# 13 Spin precession and quantum tunneling

In this chapter we will consider few basic concepts related to non-dissipative spin dynamics. Following our usual bottom-up approach, we shall focus on the dynamics of a single, atomic magnetic moment because it provides the basis to understand the dynamics of systems of coupled magnetic moments. Given the proportionality \(\hat{\mathbf \mu} = - g \mu_{\rm B}\hat{\mathbf S}\), where \(\hat{\mathbf S}\) has to be understood as an *effective* spin, this issue reduces to investigating the dynamics of a single spin.

## 13.1 Density matrix formalism for \(S=1/2\)

In quantum mechanics the time-dependent average of generic observable \(\mathcal{\hat{O}}\) at time \(t\) can be computed using the density-matrix formalism
\[\begin{equation}
\tag{13.1}
\langle\mathcal{\hat{O}}(t)\rangle = {\rm Tr} \{\hat{\rho}(t)\mathcal{\hat{O}}\}
\end{equation}\]
where \(\hat{\rho}(t)\) is the density-matrix operator^{41}
\[\begin{equation}
\label{rho_def}
\hat\rho(t) = | \psi(t) \rangle \langle \psi(t)|\,.
\end{equation}\]

For a given Hamiltonian \(\mathcal{H}\), the time evolution of the density matrix is obtained solving the equation \[\begin{equation} \label{rho-Heisenberg} \tag{13.2} i \hbar \frac{\partial \hat\rho }{\partial t} = \left[ \mathcal{H} , \hat\rho\right] \,. \end{equation}\] In the following we will focus on a single spin \(S=1/2\) and assume a Hamiltonian of the form \[\begin{equation} \tag{13.3} \mathcal{H} = \Delta \hat \sigma^z + \Omega \hat \sigma^x \,, \end{equation}\] where \(\hat \sigma^x\) and \(\hat \sigma^z\) are Pauli matrices. This Hamiltonian is simple but it contains the main physics of the problems of our interest. In fact, letting \(\Delta\) and \(\Omega\) depend on time, one can understand quite in details the origin of

- Larmor precession

- Magnetic resonance phenomena

- Resonant quantum tunneling.

On the basis of eigenstates of \(\hat \sigma^z\), \(| \uparrow \rangle\) and \(| \downarrow \rangle\), Eq.(13.2) takes the form \[\begin{equation} \tag{13.4} \begin{split} i \hbar \frac{\partial }{\partial t} \begin{pmatrix} \rho_{\uparrow \uparrow} & \rho_{\uparrow \downarrow} \\ \rho_{\downarrow \uparrow} & \rho_{\downarrow \downarrow} \end{pmatrix} = \begin{pmatrix} \Delta &\Omega \\ \Omega & -\Delta \end{pmatrix} \begin{pmatrix} \rho_{\uparrow \uparrow} & \rho_{\uparrow \downarrow} \\ \rho_{\downarrow \uparrow} & \rho_{\downarrow \downarrow} \end{pmatrix} - \begin{pmatrix} \rho_{\uparrow \uparrow} & \rho_{\uparrow \downarrow} \\ \rho_{\downarrow \uparrow} & \rho_{\downarrow \downarrow} \end{pmatrix} \begin{pmatrix} \Delta &\Omega \\ \Omega & -\Delta \end{pmatrix} \end{split} \end{equation}\] which is equivalent to the set of equations \[\begin{equation} \begin{split} i \hbar \frac{\partial }{\partial t} \rho_{\uparrow \uparrow} & = - \Omega ( \rho_{\uparrow \downarrow} - \rho_{\downarrow \uparrow} ) \\ i \hbar \frac{\partial }{\partial t} \rho_{\uparrow \downarrow} & = 2\Delta \rho_{\uparrow \downarrow} - \Omega ( \rho_{\uparrow \uparrow} - \rho_{\downarrow \downarrow}) \\ i \hbar \frac{\partial }{\partial t} \rho_{\downarrow\uparrow} & = - 2\Delta \rho_{\downarrow\uparrow} + \Omega ( \rho_{\uparrow \uparrow} - \rho_{\downarrow \downarrow}) \\ i \hbar \frac{\partial }{\partial t} \rho_{\downarrow \downarrow} & = \Omega ( \rho_{\uparrow \downarrow} - \rho_{\downarrow \uparrow} ) \,. \end{split} \end{equation}\] \[\begin{equation} \tag{13.5} \end{equation}\] The diagonal elements of the density matrix \(\rho_{\uparrow \uparrow}\) and \(\rho_{\downarrow \downarrow}\) give the probability that the system is in one of the two states \(|\uparrow\rangle\) and \(|\downarrow\rangle\), namely they represent the populations in each state of the chosen basis. The off-diagonal elements, instead, provide information about interference between the basis states and give a measure of quantum coherence of a pure state that is a linear combination of the basis states.

