Resonances

Classically, a resonance is a system where one or more variables change periodically such that when the system is no longer driven by an extended periodic process, the decay of the system's oscillation takes many (or at least several) oscillation periods. Equivalently, the response of the driven system as a function of drive frequency exhibits some form of peaked structure. The frequency corresponding to the maximum response of the system is called the resonance 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 f_0} )

\caption{A resonance in the time (left) and frequency (right) domains. The time domain picture shows the (decaying) oscillation in some variable of the undriven system, while the frequency-domain pictures shows the steady-state response amplitude of the driven system. The time units are chosen such 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 f_0 = \omega_0 / 2 \pi = 1} while the quality factor is $Q=10$.}

▼ Category:8.421

► Category:Atoms in electric fields

  Atoms

  Atoms in electric fields

  Atoms in Magnetic Fields

  Coherence

  Effects of the Nucleus on Atomic Structure

  Fine Structure and Lamb Shift

  Higher-Order Radiation Processes

  Interaction of an atom with an electromagnetic field

  Old Coherence

  Resonances

  Two-Photon Excitation

The simplest resonance systems have a single resonance frequency, corresponding to the harmonic motion of the variable. Some form of dissipation (damping) results in a finite damping 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 \Delta t<\infty} ) for the undriven system, and a finite width (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 f>0} ) of the resonance curve for the driven system. The quantities $\Delta {f}$ 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 \Delta{t}} are related by a Fourier transformation: any oscillation with time-varying amplitude must be made up of a superposition of different frequency components, giving the resonance curve a finite width in frequency space. Low dissipation results in a long decay 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 \Delta t} , and a narrow frequency width $f_{0}$. The ratio of frequency width 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 f} to resonance 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 f_0} is called the quality factor $Q$.

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 Q=\frac{f_0}{\Delta f} }

Atomic physics often deals with isolated atomic systems in a vacuum, providing resonances of high quality factor 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 Q} . For example, an optical transition, even in a room-temperature (ie Doppler-broadened) gas, will have a $Q$-factor 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 \frac{f_0}{\Delta f} \sim \frac{\unit{10^{15}}{\hertz}}{\unit{10^9}{\hertz}} \sim 10^6 }

In contrast, high 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 Q} factors in solids typically require cryogenic temperatures: typical $Q$ factors of mechanical or electrical systems 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 \sim 10^3} at room temperature 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 \sim 10^6} at Failed to parse (unknown function "\unit"): {\displaystyle \lesssim \unit{1}{\kelvin}} temperatures. Solid optical resonators are an exception to this: 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 Q\gtrsim 10^8} has been achieved in the whispering gallery modes of spheres or cylinders of high-purity glass \cite{Armani2003}. \begin{figure}

\caption{One of the whispering-gallery resonators described in \cite{Armani2003}. This particular resonator had a quality factor 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 Q=10^8} . The very high surface quality required for such low losses is achieved by having surface tension shape the rim of the disk after it has been temporarily liquefied by a laser pulse; the inset shows the resonator before this treatment.}

\end{figure} At the macroscopic scale, astronomical systems consisting of massive bodies travelling essentially undisturbed through the near-vacuum of space can also exhibit high $Q$ factors. A resonance is particularly useful for science and technology if it is reproducible by different realizations of the same system, and if it has a well-understood theoretical connection to fundamental constants. In atomic physics, quantum mechanics ensures both features: atoms of the same species are fundamentally identical (although the probing apparatus is not), and the quantum mechanical description of atomic structure is well understood and highly accurate, at least for atoms with not too many electrons. The resonance frequencies of hydrogen, and to a lesser degree helium, are directly tied via quantum mechanics to fundamental constants such as the Rydberg constant, 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 R_\infty=\unit{1.0973731568525(73)\times10^7}{\reciprocal\metre}} \cite{codata2002}, or the fine structure constant 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 \alpha=1/137.03599911(46)} \cite{codata2002}. In fact, the Rydberg constant is the most accurately known constant in all of physics, and most fundamental constants have been determined by atomic physics or optical techniques. Additional motivation for measuring fundamental constants with ever-increasing accuracy has been provided by the speculation (and claims of evidence for it \cite{Flambaum2004,Reinhold2006}), that the fundamental constants may not be so constant after all, but that their value is tied to the evolution (and therefore the age) of the universe. Furthermore unexpected resonances, or splittings of resonances, have in the past indicated the need for new theories or modifications of theories. For instance, "anomalous" effects in the Zeeman structure (magnetic-field dependance) of atomic lines led to the discovery of spin \cite{Uhlenbeck1926}. while Lamb's measurement of the splitting between the $^{2}S_{1/2}$ 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 ^2P_{1/2}} states of atomic hydrogen, on the order of \unit{1000}{\mega\hertz} instead of 0 as predicted by the Dirac theory, fostered the development of quantum electrodynamics, the quantum theory of light.

