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arXiv:1202.3903 · Communications in Mathematical Physics

Recurrence for Discrete Time Unitary Evolutions

F. Alberto Grünbaum, L. Velázquez, Albert H. Werner, Reinhard F. Werner

Published Jan 15, 2013• 123 citations
arXivHTMLPDF
DOI
10.1007/s00220-012-1645-2

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This paper develops an operator-theoretic framework for characterising a phenomenon we call phase-free recurrence: the exact return of a quantum state to itself under unitary time evolution, with no global phase change whatsoever.Starting from the unitary time-evolution operator , we construct a Hermitian operator via symmetrisation, yielding the cosine operator . Applying the real sign function to this operator defines a new operator , which we prove to be an involution — a self-inverse linear operator — whenever none of the cosines of the energy eigenvalues vanish. The involutory structure of induces a canonical decomposition of the Hilbert space into three mutually orthogonal subspaces corresponding to eigenvalues +1, −1, and 0. We study the kernel subspace in detail, characterising the discrete periodic times at which each energy component enters it.Using this structure we derive a precise algebraic criterion for phase-free recurrence. For a two-level system with energy eigenvalues and , we prove that phase-free recurrence occurs if and only if the ratio is rational. This result is generalised to arbitrary n-level systems, where it takes the form of a simultaneous proportionality condition on the integers labelling the kernel-recurrence times of each energy component.The framework is purely algebraic, relying only on the spectral theorem and elementary properties of involutions, and applies to any quantum system with a discrete non-degenerate spectrum.
Measurements on a quantum particle unavoidably affect its state, since the otherwise unitary evolution of the system is interrupted by a nonunitary projection operation. To probe measurement-induced effects in the state dynamics using a quantum simulator, the challenge is to implement controlled measurements on a small subspace of the system and continue the evolution from the complementary subspace. A powerful platform for versatile quantum evolution is represented by photonic quantum walks because of their high control over all relevant parameters. However, measurement-induced dynamics in such a platform have not yet been realized. We implement controlled measurements in a discrete-time quantum walk based on time-multiplexing. This is achieved by adding a deterministic outcoupling of the optical signal to include measurements constrained to specific positions resulting in the projection of the walker's state on the remaining ones. With this platform and coherent input light, we experimentally simulate measurement-induced single-particle quantum dynamics. We demonstrate the difference between dynamics with only a single measurement at the final step and those including measurements during the evolution. To this aim, we study recurrence as a figure of merit, that is, the return probability to the walker's starting position, which is measured in the two cases. We track the development of the return probability over 36 time steps and observe the onset of both recurrent and transient evolution as an effect of the different measurement schemes, a signature which only emerges for quantum systems. Our simulation of the observed one-particle conditional quantum dynamics does not require a genuine quantum particle but is demonstrated with coherent light.
Inside a closed many-body system undergoing the unitary evolution, a small partition of the whole system exhibits a local relaxation. If the total degrees of freedom of the whole system is a large but finite number, such a local relaxation would come across a recurrence after a certain time, namely, the dynamics of the local system suddenly appears random after a well-ordered oscillatory decay process. It is found in this paper, among a collection of N two-level systems (TLSs), the local relaxation of one TLS inside has a hierarchy structure hiding in the randomness after such a recurrence: similar recurrences appear in a periodical way, and the later recurrence brings in stronger randomness than the previous one. Both analytical and numerical results that we obtained well explains such hierarchy recurrences: the population of the local TLS (as an open system) diffuses out and regathers back periodically due to the finite-size effect of the bath [the remaining $(\mathrm{N}\ensuremath{-}1)$ TLSs]. We also find that the total correlation entropy, which sums up the entropy of all the n TLSs, approximately exhibit a monotonic increase; in contrast, the entropy of each single TLS increases and decreases from time to time, and the entropy of the whole n-body system keeps constant during the unitary evolution.