Universal Pulse Sequence to Minimize Spin Dephasing in the Central Spin Decoherence Problem

We present a remarkable finding that a recently discovered [G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007)] series of pulse sequences, designed to optimally restore coherence to a qubit in the spin-boson model of decoherence, is in fact completely model independent and generically valid for arbitrary dephasing Hamiltonians given sufficiently short delay times between pulses. The series maximizes qubit fidelity versus number of applied pulses for sufficiently short delay times because the series, with each additional pulse, cancels successive orders of a time expansion for the fidelity decay. The “magical” universality of this property, which was not appreciated earlier, requires that a linearly growing set of “unknowns” (the delay times) must simultaneously satisfy an exponentially growing set of nonlinear equations that involve arbitrary dephasing Hamiltonian operators.

Figure 1

Various UDD (left) and CDD (right) pulse sequences in side-by-side comparison where gray (red) lines mark the time of each applied $\pi$ pulse. The sequences are numbered by their order which corresponds, respectively, to the number of pulses (UDD) or the level of concatenation (CDD). Orders of UDD and CDD with the same number of pulses are lined up horizontally.

Figure 2

Numerical results showing the natural logarithm of echoes from CDD and UDD sequences for Si:P donor electrons in natural Si. Dark gray (red) curves [light gray (green) curves] denote UDD [CDD] sequence results; the Hahn echo decay is shown as a solid black curve for reference. Sequences with the same number of pulses have the same line pattern. UDD has no particular advantage (in the high fidelity regime) to CDD in Si:P, but UDD with an even number of pulses performs better than those with an odd number.

Figure 3

Numerical results showing the natural logarithm of echoes from CDD and UDD sequences for quantum dots in GaAs with a 25 nm Fock-Darwin radius, 8.5 nm quantum well thickness. Dark gray (red) curves [light gray (green) curves] denote UDD [CDD] sequence results; the Hahn echo decay is shown as a solid black curve for reference. (a) Sequences of the same order, having the same initial slope, use the same line pattern. (b) Sequences with the same number of pulses have the same line pattern. Note that the Hahn (black) and CDD [light gray (green)] curves in (a) and (b) are the same, shown for comparison with UDD curves.