Observation of Time-Invariant Coherence in a Nuclear Magnetic Resonance Quantum Simulator

The ability to live in coherent superpositions is a signature trait of quantum systems and constitutes an irreplaceable resource for quantum-enhanced technologies. However, decoherence effects usually destroy quantum superpositions. It was recently predicted that, in a composite quantum system exposed to dephasing noise, quantum coherence in a transversal reference basis can stay protected for an indefinite time. This can occur for a class of quantum states independently of the measure used to quantify coherence, and it requires no control on the system during the dynamics. Here, such an invariant coherence phenomenon is observed experimentally in two different setups based on nuclear magnetic resonance at room temperature, realizing an effective quantum simulator of two- and four-qubit spin systems. Our study further reveals a novel interplay between coherence and various forms of correlations, and it highlights the natural resilience of quantum effects in complex systems.

Figure 1

(a) Pulse sequence to prepare two-qubit BD states encoded in the ${}^1\mathrm{H}$ and ${}^{13}\mathrm{C}$ nuclear spins of chloroform. The rf pulse $(\theta {)}_{\mu }$ realizes a qubit rotation by $\theta$ around the spin-$\mu$ axis, $J$ is the scalar spin-spin coupling, and time flows from left to right. (b) Dynamics of the experimental states $\rho (t)$ (the magenta points) in the space of the spin-spin correlation triple $c_j=⟨{\sigma }_j\otimes {\sigma }_j⟩$, $j=1$, 2, 3. All of the BD states (2) fill the light blue tetrahedron, while the subclass of states spanning the inscribed green surface is predicted to have time-invariant coherence in the plus-minus basis according to any measure of Eq. (1) [25]. (c)–(f) Full tomographies of the experimental states as prepared (c),(d) at the time $t=0$ and (e),(f) after $t=0.25\mathrm{s}$ of free evolution, recorded in the computational basis (the top row) and in the plus-minus basis (the bottom row). (g) Evolution of (absolute values of) the correlation functions $|c_j|$ in the experimental states (the points), along with theoretical predictions (the solid lines) based on phase damping noise with our measured relaxation times. (h) Experimental observation of time-invariant coherence (in the plus-minus basis), measured by relative entropy (the red circles) [31], fidelity-based measure (the blue triangles) [35], and normalized trace distance (the green diamonds) [25], equal to the $l_1$ norm [31] in the BD states. The slight negative slope is due to the subdominant effect of amplitude damping. (i) Experimental dynamics of coherence and all forms of correlations [46] measured via relative entropies (the theoretical curves are omitted for graphical clarity). In (g)–(i), experimental errors due to small pulse imperfections (0.3% per pulse) result in error bars within the size of the data points.

Figure 2

(a) Modified preparation stage to engineer non-BD two-qubit states. We prepared two states, ${\rho }_1$ and ${\rho }_2$, with purity 0.92 and 0.93, respectively, setting phases $\theta \approx 0.94\mathrm{rad}$, $\alpha =\pi /3$ for ${\rho }_1$, and $\theta \approx 0.78\mathrm{rad}$, $\alpha =\pi /2$ for ${\rho }_2$. The evolution and acquisition stages were as in Fig. 1. Full tomographies of the produced states at $t=0$ are presented in (c) ${\rho }_1$, computational basis; (d) ${\rho }_1$, plus-minus basis; (e) ${\rho }_2$, computational basis; and (f) ${\rho }_2$, plus-minus basis. (b) Dynamics of the relative entropy of coherence in the prepared states, along with the lower bound inferred from the evolution of their spin-spin correlation functions. The experimental errors are estimated as in Fig. 1.

Figure 3

(a) Pulse sequence (with time flowing from left to right) to prepare four-qubit generalized Bell states encoded in the ${}^1\mathrm{H}$, ${}^{19}\mathrm{F}$, ${}^{13}\mathrm{C}$, and ${}^{31}\mathrm{P}$ nuclear spins of the ${{}^{13}\mathrm{C}}^O\text{−}{}^{15}\mathrm{N}$-diethyl-(dimethylcarbamoyl)fluoromethyl-phosphonate molecule (whose coupling topology is illustrated as an inset) by an INEPT-like procedure, where $d_{kl}=1/(4J_{kl})$ and $J_{kl}$ is the scalar coupling between the spins $k$ and $l$. The light-gray rectangles denote continuous-wave pulses, used to decouple the ${}^{15}\mathrm{N}$ nucleus. The dark gray bar denotes a variable pulse, applied to set the desired correlation triple $\{c_j(0)\}$. Thicker (red) and thinner (blue) bars denote the $\pi$ and $\pi /2$ pulses, respectively; the phases of the striped $\pi /2$ pulses were cycled to construct each density matrix element. After the preparation stage, the system was left to decohere in its environment; $\pi$ pulses were applied in the middle of the evolution to avoid $J_{kl}$ oscillations. The final $\pi /2$ pulses served to produce a detectable NMR signal in the ${}^1\mathrm{H}$ spin channel. (b) Evolution of (absolute values of) the correlation functions $|c_j|$ in the experimental states (the points), along with theoretical predictions (the solid lines) based on phase damping noise with an effective relaxation time $T_2\approx 0.04\mathrm{s}$. (c) Experimental observation of time-invariant coherence (in the plus-minus basis) in the four-qubit ensemble, measured by the relative entropy (the red circles) and the normalized trace distance (the green diamonds), along with theoretical predictions (the solid lines). In (b) and (c), experimental errors due to pulse imperfections and coupling instabilities result in error bars within the size of the data points.