Filter design for hybrid spin gates

Andreas Albrecht and Martin B. Plenio

Phys. Rev. A 92, 022340 – Published 18 August 2015

Abstract

The impact of control sequences on the environmental coupling of a quantum system can be described in terms of a filter. Here we analyze how the coherent evolution of two interacting spins subject to periodic control pulses, using the example of a nitrogen vacancy center coupled to a nuclear spin, can be described in the filter framework in both the weak- and the strong-coupling limit. A universal functional dependence around the filter resonances then allows for tuning the coupling type and strength. We show how the validity range of the filter description, originally limited to small rotation angles, can be extended to the long time limit by time-sliced evolution sequences. Based on that insight, the construction of tunable, noise decoupled, conditional gates composed of alternating pulse sequences is proposed. In particular such an approach can lead to a significant improvement in fidelity as compared to a strictly periodic control sequence. Moreover we analyze the decoherence impact, the relation to the filter for classical noise known from dynamical decoupling sequences, and we outline how an alternating sequence can improve spin sensing protocols.

Received 21 April 2015

DOI:https://doi.org/10.1103/PhysRevA.92.022340

Authors & Affiliations

Andreas Albrecht^1,2,3 and Martin B. Plenio^1,2

¹Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany
²Center for Integrated Quantum Science and Technology, Universität Ulm, 89069 Ulm, Germany
³ICFO-Institut de Ciències Fotòniques, 08860 Castelldefels, Barcelona, Spain

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Vol. 92, Iss. 2 — August 2015

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Images

Figure 1
Filter function $F$ describing the coherent spin interaction in the limit of small total evolution angles and for $N = 6$ periodic $π$ -CPMG pulses with interpulse delay $2 τ$ on the control spin. (a) Weak-coupling limit $A ≪ ω$ , which leads to a purely conditional interaction as follows out of Eq. (2). The corresponding rotation axis $σ_{ϕ}$ around the resonance peaks as given by Eq. (5) is illustrated in the inset. Amplitudes for higher-order resonances decay as $1 / (ω τ)$ (black dash-dotted), whereas the resonance peak width is inversely proportional to the total time. (b) Strong-coupling limit $A ≫ ω$ with both conditional (blue) and unconditional (green) resonances and filters. For both (a) and (b) red dashed lines denote the universal amplitude behavior of Eqs. (6) and (11) valid in the limit of large $N$ 's.
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Figure 2
Sliced evolution and long time limit extension (weak-coupling limit $A ≪ ω$ ). (a) Effective coordinate system rotation by $ϑ = c π$ after a single slice with even pulse number $n$ , tuned to the resonance parameter $c$ [Eq. (4)] and characterized by an angle $ϕ = c π / 2$ of the rotation axis $σ_{ϕ}$ [Eq. (5)]. (b) Interpretation of a large $N$ sequence on resonance $c$ beyond the filter validity by decomposition into $s n^{'}$ -pulse slices. Each slice is followed by a coordinate rotation $ϑ^{'} = c^{'} π$ with $c^{'} = c n^{'} / N$ up to a total angle $ϑ = c π$ . Thus the rotation axis [red (light gray) arrow], characterized by $ϕ = c^{'} (π / 2)$ in the local frame, is globally rotated in time. (c) Numerical simulation of a $N$ =20 $π$ -pulse sequence for $A / ω = 0.06$ , resonance order $k$ =0, and the initial state $| ↓ + 〉$ with the control spin states $| \pm 〉 = 1 / \sqrt{2} (| ↓ 〉 \pm | ↑ 〉)$ . Solid lines follow out of ten alternating $\pm c$ ( $c = 1$ ) slices as illustrated in (d) with $n_{j} \equiv 2 π$ pulses each. Dashed lines describe an equidistant $(+ c)$ CPMG sequence of 20 $π$ pulses. The total time shown corresponds to an expected $π / 2$ -pulse evolution based on the filter formalism. The difference in total time follows from the filter asymmetry in $c$ in the limit of small pulse numbers $n$ . Red arrows indicate the $π$ -pulse positions for the sliced (solid line) evolution; the different slices are shaded in different colors in analogy with the sequence coloring in (d). The final fidelity for the conditional $π / 2$ evolution can be extracted as $F = 99.9 %$ ( $\pm c$ sliced) and $F = 90 %$ (CPMG). (d) Alternating $\pm c$ control-pulse sequence composed of $s$ slices.
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Figure 3
Spin sensing (weak-coupling $A / ω = 0.05$ ). (a) Control spin coherence $〈 s_{+} 〉$ [red (light gray)] for a CPMG pulse number $N = 24$ (upper) and $N = 48$ (lower) with total time $t = 2 N τ_{0}$ . Black lines indicate the coherence evolution as expected out of the filter formalism (15). The squared filter $F^{2}$ is illustrated for comparison. (b) Coherence for an alternating $c = \pm 1 N = 48$ pulse sequence with $M$ =12, $2 n$ =4 [red (light gray)], and $M = 6, 2 n$ =8 (blue, dashed). Black lines correspond to the filter expectation for the $M$ =12 case. Lower panel: filter (17) for the $M$ =12 (left) and $M$ =6 (right) configurations and its composition (in arbitrary units) out of the grating (dotted) and block filter (dash-dotted).
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Figure 4
Comparison of the first- and second-order filter contribution in the weak-coupling limit for a number $N$ =6 of periodic pulses.
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