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Register a new userTime-dependent Hamiltonians with 100% evolution speed efficiency
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The evolution speed in projective Hilbert space is considered for Hermitian Hamiltonians and for non-Hermitian (NH) ones. Based on the Hilbert-Schmidt norm and the spectral norm of a Hamiltonian, resource-related upper bounds on the evolution speed are constructed. These bounds are valid also for NH Hamiltonians and they are illustrated for an optical NH Hamiltonian and for a non-Hermitian $mathcal{PT}-$symmetric matrix Hamiltonian. Furthermore, the concept of quantum speed efficiency is introduced as measure of the system resources directly spent on the motion in the projective Hilbert space. A recipe for the construction of time-dependent Hamiltonians which ensure 100% speed efficiency is given. Generally these efficient Hamiltonians are NH but there is a Hermitian efficient Hamiltonian as well. Finally, the extremal case of a non-Hermitian non-diagonalizable Hamiltonian with vanishing energy difference is shown to produce a 100% efficient evolution with minimal resources consumption.
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We formulate a set of conditions under which dynamics of a time-dependent quantum Hamiltonian are integrable. The main requirement is the existence of a nonabelian gauge field with zero curvature in the space of system parameters. Known solvable multistate Landau-Zener models satisfy these conditions. Our method provides a strategy to incorporate time-dependence into various quantum integrable models, so that the resulting non-stationary Schrodinger equation is exactly solvable. We also validate some prior conjectures, including the solution of the driven generalized Tavis-Cummings model.
The validity of optimized dynamical decoupling (DD) is extended to analytically time dependent Hamiltonians. As long as an expansion in time is possible the time dependence of the initial Hamiltonian does not affect the efficiency of optimized dynamical decoupling (UDD, Uhrig DD). This extension provides the analytic basis for (i) applying UDD to effective Hamiltonians in time dependent reference frames, for instance in the interaction picture of fast modes and for (ii) its application in hierarchical DD schemes with $pi$ pulses about two perpendicular axes in spin space. to suppress general decoherence, i.e., longitudinal relaxation and dephasing.
In this work we address systems described by time-dependent non-Hermitian Hamiltonians under time-dependent Dyson maps. We shown that when starting from a given time-dependent non-Hermitian Hamiltonian which is not itself an observable, an infinite chain of gauge linked time-dependent non-observable non-Hermitian Hamiltonians can be derived from it. The matrix elements of the observables associated with all these non observable Hamiltonians are, however, all linked to each other, and in the particular case where global gauges exist, these matrix elements becomes all identical to each other. In this case, therefore, by approaching whatever the Hamiltonian in the chain we can get information about any other Hamiltonian. We then show that the whole chain of time-dependent non-Hermitian Hamiltonians collapses to a single time-dependent non-Hermitian Hamiltonian when, under particular choices for the time-dependent Dyson maps, the observability of the Hamiltonians is assured. This collapse thus shows that the observability character of a non-Hermitian Hamiltonian prevents the construction of the gauge-linked Hamiltonian chain and, consequently, the possibility of approaching one Hamiltonian from another.
Given a generic time-dependent many-body quantum state, we determine the associated parent Hamiltonian. This procedure may require, in general, interactions of any sort. Enforcing the requirement of a fixed set of engineerable Hamiltonians, we find the optimal Hamiltonian once a set of realistic elementary interactions is defined. We provide three examples of this approach. We first apply the optimization protocol to the ground states of the one-dimensional Ising model and a ferromagnetic $p$-spin model but with time-dependent coefficients. We also consider a time-dependent state that interpolates between a product state and the ground state of a $p$-spin model. We determine the time-dependent optimal parent Hamiltonian for these states and analyze the capability of this Hamiltonian of generating the state evolution. Finally, we discuss the connections of our approach to shortcuts to adiabaticity.
Here we present an strategy for the derivation of a time-dependent Dyson map which ensures simultaneously the unitarity of the time evolution and the observability of a quasi-Hermitian Hamiltonian. The time-dependent Dyson map is derived through a constructed Schr{o}dinger-like equation governed by the non-Hermitian Hamiltonian itself; despite its time-dependence our scheme ensures the time-independence of the metric operator, a necessary condition for the observability of the quasi-Hermitian Hamiltonian. As an illustrative example we consider a driven Harmonic oscillator described by a time-dependent non-Hermitian Hamiltonian. After computing the Dyson map and demonstrating the time-independence of the associated metric operator, we successfully derive an eigenvalue equation for this time-dependent Hamiltonian which enable us to analyze the $mathcal{PT}$-symmetry breaking process.
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