Classical Resonances

A classical harmonic oscillator (HO) is a system described by the second order differential 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 \ddot{q}+\gamma \dot{q}+\omega_0^2 q = 0 }

such as a mass on a spring, or a charge, voltage, or currant in an electric RLC resonant circuit. For weak damping ($\gamma ^{2}<4\omega _{0}^{2}$) the undriven system decays with an exponential envelope,

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^{-\frac{\gamma}{2} t}e^{\pm{i\omega^\prime t}} }

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 \omega^\prime=\omega_0\sqrt{1-\gamma^2/4 \omega_0^2}} . In the cases we will consider, 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^2\ll{4\omega_0^2}} so 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 \omega^\prime\approx\omega_0} . The decay rate constant for the amplitude 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 \gamma/2} , while for the stored energy it is $\gamma$, i.e. the stored energy decays exponentially with a time constant 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 \tau=1/\gamma} . If the harmonic oscillator is driven at a 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} close to resonance, 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| \omega_0-\omega\right| \ll\omega_0} , the amplitude response 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 q_0} is a lorentzian:

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 q_0=\frac{\text{const.}}{\omega_0-\omega+i\frac{\gamma}{2}} =\frac{\text{const.}}{-\Delta+i\frac{\gamma}{2}} }

where $\Delta =\omega -\omega _{0}$ is the detuning as shown on the right-hand side of Figure \ref{fig:example_resonances}. The full width at half maximum (FWHM) of a lorentzian curve 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 \Delta \omega=\gamma} leading to a quality factor

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 Q = \frac{\omega_0}{\Delta \omega} = \frac{\omega_0}{\gamma} }

Note 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 \omega_0} , 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 w} , 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 \gamma} are measured in angular frequency units \unit{\radian\per\second}, or \unit{\reciprocal\second} for short, and should not be confused with frequencies $f_{0}=\omega _{0}/2\pi$, 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 f=\Delta \omega/2\pi} etc. When quoting values, we will often write explicitly 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=2\pi\times\unit{1}{\mega\hertz}} rather than Failed to parse (unknown function "\unit"): {\displaystyle \omega_0=\unit{6.28\times10^6}{\reciprocal\second}} to remind ourselves that the quantity in question is an angular frequency. We will never write Failed to parse (unknown function "\unit"): {\displaystyle \omega_0=\unit{6.28\times10^6}{Hz}} : although formally dimensionally consistent, this invites confusion with real (laboratory or Hertzian) frequencies. For brevity, we will often write or say "frequency" when we mean angular frequency, but will quote real frequencies as Failed to parse (unknown function "\unit"): {\displaystyle f_0=\unit{1}{\mega\hertz}} . Note that the natural unit for decay rate constants is that of an angular frequency, as in $e^{-i\omega _{0}t-\gamma t}$. Therefore we will express decay rates as inverse times in angular frequency units: 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=\unit{10^4}{\reciprocal\second}} for instance, but 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 \gamma=\unit{10^4}{\hertz}} , nor 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=2\pi\times\unit{1.6}{\kilo\hertz}} . Damping time constants are the inverse of decay rate constants, 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 \tau=1/\gamma=\unit{100}{\micro\second}} .

Resonance Widths and Uncertainty Relations

From $\Delta \omega =\gamma$ it follows that damping 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 \tau} and the resonance linewidth 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} of the driven system obey: 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\tau=1} or, assuming 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=\hbar\omega} , $\Delta E\tau =\hbar$: a finite number of frequency components or energies (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>0} ) is necessary to synthesize a pulse of finite (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 \tau<\infty} ) width in time.

Measurement in a limited 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 \Delta t} , modelled as an ideal monochromatic sine wave of frequency $\omega _{0}/2\pi =1$ gated by an observation window of finite width 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 t}

This is quite similar to an uncertainty relation familiar from Fourier transformation theory, according to which the measurement of a frequency in a finite 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 \Delta t} (see Figure \ref{fig:gated_pulse}) results in a frequency spread 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} obeying $\Delta \omega \Delta t\geq 1/2$. By 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 E=\hbar\omega} we can further connect this with the usual Heisenberg uncertainty relation 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 E \Delta t \geq \hbar / 2 }

Question: Does the Heisenberg uncertainty relation 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 E\Delta t\geq \hbar/2} , or rather the frequency uncertainty relation $\Delta \omega \Delta t\geq 1/2$ that follows from (classical) Fourier theory hold in classical systems? Can you measure the angular frequency of a classical harmonic oscillator in a 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 \Delta t} with an accuracy better than 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=1/2\Delta t} ?

