Time-frequency quantum information processing

Quantum Pulse sculpting

Quantum sensing

Spectral tomography

Quantum computing

Quantum communication

Phase space approach in quantum optics

Quantum Pulse sculpting

Engineering two-photon wavefunction and exchange statistics in a semiconductor chip

Francesconi S, Baboux F, Raymond A, Fabre N, Boucher G, Lemaître A, et al. Engineering two-photon wavefunction and exchange statistics in a semiconductor chip. Optica. 20 avr 2020;7(4):316.

High-dimensional entangled states of light provide novel possibilities for quantum information, from fundamental tests of quantum mechanics to enhanced computation and communication protocols. In this context, the frequency degree of freedom combines the assets of robustness to propagation and easy handling with standard telecommunication components. Here, we use an integrated semiconductor chip to engineer the wavefunction and exchange statistics of frequency-entangled photon pairs directly at the generation stage, without post-manipulation. Tailoring the spatial properties of the pump beam allows generating frequency-anticorrelated, correlated and separable states, and to control the symmetry of the spectral wavefunction to induce either bosonic or fermionic behaviors. These results, obtained at room temperature and telecom wavelength, open promising perspectives for the quantum simulation of fermionic problems with photons on an integrated platform, as well as for communication and computation protocols exploiting antisymmetric high-dimensional quantum states.

Producing a delocalized frequency-time Schrödinger-cat-like state with Hong-Ou-Mandel interferometry

N. Fabre, J. Belhassen, A. Minneci, S. Felicetti, A. Keller, M. I. Amanti, F. Baboux, T. Coudreau, S. Ducci, and P. Milman Phys. Rev. A 102, 023710 (2020)

In the late 80's, Ou and Mandel experimentally observed signal beatings by performing a non-time resolved coincidence detection of two photons having interfered in a balanced beam splitter [Phys. Rev. Lett 61, 54 (1988)]. In this work, we provide a new interpretation of the fringe pattern observed in this experiment as the direct measurement of the chronocyclic Wigner distribution of a frequency Schrodinger cat-like state produced by local spectral filtering. Based on this analysis, we also study the time-resolved HOM experiment to measure such a frequency state.

Generation of a Time-Frequency Grid State with Integrated Biphoton Frequency Combs

N. Fabre et al., Generation of a Time-Frequency Grid State with Integrated Biphoton Frequency Combs, Phys. Rev. A 102, 012607 (2020).

Encoding quantum information in continuous variables is intrinsically faulty. Nevertheless, redundant qubits can be used for error correction, as proposed by Gottesman, Kitaev and Preskill in Phys. Rev. A 64 012310, (2001). We show how to experimentally implement this encoding using time-frequency continuous degrees of freedom of photon pairs produced by spontaneous parametric down conversion. We experimentally illustrate our results using an integrated AlGaAs photon pairs source. We show how single qubit gates can be implemented and finally propose a theoretical scheme for correcting errors in a circuit-like and in a measurement-based architecture.

The major observation is that we can write either the output biphoton state outside the optical cavity either as a frequency entangled qudit state, or as a time-frequency grid state.

Anyonic Two-Photon Statistics with a Semiconductor Chip

Francesconi S, Raymond A, Fabre N, Lemaître A, Amanti MI, Milman P, et al. Anyonic Two-Photon Statistics with a Semiconductor Chip. ACS Photonics. 15 sept 2021;8(9):2764‑9.

Anyons, particles displaying a fractional exchange statistics intermediate between bosons and fermions, play a central role in the fractional quantum Hall effect and various spin−lattice models, and have been proposed for topological quantum computing schemes due to their resilience to noise. Here we use parametric down-conversion in an integrated semiconductor chip to generate biphoton states simulating anyonic particle statistics in a reconfigurable manner. Our scheme exploits the frequency entanglement of the photon pairs, which is directly controlled through the spatial shaping of the pump beam. These results, demonstrated at room temperature and telecom wavelength on a chip-integrated platform, pave the way to the practical implementation of quantum simulation tasks with tailored particle statistics.

Quantum sensing

N. Fabre, S. Felicetti., Generation of a Time-Frequency Grid State with Integrated Biphoton Frequency Combs, Phys. Rev. A 102, 012607 (2020).