## 13.2 Larmor precession about a constant magnetic field

We start considering the time evolution associated with the Hamiltonian of an isolated electron in a *constant* magnetic field, which comprises only a Zeeman contribution:
\[\begin{equation}
\tag{13.6}
\mathcal{H} = g \mu_{\rm B} \vec B\cdot \hat{\mathbf S} \,.
\end{equation}\]
This is a special case of the Hamiltonian (13.2) with \(\Omega=0\) and \(2\Delta = g \mu_{\rm B} B =\hbar\omega_{\rm L}\), with energy levels

\[\begin{equation}
\tag{13.7}
E_n= \pm \frac{1}{2} g \mu_{\rm B} B =\pm \frac{1}{2} \, \hbar \, \omega_{\rm L}
\end{equation}\]
(the choice of the subscript “L” will become clear at the end of this paragraph).
The terms on the right-hand sides of the first and the last equation of the system (13.5) vanish, which implies that the populations of the states \(|\uparrow\rangle\) and \(|\downarrow\rangle\) remain constant:
\[\begin{equation}
\label{Landau-Zener-rho-diag-L}
\begin{split}
\rho_{\uparrow \uparrow} (t) & =\rho_{\uparrow \uparrow,0} \\
\rho_{\downarrow \downarrow} (t) & = \rho_{\downarrow \downarrow,0}\,,
\end{split}
\end{equation}\]
with the subscript “0” denoting the initial values. The dynamics is constrained to the second and third equation of (13.5), namely to the off-diagonal terms of the density matrix
\[\begin{equation}
\label{Landau-Zener-rho-off-L}
\begin{split}
i \hbar \frac{\partial }{\partial t}\rho_{\uparrow \downarrow}& = \hbar\omega_{\rm L} \rho_{\uparrow \downarrow} \\
i \hbar \frac{\partial }{\partial t}\rho_{\downarrow\uparrow}& =-\hbar\omega_{\rm L}\rho_{\downarrow\uparrow} \,.
\end{split}
\end{equation}\]
The equations above are independent and their integration is straightforward:
\[\begin{equation}
\label{Landau-Zener-rho-off-L1}
\begin{split}
\rho_{\uparrow \downarrow}(t)&=\rho_{\uparrow \downarrow,0} {\rm e}^{-i \hbar\omega_{\rm L} t} \\
\rho_{\downarrow\uparrow}(t)&=\rho_{\downarrow\uparrow,0} {\rm e}^{ i \hbar\omega_{\rm L} t} \,.
\end{split}
\end{equation}\]
so that the density matrix at a generic time \(t\), expressed on the basis \(| \uparrow \rangle\) and \(| \downarrow \rangle\), reads
\[\begin{equation}
\label{rho-matrix}
\tag{13.8}
\hat\rho (t) =
\begin{pmatrix}
\rho_{\uparrow \uparrow, 0} & \rho_{\uparrow \downarrow,0} \, {\rm e }^{-i\omega_{\rm L} t} \\
\rho_{\downarrow\uparrow, 0} \, {\rm e }^{i\omega_{\rm L} t} & \rho_{\downarrow \downarrow,0}
\end{pmatrix}
\end{equation}\]
To the aim of computing the expectation values of each spin component as a function of \(t\), it is useful to recall the relations
\[\begin{equation}
\label{spin_operators}
\begin{split}
&\hat{S}^x |\uparrow \rangle = \frac{1}{2} |\downarrow \rangle \qquad \qquad \hat{S}^x |\downarrow \rangle = \frac{1}{2} |\uparrow \rangle \\
&\hat{S}^y |\uparrow \rangle = i \frac{1}{2} |\downarrow \rangle \qquad \qquad \hat{S}^y |\downarrow \rangle = - i \frac{1}{2} |\uparrow \rangle \\
