Optimal control of spin dynamics in the presence of relaxation

Khaneja, Navin; Reiss, Timo O.; Luy, Burkhard; Glaser, Steffen J.

doi:10.1016/s1090-7807(03)00003-x

Cited by 135 publications

(146 citation statements)

References 5 publications

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“…For example, if dipolar relaxation between an isolated pair of spins is the dominant relaxation mechanism, the in-phase to anti-phase transfer (I x to 2I z S x ) via analytically derived relaxation optimized pulse sequence elements (ROPE) [31,32] is up to a factor of e/2 = 1.36 more efficient than the traditional INEPT transfer. Here, we demonstrate the application of the GRAPE algorithm to the numerical optimization of ROPE-type sequences and compare the results to the analytical solutions.…”

Section: Relaxation-optimized Pulse Elements (Rope)mentioning

confidence: 99%

Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms

Khaneja¹,

Reiss²,

Kehlet³

et al. 2005

Journal of Magnetic Resonance

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1,752

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Section: Relaxation-optimized Pulse Elements (Rope)mentioning

confidence: 99%

Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms

Khaneja¹,

Reiss²,

Kehlet³

et al. 2005

Journal of Magnetic Resonance

Self Cite

1,628

1,752

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“…[22,23]. More recently, there has been vigorous effort in studying the control of systems governed by the Liouville-von Neumann (LVN) equation, where the central object is the density matrix, rather than the wavefunction [24,25,26,27,28,29,30]. The Liouville-von Neumann equation is an extension of the TDSE that allows for the inclusion of dissipative processes.…”

Section: Introductionmentioning

confidence: 99%

Optimal control of quantum dissipative dynamics: Analytic solution for cooling the three-levelΛsystem

2004

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We study the problem of optimal control of dissipative quantum dynamics. Although under most circumstances dissipation leads to an increase in entropy (or a decrease in purity) of the system, there is an important class of problems for which dissipation with external control can decrease the entropy (or increase the purity) of the system. An important example is laser cooling. In such systems, there is an interplay of the Hamiltonian part of the dynamics, which is controllable and the dissipative part of the dynamics, which is uncontrollable. The strategy is to control the Hamiltonian portion of the evolution in such a way that the dissipation causes the purity of the system to increase rather than decrease. The goal of this paper is to find the strategy that leads to maximal purity at the final time. Under the assumption that Hamiltonian control is complete and arbitrarily fast, we provide a general framework by which to calculate optimal cooling strategies. These assumptions lead to a great simplification, in which the control problem can be reformulated in terms of the spectrum of eigenvalues of ρ, rather than ρ itself. By combining this formulation with the Hamilton-Jacobi-Bellman theorem we are able to obtain an equation for the globaly optimal cooling strategy in terms of the spectrum of the density matrix. For the three-level Λ system, we provide a complete analytic solution for the optimal cooling strategy. For this system it is found that the optimal strategy does not exploit system coherences and is a 'greedy' strategy, in which the purity is increased maximally at each instant.

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“…Can inhomogeneous control [31] be extended to qudits, perhaps allowing addressable unitary maps on large arrays [32]? And finally, is it possible to optimize control in the presence of decoherence [33], and perhaps extend it to (non-unitary) completely positive maps [34]? Some of these questions can be explored in our current system, while others await the application of optimal control to scalable architectures of interacting qubits and qudits.…”

mentioning

confidence: 99%

Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System

et al. 2015

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Quantum control in large dimensional Hilbert spaces is essential for realizing the power of quantum information processing. For closed quantum systems the relevant inputoutput maps are unitary transformations, and the fundamental challenge becomes how to implement these with high fidelity in the presence of experimental imperfections and decoherence. For two-level systems (qubits) most aspects of unitary control are well understood, but for systems with Hilbert space dimension d>2 (qudits), many questions remain regarding the optimal design of control Hamiltonians 1 and the feasibility of robust implementation 2,3 . Here we show that arbitrary, randomly chosen unitary transformations can be efficiently designed and implemented in a large dimensional Hilbert space (d=16) associated with the electronic ground state of atomic 133 Cs, 4 achieving fidelities above 0.98 as measured by randomized benchmarking 5 . Generalizing the concepts of inhomogeneous control 6 and dynamical decoupling 7 to d>2 systems, we further demonstrate that these qudit unitary maps can be made robust to both static and dynamic perturbations. Potential applications include improved fault-tolerance in universal quantum computation 8 , nonclassical state preparation for high-precision metrology 9 , implementation of quantum simulations 10 , and the study of fundamental physics related to open quantum systems and quantum chaos 11 .The goal of quantum control is to perform a desired transformation through dynamical evolution driven by a control Hamiltonian H C (t) . For example, one common objective is to evolve the system from a known initial state to a desired final state. If the control task is simple or special symmetries are present, it is sometimes possible to find a high-performing control Hamiltonian through intuition, or to construct one using group theoretic methods 12 . In this letter we explore the use of "optimal control" 1 to design control Hamiltonians for tasks of varying complexity, from state-to-state maps to unitary maps on the entire accessible Hilbert space. The basic procedure is well established: the Hamiltonian H C (t) is parameterized by a set of control variables, and a numerical search is performed to find values that optimize the fidelity with which the control objective is achieved. The application of optimal control to quantum systems originated in NMR 13 and physical chemistry 1 , and has since expanded to include, e. g., ultrafast physics 14 , cold atoms 15,16 , biological molecules 17 , spins in condensed matter 18 , and superconducting circuits 19 .We study the efficacy of numerical design and the performance of the resulting control Hamiltonians using a well developed testbed consisting of the electron and nuclear spins of individual 133 Cs atoms driven by radiofrequency (rf) and microwave (µw) magnetic fields (Fig. 1) 16 . Our experiments show that the optimal control strategy is adaptable to a wide range of control tasks, and that it can generate control Hamiltonians with excellent performance even in the prese...

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Optimal control of spin dynamics in the presence of relaxation

Cited by 135 publications

References 5 publications

Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms

Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms

Optimal control of quantum dissipative dynamics: Analytic solution for cooling the three-levelΛsystem

Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System

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