Observation of quantum phase synchronization in a nuclear-spin system

We report an experimental study of phase synchronization in a pair of interacting nuclear spins subjected to an external drive in nuclear magnetic resonance architecture. A weak transition-selective radio-frequency field applied on one of the spins is observed to cause phase localization, which is experimentally established by measuring the Husimi distribution function under various drive conditions. To this end, we have developed a general interferometric technique to directly extract values of the Husimi function via the transverse magnetization of the undriven nuclear spin. We further verify the robustness of synchronization to detuning in the system by studying the Arnold tongue behavior.

Figure 1

(a) Sodium fluorophosphate molecular structure with its Hamiltonian parameters and the spin-lattice relaxation time constants $T_1$ shown in the table. The diagonal elements represent the offset, while off-diagonal element represents the scalar $J$ coupling constant. (b) Energy levels of the two-qubit system with four nondegenerate energy eigenstates. The numerical model presented here considers only the single-quantum relaxation pathways (yellow curved arrows) and ignores the zero- and double-quantum pathways (dashed gray arrows). Also shown are reference NMR spectra of (c) ${}^{31}\mathrm{P}$ and (d) ${}^{19}\mathrm{F}$ spins, each obtained with a ${90}^{\circ }$ pulse on the thermal equilibrium state, and (e) the ${}^{31}\mathrm{P}$ NMR signal after a 100 s drive on the $|2\rangle \leftrightarrow |4\rangle$ transition. The intensity of the ${\rho }_{42}$ element is indicated in the dashed box, which is used for the Arnold tongue analysis. (f) The ${}^{19}\mathrm{F}$ NMR spectrum at the end of the interferometric circuit from which we can directly extract the Husimi distribution at particular $\theta$ and $\varphi$ values.

Figure 2

Dependence of visibility of phase localization versus drive amplitude $\mathrm{\Omega }$, showing an optimum strength at around 0.1 Hz.

Figure 3

(a) IMHD circuit for direct measurement of the Husimi distribution where the drive duration $t$ is much longer than the measurement sequence time $\tau$ and (b) corresponding NMR pulse sequence.

Figure 4

Full phase-space Husimi distribution values for different drive durations to capture transient and steady-state behavior in the subspace of interest obtained by (a) experiments and (b) the numerical model. The values above each panel indicate the visibility factors of the Husimi distributions. We can see that in the absence of a drive, there is no localization in the phase space. Upon applying the drive, in the transient regime (up to 10 s), the phase begins to localize gradually, showing the strongest localization experimentally at a drive duration of 1 s and saturating after 10 s.

Figure 5

Arnold tongue behavior of the system. The system shows strongest synchronization for on-resonant drive and strong drive strengths, while off-resonant drives are unable to synchronize with the transition of interest even for large driving strengths.

Figure 6

TMP molecule and Hamiltonian parameters.

Figure 7

Low-power calibration using trimethylphosphite. The intensity of the ${}^{31}\mathrm{P}$ spin after decoupling protons is shown as a function of time for various low-amplitude pulses. We can see that the response is linear in this regime. The expected and exactly obtained values from backcalculation of the linear fit are shown in the legend.