Answer: Yes - if you have a good model for the signal, for instance if you know that it is a pure sine wave, and if the signal to noise ratio (SNR) of the measurement is good enough, then you can fit the sine wave's frequency to much better accuracy than its spectral width 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} (see Fig. \ref{fig:line_splitting}). As a general rule, the line center can be found to within an uncertainty $\delta \omega \approx \Delta \Omega /{\text{SNR}}$. This is known as splitting the line. Splitting the line by a factor of 100 is fairly straightforward, splitting it by a factor of a 1000 is a real challenge. As an extreme example of this, Caesium fountain clocks interrogate a \unit{9}{\giga\hertz} transition for a typical measurement time of \unit{1}{\second}, yielding 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\approx 10^{-10}} , but the best ones can nonetheless achieve relative accuracies 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 10^{-16}} \cite{Santarelli1999}. Of course, this sort of performance requires a very good understanding of the line shape and of systematics.

Splitting the line in time and frequency domains. Given a sufficiently good model of the signal or line shape, and adequate signal-to-noise, one can find the center 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} of the underlying signal to much better than $1{\text{ cycle}}/\Delta t$ or the line width.

Question: Can you measure the angular frequency of a quantum mechanical harmonic oscillator in a 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 \Delta t} to better than 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=1/2\Delta t} ?

Question: Can you measure the angular frequency of a laser pulse lasting a 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 \Delta t} to better than 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=1/2\Delta t} ?

If you answered \ref{q:quantum_line_splitting} and \ref{q:classical_line_splitting} differently, you may have a problem with \ref{q:laser_line_splitting}. Is a laser pulse quantum or classical? Clearly it is possible to beat $\Delta \omega =1/2\Delta t$ for a laser pulse. For instance, you can superimpose the laser pulse with a laser of fixed and known frequency on a photodiode, and fit the observed beatnote. The stronger the probe pulse, the better the SNR of the beatnote, and the more accurately you can extract the probe frequency. So how are we to reconcile quantum mechanics and classical mechanics, and what does the Heisenberg uncertainty really state?

The Heisenberg uncertainty relation makes a statement about predicting the outcome of a single measurement on a single system. For a single photon, the limit 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\Delta t} does indeed hold. However, nothing in quantum mechanics says that you cannot improve your knowledge about the average values of the quantities characterizing the system by repeated measurements on identical copies of it. In the laser example above, the frequency resolution improves with probe pulse strength because we are measuring many photons - for the purposes of this discussion we could have measured the frequencies of the photons sequentially.

For independent measurement 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 n} uncorrelated photons the SNR is 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 \sqrt{n}} : the signal grows as $n$, while the noise is the shot noise of the photon number, which grows 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 \sqrt{n}} . This SNR allows a frequency uncertainty 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 1/2\Delta t\sqrt{n}} , known as the standard quantum limit. If we allow the use of entangled or correlated states of the photons, then the uncertainty can be as low as the Heisenberg limit 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\Delta t n} \cite{Wineland1992,Wineland1994}.

Let us now revisit \ref{q:quantum_line_splitting}. You may have heard that the quantum mechanical description of electromagnetism, quantum electrodynamics (QED), represents any single mode of the electromagnetic field by a harmonic oscillator of the same frequency. The ground state $n=0$ corresponds to no photons in the mode, the first excited 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 n=1} to one photon and so forth. This description is valid because photons to a very good approximation do not interact: the addition of a photon changes the system's energy always by the same increment. Since the many-photon pulse from \ref{q:laser_line_splitting} maps onto a quantum harmonic oscillator, it seems to follow that \ref{q:quantum_line_splitting} must also be answered in the affirmative.

On the other hand, a harmonic oscillator is a single quantum system to which the Heisenberg limit must apply. The resolution of this apparent contradiction is that \ref{q:quantum_line_splitting} is asking about a measurement of the fundamental frequency of the harmonic oscillator, not the energy of a particular level. Therefore if you measure the energy of the n-th level 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 \Delta E=\hbar/2\Delta t} , which can be done by exciting 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} -th level from the ground state through a nonlinear process, then the uncertainty on the harmonic oscillator resonance frequency (that corresponds to the laser frequency in \ref{q:quantum_line_splitting}) is given by $\Delta \omega _{0}=\Delta E/n=1/n2\Delta t$.

