NMR Time Reversal as a Probe of Incipient Turbulent Spin Dynamics

We demonstrate time reversal of nuclear spin dynamics in highly magnetized dilute liquid ${}^3\mathrm{He}\mathrm{\text{−}}{}^4\mathrm{He}$ mixtures through effective inversion of long-range dipolar interactions. These experiments, which involve using magic sandwich NMR pulse sequences to generate spin echoes, probe the spatiotemporal development of turbulent spin dynamics and promise to serve as a versatile tool for the study and control of dynamic magnetization instabilities. We also show that a repeated magic sandwich pulse sequence can be used to dynamically stabilize modes of nuclear precession that are otherwise intrinsically unstable. To date, we have extended the effective precession lifetimes of our magnetized samples by more than three orders of magnitude.

Figure 1

FID response of polarized ${}^3\mathrm{He}$ in liquid ${}^4\mathrm{He}$ (a) to a 90° rf tipping pulse and (b) to a $90°\mathrm{\text{−}}\tau \mathrm{\text{−}}180°$ sequence. The time at which the FID amplitude $S$ drops to half of its initial value $S_0$ is denoted $T_{1/2}$, which is of order 70 ms for these examples. Trace (b) has been displaced upward by 0.4 mV for the sake of clarity. The inset shows the deviation of trace (b) from its initial amplitude. These data were acquired at $T=1.14\mathrm{K}$, with $X\sim 2\%$ and $B_{\mathrm{dip}}\sim 0.9\mu \mathrm{T}$ for trace (a) and $X\sim 5\%$ and $B_{\mathrm{dip}}=0.7\mu \mathrm{T}$ for trace (b).

Figure 2

Spin echo formation: A 90° pulse is applied at $t=0$, after which the magnetization is allowed to evolve for $\tau =70\mathrm{ms}$. MS sequences of various durations are then used to progressively refocus the magnetization. For traces (a)–(c), the amplitude of the FID immediately after the MS probes the rotary echo formed in the doubly-rotating frame. The symbols ▪, $●$, ▴ indicate pairings between points on the initial FID to which the state of the magnetization is nominally returned and the ensuing free precession signal. For traces (d)–(f), the amplitude of the FID probes the magic sandwich echo formed in the rotating frame after the magnetization is first “unwound” back to $t=-\tau$ before being “released” and allowed to undergo further free evolution. These data were acquired under conditions similar to those outlined in Fig. 1. Inset: A classic magic sandwich [15] consists of a continuous train of ($180{°}_x$, $-180{°}_x$) rotation pairs sandwiched between a leading $90{°}_y$ and a trailing $-90{°}_y$ rotation. We have used 360° (rather than 180°) rotations to reduce the accumulation of phase errors associated with offsets between the rf and Larmor frequencies. Each 360° rotation is accomplished in 2 ms, corresponding to $B_1=15\mu \mathrm{T}$. Note that for trace (c), the sense of the terminal 90° rotation was reversed (i.e., $90{°}_y$ instead of $-90{°}_y$) in order to also refocus interactions with a linear dependence on $\mathbf{M}$. This does not significantly alter the initial amplitude of the echo or its subsequent growth.

Figure 3

Spin echo trains: Traces (a)–(d) correspond to echo trains that are observed when RMS sequences with delay times $\tau =16$, 24, 32, and 40 ms, respectively, are applied to the magnetized liquid. The MS cycles used to obtain these data are identical to those described in the caption of Fig. 2 except for the facts that each one terminates with an additional $180{°}_y$ rotation (cf. trace (c) of Fig. 2) to refocus interactions linear in $\mathbf{M}$, and that the symmetry of alternate MS cycles is inverted to reduce the accumulation of phase errors. That is, odd MS cycles are of the form [$90{°}_y$, $n\times (360{°}_x,-360{°}_x)$, $90{°}_y$] while even MS cycles are of the form $-90{°}_y$, $n\times (-360{°}_x,360{°}_x)$, $-90{°}_y$]. These data were acquired at $T\sim 1.1\mathrm{K}$, with $X\sim 4.5\%$ and $B_{\mathrm{dip}}\sim 0.8\mu \mathrm{T}$. In the limit $B_{\mathrm{dip}}\to 0$, CPMG-based measurements yield $D\sim 2\times {10}^{-7}{\mathrm{m}}^2/\mathrm{s}$.