Vector spin polarization evolution determined in an entangled muon-fluorine system under pulsed excitation

Abstract

A spin-polarized muon implanted into a fluoride forms a coupled F--$μ$--F complex in which the muon spin and neighbouring fluorine nuclear spins become entangled. Here we apply radio-frequency (RF) excitation to this coupled system and use the three-dimensional distribution of emitted positrons to reconstruct the time-dependent evolution of the muon spin polarization. This three-dimensional readout, using single spin detection, is not possible in a single NMR experiment and demonstrates significant advantages that are achieved by using RF muon techniques. We demonstrate the application of this vector-readout method to the experimental observation of a muon spin echo signal that is controlled by the dipolar coupling to fluorine, as well as to a double resonance experiment, in which we use pulses tuned to separate frequencies to address both the muon and fluorine spins. This targeted approach, in which selective RF pulses can control the muon spin and other spins to which it is coupled, provides a novel route for probing systems of entangled spins.

Figures (5)

• Figure 1: Schematic of experiment. The muon is implanted with its spin antiparallel to its momentum, with the pulse of the muon synchronised with the pulse generator that controls the radio-frequency (RF) excitation. A steady magnetic field $B_0$ is applied longitudinally (aligned with the initial muon spin direction). The RF amplifier and tuned circuit result in an oscillating field $B_1$ which is applied in a direction transverse to the initial muon spin direction. The detectors are segmented into eight octants. Their outputs can be combined in various ways in software, thereby realising a three-dimensional readout of the muon spin polarization as a function of time. The forward (F) detector is assembled using FLU+FRU+FLD+FRD; the left (L) detector is FLU+FRU+BLU+BRU; the down (D) detector by FLD+FRD+BLD+BRD, etc.
• Figure 2: The components of both measured and simulated vector polarization $[\vec{P}_x(t)), \vec{P}_y(t), \vec{P}_x(t)]$ under continuous RF excitation for five input power levels parametrised by $B_1$. The amplitude of the traces have been normalized.
• Figure 3: (a--c) The vector polarization $[{P}_z(t)), {P}_y(t), {P}_x(t)]$ following a Hahn-echo sequence (a $\pi/2$ pulse followed by a $\pi$ pulse. The experimental data (red) are shown together with simulations that ignore (blue) and include (pink) the dipolar coupling with the fluorine nuclei. (d) The Breit-Rabi diagram for the F--$\mu$--F state in LaF$_3$ with various transitions highlighted. The grey dotted vertical line at $B=13.5~\mathrm{mT}$ marks the longitudinal field used for the Hahn-echo measurements shown in panels (a--c).(e) Echo position tracks the interpulse delays $\tau$. Red traces show $P_x(t)$ for successive $\tau$; shifting the muon RF $\pi$ pulse (orange) to time $\tau$ after the $\pi/2$ pulse moves the refocused echo (marked in yellow) to $\approx 2\tau$ after the $\pi/2$ pulse. Purple traces show the pulse sequence that is applied. The top (green) trace shows the effect of no $\pi$ pulse being applied, so that there is no refocusing.
• Figure 4: (a--c) Vector-resolved muon polarization $\mathbf P(t)=(P_z,P_y,P_x)$ at the working field of 13.5 mT where $\mu$-only reference [green; single $(\pi/2)_\mu$] versus double resonance [red; $(\pi/2)_\mu$ followed by fluorine $\pi_\text{F}$]. The fluorine $\pi$ pulse produces a discernible amplitude/phase perturbation of the muon signal that becomes pronounced in the region after a time 2$\tau$ has elapsed since the original $(\pi/2)_\mu$ pulse. These plots use variable binning to help resolve the features at late times where the statistics are poor, due to the fact that most muons have decayed at late times. (d--f) Our simulations including the F--$\mu$--F dipolar Hamiltonian reproduce the observed modification of the red trace relative to the green trace (but of course do not suffer from the effect of the finite muon lifetime). (g--i) If the F--$\mu$--F dipolar term is removed from the Hamiltonian in the simulations, the fluorine inversion has no effect and the two traces are indistinguishable, as well as failing to reproduce the main features of the data, confirming the important role of the $\mu$–F coupling in these experiments.
• Figure S1: Linear dependence of $B_{1}$ on the square root of applied RF power delivered to the power amplifier.