&\hat{S}^z |\uparrow \rangle = \frac{1}{2} |\uparrow \rangle \qquad \qquad \hat{S}^z |\downarrow \rangle = - \frac{1}{2} |\downarrow \rangle \,,
\end{split}
\end{equation}\]
and the obvious orthogonality relations \(\langle \uparrow|\uparrow \rangle=\langle \downarrow|\downarrow \rangle=1\) and \(\langle \downarrow|\uparrow \rangle=\langle \uparrow|\downarrow \rangle=0\).
Then, combining Eq.(13.1) and Eq.(13.8), the time-dependent average of each spin component can be computed:
\[\begin{equation}
\tag{13.9}
\begin{split}
\langle \hat{S}^x (t) \rangle &=
\frac{1}{2} \, {\rm Tr}
\left\{
\begin{pmatrix}
\rho_{\uparrow \uparrow, 0} & \rho_{\uparrow \downarrow,0} \, {\rm e }^{-i\omega_{\rm L} t} \\
\rho_{\downarrow\uparrow, 0} \, {\rm e }^{i\omega_{\rm L} t} & \rho_{\downarrow \downarrow,0}
\end{pmatrix}
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\right \} \\
& =
\frac{1}{2} \, {\rm Tr}
\begin{pmatrix}
\rho_{\uparrow \downarrow,0} \, {\rm e }^{-i\omega_{\rm L} t} & \rho_{\uparrow \uparrow, 0} \\
\rho_{\downarrow \downarrow,0} & \rho_{\downarrow\uparrow, 0} \, {\rm e }^{i\omega_{\rm L} t}
\end{pmatrix}
= \frac{1}{2} \, 2 \, \Re{\rm e}\left(\rho_{\uparrow \downarrow,0} \, {\rm e }^{-i\omega_{\rm L} t}\right) \,,
\end{split}
\end{equation}\]
\[\begin{equation}
\tag{13.10}
\begin{split}
\langle \hat{S}^y (t) \rangle &=
\frac{1}{2} \, {\rm Tr}
\left\{
\begin{pmatrix}
\rho_{\uparrow \uparrow, 0} & \rho_{\uparrow \downarrow,0} \, {\rm e }^{-i\omega_{\rm L} t} \\
\rho_{\downarrow\uparrow, 0} \, {\rm e }^{i\omega_{\rm L} t} & \rho_{\downarrow \downarrow,0}
\end{pmatrix}
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}
\right \} \\
& =
\frac{1}{2} \, {\rm Tr}
\begin{pmatrix}
i\,\rho_{\uparrow \downarrow,0} \, {\rm e }^{-i\omega_{\rm L} t} & -i\, \rho_{\uparrow \uparrow, 0} \\
i\, \rho_{\downarrow \downarrow,0} & -i\, \rho_{\downarrow\uparrow, 0} \, {\rm e }^{i\omega_{\rm L} t}
\end{pmatrix}
= - \frac{1}{2} \, 2 \, \mathbb{I}{\rm m}\left(\rho_{\uparrow \downarrow,0} \, {\rm e }^{-i\omega_{\rm L} t}\right) \,,
\end{split}
\end{equation}\]
\[\begin{equation}
\tag{13.11}
\begin{split}
\langle \hat{S}^z (t) \rangle &=
\frac{1}{2} \, {\rm Tr}
\left\{
\begin{pmatrix}
\rho_{\uparrow \uparrow, 0} & \rho_{\uparrow \downarrow,0} \, {\rm e }^{-i\omega_{\rm L} t} \\
\rho_{\downarrow\uparrow, 0} \, {\rm e }^{i\omega_{\rm L} t} & \rho_{\downarrow \downarrow,0}
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
\right \} \\
& =
\frac{1}{2} \, {\rm Tr}
\begin{pmatrix}
\rho_{\uparrow \uparrow, 0} & 0 \\
0 &-\rho_{\downarrow \downarrow,0}
\end{pmatrix}
= \frac{1}{2} \,\left(\rho_{\uparrow \uparrow, 0}-\rho_{\downarrow \downarrow,0}\right)
\end{split}
\end{equation}\]
\[\begin{equation}
\tag{13.12}
\end{equation}\]
If at time \(t=0\) the system is in a generic, pure state^{42}

