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QuantumRotatingFrame

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Quantum treatment of the rotating frame

(From Slichter translated into density matrix formalism).

The aim is to describe a nutation quantum mechanically with the use of the density matrix formalism. The density matrix of an ensemble of non interacting spin I with gyromagnetic ratio γ in an external field B₀ =B₀ z is ρ₀=I_z . The equation of motion for the density matrix is:

{$$ \begin{equation} \frac {d \rho} {dt} = \frac i \hbar [H,\rho] \end{equation}$$}

The standard formal solution of this equation is:

{$$ \begin{equation} \rho(t) = e^{iHt/\hbar} \rho_0 e^{-iHt/\hbar} \end{equation}$$}

When the Hamiltonian is only the Zeeman term appropriate for the steady field B₀ the evolution exponential operator commutes with I_z and the solution is the stationary ρ₀ .

If however we add a rotating field B₁ in the xy plane, i.e. at resonance (ω=ω₀ ), the Hamiltonian becomes time dependent. Its quantum operator may be written with {$I_{x'}(t)$}, as introduced in the previous section

{$$ \begin{equation} H = - \hbar (\gamma B I_z + \gamma B_1 e^{i\omega_0 I_z t} I_x e^{-i\omega_0 I_z t}) = - \hbar (\omega_0 I_z + \omega_1 e^{i\omega_0 I_z t} I_x e^{-i\omega_0 I_z t}) \end{equation}$$}

A very straightforward way of solving the problem is to transform the equations to the rotating frame, by using the Interaction representation. Its density matrix is:

{$$ \begin{equation} \rho_I = e^{-i \omega_0 t I_z} \rho e^{i \omega_0 t I_z} \end{equation}$$}

and its evolution is governed by

{$$ \begin{equation} \frac {d\rho_I} {dt} = \frac i \hbar [H_{1I},\rho_I ] \end{equation}$$}

where the Hamiltonian is the perturbation in the Interaction representation, i.e.:

{$$ \begin{equation} H_{1I} = e^{-i \omega_0 t I_z}\hbar\omega_1 e^{i\omega_0 I_z t} I_x e^{-i\omega_0 I_z t}e^{i \omega_0 t I_z}= \hbar\omega_1 I_x\end{equation}$$}

The evolution is straightforward, since the Hamiltonian that drives it is now time independent. Also {$\rho_I$}, like {$\rho$} coincides with {$\rho_0$} at time t=0, which must represent the nuclear magnetization along {$\hat z$}. That is

{$$ \rho_0= \begin{bmatrix} 1 \quad 0 \\ 0 \quad 0 \end{bmatrix}=\frac 1 2 (\sigma_z+1)$$}

The identity may be dropped (see here) to leave {$ \rho_0= I_z$}, so that eq. (5) and (6) are easily solved, like in the case of the spin precession with the density matrix) to give a precession around {$\hat x$}. However this simple evolution is taking place in the Interaction representation, which turns out to be the quantum analogue of the rotating frame. The evolution under {$H_0$} is hidden, and it corresponds to a precession around {$\hat z$} that must be composed with that around {$\hat x$} to yield the classical nutation.

< A note on rotations | Index | Quantum rotating frame in matlab >