Quantized spin in a magnetic field

Equation of Motion for the Expectation Value

For the system we have been considering, the Hamiltonian is

${\hat {H}}=-{\bf {\hat {\mu }}}\cdot {\bf {B}}_{0}=-\gamma {\bf {\hat {L}}}\cdot {\bf {B}}_{0}=-\gamma {\hat {L}}_{z}B_{0}{\text{ for }}{\bf {B}}=B_{0}{\bf {\hat {e}}}_{z}$

Recalling the Heisenberg equation of motion for any operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{O}} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}\hat{O} =\frac{i}{\hbar}\left[\hat{H},\hat{O}\right]+\frac{\partial\hat{O}}{\partial t} }

where the last term refers to operators with an explicit time dependence, we have in this instance

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}\hat{\mu}_k =\gamma\frac{d}{dt}\hat{L}_k =\frac{i\gamma}{\hbar}\left[\hat{H},\hat{L}_k\right] =-\frac{i\gamma^2}{\hbar}B_0\left[\hat{L}_z,\hat{L}_k\right] }

Using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\hat{L}_k,\hat{L}_l\right]=i\hbar\epsilon_{klm}\hat{L}_m} with the Levi-Civita symbol $\epsilon _{klm}$, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{d}{dt} \hat{\mu}_x &= - \frac{i \gamma^2}{\hbar} B_0 i \hbar \hat{L}_y = \gamma^2 \hat{L}_y B_0 = \hat{\mu}_y \gamma B_0 \\ \frac{d}{dt} \hat{\mu}_y &= - \mu_x \gamma B_0 \\ \frac{d}{dt} \hat{\mu}_z &= 0 \end{align}}

or in short

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{d}{dt} {\bf{\hat\mu}} &= \gamma {\bf{\hat\mu}} \times {\bf{B}} \\ \frac{d}{dt} {\bf{\hat L}} &= \gamma {\bf{\hat L}} \times {\bf{B}} \end{align}}

These are just like the classical equations of motion \ref{eq:classical_precession_in_static_field}, but here they describe the precession of the operator for the magnetic moment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\hat\mu}} } or for the angular momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\hat L}} } about the magnetic field at the (Larmor) angular frequency $\omega _{L}=\gamma B_{0}$. Note that \begin{itemize} \item Just as in the classical model, these operator equations are exact; we have not neglected any higher order terms. \item Since the equations of motion hold for the operator, they must hold for the expectation value

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle{\dot{ {\bf{\hat L}} }}\rangle =\gamma\langle{ {\bf{\hat L}} }\rangle\times {\bf{B}} }

\item We have not made use of any special relations for a spin-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2} system, but just the general commutation relation for angular momentum. Therefore the result, precession about the magnetic field at the Larmor frequency, remains true for any value of angular momentum $l$. \item A spin-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2} system has two energy levels, and the two-level problem with coupling between two levels can be mapped onto the problem for a spin in a magnetic field, for which we have developed a good classical intuition. \item If coupling between two or more angular momenta or spins within an atom results in an angular momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F>1/2} , the time evolution of this angular momentum in an external field is governed by the same physics as for the two-level system. This is true as long as the applied magnetic field is not large enough to break the coupling between the angular momenta; a situation known as the Zeeman regime. Note that if the coupled angular momenta have different gyromagnetic ratios, the gyromagnetic ratio for the composite angular momentum is different from those of the constituents. \item For large magnetic field the interaction of the individual constituents with the magnetic field dominates, and they precess separately about the magnetic field. This is the Paschen-Back regime. \item An even more interesting composite angular momentum arises when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} two-level atoms are coupled symmetrically to an external field. In this case we have an effective angular momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=N/2} for the symmetric coupling (see Figure \ref{fig:dicke_states}). \begin{figure}

\caption{Level structure diagram for $n$ two-level atoms in a basis of symmetric states \cite{Dicke1954}. The leftmost column corresponds to an effective spin-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n/2} object. Other columns correspond to manifolds of symmetric states of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} atoms with lower total effective angular momentum.}

\end{figure} Again the equation of motion for the composite angular momentum ${\bf {\hat {L}}}$ is a precession. This is the problem considered in Dicke's famous paper \cite{Dicke1954}, in which he shows that this collective precession can give rise to massively enhanced couplings to external fields ("superradiance") due to constructive interference between the individual atoms. \end{itemize}

The Two-Level System: Spin-1/2

Let us now specialize to the two-level system and calculate the time evolution of the occupation probabilities for the two levels.

