Time-frequency Talbot effect as Clifford operations on entangled time-frequency GKP states

Abstract

The Talbot effect -- a near-field diffraction phenomenon in which a periodic wavefront self-images at regular distances -- can be transposed to the time--frequency domain via the space--time duality between diffraction and dispersive broadening. We exploit this analogy to define the time--frequency (TF) Talbot effect and show that it implements different Clifford operations on TF Gottesman-Kitaev-Preskill (TF-GKP) qubits (Phys. Rev. 102, 012607), a class of qubit states encoded in the discretised frequency and time-of-arrival degrees of freedom of entangled photon pairs, whose logical basis corresponds to even and odd components of an entangled frequency combs. These states are intrinsically robust against small frequency and temporal displacements, which can be further corrected by linear or nonlinear quantum error-correction schemes. We analyse the role of the comb envelope and peak width relative to the free spectral range, and show that a compromise must be made between the gate fidelity of the Clifford gates induced by TF-Talbot operation and the error-correction capacity of the code. We then demonstrate that the signature of the TF-Talbot effect is directly accessible via the generalised Hong-Ou-Mandel interferometer: all six logical GKP states can be unambiguously distinguished by introducing a frequency shift of half the comb periodicity in one interferometer arm. We conclude with a feasibility analysis based on current experimental technology, identifying the comb finesse as the key figure of merit for both gate performance and correctability. This conclusion extends naturally to quadrature GKP states, where a shear in quadrature phase space is precisely a Talbot effect.

Figures (10)

• Figure 1: Frequency-domain (top) and time-domain (bottom) amplitudes of the $\ket{0_{\omega}}=\ket{+_{t}}$ and $\ket{1_{\omega}}=\ket{-_{t}}$ logical codewords, together with the $\ket{+i_{\omega}}$ state, which arises naturally in the TF-Talbot evolution. The remaining logical states can be straightforwardly deduced from these.
• Figure 2: Overlap between the (a) $\ket{0_\omega}$ and $\ket{1_\omega}$ states and (b) $\ket{0_t}$ and $\ket{1_t}$ states. The physical TF-GKP states can be considered orthogonal only in the limit of a large envelope and narrow peaks.
• Figure 3: Joint spectral intensity of the entangled TF-GKP state $\frac{\ket{R0_{\omega}0_{\omega}}_{ab}+\ket{R1_{\omega}1_{\omega}}_{ab}}{\sqrt{2}}$, with comb periodicity $\overline{\omega}=1$ and $\omega_{p}$ is the pump degeneracy. The frequency units are set with respect to the free spectral range.
• Figure 4: Joint temporal intensity along the $t_{-}$ collective variable, alongside dispersion with (a) $(\kappa = 10\overline\omega, \sigma = 0.05\overline\omega)$, (b) $(\kappa = 2\overline\omega, \sigma= 0.05\overline\omega)$ and (c) $(\kappa = 30\overline\omega, \sigma = 0.05\overline\omega)$. The input state at $\beta = 0$ dispersion is a $\ket{0_t}$ state. The state is transformed into a $\ket{-i_t}$ state, a $\ket{1_t}$ state then a $\ket{+i_t}$ state and finally returns to being a $\ket{0_t}$ state. The smaller the envelope, the lesser the effect of dispersion for fixed propagation. Hence, the transformation reaches higher fidelity for narrower envelope.
• Figure 5: Error probability after Steane-type stabilisation. Probability that a noisy TF-GKP state undergoes a logical error after one stabilisation cycle using a physical TF-GKP ancilla in a Steane circuit. A higher error probability indicates reduced robustness of the TF-GKP state against displacement errors, and therefore a diminishing return from further stabilisation cycles.