon and two-photon input states during the evolution provided by a 8-mode Fourier transformation, implemented by exploiting a three-dimensional architecture enabled by the femtosecond laser micromachining technology. In such a way, we fabricated three devices to study the evolution after each step of the decomposition. We observe that the probability distributions obey a step-by-step majorization relationship, where the quantum state occupies a progressively larger portion of the Hilbert space. Such behaviour can be related to the majorization principle, which has been conjectured as a necessary condition for quantum speedup.
The Majorization Principle is a fundamental statement governing the dynamics of information processing in optimal and efficient quantum algorithms. While quantum computation can be modeled to be reversible, due to the unitary evolution undergone by the system, these quantum algorithms are conjectured to obey a quantum arrow of time dictated by the Majorization Principle: the probability distribution associated to the outcomes gets ordered step-by-step until achieving the result of the computation. Here we report on the experimental observation of the effects of the Majorization Principle for two quantum algorithms, namely the quantum fast Fourier transform and a recently introduced validation protocol for the certification of genuine many-boson interference. The demonstration has been performed by employing integrated 3-D photonic circuits fabricated via femtosecond laser writing technique, which allows to monitor unambiguously the effects of majorization along the execution of the algorithms. The measured observables provide a strong indication that the Majorization Principle holds true for this wide class of quantum algorithms, thus paving the way for a general tool to design new optimal algorithms with a quantum speedup.
The investigation of multi-photon quantum interference in symmetric multi-port splitters has both fundamental and applicative interest. Destructive quantum interference in devices with specific symmetry leads to the suppression of a large number of possible output states, generalizing the Hong-Ou-Mandel effect; simple suppression laws have been developed for interferometers implementing the Fourier or the Hadamard transform over the modes. In fact, these enhanced interference features in the output distribution can be used to assess the indistinguishability of single-photon sources, and symmetric interferometers have been envisaged as benchmark or validation devices for Boson-Sampling machines. In this work we devise an innovative approach to implement symmetric multi-mode interferometers that realize the Fourier and Hadamard transform over the optical modes, exploiting integrated waveguide circuits. Our design is based on the optical implementations of the Fast-Fourier and Fast-Hadamard transform algorithms, and exploits a novel three-dimensional layout which is made possible by the unique capabilities of femtosecond laser waveguide writing. We fabricate devices with m = 4 and m = 8 modes and we let two identical photons evolve in the circuit. By characterizing the coincidence output distribution we are able to observe experimentally the known suppression laws for the output states. In particular, we characterize the robustness of this approach to assess the photons' indistinguishability and to rule out alternative non-quantum states of light. The reported results pave the way to the adoption of symmetric multiport interferometers as pivotal tools in the diagnostics and certification of quantum photonic platforms.
The identification of phenomena able to pinpoint quantum interference is attracting large interest. Indeed, a generalization of the Hong-Ou-Mandel effect valid for any number of photons and optical modes would represent an important leap ahead both from a fundamental perspective and for practical applications, such as certification of photonic quantum devices, whose computational speedup is expected to depend critically on multi-particle interference. Quantum distinctive features have been predicted for many particles injected into multimode interferometers implementing the Fourier transform over the optical modes. Here we develop a scalable approach for the implementation of the fast Fourier transform algorithm using three-dimensional photonic integrated interferometers, fabricated via femtosecond laser writing technique. We observe the suppression law for a large number of output states with four- and eight-mode optical circuits: the experimental results demonstrate genuine quantum interference between the injected photons, thus offering a powerful tool for diagnostic of photonic platforms.
The identification of phenomena able to pinpoint quantum interference is attracting large interest. Indeed, a generalization of the Hong-Ou-Mandel effect valid for any number of photons and optical modes would represent an important leap ahead both from a fundamental perspective and for practical applications, such as certification of photonic quantum devices, whose computational speedup is expected to depend critically on multiparticle interference. Quantum distinctive features have been predicted for many particles injected into multimode interferometers implementing the Fourier transformation in the Fock space. In this work we develop a scalable approach for the implementation of quantum fast Fourier transform using 3-D photonic integrated interferometers, fabricated via femtosecond laser writing technique. We observe the quantum suppression of a large number of output states with 4- and 8-mode optical circuits: the experimental results demonstrate genuine quantum interference between the injected photons, thus offering a powerful tool for diagnostic of photonic platforms.
Certification is important to guarantee the correct functioning of quantum devices. A key certification task is verifying that a device has produced a desired output state. In this work, we study this task in the context of photonic platforms, where single photons are propagated through linear optical interferometers to create large, entangled resource states for metrology, communication, quantum advantage demonstrations and for so-called linear optical quantum computing (LOQC). This setting derives its computational power from the indistinguishability of the photons, i.e., their relative overlap. Therefore, standard fidelity witnesses developed for distinguishable particles (including qubits) do not apply directly, because they merely certify the closeness to some fixed target state. We introduce a measure of fidelity suitable for this setting and show several different ways to witness it, based on earlier proposals for measuring genuine multi-photon indistinguishability. We argue that a witness based upon the discrete Fourier transform is an optimal choice. We experimentally implement this witness and certify the fidelity of several multi-photon states.
Most of the experimental interferometric setup are based on the Mach-Zehnder geometry. Usually in the experimental configurations the photons travel in air, at a speed v independent of λ. An integrated optics approach appears necessary both for practical applications and because it can be useful for fundamental optics studies because the transmitted photons are in a spatial coherent state. We exploited a hybrid setup integrating optical fibres and monolithic optical waveguides to implement an asymmetric nonlinear Mach-Zehnder Interferometer (ANLI). One arm of the interferometer was equipped with a single mode Er:LiNbO3 waveguide, acting as nonlinear component whereas the other arm was an undoped LiNbO3 single mode waveguide, used to obtain a phase shift through the application of a voltage ramp. We injected in the ANLI a 980 nm laser radiation to collect interferograms that could be ascribed exclusively to the pump photons, as all frequency conversion events are localized only in one arm of the interferometer. The Fourier Transform analysis of the experimental data allowed us to perform a n Interaction-Free Quantum Optics experiment. This is a novel approach that can be used because in quantum mechanics the wave functions of a state, expressed in terms of two conjugate variables, are Fourier Transform pairs. In fact, operating in the Fourier Transform time-frequency space-domain, we recovered the spectral composition, in the wavenumber space-domain, of the photons generated by the excitation of an Er3+ doped crystal placed in one arm of the interferometer.