Experimental Quantum-Walk Revival with a Time-Dependent Coin

Abstract

We demonstrate a quantum walk with time-dependent coin bias. With this technique we realize an experimental single-photon one-dimensional quantum walk with a linearly-ramped time-dependent coin flip operation and thereby demonstrate two periodic revivals of the walker distribution. In our beam-displacer interferometer, the walk corresponds to movement between discretely separated transverse modes of the field serving as lattice sites, and the time-dependent coin flip is effected by implementing a different angle between the optical axis of half-wave plate and the light propagation at each step. Each of the quantum-walk steps required to realize a revival comprises two sequential orthogonal coin-flip operators, with one coin having constant bias and the other coin having a time-dependent ramped coin bias, followed by a conditional translation of the walker. [4][5][6] plus the fundamental interest of being a natural quantized version of the ubiquitous random walk that appears in statistics, computer science, finance, physics, and chemistry. QW research has focused on evolution due to repeated applications of a time-independent unitary step operator U , but a QW with time-dependent unitary steps U (t), with discrete time t ∈ N := {0, 1, 2, . . . }, opens a much richer array of phenomena including localization and quasiperiodicity [7,8]. Here we demonstrate a time-dependent QW and use this technique to demonstrate a revival of the walker's position distribution.Rather than employing direct time-dependent control, we simulate time-dependent coin control by setting different coin parameters for different steps, which are effected in different locations along the longitudinal axis within our photonic beam-displacer interferometer (BDI) [9]. The quantum walker within the BDI is a single heralded photon produced by spontaneous parametric down conversion, and its walking degree of freedom is the set of discretely spaced transverse beam modes. The coin flip is effected by employing quarter-and half-wave plates.Our method for realizing the first time-dependent QW demonstrates the phenomenon of revivals and also opens the door to realizing a multitude of time-dependent QWs experimentally. Compared to prior work employing position-dependent control [10][11][12], our new technique decreases experimental complexity by relaxing the requirement of optical compensation. Our QW revival displays a different characteristic than typical QW properties such as ballistic spreading and localization of the walker distribution.The QW with a coin proceeds as a sequence of coin flips and then walker-coin entangling operations whereby the walker's position is displaced according to the coin state. We explain the QW now in full generality so the coin operator admits both spatial and temporal dependence. Spatially-dependent coin operations have dramatically demonstrated the realization of topological phases by QWs [4][5][6], but the time-dependent QW is, until now, only a theoretical construct and not yet explored experimentally.We employ a two...

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