If the interaction with the oscillator is restricted to classical fields or forces, then there is an additional uncertainty of ${\sqrt {n}}$ on which particular level of the oscillator is being measured and so we recover the standard quantum limit $1/2\Delta t{\sqrt {n}}$ mentioned previously.

Magnetic Resonance: The Classical Magnetic Moment in a Spatially Uniform Field

Harmonic oscillator versus two-level systems

Now, unlike in classical mechanics, most resonances studied in atomic physics are not harmonic oscillators, but two-level systems. Unlike harmonic oscillators, two-level systems show saturation. When a harmonic oscillator is driven longer or faster, higher and higher excited states are populated: the oscillator amplitude can become arbitrarily large. In contrast, the amplitude of oscillation in a two-level system is limited to one half in the appropriate dimensionless units.\footnote{An analogous dimensionless amplitude for the harmonic oscillator would be the amplitude measured in units of the oscillator ground state size.} Why, and under what conditions, do classical harmonic oscillator models of a two-level system work? A two-level system of energy difference $E$ can be approximated by a harmonic oscillator of frequency $\omega _{o}=E/\hbar$ when saturation effects in the two-level system are negligible, i.e. when the population of the second excited state in the harmonic oscillator problem is negligible, or equivalently when the population ratio of the first excited state to the the ground state is small: 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_1/P_0\ll 1} . This is the basis for the classical Lorentz model of the electron bound in the atom that describes many linear atomic properties (for instance the refractive index) very well. See "The origin of the refractive index" in chapter 31 of the Feynman Lectures on Physics \cite{Feynman1963}. When saturation comes into play, i.e. when the initial ground state is appreciably depleted, the harmonic oscillator ceases to be a good model. The limit on the oscillation amplitude in a two-level system suggests that a classical system with periodic evolution and a limit on the amplitude, namely rotation, could provide a better classical model of a two-level system. Indeed, the motion of a classical magnetic moment in a uniform field provides a model that captures almost all features of the quantum-mechanical two-level system, the exception being the projection onto one of the two possible outcomes in a measurement.

Magnetic Moment in a Static Field

The interaction energy of a 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{\mu}} } with a magnetic 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}} } is given by

$W=-{\bf {\mu }}\cdot {\bf {B}}$

In a uniform field the force

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{F}} =- {\bf{\nabla}} W=0 }

vanishes, but the torque

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{T}} = {\bf{\mu}} \times {\bf{B}} }

does not. For 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 {\bf{L}} } the equation of motion is then

${\dot {\bf {L}}}={\bf {T}}={\bf {\mu }}\times {\bf {B}}=\gamma {\bf {L}}\times {\bf {B}}$

where we have introduced the gyromagnetic ratio 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} as the proportionality constant between angular momentum and magnetic moment, as shown in Figure \ref{fig:static_precession}.

{fig:static_precession} Precession of a the magnetic moment and associated angular momentum about a 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}} } .

This describes the precession of the magnetic moment about the magnetic field with angular 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_L=-\gamma B }

$\Omega _{L}$ is known as the Larmor frequency. For electrons 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 \gamma_e=2\pi\times\unit{2.8}{\mega\hertz\per G}} , for protons 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_e=2\pi\times\unit{4.2}{\kilo\hertz\per G}} .

Transformation to a Rotating Coordinate System

A vector 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{A}} } rotating at constant angular velocity ${\bf {\Omega }}$ is described 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 \dot{ {\bf{A}} } = {\bf{\Omega}} \times {\bf{A}} }

Then the rates of change 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 {\bf{A}} } measured in a coordinate system rotating at 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{\Omega}} } and in an inertial system are related by

${\dot {\bf {A}}}_{\text{in}}={\dot {\bf {A}}}_{\text{rot}}+{\bf {\Omega }}\times {\bf {A}}_{\text{in}}$

This follows immediately from the following facts:

• If 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{A}} } is constant in the rotating system then 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 \dot{ {\bf{A}} }_\text{in}= {\bf{\Omega}} \times {\bf{A}} _\text{in}} .
• If 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{\Omega}} =0} then ${\dot {\bf {A}}}_{\text{in}}={\dot {\bf {A}}}_{\text{rot}}$.
• Coordinate rotation is a linear transformation.