Hong-Ou-Mandel interferometry takes advantage of the quantum nature of two-photon interference to increase the resolution of precision measurements of time delays. Relying on few-photon probe states, this approach is applicable also in cases of extremely sensible samples and it achieves attosecond-scale resolution, which is relevant to cell biology and two-dimensional materials. Here, we theoretically analyze how the precision of Hong-Ou-Mandel interferometers can be significantly improved by engineering the spectral distribution of two-photon probe states. In particular, we assess the metrological power of different classes of biphoton states with non-Gaussian time-frequency spectral distributions, considering the estimation of both time and frequency shifts. We find that grid states, characterized by a periodic structure of peaks in the chronocyclic Wigner function, can outperform standard biphoton states in sensing applications.

E. Descamps, N. Fabre, A. Keller, and P. Milman, Quantum Metrology Using Time-Frequency as Quantum Continuous Variables: Sub Shot-Noise Precision and Phase Space Representation, Phys. Rev. Lett. 131, 030801, arXiv:2210.05511 (2022).

After discussing the spectral engineering of photon pairs, we will discuss the use of more general quantum states possessing a higher number of photons for estimating time shifts using the presented intrinsic multimode quantum metrology approach. We will show that the particle-number and time-frequency degree of freedom are intertwined for quantifying the ultimate precision achievable by quantum means. Increasing the number of photons of a large entangled EPR probe state actually increase the noise coming from the frequency continuous variable, and hence deteriorating the precision over the estimation of a time shift.

Spectral tomography

Spectral single photons characterization using generalized Hong, Ou and Mandel interferometry

Fabre N. Spectral single photons characterization using generalized Hong, Ou and Mandel interferometry. Journal of Modern Optics. 2022 ;69(12):653‑64.

New methods are proposed to characterize the spectral-temporal distribution of single photons. The presented protocols take advantage of spectral filtering, frequency entanglement between two single photons the one of interest and a reference, followed by the generalized Hong, Ou and Mandel interferometer. The measurement of the coincidence probability in these different schemes reveals the chronocyclic Wigner, the pseudo-Wigner distribution and the spectrogram at the single-photon level.

Interferometric signature of different spectral symmetries of biphoton states

N. Fabre Phys. Rev. A 105, 053716 (2022)

In this paper, we investigate the influence of the symmetry of the biphoton wave function on the coincidence measurement of the generalized Mach-Zehnder (MZ) interferometer, where there are a temporal and frequency shift operations between the two beam splitters. We show that the generalized MZ interferometer allows the measurement of the short-time Fourier transform of the function modeling the energy conservation of a spontaneous parametric down-conversion process if the full biphoton state is symmetric, and of the symmetric characteristic distribution of the phase-matching function if the state is antisymmetric. Thus, this technique is phase sensitive to the spectral distribution of the photon pairs. Finally, we investigate in detail the signature of a pair of anyons whose peculiar statistics can be simulated by engineering the spectrum of photon pairs.

Variable electro-optic shearing interferometry for ultrafast single-photon-level pulse characterization

Kurzyna S, Jastrzębski M, Fabre N, Wasilewski W, Lipka M, Parniak M. Variable electro-optic shearing interferometry for ultrafast single-photon-level pulse characterization. Opt Express. 24 oct 2022;30(22):39826.

Despite the multitude of available methods, the characterization of ultrafast pulses remains a challenging endeavor, especially at the single-photon level. We introduce a pulse characterization scheme that maps the magnitude of its short-time Fourier transform. Contrary to many well-known solutions it does not require nonlinear effects and is therefore suitable for single-photon-level measurements. Our method is based on introducing a series of controlled time and frequency shifts, where the latter is performed via an electro-optic modulator allowing a fully-electronic experimental control. We characterized the full spectral and temporal width of a classical and single-photon-level pulse and successfully tested the applicability of the reconstruction algorithm of the spectral phase and amplitude. The method can be extended by implementing a phase-sensitive measurement and is naturally well-suited to partially-incoherent light.

