Observation of interaction induced blockade and local spin freezing in a NMR quantum simulator

We experimentally emulate interaction induced blockade and local spin freezing in two and three qubit NMR architecture. These phenomena are identical to the Rydberg blockade and Rydberg biased freezing. In Rydberg blockade, the simultaneous excitation of two or more atoms is blocked due to the level shift induced by the strong van der Waal's interaction. In such a strong interaction regime, one can also observe Rydberg biased freezing, wherein the dynamics is confined to a subspace, with the help of multiple drives with unequal amplitudes. Here we drive NMR qubits with specific transition-selective radio waves, while intermittently characterizing the quantum states via quantum state tomography. This not only allows us to track the population dynamics, but also helps to probe quantum correlations, by means of quantum discord, evolving under blockade and freezing phenomena. Our work constitutes experimental simulations of these phenomena in the NMR platform. Moreover, these studies open up interesting quantum control perspectives in exploiting the above phenomena for entanglement generation as well as subspace manipulations.

Figure 1

The energy-level diagram and allowed transitions under (a) blockade and (b) Rydberg biased freezing of two interacting Rydberg atoms. The ground state of each atom is coupled to the excited state by laser fields with Rabi frequencies ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$, respectively. Under strong interaction, the doubly excited state shifts out of resonance. In the regime ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2$ (a), the system exhibits dynamics between the ground state $|gg\rangle$ and the entangled state $|+\rangle =\left(\right|eg\rangle +|ge\rangle )/\sqrt{2}$ with enhanced frequency $\sqrt{2}{\mathrm{\Omega }}_1$. The entangled state $|-\rangle =\left(\right|eg\rangle -|ge\rangle )/\sqrt{2}$ is also cut-off from the system's dynamics. In the regime where second atom is driven by a much weaker drive than the first atom (b), the dynamics of the second atom is suppressed and that of the first atom is unhindered, resulting in freezing of the second atom.

Figure 2

Sodium fluorophosphate molecule (a), the parameters of the rotating-frame Hamiltonian $H_0$ [Eq. (6)] (b) forming the two-qubit register, and the corresponding energy eigenstates and eigenvalues (c). Dibromofluoromethane molecule (d), the parameters of rotating-frame Hamiltonian (e) forming the three-qubit register, and the corresponding energy eigenstates and eigenvalues (f). In tables [(b), (e)] the diagonal elements represent tunable off-set frequencies ${\nu }_i$ set according to Eq. (7), and off-diagonal elements show scalar coupling constants $J_{ij}$ between the respective qubits.

Figure 3

Population dynamics (left column) and discord (right column) vs driving time for the two-qubit register under Rabi drive (a), interaction induced blockade [(c),(d)], freezing of the second qubit [(e),(f)] and of the first qubit [(g),(h)]. Plot (b) shows the population dynamics of a pair of noninteracting spins with the same driving parameters as in freezing on second qubit case (g), illustrating the importance of spin-spin interaction to realize freezing. Experimental data points are shown by dots with error bars (indicating random errors), theoretically expected dynamics are shown by dashed lines, and realistic numerical models are shown by solid lines. In each case, the corresponding energy-level diagram is also shown (central column) with the same color coding as the legend shown at the top of the figure. The energy-level diagram of (a) corresponds to uncoupled spins, while those of (c)–(g) show eigenstates of the rotating frame Hamiltonian ($H_0$) in a relevant basis (up to the freedom in degenerate subspace) along with prominent transitions. The discord values D($\mathrm{A}|\mathrm{B}$) are expressed in units of $ln2/{\epsilon }^2$ [30], where $\epsilon$ is the purity factor as described in Sec. 3c.

Figure 4

Population dynamics (left column) and discord (right column) versus driving time for the three-qubit register under Rabi drive (a), interaction induced blockade [(c),(d)], freezing of the second and third qubits [(e),(f)] and of the first and third qubits [(g),(h)] and of only the last qubit [(i),(j)]. Plot (b) shows the population dynamics for three noninteracting spins with the same driving parameters as freezing on second and third qubits [(e),(f)], illustrating the importance of spin-spin interactions to realize freezing. Experimental data points are shown by dots with error bars (indicating random errors), theoretically expected dynamics are shown by dashed lines, and realistic numerical models are shown by solid lines. In each case, the corresponding energy-level diagram is also shown (central column) with the same color coding as the legend shown at the top of the figure. The energy-level diagram of (a) corresponds to uncoupled spins, while those of (c)–(g) show eigenstates of the rotating frame Hamiltonian ($H_0$) in a relevant basis (up to the freedom in degenerate subspace) along with prominent transitions. The discord values D($\mathrm{X}|\mathrm{YZ}$) are expressed in units of $ln2/2{\epsilon }^2$ [30], where $\epsilon$ is the purity factor as described in Sec. 3c.

Figure 5

(a) Population in $|ge\rangle$ and $|eg\rangle$ as the driving amplitude of the second qubit (${\nu }_2^{RF}$) is gradually reduced from Rydberg blockade condition (${\nu }_1^{RF}=217$ Hz) to Rydberg biased freezing condition (54.2 Hz) (b) the corresponding discord values $D\left(A\right|B)$ in units of $ln2/{\epsilon }^2$. Experimental data are recorded at half the effective Rabi period (for each value of ${\nu }_2^{RF}$) and shown by filled circles, which are overlaid on theoretical simulations shown by dashed lines.

Figure 6

PPS sequences. Pulse sequence for preparing PPS $|00\rangle \langle 00|$ and $|000\rangle \langle 000|$ from thermal equilibrium state for (a) two-qubit register in sodium fluorophosphate molecule with ${\tau }_{PF}=1/\left(2J_{PF}\right)$ and (b) three qubit register in dibromofluoromethane with ${\tau }_{HC}=1/\left(2J_{HC}\right)$, ${\tau }_{FC}=2/|J_{FC}|-1/|2J_{FC}|$ (that takes care of the negative sign of $J_{FC}$) and ${\tau }_{HF}=1/\left(2J_{HF}\right)$, respectively. The solid bars represent rotations by an angle and about a direction as indicated over them. The blank rectangles represent $\pi$ pulses and the black half ellipsoids represent PFG along $+\mathrm{z}$ axis to destroy coherences.

Figure 7

Energy levels of a coupled two-qubit system AB. The dashed ellipses show the $B_{\uparrow }$ and $B_{\downarrow }$ subspaces. The two transitions of qubit A are displayed at the center with the corresponding spin orientation of qubit B labeled under each peak.