Rotary Spin Echoes

Abstract

Torrey 1 has observed the free precession of nuclear spins around an rf field H lf fixed in a frame rotating at the Larmor frequency a) 0 =yH 0 around a large dc magnetic field H 0 . He showed that, for an H 1 much larger than the inhomogeneity of H 0 , the latter has a negligible effect on the decay of the spin magnetization which is mainly due to the inhomogeneity of H x . We report here on a method of overcoming the inhomogeneity of H x by the production of echoes in the rotating frame ("rotary echoes") which are very similar to the usual spin echoes. 2 The rotary echoes, however, have some additional specific features that make them particularly suitable for the measurement of long relaxation times.Consider the rotating frame with the z -axis along H 0 and the x -axis along the rf field H x . If at the time £ = 0 we suppose that the spin magnetization M is along the z -axis, M will precess in the yz plane and, at t = r, the angle of precession is <p =yH x T. In practice H x is inhomogeneous, with a half-width AH X , and will vary within the sample. Therefore, the total magnetization, sum of magnetization vectors at different points of the sample, will fan out in the yz plane in a time of the order of l/yA^. At t = T, we perform a 180° phase shift on the rf field so that H 1 is suddenly reversed in the rotating frame. From then on, the magnetization vector, at each point, will precess at the same rate as before, but in the opposite direction. Therefore, at time * = 2r, the angle of precession will cancel out and all the magnetization vectors will be in phase again on the z -axis producing an "echo, " as shown in Fig. 1.In that way, we overcome the effect of the inhomogeneity of H x and we measure only the effect of spin relaxation. For example, in a liquid where the Bloch equations 8 are valid, the envelope of the echoes, when T is varied, is an exponential curve with a time constant T given

Decay of the rotary echoes for the spin of a nitrogen-vacancy center in diamond

“…Robustness of the continuous decoupling with respect to these experimental imperfections can be greatly improved by periodically inverting the driving field (switching the driving field phase by 180 • ). 48,49,[58][59][60] The resulting signal demonstrates a series of so-called rotary echoes, which decay much slower than the regular Rabi oscillations, and may be utilized for precise nanoscale sensing. 48,49,58 Although the rotary echo protocol requires constant driving of the NV spin, and therefore much larger dissipated power, it has an additional advantage over the pulse-based dynamical decoupling: an experimentalist can independently control both the strength of driving and the driving reversal time.…”

Driven Dynamics and Rotary Echo of a Qubit Tunably Coupled to a Harmonic Oscillator

“…Similarly, the fluctuations in drive field that cause decay of the Rabi oscillations in Fig. 4(b) can be mitigated with the rotary-echo pulse sequence [21], depicted in Fig. 4(c), which for driven systems is analogous to the Hahn-echo sequence.…”

“…Fluctuations in the coupling parameter translate into effective driving-field amplitude noise, which reduces the qubit coherence during driven evolution. We show that the qubit dephasing due to amplitude noise (whether due to tunable coupling or otherwise) can be mitigated by a rotary echo [21], a pulse sequence originally developed for nuclear magnetic resonance.The device, shown in Fig. 1(a), consists of a flux qubit and a SQUID embedded in a two-mode LC resonant circuit.…”

Atom–molecule coherence for ultralong-range Rydberg dimers

“…As a proof of coherent photoassociation, we apply a rotary echo sequence 13,14 that we have already successfully used to investigate the 43s state of rubidium atoms 10 . A square laser pulse of fixed length τ and Rabi frequency Ω on resonance ∆ = 0 excites pairs of suitably spaced rubidium ground-state atoms to the molecular ground state.…”