\[\begin{equation}\label{psi_in}
| \psi(0) \rangle = a | \uparrow \rangle + b| \downarrow \rangle \,,
\end{equation}\]
the initial values of the matrix elements \(\rho_{\uparrow \uparrow,0}\), \(\rho_{\downarrow \downarrow,0}\), \(\rho_{\uparrow \downarrow,0}\), \(\rho_{\downarrow\uparrow,0}\), are related to the amplitudes \(a\) and \(b\) as follows
\[\begin{equation}
\label{rho-matrix-0}
\tag{13.13}
\hat\rho (0) =
\begin{pmatrix}
a \, \bar a & a \, \bar b \\
b \, \bar a & b \, \bar b
\end{pmatrix}
\end{equation}\]
where \(\bar a\) and \(\bar b\) denote the complex conjugated of \(a\) and \(b\). Choosing \(a\) and \(b\) as real and parameterized through an angle \(\theta\)
\[\begin{equation}
\tag{13.14}
\begin{cases}
&a= \cos\left(\frac{\theta}{2}\right) \\
&b= \sin\left(\frac{\theta}{2}\right) \, ,
\end{cases}
\end{equation}\]
the expectations values of the spin projections (13.12) can readily be mapped into the components of a classical spin
\[\begin{equation}
\tag{13.15}
\begin{cases}
&\langle \hat{S}^x(t) \rangle = S\,\sin\theta \cos(\omega_{\rm L} t) \\
&\langle \hat{S}^y(t) \rangle = S\,\sin\theta \sin(\omega_{\rm L} t) \\
&\langle \hat{S}^z(t) \rangle = S\,\cos\theta
\end{cases}
\end{equation}\]
with \(S=1/2\) in this case. This result clarify the common misconception that *a quantum spin 1/2 can can only “point” up or down*.
On the contrary, Eq.(13.15) shows that a proper linear combinations of the states \(|\uparrow\rangle\) and \(|\downarrow\rangle\) can produce *any* expectation value of spin projections, which can readily be mapped into the projections of a classical spin. Later on we will see that this has nothing to do with the number of configurations that need to be considered to perform thermal averages.
The last ones are indeed just two for a quantum spin 1/2, while they are infinite for a classical spin (i.e. all the points lying on the unit sphere), which is the reason why the Brillouin function is different for quantum spins with different multiplicity and from the Langevin function (holding for a classical spin).

Equation(13.15) describes a *counterclockwise* precession of the spin about the direction of the applied field. Note that the direction of precession is closely related to the fact that for an electron (considered so far) the magnetic moment and the spin are antiparallel. For a proton one obtains a *clockwise* precession about the axis of the applied field. The precession frequency \(\omega_{\rm L} =g \mu_{\rm B} B\) is called Larmor frequency and represents a characteristic frequency of the system spin plus applied field. All magnetic-resonance techniques are essentially based on the excitation of the Larmor frequency by means of an external stimulus, typically an oscillating magnetic field superimposed to the static one^{43}. To all these purposes, the fact that the Larmor frequency is proportional to the applied field and to the magnetic moment of the precessing spin is fundamental. The careful reader shall have noted that for a given field \(B\) the splitting of the two spin levels, \(| \uparrow \rangle\) and \(| \downarrow \rangle\), is right \(\hbar \omega_{\rm L}\), consistently with Eq.(13.7). This is – of course – not accidental and one may have concluded without too much calculus that a characteristic frequency is always associated with an energy gap. However, the explicit dynamic calculation provides a deeper physical insight and helps us set the formalism for more advanced calculations, such as the specific transition probabilities with corresponding selections rules.