This transformation is sometimes abbreviated as the schematic rule

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(\frac{d}{dt}\right)_\text{rot}= \left(\frac{d}{dt}\right)_\text{in}- {\bf{\Omega}} \times\big(\big)_\text{in} }

It follows that the angular momentum in a rotating frame obeys

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 \dot{ {\bf{L}} }_\text{rot} = \dot{ {\bf{L}} }_\text{in}- {\bf{\Omega}} \times {\bf{L}} _\text{in} = {\bf{L}} \times(\gamma {\bf{B}} + {\bf{\Omega}} ) }

If we choose 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{\Omega}} = {\bf{\Omega_L}} =-\gamma {\bf{B}} } , then ${\bf {L}}_{\text{rot}}$ is constant in the rotating frame. Often it is useful to think of a fictitious magnetic 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}} _\text{fict}= {\bf{\Omega}} /\gamma} that appears in a rotating frame.

Larmor's Theorem for a Charged Particle in a Magnetic Field

The vanishing of the torque on a magnetic moment when viewed in the correct rotating frame is reminiscent of Larmor's theorem for the motion of a charged particle in a magnetic field, which we now present.

According to answers.com, Lamor's theorem can be stated as For a system of charged particles, all having the same ratio of charge to mass, moving in a central field of force, the motion in a uniform magnetic induction B is, to first order in B, the same as a possible motion in the absence of B except for the superposition of a common precession of angular frequency equal to the Larmor frequency.

If the Lorentz force acts in an inertial frame,

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{F}} _\text{in}=q {\bf{v}} _\text{in}\times {\bf{B}} }

then in the rotating frame, according to the rule \ref{eqn:rot_frame_transformation} 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} \dot{ {\bf{r}} }_\text{rot} &= \dot{ {\bf{r}} }_\text{in} - {\bf{\Omega}} \times {\bf{r}} _\text{in} \\ \ddot{ {\bf{r}} }_\text{rot} &= \ddot{ {\bf{r}} }_\text{in} - 2 {\bf{\Omega}} \times \dot{ {\bf{r}} }_\text{in} + {\bf{\Omega}} \times \left( {\bf{\Omega}} \times {\bf{r}} _\text{in} \right) \end{align}}

resulting in a force ${\bf {F}}_{\text{rot}}=m{\ddot {\bf {r}}}_{\text{rot}}$ in the rotating frame 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} {\bf{F}} _\text{rot} &= q {\bf{v}} _\text{in} \times {\bf{B}} - 2 m {\bf{\Omega}} \times {\bf{v}} _\text{in} + m {\bf{\Omega}} \times \left( {\bf{\Omega}} \times {\bf{r}} _\text{in} \right) \\ &= q {\bf{v}} _\text{in} \times \left( {\bf{B}} + 2 \frac{m}{q} {\bf{\Omega}} \right) + m {\bf{\Omega}} \times \left( {\bf{\Omega}} \times {\bf{r}} _\text{in} \right) \end{align}}

where we have used 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 m\ddot{r}_\text{in}= {\bf{F}} _\text{in}=q {\bf{v}} _\text{in}\times {\bf{B}} } . Choosing

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 2\frac{m}{q} {\bf{\Omega}} =- {\bf{B}} =-B {\bf{\hat e}} _z }

yields

${\bf {F}}_{\text{rot}}={\frac {q^{2}}{4m}}B^{2}{\bf {\hat {e}}}_{z}\times \left({\bf {\hat {e}}}_{z}\times {\bf {r}}_{\text{in}}\right)\approx 0$

if 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 B} field is not too large. Thus the Lorentz force approximately disappears in the rotating frame.

Although Larmor's theorem is suggestive of the rotating co-ordinate transformation, it is important to realize that the two transformations, though identical in form, apply to fundamentally different systems. A magnetic moment is not necessarily charged- for example a neutral atom can have a net magnetic moment, and the neutron possesses a magnetic moment in spite of being neutral - and it experiences no net force in a uniform magnetic field. Furthermore, the rotating co-ordinate transformation is exact for a magnetic moment, whereas Larmor's theorem for the motion of a charged particle is only valid when 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 B^2 } term is neglected.

Note: A breakdown of Larmor's theorem occurs for free electrons. Free electrons orbit in a magnetic field at the cyclotron 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_c=qB/m} which is twice Larmor's frequency. The reason is that the orbiting velocity is now given by $v=\omega _{c}r$. As can be seen in the equation above, in this case, the centrifugal force equals minus one half of the Coriolis force, and therefore twice the frequency of the rotating frame is needed to cancel the effects of the applied magnetic field.

Rotating Magnetic Field on Resonance

{fig:rotating_frame} Field and moment vectors in the static and rotating frames for the case of resonant drive.

