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Zeno subspaces and interaction-free evolutions with non-Hermitian Hamiltonians

Benedetto Militello, Anna Napoli

Published Jul 3, 2019• 0 citations
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Abstract

Effective non-Hermitian Hamiltonians describing decaying systems are considered and analyzed in connection with the occurrence of possible Hilbert space partitioning. This fact can be interpreted as Zeno effect or Zeno dynamics, according to the dimension of the subspace one focuses on. Depending on the complex phases of the diagonal terms of the Hamiltonian, the system reacts in different ways, requiring larger moduli for the Zeno phenomena to occur when the complex phase is close to $\pi/2$. Unperturbed dynamics are also shown to persist in some especial conditions where interaction-free subspaces are present.

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Effective non-Hermitian Hamiltonians describing decaying systems are considered and analyzed in connection with the occurrence of possible Hilbert space partitioning. This fact can be interpreted as Zeno effect or Zeno dynamics, according to the dimension of the subspace one focuses on. Depending on the complex phases of the diagonal terms of the Hamiltonian, the system reacts in different ways, requiring larger moduli for the Zeno phenomena to occur when the complex phase is close to π/2\pi/2π/2. Unperturbed dynamics are also shown to persist in some especial conditions where interaction-free subspaces are present.
The quantum Zeno evolution of a quantum system takes place in a proper subspace of the total Hilbert space. The physical and mathematical features of the "Zeno subspaces" depend on the measuring apparatus: when this is included in the quantum description, the Zeno effect becomes a mere consequence of the dynamics and, remarkably, can be cast in terms of an adiabatic theorem, with a dynamical superselection rule. We look at several examples and focus on quantum computation and decoherence-free subspaces.
Non-Hermitian Hamiltonians are a fascinating class of many-body models that describe the effective dynamics of quantum systems interacting with the environment through particle, energy, or information exchange. Although their theoretical framework is well established, the controlled engineering of such Hamiltonians in the context of quantum simulations remains challenging, even more so when the non-Hermitian part describes a kkk-body interaction. Qudit quantum simulators offer a compelling framework to implement such models. We theoretically investigate the dynamics of a one-dimensional chain of qudits undergoing hybrid unitary-projective evolution, where suitably designed measurements constrain the dynamics to a Zeno subspace. As we illustrate for the case of qutrits, within the Zeno subspace the dynamics is governed by an effective non-Hermitian Hamiltonian for an ensemble of pseudo-spins 1/21/21/2, which can inherit non-Hermitian two-body interactions with the same connectivity as the full qutrit chain. We derive an analytical relation linking the monitored qutrits' evolution to a desired target non-Hermitian Hamiltonian and validate the effective description through numerical simulations of a representative model. Our scheme provides a constructive route for the realization of a large class of interacting non-Hermitian many-body Hamiltonians in experimentally relevant multilevel quantum platforms, including trapped ions and superconducting circuits.