In Fig.13.1 the mapping between the components of a generic ket \(| \psi(t)\rangle\) in the Hilbert space and the points on the unit sphere is visualized. The convention is the same adopted in quantum computing to identify a generic Q-bit on the Bloch sphere: \[\begin{equation} \tag{13.16} |\psi\rangle=\cos\left(\frac{\theta}{2}\right)|\uparrow\rangle + e^{i \phi} \sin\left(\frac{\theta}{2}\right)|\downarrow\rangle \end{equation}\] Note that the parametrization (13.14) is used for the polar angle \(\theta\) but the phase factor \(\phi\) is defined differently with respect to the azimuthal angle of a generic rotation in SU(2) (see Eq.(13.19)). In the next section we will see that a Larmor precession is directly related to the latter.

## 13.3 Tunneling in a time-dependent magnetic field

Another typical problem in magnetism is the response of a magnetic atom or molecule to an applied field which varies linearly with time. Associating the terms that are different from this *time-dependent* Zeeman contribution with a Hamiltonian \(\mathcal{H} = \mathcal{H}_0\), the total Hamiltonian reads

\[\begin{equation}
\tag{13.17}
\mathcal{H} = \mathcal{H}_0 + g \mu_{\rm B} v_B(t-t_0) \hat S^z \,,
\end{equation}\]
with \(v_B\) representing the sweeping rate of the magnetic field (T/s). Generally \(| \uparrow\rangle\) and \(| \downarrow\rangle\) shall not be eigenstates of the Hamiltonian above.
The Hamiltonian (13.3) is a simplified version of Eq.(13.17) that still retains the essential physics of the problem. Referring to the set of equations (13.5) with \(2\Delta(t) = g \mu_{\rm B} v_B(t-t_0)\), let us develop some qualitative considerations.

If \(\Omega= 0\) and the system has been prepared in the state \(|\uparrow\rangle\) (i.e. with \(\rho_{\uparrow \uparrow, 0} =1\)), it will remain in the same state at any future time and for any value of the Zeeman energy \(\Delta(t)\). In particular, the spin does not leave this state even for positive fields \(B>0\), i.e., when \(|\uparrow\rangle\) is not the ground state but the excited state. In other words, based on this dynamics, the spin would never be able to align its magnetic moment to the external field, not even when this becomes energetically inconvenient.

When \(\Omega \ne0\) the situation is significantly different. It is instructive to start focusing on the limiting cases obtained for zero and very large positive or negative fields corresponding to \(|\Delta|\gg \Omega\). The eigenstates associated with these situations are

- for \(B=0\): \(|\psi_{\pm} \rangle \propto |\uparrow\rangle \pm |\downarrow\rangle\)
- for large fields: \(|\uparrow\rangle\) and \(|\downarrow\rangle\).

Therefore, while sweeping the field from large negative values to positive values, one expects the two states \(|\uparrow\rangle\) and \(|\downarrow\rangle\) to get admixed, as a result of the passage through the value \(B=0\). At this point, \(|\uparrow\rangle\) and \(|\downarrow\rangle\) are not the true eigenstates because of the presence of a finite \(\Omega\), which transforms the level crossing (realized rigorously for \(\Omega= 0\)) into an *avoided* level crossing. Even if we concentrate only on the behavior of the diagonal terms of the density matrix, the problem is complex to be tackled analytically. However, an analytic result can be obtained for the limit \(t\rightarrow \infty\)
\[\begin{equation}
\tag{13.18}
\hat \rho =
\begin{pmatrix}
{\rm e}^{-\pi\gamma_{\rm LZ} } & {\rm X} \\
\bar{\rm X} & 1- {\rm e}^{-\pi\gamma_{\rm LZ} }
\end{pmatrix}
\end{equation}\]
assuming an external \(B\) varying linearly with \(t\) and the initial condition \(\rho_{\uparrow \uparrow,0} =1\).
The diagonal terms of the matrix are given by the Landau-Zener formula where
\[\begin{equation}
\gamma_{\rm LZ}= \frac{\Omega^2}{\hbar \dot\Delta }
\end{equation}\]
and \(\dot\Delta = g \mu_{\rm B} v_B/2\) represents the rate at which the Zeeman energy is varied. This first term of the matrix \(\rho_{\uparrow\uparrow}\) expresses the probability of permanence in the state in which the two-level system was prepared at the time \(t_0 \rightarrow - \infty\), after having crossed the avoided level crossing. From the dependence of \(\gamma_{\rm LZ}\) on the sweeping rate \(v_B\), one can appreciate that the slower the field is varied the larger is the probability to find the system in the actual ground state for large positive fields (\(t\rightarrow \infty\)). In the next chapter we will see that Boltzmann statistics prescribes that \(\rho_{\uparrow \uparrow}\) and \(\rho_{\downarrow \downarrow}\) should take some precise values at thermal equilibrium, independently of the previous history. Thus, this tunneling mechanism alone does not generally bring a system into the state of thermal equilibrium.