Consider a 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{\mu}} } precessing about a 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}} =B_0 {\bf{\hat e}} _z} 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 {\bf{\mu}} =(\mu, \theta, \phi=-\omega_0 t)} in spherical coordinates, where 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=-\gamma B_0} . Assume that we now apply an additional field ${\bf {B}}_{1}$, in the $xy$-plane rotating at $\omega _{0}$. To solve the resulting problem it is simplest to go into the rotating frame (Figure \ref{fig:rotating_frame}). Then ${\bf {B}}_{1}$ is stationary, say along ${\bf {\hat {e}}}_{x}$, and there is an additional fictitious field ${\bf {B}}_{\text{fict}}={\bf {\omega _{0}}}/\gamma =-{\bf {B}}_{0}$ which cancels the field ${\bf {B}}_{0}$. So in the rotating frame we are left just 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}} _1} , and the magnetic moment precesses about 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}} _1} at the Rabi frequency

$\omega _{R}=\gamma B_{1}$

A magnetic moment initially along 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 - {\bf{\hat e}} _z} axis will point along 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 + {\bf{\hat e}} _z} axis at a time $T$ 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 \omega_r T=\pi} , while a magnetic moment parallel or antiparallel to the applied magnetic 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}} _1} does not precess in the rotating frame.

Question: {q:rabi_freq_blue_detuned} Assume the magnetic moment is initially pointing along the $-{\bf {\hat {e}}}_{z}$ axis. Assume that a rotating 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\ll B_0} is applied, but that it rotates at a 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_1>\omega_0} , where $\omega _{0}$ is the Larmor frequency for 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}} =B_0 {\bf{\hat e}} _z} . Compared to the on-resonant case, 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_1=\omega_0} , is the oscillation frequency of the magnetic moment.

• larger
• the same
• smaller

Question: Same question as \ref{q:rabi_freq_blue_detuned} but for $\omega _{1}<\omega _{0}$.

Rotating Magnetic Field Off-Resonance

If the rotation 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 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}} _1} does not equal the Larmor frequency $\omega _{0}=\gamma B_{0}$ 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} , then in the frame rotating 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 {\bf{B}} _1} at frequency $\omega$ the static field is not completely cancelled by the fictitious 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 \omega/\gamma} , but a difference 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} remains, giving rise to a total effective field in the rotating frame

${\bf {B}}_{\text{eff}}=B_{1}{\bf {\hat {e}}}_{x}+\left(B_{0}-{\frac {\omega }{\gamma }}\right){\bf {\hat {e}}}_{z}$

The effective field is static, lies at an angle 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 \theta} with the 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 \tan\theta=\frac{B_1}{B_0-\frac{\omega}{\gamma}} }

and is of magnitude

$\left|B_{\text{eff}}\right|={\sqrt {B_{1}^{2}+\left(B_{0}-{\frac {\omega }{\gamma }}\right)^{2}}}$

The magnetic moment precesses around it with an effective (sometimes called generalized) Rabi 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_R =\gamma B_\text{eff} =\sqrt{(\omega_0-\omega)^2+\omega_R^2}=\sqrt{\omega_R^2+\delta^2} }

where 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=\gamma B_1} is the Rabi frequency associated 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 B_1} , and $\delta =\omega -\omega _{0}$ is the detuning from resonance with the Larmor 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=\gamma B_0} .

Geometrical Solution for the Classical Magnetic Moment in Static and Rotating Fields

{fig:rotating_coord_construction} Geometrical relations for the spin in combined static and rotating magnetic fields, viewed in the frame co-rotating with the drive 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}} _1} . At lower right is a view looking straight down 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 {\bf{B}} _\text{eff}} axis.

Referring to Figure \ref{fig:rotating_coord_construction}, we have

${\begin{aligned}\phi &=\Omega _{R}t\\C&=\mu \sin \alpha \\A^{2}&=\mu ^{2}(1-\cos \alpha )^{2}+C^{2}\\&=\mu ^{2}(1-2\cos \alpha +\cos ^{2}\alpha +\sin ^{2}\alpha )\\&=2\mu ^{2}(1-\cos \alpha )\\\Rightarrow \cos \alpha &=1-{\frac {A^{2}}{2\mu ^{2}}}\end{aligned}}$

On the other hand

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{A}{2} &= \mu \sin \theta \sin \frac{\phi}{2} \\ A &= 2 \mu \sin \theta \sin \frac{\Omega_R t}{2} \end{align}}

so that

{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} \cos \alpha &= 1 - \frac{4 \mu^2 \sin^2 \theta \sin^2 \frac{\Omega_R t}{2}}{2 \mu^2} \\ &= 1 - 2 \sin^2 \theta \sin^2 \frac{\Omega_R t}{2} \\ \mu_z(t) &= \mu \cos \alpha \\ &= \mu \left( 1 - 2 \frac{\omega_R^2}{\delta^2 + \omega_R^2} \sin^2 \frac{\Omega_R t}{2} \right) \\ \mu_z(t) &= \mu \left( 1 - 2 \frac{\omega_R^2}{\Omega_R^2} \sin^2 \frac{\Omega_R t}{2} \right) \end{align}}

