Temporal Order in Periodically Driven Spins in Star-Shaped Clusters

We experimentally study the response of star-shaped clusters of initially unentangled $N=4$, 10, and 37 nuclear spin-$1/2$ moments to an inexact $\pi$-pulse sequence and show that an Ising coupling between the center and the satellite spins results in robust period-2 magnetization oscillations. The period is stable against bath effects, but the amplitude decays with a timescale that depends on the inexactness of the pulse. Simulations reveal a semiclassical picture in which the rigidity of the period is due to a randomizing effect of the Larmor precession under the magnetization of surrounding spins. The timescales with stable periodicity increase with net initial magnetization, even in the presence of perturbations, indicating a robust temporal ordered phase for large systems with finite magnetization per spin.

Figure 1

Molecules used in the experiments: (a) acetonitrile, (b) trimethyl phosphite, and (c) tetrakis(trimethylsilyl)silane, with the 4, 10, and 37 NMR active nuclei encircled. (d) Experimentally measured magnetization $⟨S_i^z⟩$ of satellite spins of TMP for the pulse sequence in Eq. (1) with $JT/\hslash =6.5$ and $\theta =\pi -0.1$. Red (green) dots show the magnetization at odd (even) time steps. For visibility in the plot, the $y$ axis has been rescaled at every 100th time step. (e) Blue line shows experimentally measured magnetization oscillations of free or noninteracting spins of protons in acetonitrile that contain a spinless C-12 central spin, at a pulse angle $\theta =\pi -0.27$. Gray lines indicate the expected response in the absence of a bath.

Figure 2

Numerical simulations of spins in the rotating basis. (a),(b) Time dependence of the expectation values of the three spin components of a satellite (b) and central (a) spin for a system with $N=8$ spins, $[(JT)/\hslash ]=4$, and pulse angle $\theta =e=0.4$. Initial state is the fully $z$ polarized state. (c) Entanglement entropy of the central spin. (d) Bloch sphere representing the spin components of a central spin (of a six spin cluster) at times $t=0$, $T^{+}$, $2T^{-}$, and $2T^{+}$; $+$ ($-$) labels the time just after (before) the pulse. Sequence of intermediate dots track the evolution between time $t=T$ and $2T$. (e) Same as (d) but for a satellite spin. (f) Bloch vectors for a single isolated spin at successive time steps.

Figure 3

(a),(b) Time dependence of cross correlation [multiplied by $(-1{)}^t$] between the central spin $S_0^z$ and a satellite spin $S_i^z$ from simulations of systems of different sizes [$JT/\hslash =4$, $e=0.05$, $\psi =R_x(\pi /8)|\uparrow \uparrow \dots ⟩$, $R_x(\pi /8)$ being the rotation of all spins by $\pi /8$ about $x$]. Disorder strengths are 0 (a),(c) and 0.5 (b),(d). (c),(d) Entanglement entropy of the central spin. Disorder averaging has been performed in (b),(d).

Figure 4

Experimentally measured satellite spin magnetization $\sum _{i=1}^{N-1}⟨S_i^z⟩$. (a),(d) Magnitude of the subharmonic peak upon varying $e$ in TMP and acetonitrile. Solid continuous lines show results from simulations. Different markers indicate Fourier transforms of experimental measurements in different time windows. (b) Waterfall plot of the Fourier spectrum (time window $0<t<80T$) of the experimentally observed magnetization of TMP at different deviations $e$. Dashed blue lines indicate the location of peaks expected for a free spin. (c) Variation of the decay time of the experimentally observed magnetization amplitude with $e$ for TMP. (e),(f) Same as (b) but for acetonitrile with a spinful C-13 (e) and spinless C-12 (f) atom at the center.

Figure 5

Experimental values of central spin magnetization $⟨S_0^z⟩$ in TTSS. (a) Subharmonic peak strength as a function of the deviation $e$. Different markers indicate Fourier transforms in different time windows. (b) Waterfall plot of the Fourier spectrum of the experimentally observed central spin magnetization at different $e$. Blue dashed line shows the location of the Fourier peaks expected for free spins. (c) Decay timescale as a function of $e$. (Inset) A semilog plot of the amplitude of magnetization as a function of time.