In Eq.(13.18) we replaced the off-diagonal terms with the symbol “X” to make it clear that we overlook them because they are beyond the scope of the present discussion. Non-vanishing off-diagonal terms in the density matrix indicate that the system has some probability of being in some linear combination of the two basis states \[\begin{equation} \label{psi-off-diag} \tag{13.19} |\psi \rangle = \cos\left(\frac{\theta}{2}\right) \, {\rm e }^{-i\varphi/2}| \uparrow \rangle + \sin\left(\frac{\theta}{2}\right) \, {\rm e }^{i\varphi/2}| \downarrow \rangle \,, \end{equation}\] with a well-defined phase \(\varphi\). When the system is coupled to the environment – as it typically happens for a Q-bit – this phase gets randomized with time, which leads to a decay of quantum coherence (dephasing). At thermal equilibrium, all values of \(\varphi\) occur with the same probability and therefore off-diagonal terms of the density matrix vanish (of course, this also applies to a pure state defined with the Bloch-sphere convention (13.16) on the phase factor \(\phi\)). This condition only indicates the loss of quantum coherence but does not suffice to guarantee thermal equilibrium.

## 13.4 Appendix: Magnetic resonance and hyperfine structure

In this section we give a closer look at the basic energy splitting associated with two relevant magnetic-resonance techniques: electron-spin resonance (ESR) and nuclear-spin resonance (NMR). As the reader may know from previous courses, the hyperfine interaction is the interaction between the magnetic moment of the nucleus \(\hat{\boldsymbol \mu}_N\) with the magnetic field produced by electrons. The latter encompasses two contributions: the field produced by the orbital motion of electrons and the dipolar field generated by the nuclear magnetic moment on the effective spin of electrons.

Generally, the hyperfine interaction will be linear in the electron spin \(\hat{\mathbf S}\) and in the nuclear spin \(\hat{\mathbf I}\) operators: \(\hat{\mathbf I}\, \underline{A} \,\hat{\mathbf S}\). The spin coordinate \(\hat{\mathbf S}\) may be regarded as the *true* total spin of electrons, under the assumption that the magnetic orbital contribution is quenched in the solid-state phase; alternatively, \(\hat{\mathbf S}\) may represent an *effective* spin. The coupling between nuclear and electron spin variables is generally expressed by a tensor \(\underline{A}\). Thus, the basic spin Hamiltonian to describe electron and nuclear spin resonance is
\[\begin{equation}
\label{Spin-ham-HF}
\tag{13.20}
\mathcal{H}_{\rm spin} = g \mu_{\rm B} \vec B \cdot \hat{\mathbf S} - g_{N} \mu_N \vec B \cdot \hat{\mathbf I} +\hat{\mathbf I}\, \underline{A} \,\hat{\mathbf S} \,,
\end{equation}\]
where \(g\) is the electronic Landé factor and \(g_{N}\) the nuclear one^{44}. Since the electronic Zeeman energy is often dominating, we shall assume that the eigenfunctions of \(\hat S^z\) are good quantum numbers (as a reasonable approximation). Under these assumptions, the spin Hamiltonian(13.20) can be simplified as follows:

