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New Journal of Physics

Quantum simulation of interaction blockade in a two-site Bose–Hubbard system with solid quadrupolar crystal

Xinfang Nie, Jun Li, Jiangyu Cui, Zhihuang Luo, Jiahao Huang, Hongwei Chen, Chaohong Lee, Xinhua Peng, Jiangfeng Du

Published May 20, 2015• 7 citations

Abstract

The Bose–Hubbard model provides an excellent platform for exploring exotic quantum coherence. Interaction blockade is an important fundamental phenomenon in the two-site Bose–Hubbard system (BHS), which gives a full quantum description for the atomic Bose–Josephson junction. Using the analogy between the two-site BHS and the quadrupolar nuclear magnetic resonance (NMR) crystal, we experimentally simulate a two-site Bose–Hubbard system in a NMR quantum simulator composed of the quadrupolar spin-3/2 sodium nuclei of a NaNO 3 single crystal, and observe the interesting phenomenon of interaction blockade via adiabatic dynamics control. To our best knowledge, this is the first experimental implementation of the quantum simulation of the interaction blockade using quadrupolar nuclear system. Our work exhibits important applications of quadrupolar NMR in the quantum information science, i.e. a spin-3/2 system can be used as a full 2-qubit su(4) system, if the quadrupole moment is not fully averaged out by fast tumbling in the liquid phase.

DOI: 10.1088/1367-2630/17/5/053028

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The Bose–Hubbard model provides an excellent platform for exploring exotic quantum coherence. Interaction blockade is an important fundamental phenomenon in the two-site Bose–Hubbard system (BHS), which gives a full quantum description for the atomic Bose–Josephson junction. Using the analogy between the two-site BHS and the quadrupolar nuclear magnetic resonance (NMR) crystal, we experimentally simulate a two-site Bose–Hubbard system in a NMR quantum simulator composed of the quadrupolar spin-3/2 sodium nuclei of a NaNO 3 single crystal, and observe the interesting phenomenon of interaction blockade via adiabatic dynamics control. To our best knowledge, this is the first experimental implementation of the quantum simulation of the interaction blockade using quadrupolar nuclear system. Our work exhibits important applications of quadrupolar NMR in the quantum information science, i.e. a spin-3/2 system can be used as a full 2-qubit su(4) system, if the quadrupole moment is not fully averaged out by fast tumbling in the liquid phase.

The Josephson junction model is applied to the experimental implementation of classical bifurcation in a quadrupolar nuclear magnetic resonance system. There are two regimes, one linear and one nonlinear, which are implemented by the radio-frequency and the quadrupolar terms of the Hamiltonian of a spin system, respectively. These terms provide an explanation of the symmetry breaking due to bifurcation. Bifurcation depends on the coexistence of both regimes at the same time in different proportions. The experiment is performed on a lyotropic liquid crystal sample of an ordered ensemble of

{}^{133}

Cs nuclei with spin

I=7/2

at room temperature. Our experimental results confirm that bifurcation happens independently of the spin value and of the physical system. With this experimental spin scenario, we confirm that a quadrupolar nuclei system could be described analogously to a symmetric two-mode Bose-Einstein condensate.

This paper presents a description of nuclear magnetic resonance (NMR) of quadrupolar systems using the Holstein-Primakoff (HP) formalism and its analogy with a Bose-Einstein condensate (BEC) system. Two nuclear spin systems constituted of quadrupolar nuclei I=3/2 ((23)Na) and I=7/2 ((133)Cs) in lyotropic liquid crystals were used for experimental demonstrations. Specifically, we derived the conditions necessary for accomplishing the analogy, executed the proper experiments, and compared with quantum mechanical prediction for a Bose system. The NMR description in the HP representation could be applied in the future as a workbench for BEC-like systems, where the statistical properties may be obtained using the intermediate statistic, first established by Gentile. The description can be applied for any quadrupolar systems, including new developed solid-state NMR GaAS nanodevices.

NMR quantum computing with qubit systems represented by nuclear spins (I=12) in small molecules in liquids has led to the most successful experimental quantum information processors so far. We use the quadrupolar spin-32 sodium nuclei of a NaNO3 single crystal as a virtual two-qubit system. The large quadrupolar coupling in comparison with the environmental interactions and the usage of strongly modulating pulses allow us to manipulate the system fast enough and at the same time keeping the decoherence reasonably slow. The experimental challenge is to characterize the "calculation" behavior of the quantum processor by process tomography which is here adapted to the quadrupolar spin system. The results of a selection of quantum gates and algorithms are presented as well as a detailed analysis of experimental results.

We have produced and characterized spin-squeezed states at a temperature of 26 °C in a nuclear magnetic resonance quadrupolar system. The experiment was carried out on 133Cs nuclei of spin I=7/2 in a sample of lyotropic liquid crystal. The source of spin squeezing was identified as the interaction between the quadrupole moment of the nuclei and the electric field gradients present within the molecules. We use the spin angular momentum representation to describe formally the nonlinear operators that produce the spin squeezing on a Hilbert space of dimension 2I+1=8. The quantitative and qualitative characterization of this spin-squeezing phenomenon is expressed by a squeezing parameter and squeezing angle developed for the two-mode Bose-Einstein condensate system, as well as by the Wigner quasiprobability distribution function. The generality of the present experimental scheme points to potential applications in solid-state physics.

Physical implementation of quantum-information processing by liquid-state nuclear magnetic resonance, using weakly coupled spin-

\frac{1}{2}

nuclei of a molecule, is well established. Nuclei with

\mathrm{spin}>1/2

oriented in liquid-crystalline matrices is another possibility. Such systems have multiple qubits per nuclei and large quadrupolar couplings resulting in well separated lines in the spectrum. So far, creation of pseudopure states and logic gates has been demonstrated in such systems using transition selective radio-frequency pulses. In this paper we report two developments. First, we implement a quantum algorithm that needs coherent superposition of states. Second, we use evolution under quadrupolar coupling to implement multiqubit gates. We implement the Deutsch-Jozsa algorithm on a spin-

\frac{3}{2}

(2 qubit) system. The controlled-NOT operation needed to implement this algorithm has been implemented here by evolution under the quadrupolar Hamiltonian. To the best of our knowledge, this method has been implemented for the first time in quadrupolar systems. Since the quadrupolar coupling is several orders of magnitude greater than the coupling in weakly coupled spin-

\frac{1}{2}

nuclei, the gate time decreases, increasing the clock speed of the quantum computer.