Equivalence of two-level system with spin-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2} . Note that for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma>0} the spin-up state (spin aligned with field) is the ground state. For an electron, with $\gamma <0$, spin-up is the excited state. Be careful, as both conventions are used in the literature.

We have that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle S_z\right\rangle =\frac{\hbar}{2}\left(P_\uparrow-P_\downarrow\right) =\frac{\hbar}{2}\left(P_g-P_e\right) =\frac{\hbar}{2}\left(1-2P_e\right) }

where in the last equation we have used the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_e+P_g=1} . The signs are chosen for a spin with $\gamma >0$, such as a proton (Figure \ref{fig:two_level_spin_half}). For an electron, or any other spin with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma<0} , the analysis would be the same but for the opposite sign of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_z} and the corresponding exchange of $\uparrow$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \downarrow} . If the system is initially in the ground state, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_g(t=0)=1} (or the spin along $+{\bf {\hat {e}}}_{z}$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_z(0)=+\hbar/2} ), the expectation value obeys the classical equation of motion \ref{eq:classical_rabi_flopping}:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} P_e(t) &= \frac{1}{2} - \frac{1}{\hbar} \left\langle S_z\right\rangle = \frac{1}{2} - \frac{ \left\langle S_z(0)\right\rangle }{\hbar} \left(1 - 2 \frac{\omega_R^2}{\Omega_R^2} \sin^2 \frac{\Omega_R t}{2}\right) \\ P_e(t) &= \frac{\omega_R^2}{\Omega_R^2} sin^2 \left(\frac{\Omega_R t}{2}\right) \end{align}}

Equation \ref{eqn:rabi_transition_probability} is the probability to find system in the excited state at time $t$ if it was in ground state at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=0} . Figure \ref{fig:rabi_signal} shows a real-world example of such an oscillation.

Rabi oscillation signal taken in the Vuleti\'{c} lab shortly after this topic was covered in lecture in 2008. The amplitude of the oscillations decays with time due to spatial variations in the strength of the drive field (and hence of the Rabi frequency), so that the different atoms drift out of phase with each other.

Matrix form of Hamiltonian

With the matrix representation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left|e\right\rangle &= \left|S_z = - \frac{1}{2}\right\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\ \left|g\right\rangle &= \left|S_z = + \frac{1}{2}\right\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \end{align}}

we can write the Hamiltonian $H_{0}$ associated with the static field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _0 = B_0 {\bf{\hat e}} _z} as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}_0 = - {\bf{\hat\mu}} \cdot {\bf{B}} _0 = - \gamma \hat{S}_z B_0 = - \hbar \omega_0 \frac{\hat{S}_z}{\hbar} = \frac{1}{2} \hbar \omega_0 \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \frac{1}{2} \hbar \omega_0 \hat{\sigma}_z }

where $\omega _{0}=\gamma B_{0}$ is the Larmor frequency, and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\sigma}_z= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} }

is a Pauli spin matrix. The eigenstates are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|e\right\rangle } , $\left|g\right\rangle$ with eigenenergies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\hbar\omega_0/2} . A spin initially along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\hat e}} _x} , corresponding to

$\left|\psi (t=0)\right\rangle ={\frac {1}{\sqrt {2}}}\left(\left|e\right\rangle +\left|g\right\rangle \right)={\frac {1}{\sqrt {2}}}\left(\left|-{\frac {1}{2}}\right\rangle +\left|+{\frac {1}{2}}\right\rangle \right)$

evolves in time as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\psi(t)\right\rangle =\frac{1}{\sqrt{2}} \left(e^{-i\omega_0 t / 2} \left|e\right\rangle +e^{+i\omega_0 t / 2} \left|g\right\rangle \right) =\frac{e^{-i\omega_0 t/2}}{\sqrt{2}}\left( \left|e\right\rangle +e^{i\omega_0 t} \left|g\right\rangle \right) }

which describes a precession with angluar frequency $\omega _{0}$. The field ${\bf {B}}_{1}$, rotating at $\omega$ in the $X-Y$ plane corresponds to