\[\begin{equation}
\label{Spin-ham-HF-1}
\tag{13.21}
\mathcal{H}_{\rm spin} = g \mu_{\rm B} B_0 \hat{S}^z - g_{N} \mu_N B_0 \hat{I}^z + \hat{I}^z A_z \hat{S}^z \,,
\end{equation}\]

whose eigenvalues are
\[\begin{equation}
\label{Spin-ham-HF-2}
\tag{13.22}
E_{\rm spin} = g \mu_{\rm B} B_0 m_S - g_{N} \mu_N B_0 m_I + A_z m_S \,m_I\,.
\end{equation}\]

Note that in the last two equations it was assumed \(\vec B_0 = B_0 \mathbf{z}\), the subscript “0” indicating the static character of this field. The possible transitions produced by an alternating field \(\vec B_1 = B_1(t) \mathbf{x}\) applied along \(x\) are given by the matrix elements of the operator
\[\begin{equation}
\label{Spin-Ham-pert}
\tag{13.23}
\mathcal{H}_{\rm ac} = \left(g \mu_{\rm B} \hat{S}^x - g_{N}\mu_N \hat{I}^x \right) B_1(t)
\end{equation}\]

evaluated on the states \(|S,\,m_S;\, I,\,m_I\rangle\), on which Hamiltonian(13.21) is diagonal.

Such calculation produces the following selection rules:

electron resonance (ESR) | nuclear resonance (NMR) |
---|---|

\(\Delta m_S=\pm 1\quad \Delta m_I=0\) | \(\Delta m_S=0\quad \Delta m_I=\pm 1\) |

with \(\quad \hbar\omega_e = g \mu_{\rm B} B_0 + A_z m_I\) | with \(\quad \hbar\omega_N = g_N \mu_N B_0 + A_z m_s\) |

## 13.5 Appendix: Rotating frame formalism (NMR)

It is useful to consider first the effect of a rotation of coordinates on the density matrix. Like any other operator, the density matrix transforms under a rotation about the \(z\) axis as follows: \[\begin{equation} \tag{13.24} \hat\rho_\mathcal{R} = \mathcal{R} \, \hat\rho \, \mathcal{R}^\dagger \end{equation}\] where \(\mathcal{R}\) is the rotation operator for a spin 1/2 \[\begin{equation} \tag{13.25} \mathcal{R} = \begin{pmatrix} {\rm e }^{i\alpha_z/2} & 0 \\ 0 & {\rm e }^{-i\alpha_z/2} \end{pmatrix} \end{equation}\] \(\mathcal{R}^\dagger\) its complex conjugated, and \(\alpha_z\) is the angle by which the reference frame (i.e. the \(x\) and \(y\) axes in this case) is rotated. The subscript \(\mathcal{R}\) indicates that the considered operator or direction is expressed in the rotated frame. The reader can easily verify that, assuming \(\hat\rho = \hat\rho(0)\), after this transformation the density matrix in the new reference frame reads: \[\begin{equation} \label{rho-matrix-Rot} \tag{13.26} \hat\rho_\mathcal{R} = \begin{pmatrix} \rho_{\uparrow \uparrow, 0} & \rho_{\uparrow \downarrow,0} \, {\rm e }^{i\alpha_z} \\ \rho_{\downarrow\uparrow, 0} \, {\rm e }^{-i\alpha_z} & \rho_{\downarrow \downarrow,0} \end{pmatrix} \,. \end{equation}\] This result is equivalent to Eq.(13.8) with \(\alpha_z =- \omega_{\rm L} t\). In other words, changing reference system has a similar effect on averaged spin components as applying a static field along the \(z\) axis. Conversely, if we let the rotation angle depend linearly on time \(\alpha_z =\omega \, t\), the averaged spin components shall obey the following equation of motion in the rotating frame \[\begin{equation} \label{S-Rot} \tag{13.27} \frac{d}{dt}\langle \hat{\mathbf S}_\mathcal{R} \rangle = \langle \hat{\mathbf S}_\mathcal{R}\rangle \times (\omega\,\hat{z} ) \end{equation}\] where \(\hat{z}\) is the director of the \(z\) axis, which is common to the static and the rotating frame. Note that the time evolution in Eq.(13.27) is only due to the transformation of coordinates and no external applied field has been considered yet.

