Boundary of quantum evolution under decoherence

Abstract

Relaxation effects impose fundamental limitations on our ability to coherently control quantum mechanical phenomena. In this article, we use principles of optimal control theory to establish physical limits on how closely a quantum mechanical system can be steered to a desired target state in the presence of relaxation. In particular, we explicitly compute the maximum amplitude of coherence or polarization that can be transferred between coupled heteronuclear spins in large molecules at high magnetic fields in the presence of relaxation. Very general decoherence mechanisms that include cross-correlated relaxation have been included in our analysis. We give analytical characterization for the pulse sequences (control laws) that achieve these physical limits and provide supporting experimental evidence. Exploitation of crosscorrelation effects has recently led to the development of powerful methods in NMR spectroscopy to study very large biomolecules in solution. For two heteronuclear spins, we demonstrate with experiments that cross-correlated relaxation optimized pulse (CROP) sequences provide significant gains over the state-of-the-art methods. It is shown that despite large relaxation rates, coherence can be transferred between coupled spins without any loss in special cases where cross-correlated relaxation rates can be tuned to autocorrelated relaxation rates. T he control of quantum ensembles has many applications, ranging from coherent spectroscopy to quantum information processing. In practice, the quantum system of interest interacts with its environment, which leads to the phenomenon of relaxation. This results in signal loss and ultimately limits the range of applications. Relaxation is also a major road block standing in the way of practical quantum computing. Manipulating quantum systems in a manner that minimizes relaxation losses is a fundamental challenge of utmost practical importance. What is the ultimate limit on how close an ensemble of quantum systems can be steered from an initial state to a desired target state in the presence of relaxation? Until now there existed no theory that answers this question. This situation is comparable to the time before the fundamental efficiency limits of heat engines were known: More than 100 years after the invention of the steam engine, the physical limits for the maximum amount of work a steam engine could produce were unclear, despite decades of advances in its design. ''The theory of its operation is rudimentary and attempts to improve its performance are still made in an almost haphazard way'' (1). Of course, the maximum efficiency of a heat engine is not given by the cleverness of the engineer who attempts to build such a machine, but by the fundamental law of thermodynamics as captured in Carnot's Theorem.In this article we derive fundamental limits on how close an ensemble of nuclear spins can be driven from its initial state to a desired target state in the presence of relaxation. In particular, we derive the maximum efficiency of polarization and c...

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

“…In the last phase, 2I y S z is lifted in a an optimal way to 2I z S z , which is again protected against relaxation [31]. Although for simplicity, CSA relaxation was not considered in this example, it is straight-forward to include CSA relaxation as well as the effects of cross-correlation in the relaxation matrix and to numerically optimize corresponding pulses (data not shown) [33]. Furthermore, the algorithm is not limited to two coupled spins and more complicated relaxation networks can be taken into account.…”

Controlling open quantum systems: tools, achievements, and limitations

“…For a few exceptional cases, for example one or two spins (or qubits) [144][145][146][147][148][149], a harmonic oscillator [150,151], or a sequence of Λ-systems subject to decay [152,153], the external controls can be determined using geometric techniques based on Pontryagin's maximum principle [4]. Typically, however, the control problem cannot be solved in closed form, and one needs to resort to numerical optimization.…”

“…Indeed, optimization of an open system's dynamics for targets that rely on quantum coherence is intrinsically biased toward those subspaces in Hilbert (or Liouville) space that are least affected by decoherence [170]. For example, transfer of coherence and polarization between coupled heteronuclear spins was improved by cross-correlated relaxation optimized pulse (CROP) sequences and relaxation optimized pulse elements (ROPE) [144,145,171,172]. Optimization can take both longitudinal and transveral relaxation into account [171].…”

Symmetry constraints on spin dynamics: Application to hyperpolarized NMR

“…These results were applied to a variety of NMR problems, including polarization transfer between spin species [1,2,3,4,5,6], and cross-polarization in solids [7,8,6]. Extensions to non-Hermitian operators and non-unitary dissipative evolution were developed [9,10,11].…”