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Control-free state engineering by hybrid measurement-unitary bang-bang

Mads Kock Pedersen, Jens Jakob Sørensen, Malte C. Tichy, Jacob Friis Sherson

Published Jun 3, 2014• 0 citations
OpenAlex

Abstract

State preparation in high-dimensional Hilbert-spaces does not require control over a system Hamiltonian or over applicable measurement operators: We show how to prepare a desired state or subspace, given a static projection operator onto the desired target that is applied repeatedly at optimised moments in time. Benchmarks against other schemes, performed on random Hamiltonians and on Bose-Hubbard systems, establish the competitiveness of the method.

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State preparation in high-dimensional Hilbert-spaces does not require control over a system Hamiltonian or over applicable measurement operators: We show how to prepare a desired state or subspace, given a static projection operator onto the desired target that is applied repeatedly at optimised moments in time. Benchmarks against other schemes, performed on random Hamiltonians and on Bose-Hubbard systems, establish the competitiveness of the method.
We develop a scheme to prepare a desired state or subspace in high-dimensional Hilbert-spaces using repeated applications of a single static projection operator onto the desired target and fixed unitary dynamics. Benchmarks against other control schemes, performed on generic Hamiltonians and on Bose--Hubbard systems, establish the competitiveness of the method. As a concrete application of the control of mesoscopic atomic samples in optical lattices we demonstrate the near deterministic preparation of Schrödinger cat states of all atoms residing on either the odd or the even sites.
We introduce an efficient iterative method to prepare a target state in Hilbert spaces with high dimensionality using a combination of unitary evolution, measurements, and quantum Zeno dynamics. The latter confines the evolution within Zeno subspaces of decreasing size. This gives an exponential speedup relative to the case of states evolving in the full Hilbert space between projective measurements. We demonstrate our approach on the control problem of rapidly transferring a superfluid into the Mott insulator in the Bose-Hubbard model. We discuss the general applicability of the method by preparing arbitrary superpositions with random Hamiltonians.
Aiming on the preparation of a given physical system in a particular quantum state, we can follow one of several standard approaches. The most straightforward one is to expose the system to a specific coherent evolution that will deterministically bring it into the desired final state. However, in case of microscopic few-particle systems this approach requires very good control of physical parameters at the single particle level. Another method of the state production is based on projective quantum measurements of the system: Quantum Mechanics postulates that the system is randomly projected onto one of the eigenstates of the measurement operator M. By a proper choice of M, the system can end up in the desired state after the measurement. Being less demanding this method is however inherently non-deterministic. To steer the system towards the target state and thus to overcome the randomness of the measurement, we propose to use an active feedback on the system while the measurement is being performed.
It is notoriously known that in contrast to the measure of classical macro systems, the quantum measurement will in general cause quantum state collapse. In this note, we will present some further results on stabilizing control of quantum systems with measurement being involved. To be precise, we will study the effects of the open-loop control and the feedback control in preparing an arbitrarily desired eigenstate of an effective Hamiltonian, respectively. For the open-loop control case, it is shown that no matter how to select the measurement channel, control channel and the control law, one cannot prepare all the eigenstates of the effective Hamiltonian with arbitrary high fidelity for all sufficient large time. While for the feedback case, we not only demonstrate how to select the measurement channel properly to achieve the control target, but also show if not, what cannot be done by the measurement-based feedback control.
Abstract Equidistant and non-equidistant single pulse ‘bang–bang’ dynamical controls are investigated in the context of mean ergodic theorems. We show the requirements in which the limit of infinite pulse control for both the equidistant and the non-equidistant dynamical control converges to the same unitary evolution. It is demonstrated that the generator of this evolution can be obtained by projecting the generator of the free evolution onto the commutant of the unitary operator representing the pulse. Inequalities are derived to prove this statement and in the case of non-equidistant approach these inequalities are optimised as a function of the time intervals.
In our model a fixed Hamiltonian acts on the joint Hilbert space of a quantum system and its controller. We show conditions under which measurements, state preparations, and unitary implementations on the system can be performed by quantum operations on the controller only. It turns out that a measurement of the observable AAA and an implementation of the one-parameter group exp(iAr)\mathrm{exp}(\mathrm{iAr})exp(iAr) can be performed by almost the same sequence of control operations. Furthermore, measurement procedures for A+B,A+B,A+B, for (AB+BA),(AB+BA),(AB+BA), and for $i\ensuremath{\lfloor}A,B\ensuremath{\rfloor}$ can be constructed from measurements of AAA and B.B.B. This shows that the algebraic structure of the set of observables can be explained by the Lie group structure of the unitary evolutions on the joint Hilbert space of the measuring device and the measured system. A spin chain model with nearest-neighborhood coupling shows that the border line between controller and system can be shifted consistently.