${\begin{aligned}{\hat {H}}_{1}&=-{\bf {\hat {\mu }}}\cdot {\bf {B}}_{1}(t)=-{\bf {\hat {\mu }}}\cdot {\frac {\omega _{R}}{\gamma }}\left(-{\bf {\hat {e}}}_{x}\cos \omega t-{\bf {\hat {e}}}_{y}\sin \omega t\right)\\&=\omega _{R}\left({\hat {S}}_{x}\cos \omega t+{\hat {S}}_{y}\sin \omega t\right)\\&={\frac {\hbar \omega _{R}}{2}}\left({\hat {\sigma }}_{x}\cos \omega t+{\hat {\sigma }}_{y}\sin \omega t\right)\\&={\frac {\hbar \omega _{R}}{2}}\left({\begin{pmatrix}0&1\\1&0\end{pmatrix}}\cos \omega t+{\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\sin \omega t\right)\\{\hat {H}}_{1}&={\frac {\hbar \omega _{R}}{2}}{\begin{pmatrix}0&e^{-i\omega t}\\e^{i\omega t}&0\end{pmatrix}}\end{aligned}}$

where have used the Pauli spin matrices $\sigma _{x}$, $\sigma _{y}$. The full Hamiltonian is thus given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}=\frac{\hbar}{2} \begin{pmatrix} +\omega_0 & \omega_R e^{-i \omega t} \\ \omega_R e^{-i \omega t} & - \omega_0 \end{pmatrix} }

This is the famous "dressed atom" Hamiltonian in the so-called "rotating wave approximation". Its eigenstates and eigenvalues provide a very elegant, very intuitive solution to the two-state problem.

Solution of the Schrodinger Equation for Spin-1/2 in the Interaction Representation

The interaction representation consists of expanding the state $\left|\psi (t)\right\rangle$ in terms of the eigenstates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|e\right\rangle } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|g\right\rangle } of the Hamiltonian $H_{0}$, including their known time dependence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{iH_0 t/\hbar}} due to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} . That means we write here

$\left|\psi (t)\right\rangle =a_{e}(t)\left|e\right\rangle e^{-i\omega _{0}t/2}+a_{g}(t)\left|g\right\rangle e^{i\omega _{0}t/2}$

Substituting this into the Schrodinger equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\hbar\frac{d}{dt} \left|\psi(t)\right\rangle =\left(H_0+H_1\right) \left|\psi(t)\right\rangle }

then results in the equations of motion for the coefficients

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} i\dot{a}_e&= \frac{\omega_R}{2}e^{i\left(\omega_0-\omega\right)t}a_g \\ i\dot{a}_g&= \frac{\omega_R}{2}e^{-i\left(\omega_0-\omega\right)t}a_e \end{align}}

Where we have used the matrix form of the Hamiltonian, \ref{eq:dressed_atom_hamiltonian}. Introducing the detuning $\delta =\omega -\omega _{0}$, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} i\dot{a}_g&= \frac{\omega_R}{2}e^{i\delta t}a_e \\ i\dot{a}_e&= \frac{\omega_R}{2}e^{-i\delta t}a_g \end{align}}

The explicit time dependence can be eliminated by the sustitution

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} b_g&= e^{-i\delta t/2}a_g \\ b_e&= e^{i\delta t/2}a_e \end{align}}

As you will show (or have shown) in the problem set, this leads to solutions for $b_{g,e}$ given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} b_g(t)&= e^{i\Omega_R t/2}B_1+e^{-i\Omega_R t/2}B_2 \\ b_e(t)&= \frac{\omega_R}{\delta-\Omega_R}e^{i\Omega t/2}B_1 +\frac{\omega_R}{\delta+\Omega_R}e^{-i\Omega_R t/2}B_2 \end{align}}

with two constants that are determined by the initial conditions. For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_g(t=0)=1} we find

$\left|a_{e}(t)\right|^{2}={\frac {\omega _{R}^{2}}{\Omega _{R}^{2}}}\sin ^{2}{\frac {\Omega _{R}t}{2}}$

as already derived from the fact that the expectation value for the magnetic moment obeys the classical equation.

