Gottesman-Kitaev-Preskill Encoding in Continuous Modal Variables of Single Photons.

GKP states, introduced by Gottesman, Kitaev, and Preskill, are continuous variable logical qubits that can be corrected for errors caused by phase space displacements. Their experimental realization is challenging, in particular, using propagating fields, where quantum information is encoded in the quadratures of the electromagnetic field. However, traveling photons are essential in many applications of GKP codes involving the long-distance transmission of quantum information. We introduce a new method for encoding GKP states in propagating fields using single photons, each occupying a distinct auxiliary mode given by the propagation direction. The GKP states are defined as highly correlated states described by collective continuous modes, as time and frequency. We analyze how the error detection and correction protocol scales with the total photon number and the spectral width. We show that the obtained code can be corrected for displacements in time-frequency phase space, which correspond to dephasing, or rotations, in the quadrature phase space and to photon losses. Most importantly, we show that generating two-photon GKP states is relatively simple, and that such states are currently produced and manipulated in several photonic platforms where frequency and time-bin biphoton entangled states can be engineered.

Quantum Metrology Using Time-Frequency as Quantum Continuous Variables: Resources, Sub-Shot-Noise Precision and Phase Space Representation.

We study the role of the electromagnetic field's frequency on the precision limits of time measurements from a quantum perspective, using single photons as a paradigmatic system. We demonstrate that a quantum enhancement of precision is possible only when combining both intensity and spectral resources and, in particular, that spectral correlations enable a quadratic scaling of precision with the number of probes. We identify the general mathematical structure of nonphysical states that achieve the Heisenberg limit and show how a finite spectral variance may cause a quantum-to-classical-like transition in precision scaling for pure states similar to the one observed for noisy systems. Finally, we provide a clear and consistent geometrical time-frequency phase space interpretation of our results, identifying what should be considered as spectral classical resources.