A typical configuration that is realized in magnetic-resonance experiments is described by the Hamiltonian
\[\begin{equation}
\tag{13.28}
\mathcal{H} = g \mu_{\rm B} \vec B\cdot \hat S^z + g \mu_{\rm B} \vec B'\cdot \hat{\mathbf S}
\end{equation}\]
which is equivalent to the Hamiltonian (13.6) with the addition of a time-dependent magnetic field rotating on the \(xy\) plane with frequency \(\omega\)
\[\begin{equation}
\tag{13.29}
\vec B'(t) = B' \left[\cos(\omega t) \hat{x} + \sin(\omega t) \hat{y} \right]\,.
\end{equation}\]
Viewed from a reference frame rotating right with frequency \(\omega\), the field \(\vec B'(t)\) appears as a static field directed, say, along the \(x\) axis of the rotated system, formally indicated with \(x_\mathcal{R}\). The equation of motion for the averaged spin components expressed in the rotating frame is
\[\begin{equation}
\label{S-Rot-MR}
\tag{13.30}
\frac{d}{dt}\langle \hat{\mathbf S}_\mathcal{R} \rangle = -\langle \hat{\mathbf S}_\mathcal{R}\rangle\times\left[(\omega_{\rm L} -\omega) \hat{z}+ \gamma B' \hat{x}_\mathcal{R} \right]
\end{equation}\]
with \(\omega_{\rm L}\) and \(\gamma\) having the same meaning as in the previous section. The spin components evolve in the rotating frame as if they experienced an *effective* static field equal to
\[\begin{equation}
\label{B-eff-MR}
\tag{13.31}
\vec B_{\rm eff} = (\omega_{\rm L} -\omega) \hat{z} + B' \hat{x}_\mathcal{R}\,,
\end{equation}\]
namely they will precess about this field. Note that when the frequency of the rotating field exactly equals the Larmor frequency the spin will precess in the plane perpendicular to the oscillating field \(\vec B'(t)\), that is the plane defined by \(y_\mathcal{R}\) and \(z\).

This description of the combined effect of a static and dynamic field on a spin provides the starting point for understanding any type of magnetic-resonance experiment^{45}.

The validity of this equivalence can be checked by computing the trace of the product \(\hat\rho(t) \mathcal{\hat O}\) over a generic basis \(| n \rangle\) \[\begin{equation*} \begin{split} \langle\mathcal{\hat O}(t)\rangle &= {\rm Tr} \{\hat\rho(t)\mathcal{\hat O}\} =\sum_n \langle n | \psi(t) \rangle \langle \psi(t)| \mathcal{\hat O} | n \rangle \\ &= \sum_n \langle \psi(t)| \mathcal{\hat O} | n \rangle \langle n | \psi(t) \rangle = \langle \psi(t)| \mathcal{\hat O} | \psi(t) \rangle \end{split} \end{equation*}\] where the completeness relation \(\sum_n | n \rangle \langle n | = \mathbb{I}\) (identity operator) was used.↩︎

These notes taken from the web pinpoint the difference between a

*pure*and a mixed*quantum*state↩︎In recent research developments this is not the only possibility: in multiferroics exciting the Larmor frequency by means of an electric field is a tool to quantify magneto-electric coupling; the splitting of electron levels of a nitrogen vacancy in diamond produced by the local field of a magnetic nanoparticle is used as a very local and sensitive probe of the magnetization of the nanoparticle itself.↩︎

The electronic \(g\) is often anisotropic and therefore it would be more appropriate to write it as a tensor. Nonetheless, here we prefer to keep \(g\) as scalar in order to avoid useless complications.↩︎

A classical textbook on the topic is

*Principles of Magnetic Resonance*, C.P. Slichter, Springer-Verlag Berlin (1990).↩︎