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 \omega_0 = \gamma B_0} the Larmor frequency of 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 \delta=\omega-\omega_0} the detuning, $\omega _{R}=\gamma B_{1}$ the resonant 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 \Omega_R=\sqrt{\omega_R^2+\delta^2}} the generalized Rabi frequencies. Note that the precession is faster, but the amplitude smaller for an off-resonant field than for the resonant case. The above result is also the correct quantum-mechanical result.

If we define the quantity 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= \frac{\mu_z(t)-\mu_z(0)}{2\mu_z(0)} } , we obtain

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= \frac{\omega_R^2}{\Omega_R^2} \sin^2 \frac{\Omega_R t}{2} }

In the quantum mechanical treatment (see below), $P$ is the probability to find the system in the second state, or the spin flip probability.

"Rapid" Adiabiatic Passage (classical treatment)

Rapid adiabatic passage is a technique for inverting a spin by (slowly) sweeping the detuning of a drive field through resonance. "Slowly" means slowly compared to the Larmor frequency $\gamma B_{\text{eff}}$ about the effective static field in the rotating frame for all times ("Rapid" means fast compared to relaxation processes which we neglect here).

The physical picture is as follows. Assume the detuning is initially negative ($\delta <0$, $\left|\delta \gg \omega _{R}\right|$). Since

$\tan \theta ={\frac {B_{1}}{B_{0}-{\frac {\omega }{\gamma }}}}={\frac {\omega _{R}}{\omega _{0}-\omega }}=-{\frac {\omega _{R}}{\delta }}$

the effective magnetic field initially points of a small angle relative to the ${\bf {\hat {e}}}_{z}$ axis. If the detuning is increased slowly compared to the Larmor frequency, the spin will continue to precess tightly around 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}} _\text{eff}} , which 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} points along the x axis, and 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\gg\omega_R} along 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 - {\bf{\hat e}} _z} axis (see Figure \ref{fig:rapid_adiabatic_passage}).

{fig:rapid_adiabatic_passage} {Motion of the spin during rapid adiabatic passage, viewed in the frame rotating with ${\bf {B}}_{1}$. The spin's rapid precession locks it to the direction 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 {\bf{B}} _\text{eff}} and thus it is dragged through an angle 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} as the frequency is swept through resonance.

Thus the magnetic moment, starting out 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{B}} _0=B_0 {\bf{\hat e}} _z} , ends up pointing along $-{\bf {B}}_{0}=B_{0}{\bf {\hat {e}}}_{z}$. Note that in the rotating frame 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{\mu}} } remains always (almost) parallel to the effective 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}} _\text{eff}} . A similar precess is used in magnetic traps for atoms, but there 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}} _\text{eff}} is a real, spatially dependent field constant in time. As the atom moves in this field, the fast precession of the magnetic moment about the local field keeps its direction locked to the local field, whose direction varies in the lab frame. Returning to rapid adiabatic passage, since the generalized Rabi frequency is smallest and equal to the resonant Rabi frequency $\omega _{R}$ at 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} , the adiabatic requirement is most severe there, i.e. 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 \theta=\pi/2} . 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 \theta=\pi/2} we have, with $B_{z}=B_{0}-\omega (t)/\gamma$,

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| \dot\theta\right| =\frac{ \left| \dot{B}_z\right| }{B_1}=\frac{ \left| \dot\omega\right| }{\gamma B_1} =\frac{ \left| \dot\omega\right| }{\omega_R}\overset{!}{\ll}\omega_R }

where the exclamation point 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 \overset{!}{\ll}} indicates a requirement which we impose. Consequently, if the evolution is to be adiabatic, we must 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 \left| \dot{\omega}\right| \ll\omega_R^2} . This means that the change $\Delta \omega$ of rotation 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} per Rabi period 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=2\pi/\omega_R} , 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| \Delta\omega\right| = \left| \dot\omega\right| T= \left| \dot\omega\right| /\omega_R 2\pi} , must be small compared to the Rabi frequency $\omega _{R}$.

Note that the inversion of the spin is independent of whether 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} is swept up or down through the resonance.

