# General tracking control of arbitrary N-level quantum systems using piecewise time-independent potentials

General tracking control of arbitrary N-level quantum systems using piecewise time-independent... Here we propose a tracking quantum control protocol for arbitrary N-level systems. The goal is to make the expected value of an observable $${\mathcal O}$$ O to follow a predetermined trajectory S(t). For so, we drive the quantum state $$|\varPsi (t) \rangle$$ | Ψ ( t ) ⟩ evolution through an external potential V which depends on $$M_V$$ M V tunable parameters (e.g., the amplitude and phase (thus $$M_V = 2$$ M V = 2 ) of a laser field in the dipolar condition). At instants $$t_n$$ t n , these parameters can be rapidly switched to specific values and then kept constant during time intervals $$\Delta t$$ Δ t . The method determines which sets of parameters values can result in $$\langle \varPsi (t) | {\mathcal O} |\varPsi (t) \rangle = S(t)$$ ⟨ Ψ ( t ) | O | Ψ ( t ) ⟩ = S ( t ) . It is numerically robust (no intrinsic divergences) and relatively fast since we need to solve only nonlinear algebraic (instead of a system of coupled nonlinear differential) equations to obtain the parameters at the successive $$\Delta t$$ Δ t ’s. For a given S(t), the required minimum $$M_V = M_{\min }$$ M V = M min ‘degrees of freedom’ of V attaining the control is a good figure of merit of the problem difficulty. For instance, the control cannot be unconditionally realizable if $$M_{\min } > 2$$ M min > 2 and V is due to a laser field (the usual context in real applications). As it is discussed and exemplified, in these cases a possible procedure is to relax the control in certain problematic (but short) time intervals. Finally, when existing the approach can systematically access distinct possible solutions, thereby allowing a relatively simple way to search for the best implementation conditions. Illustrations for 3-, 4-, and 5-level systems and some comparisons with calculations in the literature are presented. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# General tracking control of arbitrary N-level quantum systems using piecewise time-independent potentials

, Volume 15 (5) – Jan 25, 2016
24 pages

/lp/springer_journal/general-tracking-control-of-arbitrary-n-level-quantum-systems-using-SEWF7xwwau
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-016-1241-z
Publisher site
See Article on Publisher Site

### Abstract

Here we propose a tracking quantum control protocol for arbitrary N-level systems. The goal is to make the expected value of an observable $${\mathcal O}$$ O to follow a predetermined trajectory S(t). For so, we drive the quantum state $$|\varPsi (t) \rangle$$ | Ψ ( t ) ⟩ evolution through an external potential V which depends on $$M_V$$ M V tunable parameters (e.g., the amplitude and phase (thus $$M_V = 2$$ M V = 2 ) of a laser field in the dipolar condition). At instants $$t_n$$ t n , these parameters can be rapidly switched to specific values and then kept constant during time intervals $$\Delta t$$ Δ t . The method determines which sets of parameters values can result in $$\langle \varPsi (t) | {\mathcal O} |\varPsi (t) \rangle = S(t)$$ ⟨ Ψ ( t ) | O | Ψ ( t ) ⟩ = S ( t ) . It is numerically robust (no intrinsic divergences) and relatively fast since we need to solve only nonlinear algebraic (instead of a system of coupled nonlinear differential) equations to obtain the parameters at the successive $$\Delta t$$ Δ t ’s. For a given S(t), the required minimum $$M_V = M_{\min }$$ M V = M min ‘degrees of freedom’ of V attaining the control is a good figure of merit of the problem difficulty. For instance, the control cannot be unconditionally realizable if $$M_{\min } > 2$$ M min > 2 and V is due to a laser field (the usual context in real applications). As it is discussed and exemplified, in these cases a possible procedure is to relax the control in certain problematic (but short) time intervals. Finally, when existing the approach can systematically access distinct possible solutions, thereby allowing a relatively simple way to search for the best implementation conditions. Illustrations for 3-, 4-, and 5-level systems and some comparisons with calculations in the literature are presented.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Jan 25, 2016

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