Reconstructing the full modal structure of photonic states by stimulated emission tomography

A. Keller, A. Z. Khoury, N. Fabre, M. Amanti, F. Baboux, S. Ducci, and P. Milman Phys. Rev. A 106, 063709  (2022)

Stimulated emission tomography is a powerful and successful technique to both improve the resolution and experimentally simplify the task of determining the modal properties of biphotons. In the present manuscript we provide a theoretical description of SET valid for any quadratic coupling regime between a non-linear medium and pump fields generating photons by pairs. We use our results to obtain not only information about the associated modal function modulus but also its phase, for any mode, and we discuss the specific case of time-frequency variables as well as the quantities and limitations involved in the measurement resolution.

Quantum computing

Time-frequency as quantum continuous variables

Nicolas Fabre, Arne Keller, and Pérola Milman Phys. Rev. A 105, 052429 (2022)

We present a second quantization description of frequency-based continuous variables quantum computation in the subspace of single photons. For this, we define frequency and time operators using the free field Hamiltonian and its Fourier transform, and show that these observables, when restricted to the one photon per mode subspace, reproduce the canonical position-momentum commutation relations. As a consequence, frequency and time operators can be used to define a universal set of gates in this particular subspace. We discuss the physical implementation of these gates as well as their effect on single photon states, and show that frequency and time variables can also be used to implement continuous variables quantum information protocols, in the same way than polarization is currently used as a two-dimensional quantum variable.

Quantum communication

Teleportation-based error correction protocol of time-frequency qubits states

N. Fabre, Teleportation-Based Error Correction Protocol of Time-Frequency Qubits States, arXiv:2302.06940. (2023)

We present a linear optical protocol for teleporting and correcting both temporal and frequency errors in two time-frequency qubit states. The first state is the frequency (or time-of-arrival) cat qubit, which is a single photon in a superposition of two frequencies (or time-of-arrival), while the second is the time-frequency Gottesman-Kitaev-Preskill (GKP) state, which is a single photon with a frequency comb structure. The proposed optical scheme could be valuable for reducing error rate in quantum communication protocols involving one of these qubits.

Phase space approach in quantum optics

E(2) group

Wigner Distribution on a Double-Cylinder Phase Space for Studying Quantum Error-Correction Protocols

N. Fabre, A. Keller, and P. Milman, Wigner Distribution on a Double-Cylinder Phase Space for Studying Quantum Error-Correction Protocols, Phys. Rev. A 102, 022411 (2020). 

Majorana stellar representation, which visualizes a quantum spin as points on the Bloch sphere, allows quantum mechanics to accommodate the concept of trajectory, the hallmark of classical physics. We extend this notion to the discrete cylinder, which is the phase space of the canonical pair angle and orbital angular momentum. We demonstrate that the geometrical properties of the ensuing constellations aptly encapsulate the quantumness of the state.

Functional Phase space

Wigner functional theory for quantum optics

Roux FS, Fabre N. Wigner functional theory for quantum optics. arXiv; (2020): http://arxiv.org/abs/1901.07782

Using the quadrature bases that incorporate the spatiotemporal degrees of freedom, we develop a Wigner functional theory for quantum optics, as an extension of the Moyal formalism. Since the spatiotemporal quadrature bases span the complete Hilbert space of all quantum optical states, it does not require factorization as a tensor product of discrete Hilbert spaces. The Wigner functions associated with such a space become functionals and operations are expressed by functional integrals -- the functional version of the star product. The resulting formalism enables tractable calculations for scenarios where both spatiotemporal degrees of freedom and particle-number degrees of freedom are relevant. To demonstrate the approach, we compute examples of Wigner functionals for a few well-known states and operators.

SU(1,1) group

 Local Sampling of the SU(1,1) Wigner Function

N. Fabre, A. B. Klimov, G. Leuchs, and L. L. Sánchez-Soto, Local Sampling of the SU(1,1) Wigner Function, AVS Quantum Sci. 5, 014404 (2023).

Despite its indisputable merits, the Wigner phase-space formulation has not been widely explored for systems with SU(1,1) symmetry, as a simple operational definition of the Wigner function has proved elusive in this case. We capitalize on unique properties of the parity operator, to derive in a consistent way a bona fide SU(1,1) Wigner function that faithfully parallels the structure of its continuous-variable counterpart. We propose an optical scheme, involving a squeezer and photon-number-resolving detectors, that allows for direct point-by-point sampling of that Wigner function. This provides an adequate framework to represent SU(1,1) states satisfactorily.