Atomic Clocks and the Ramsey Method

When comparing the Hamiltonian for a spin-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2} in a magnetic field to that of a two-level system with a coupling between the two levels characterized by the strength Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R} and frequency $\omega$, we see that the energy spacing between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|e\right\rangle } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|g\right\rangle } corresponds to the Larmor frequency $\omega _{0}$ in the static field. This spacing can provide a frequency or time reference if perturbations affecting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0} are sufficiently well controlled. For instance, the time unit "second" is defined via the transition frequency between two hyperfine states in the electronic ground state of the caesium atom, which is near \unit{9.2}{\giga\hertz} in the microwave domain. The task of an atomic clock is then to measure this frequency accurately by trying to tune a frequency source (the frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} of the rotating field $B_{1}$ in the spin picture) to the atomic frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0} . Equivalently, we want to find the frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} such that the detuning $\delta =\omega -\omega _{0}$ is equal to zero. Starting with an atom in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|g\right\rangle } (spin along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + {\bf{\hat e}} _z} for $\gamma >0$), we could try to find the resonance frequency by noting that according to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_e(t)=\frac{\omega_R^2}{\omega_R^2+\delta^2} sin^2\left(\sqrt{\omega_R^2+\delta^2}\,\frac{t}{2}\right) }

the population of the upper state is maximized for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=0} (i.e. the precession of the spin to the $-{\bf {\hat {e}}}_{z}$ direction is only complete on resonance). This is the so-called Rabi method. It suffers from a number of drawbacks. For one, the signal is only quadratic in the detuning Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} , i.e. the method is relatively insensitive near Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=0} . Furthermore, the optimum time $t$ depends on the strength Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R} of the coupling (i.e. the strength Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_1} of the rotating field), so fluctuations in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R} can be mistaken for changes in $\delta$. Finally the coupling by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R} to other levels can lead to level shifts that are not intrinsic to the atom, but depend on the applied drive $\omega _{R}$ (Figure \ref{fig:rabi_third_level}). \begin{figure}

\caption{The drive used for Rabi flopping within the $\left|e\right\rangle$ , $\left|g\right\rangle$ system can also off-resonantly couple one or both levels to other states, perturbing the transition frequency $\omega _{0}$.}

\end{figure} Norman Ramsey invented an alternative method (the so-called "separated oscillatory fields method", known for short as the "Ramsey method" \cite{Ramsey1949,Ramsey1950}, for which he received the Nobel prize), that fixes all of these problems. It leads to a signal that is linear rather than quadratic in the detuning $\delta$, does not require tuning the measurement time to match the applied field strength $\omega _{R}=\gamma B_{1}$, and, most importantly, eliminates level shifts due to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R} altogether. The method is as follows. Instead of applying a pulse for a time t that corresponds to Rabi rotation of the spin by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} (called a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} pulse), the pulse is applied for half that time, corresponding to the Rabi rotation of the spin by $\pi /2$ into the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X-Y} plane (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/2} pulse). Then the applied field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_1} is turned off and the system is left to precess in the static field $B_{0}$ (or at its natural frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0} ) for a measurement time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} . Finally, a second Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/2} pulse, identical to the original one, is applied (see Figure \ref{fig:ramsey_sequence}). \begin{figure}

\caption{Ramsey sequence}

\end{figure} The signal is the ${\bf {\hat {e}}}_{z}$ component of the spin after the second interaction. The signal after the second Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/2} pulse is an oscillating signal in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\delta} , depending on how much phase the spin has acquired relative to the local oscillator (the microwave signal generator at frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} ). Examples of such curves are shown in Figures \ref{fig:ramsey_signal} and \ref{fig:ramsey_vs_freq}. \begin{figure}

\caption{Ramsey oscillation signal as a function of time taken in the Vuleti\'{c} lab in 2007. The drive field was deliberately detuned from resonance so that the oscillation at the detuning frequency would be visible.}

\end{figure} \begin{figure}

\caption{Experimental data from Ramsey's original paper \cite{Ramsey1950}, showing the signal as a function of frequency. Note the narrow oscillation, whose width is set by the measurement time $T$, superimposed on the much broader background set up by the inhomogeneously broadened Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/2} pulses.}

\end{figure} At the zero crossings we have maximum sensitivity of the signal with respect to frequency changes. Note that the signal as a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} looks similar to Rabi flopping. However, there the zero crossing measure the Rabi frequency, not Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=\omega-\omega_0} .