Quantized Spin in a Magnetic Field

Equation of Motion for the Expectation Value

For the system we have been considering, the Hamiltonian 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 \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

${\frac {d}{dt}}{\hat {O}}={\frac {i}{\hbar }}\left[{\hat {H}},{\hat {O}}\right]+{\frac {\partial {\hat {O}}}{\partial t}}$

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 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 \epsilon_{klm}} , we have

${\begin{aligned}{\frac {d}{dt}}{\hat {\mu }}_{x}&=-{\frac {i\gamma ^{2}}{\hbar }}B_{0}i\hbar {\hat {L}}_{y}=\gamma ^{2}{\hat {L}}_{y}B_{0}={\hat {\mu }}_{y}\gamma B_{0}\\{\frac {d}{dt}}{\hat {\mu }}_{y}&=-\mu _{x}\gamma B_{0}\\{\frac {d}{dt}}{\hat {\mu }}_{z}&=0\end{aligned}}$

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 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_L=\gamma B_0} .

Note that:

• Just as in the classical model, these operator equations are exact; we have not neglected any higher order terms.

• Since the equations of motion hold for the operator, they must hold for the expectation value

$\langle {\dot {\bf {\hat {L}}}}\rangle =\gamma \langle {\bf {\hat {L}}}\rangle \times {\bf {B}}$

• 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 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} .

• 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.

• 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.

• 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.

• An even more interesting composite angular momentum arises when $N$ 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:

Level structure diagram 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 n} two-level atoms in a basis of symmetric states \cite{Dicke1954}. The leftmost column corresponds to an effective spin-$n/2$ object. Other columns correspond to manifolds of symmetric states of the $n$ atoms with lower total effective angular momentum.

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.

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.

{fig:two_level_spin_half} Equivalence of two-level system with spin-$1/2$. Note that for $\gamma >0$ the spin-up state (spin aligned with field) is the ground state. For an electron, 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} , 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 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} , such as a proton (Figure \ref{fig:two_level_spin_half}). For an electron, or any other spin with $\gamma <0$, 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 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 \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}:

{eqn:rabi_transition_probability}

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 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} 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.

{fig:rabi_signal} 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

${\begin{aligned}\left|e\right\rangle &=\left|S_{z}=-{\frac {1}{2}}\right\rangle ={\begin{pmatrix}1\\0\end{pmatrix}}\\\left|g\right\rangle &=\left|S_{z}=+{\frac {1}{2}}\right\rangle ={\begin{pmatrix}0\\1\end{pmatrix}}\end{aligned}}$

we can write the Hamiltonian 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} 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 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=\gamma B_0} is the Larmor frequency, and

${\hat {\sigma }}_{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$

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 } , 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 } 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 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} . The 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}} _1} , rotating at 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} in the $X-Y$ plane corresponds 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 \begin{align} \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{align}}

where have used the Pauli spin matrices 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 \sigma_x} , 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 \sigma_y} . The full Hamiltonian is thus given by

{eq:dressed_atom_hamiltonian}

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". However, note that in the treatment here there are no approximations; it is exact. 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 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} , 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 $H_{0}$. That means we write here

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 =a_e(t) \left|e\right\rangle e^{-i\omega_0t/2}+a_g(t) \left|g\right\rangle e^{i\omega_0t/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 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} , we have

${\begin{aligned}i{\dot {a}}_{g}&={\frac {\omega _{R}}{2}}e^{i\delta t}a_{e}\\i{\dot {a}}_{e}&={\frac {\omega _{R}}{2}}e^{-i\delta t}a_{g}\end{aligned}}$

The explicit time dependence can be eliminated by the sustitution

${\begin{aligned}b_{g}&=e^{-i\delta t/2}a_{g}\\b_{e}&=e^{i\delta t/2}a_{e}\end{aligned}}$

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

${\begin{aligned}b_{g}(t)&=e^{i\Omega _{R}t/2}B_{1}+e^{-i\Omega _{R}t/2}B_{2}\\b_{e}(t)&={\frac {\omega _{R}}{\delta -\Omega _{R}}}e^{i\Omega t/2}B_{1}+{\frac {\omega _{R}}{\delta +\Omega _{R}}}e^{-i\Omega _{R}t/2}B_{2}\end{aligned}}$

with two constants that are determined by the initial conditions. For $a_{g}(t=0)=1$ 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.

"Rapid" Adiabiatic Passage (quantum treatment)

(in the works)

The quantum mechanical treatment yields a probability for non-adiabatic transition (probability for the magnetic moment not following the magnetic field) 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 P_\text{na}=exp\left(-\frac{\pi}{2}\frac{\omega^2_R}{ \left| \dot\omega\right| }\right) }

in agreement with the qualitative classical criterion derived above.