Many-Spin Entanglement in Multiple Quantum NMR with a Dipolar Ordered Initial State

Abstract

Many-spin entanglement is investigated in a gas of spin-carrying molecules (atoms) in a nanopore under NMR conditions with a dipolar ordered initial state. To estimate the number of entangled spins, the second moment of the distribution of the intensities of multiple quantum (MQ) NMR coherences is used, which provides a lower bound for the quantum Fisher information. Many-spin entanglement is investigated at different temperatures and different numbers of spins.

APPENDIX

1.1 Two-Pulse Broekaert–Jeener Experiment at Low Zeeman Temperature and High Dipolar Temperature

Initially, the system is in a state of thermodynamic equilibrium in a strong external magnetic field with the density matrix

$${{\sigma }_{i}} = \frac{{\exp ({{\beta }_{L}}{{\omega }_{0}}{{I}_{z}})}}{{{{Z}_{i}}}},\quad {{Z}_{i}} = {\text{Tr}}\{ \exp ({{\beta }_{L}}{{\omega }_{0}}{{I}_{z}})\} ,$$

(A.1)

where β_L is proportional to the inverse temperature of the lattice. After the first resonant x-pulse, we obtain

$$\begin{gathered} \sigma '(0) = \exp \left( {i\frac{\pi }{2}{{I}_{x}}} \right){{\sigma }_{i}}\exp \left( { - i\frac{\pi }{2}{{I}_{x}}} \right) \\ = \frac{{\exp ({{\beta }_{L}}{{\omega }_{0}}{{I}_{y}})}}{{{{Z}_{i}}}}, \\ \end{gathered} $$

(A.2)

where I_α is the operator of the projection of the total spin angular momentum onto the axis α = x, y, z. Then the system freely evolves over time τ, and after that a second resonant y-pulse is applied, which rotates the spins through angle θ around the y-axis of the RRF. As a result, we find that

$$\sigma '(\tau ) = \frac{{\exp ( - i\theta {{I}_{y}})\exp ( - i{{H}_{{dz}}}\tau )\exp ({{\beta }_{L}}{{\omega }_{0}}{{I}_{y}})\exp (i{{H}_{{dz}}}\tau )\exp (i\theta {{I}_{y}})}}{{{{Z}_{i}}}}.$$

(A.3)

After time T₂ (T₂ is the spin relaxation time [17]), the system reaches a state of thermodynamic equilibrium,

$${{\sigma }_{f}} = \frac{{\exp ({{\alpha }_{Z}}{{\omega }_{0}}{{I}_{z}} + {{\beta }_{d}}{{H}_{{dz}}})}}{{{{Z}_{f}}}},$$

(A.4)

where α_Z and β_d are the inverse Zeeman and dipolar temperatures. It is obvious that the system has a single equilibrium state, and the temperatures α_Z and β_d in the equilibrium state are determined from the conservation laws:

$${\text{Tr}}\{ {{I}_{z}}\sigma '(\tau )\} = {\text{Tr}}\{ {{I}_{z}}{{\sigma }_{f}}(\tau )\} ,$$

(A.5)

$${\text{Tr}}\{ {{H}_{{dz}}}\sigma '(\tau )\} = {\text{Tr}}\{ {{H}_{{dz}}}{{\sigma }_{f}}(\tau )\} .$$

(A.6)

We can rewrite Tr{I_zσ'(τ)} as

$$\begin{gathered} {\text{Tr}}\{ {{I}_{z}}\sigma '(\tau )\} = \frac{1}{{{{Z}_{i}}}}{\text{Tr}}\{ \exp (i\theta {{I}_{y}}){{I}_{z}}\exp ( - i\theta {{I}_{y}}) \\ \times \exp ( - i{{H}_{{dz}}}\tau )\exp ({{\beta }_{L}}{{\omega }_{0}}{{I}_{y}})\exp (i{{H}_{{dz}}}\tau )\} \\ = \frac{1}{{{{Z}_{i}}}}{\text{Tr}}\{ ({{I}_{z}}\cos \theta - {{I}_{x}}\sin \theta )\exp ( - i{{H}_{{dz}}}\tau ) \\ \times \exp ({{\beta }_{L}}{{\omega }_{0}}{{I}_{y}})\exp (i{{H}_{{dz}}}\tau )\} \\ = \frac{1}{{{{Z}_{i}}}}{\text{Tr}}\{ \exp ( - i\pi {{I}_{y}})({{I}_{z}}\cos \theta - {{I}_{x}}\sin \theta ) \\ \times \exp ( - i{{H}_{{dz}}}\tau )\exp ({{\beta }_{L}}{{\omega }_{0}}{{I}_{y}})\exp (i{{H}_{{dz}}}\tau )\exp (i\pi {{I}_{y}})\} \\ = - \frac{1}{{{{Z}_{i}}}}{\text{Tr}}\{ ({{I}_{z}}\cos \theta - {{I}_{x}}\sin \theta )\exp ( - i{{H}_{{dz}}}\tau ) \\ \times \exp ({{\beta }_{L}}{{\omega }_{0}}{{I}_{y}})\exp (i{{H}_{{dz}}}\tau )\} = 0. \\ \end{gathered} $$

(A.7)

In expression (A.7), we took into account that [exp(–iπI_y), H_dz] = 0. Since we consider the case of a high dipolar temperature, Eq. (A.5) can be rewritten as

$$\begin{gathered} 0 = \frac{1}{{{{Z}_{f}}}}{\text{Tr}}\{ {{I}_{z}}\exp ({{\alpha }_{Z}}{{\omega }_{0}}{{I}_{z}})\} \\ + \frac{\beta }{{{{Z}_{f}}}}{\text{Tr}}\{ {{I}_{z}}\exp ({{\alpha }_{Z}}{{\omega }_{0}}{{I}_{z}}){{H}_{{dz}}}\} . \\ \end{gathered} $$

(A.8)

Notice that Tr{I_z} = Tr{I_zH_dz} = 0. In such a case, α_Z = 0 satisfies Eq. (A.5). Thus, in this case we obtain a dipolar ordered state.