NMR Hamiltonian as an effective Hamiltonian to generate Schrödinger’s cat states

Abstract

This report experimentally demonstrates that the theoretical background of the atom–field scenario points out that the NMR quadrupolar Hamiltonian works as an effective Hamiltonian to generate Schrödinger’s cat states in a $2I+1$ low-dimensional Hilbert space. The versatility of this nuclear spin setup is verified by monitoring the $^{23}$Na nucleus of a lyotropic liquid crystal sample at the nematic phase. The quantum state tomography and the Wigner quasiprobability distribution function are performed to characterize the accuracy of the experimental implementation.

Appendix A theoretical procedures

1.1 A.1 The quadrupolar Hamiltonian

The quadrupolar Hamiltonian is the generator of the cat state. In this appendix, more details and explanations about the origin of this important quantum state are presented. The nature of the quadrupolar coupling has an electrical source [21, 22], because it arises from the interaction of any distribution of positive charges on the atomic nuclei (the black nonregular geometric volume in Fig. 1d) with an effective electric field gradient generated by the charges of the molecule itself or its neighbors (the green ellipsoid in Fig. 1d). The best formalism to explain this type of interaction is detailed in Chapter 10 of Ref. [21] and here is presented the main aspects of this theory.

From the fundamentals of the electromagnetism, the electric potential energy of any charge distribution $\varrho \left( \mathbf {r}\right)$ submitted to an electric potential $V\left( \mathbf {r}\right)$ is denoted by:

$$\begin{aligned} E=\int \varrho \left( \mathbf {r}\right) V\left( \mathbf {r}\right) d^{3} \mathbf {r}\text {.} \end{aligned}$$

(21)

As a matter of fact, the phenomenon is reduced to a small spatial arrangement of particles (the nucleus); it enables applying the Taylor’s series definition and rewriting the electric potential around the origin of any coordinate system up to second-order expansion and neglecting the higher ones. From this, expansion emerges the physical interpretation of each term, such that the zero-order term represents an energy offset; the first-order term represents the energy of the electrical dipole moment of the nucleus such that at the stationary equilibrium the average electric field around the nucleus is null; the second-order term defines the quadrupolar energy contribution as denoted by:

$$\begin{aligned} E_{Q} = \sum _{x_{j},x_{k}=x,y,z}\left. \frac{\partial ^{2}\left( V\left( \mathbf {r}\right) \right) }{\partial x_{j}\partial x_{k}} \right| _{\mathbf {r}=\mathbf {0}}\int x_{j}x_{k}\varrho \left( \mathbf {r} \right) d^{3}\mathbf {r}\text {.} \end{aligned}$$

(22)

This energy contribution is described in its operator representation applying the quantum mechanical notation of operators, the irreducible tensor operators, the Clebsch–Gordan coefficients and the Wigner–Eckart theorem such that the quadrupolar energy contribution of Eq. (22) is represented by:

$$\begin{aligned} \hat{ \mathcal {H}}_{Q} = \sum _{x_{j},x_{k}}\frac{eQV_{x_{j},x_{k}}}{6I\left( 2I-1\right) } \left( \frac{3}{2}\left( {\hat{\mathbf {I}}}_{x_{j}}{\hat{\mathbf {I}}}_{x_{k}}+ {\hat{\mathbf {I}}}_{x_{k}}{\hat{\mathbf {I}}}_{x_{j}}\right) -\delta _{x_{j},x_{k}}{\hat{\mathbf {I}}}^{2}\right) \text {,} \end{aligned}$$

where e is the elemental charge, Q is the quadrupole moment of the nucleus, $V_{x_{j},x_{k}}=\left. \frac{\partial ^{2}\left( V\left( \mathbf {r} \right) \right) }{\partial x_{j}\partial x_{k}}\right| _{\mathbf {r}= \mathbf {0}}$ with $x_{j},x_{k} = x,y,z$ and ${\hat{\mathbf {I}}}^{2} = {\hat{\mathbf {I}}}_{x}^{2} + \hat{\mathbf { I}}_{y}^{2}+{\hat{\mathbf {I}}}_{z}^{2}$. This Hamiltonian related at any set of principal axes system of coordinates satisfies the property of $V_{x_{j},x_{k}}=0$ for $x_{j} \ne x_{k}$, and using the Laplace’s equation $V_{x,x}+V_{y,y}+V_{z,z}=0$, the Hamiltonian is rewritten as:

$$\begin{aligned} {\hat{\mathcal {H}}}_{Q}=\frac{\hbar \omega _{Q}}{6 }\left( \left( 3 {\hat{\mathbf {I}}}_{z}^{2}-{\hat{\mathbf {I}}}^{2}\right) +\eta \left( \hat{ \mathbf {I}}_{x}^{2}-{\hat{\mathbf {I}}}_{y}^{2}\right) \right) \text {,} \end{aligned}$$

(23)

where $\ \omega _{Q}=\frac{eQV_{z,z}}{I\left( 2I-1\right) \hbar }$ defines the quadrupolar angular frequency and $\eta =\frac{V_{x,x}-V_{y,y}}{V_{z,z}}$ defines an asymmetry parameter.

The lyotropic liquid crystal used in this experimental implementation (see Fig. 1b) follows a spatial arrangement with the molecules axis oriented along the strong static magnetic field. For this setup, the value of the asymmetry parameter is null, and the Hamiltonian of the quadrupolar contribution is:

$$\begin{aligned} {\hat{\mathcal {H}}}_{Q}=\hbar \frac{\omega _{Q}}{6}\left( 3{\hat{\mathbf {I}}} _{z}^{2}-{\hat{\mathbf {I}}}^{2}\right) \text {.} \end{aligned}$$

(24)

This Hamiltonian characterizes the quadrupolar energy contribution at the NMR Hamiltonian of Eq. (7) of the main text.