Geometric driving of two-level quantum systems

Abstract

We investigate a class of cyclic evolutions for driven two-level quantum systems (effective spin-1/2) with a particular focus on the geometric characteristics of the driving and their specific imprints on the quantum dynamics. By introducing the concept of geometric field curvature for any field trajectory in the parameter space we are able to unveil underlying patterns in the overall quantum behavior: the knowledge of the field curvature provides a non-standard and fresh access to the interrelation between field and spin trajectories, and the corresponding quantum phases acquired in non-adiabatic cyclic evolutions. In this context, we single out setups in which the driving field curvature can be employed to demonstrate a pure geometric control of the quantum phases. Furthermore, the driving field curvature can be naturally exploited to introduce the geometrical torque and derive a general expression for the total quantum phase acquired in a cycle. Remarkably, such relation allows to access the mechanisms controlling the changeover of the quantum phase across a topological transition and to disentangle the role of the spin and field topological windings. As for implementations, we discuss a series of physical systems and platforms to demonstrate how the geometric control of the quantum phases can be realized for pendular field drivings. This includes setups based on superconducting islands coupled to a Josephson junction and inversion asymmetric nanochannels with suitably tailored geometric shapes.

Dynamically corrected gates from geometric space curves

“…This includes pulse-amplitude errors that alter the rate at which the Bloch sphere trajectory is traversed while leaving its shape intact. Although geometric phases were originally defined in the context of adiabatic evolution [126], this concept was later generalized to non-adiabatic evolution [134], enabling fast holonomic gates [135][136][137][138][139][140][141][142][143][144]. Such gates have been experimentally implemented in a number of qubit platforms [145][146][147][148][149][150][151].…”

Spin dynamics under the influence of elliptically rotating fields: Extracting the field topology from time-averaged quantities

“…This critical region has recently been explored in Ref. [33] using another type of adiabatic approximation.…”

Doubly geometric quantum control

“…Using geometric rather than dynamical phases to implement quantum gates can mitigate the effect of noise that leaves holonomy loops in the control space unperturbed. Geometric phases can be accrued using either adiabatic [9][10][11] or non-adiabatic driving [12][13][14][15][16][17][18][19][20][21][22]; the latter alleviates decoherence by reducing the operation time. Non-adiabatic holonomic (geometric) gates have been successfully realized in superconducting systems [23,24], trapped ions [25,26], and NV centers in diamond [27][28][29].…”