Coherence

We speak of coherence if there exist well defined phases between two or more amplitudes that can interfere. These can be, e.g., the relative phase of the electric field in two arms of an interferometer, the relative phase of two or more states within one atom, or the relative phase of oscillating dipole moments in different atoms. Coherence is often a measurement tool in that the relative phase is the time integral over the energy difference between the states. For instance, if the atomic states have different magnetic moments, coherence between them provides a very sensitive measurement tool for magnetic field, or if the states have different spatial wavefunctions, the relative phase is a sensitive measure of the gravitational energy difference. In atomic clocks, coherence allows one to transform as energy difference between two internal atomic states into a frequency and time standard.

▼ 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

Coherence in two-level systems

A two-level atom that has been prepared by a ${\frac {\pi }{2}}$ pulse in a superposition ${\frac {1}{\sqrt {2}}}(|g\rangle +|e\rangle )$ can be viewed as exhibiting coherence, since the phase between $|e\rangle$ and $|g\rangle$ is well defined, and the system will evolve coherently 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 \frac{1}{\sqrt{2}}(|g\rangle+e^{-i\omega_ot}|e\rangle)} for times Failed to parse (MathML with SVG or PNG fallback (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\leq\frac{1}{\Gamma}} , 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 \Gamma} is the decay rate constant for the excited state. Figure \ref{fig:single-atom-coherence} displays a conceptual experiment that can be used to test this. \begin{figure} \centering

\caption{Measurement of definite phase for light emitted by a two-level atom prepared with a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/2} pulse. The laser light is sent through both arms of an interferometer; a switch in one arm selects a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/2} pulse of light from the laser with which to excite the atom. The light the atomic dipole emits is then mixed with the other interferometer arm at the output. Averaging the output signal over many repetitions of the experiment, the interferometer measures a definite phase for the light emitted by the atom, defined relative to the phase of the exciting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/2} pulse.}

\end{figure} What about an atom prepared by a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} pulse 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 |e\rangle} ? There is no coherence 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 t=0} , since the atom is in a single state, but what 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 t=\frac{1}{\frac{\Gamma}{2}}} ? Then the atom is in a superposition of states Failed to parse (MathML with SVG or PNG fallback (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|g\rangle+e^{-i\omega_0t+\phi_0}|e\rangle} . Obviously some phase must exist, because otherwise no dipole 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{d}} (t)=\langle e(t)|q {\bf{r}} |g(t)\rangle} exists that can emit (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int^{2\pi}_{0}d\phi\langle e|e^{i\phi}q {\bf{r}} |g\rangle=0} ), but the phase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_0} is completely unpredictable, so the experiment pictured above (or an emsemble average, as in Figure \ref{fig:ensemble-average-coherence}) would yield no definite phase. We conclude that an atom prepared 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 |e\rangle} does not exhibit coherence. \begin{figure} \centering

\caption{Ensemble average of the phase measurement.}

\end{figure} The ensemble average (parallel setups) or time average (repeated experiment at same location) yields no definite phase, so we conclude that the expectation value of the dipole 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 \langle d(t)\rangle=0} is zero at all times. What is the origin of uncertain phase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_o} ? Vacuum fluctuations. What then happens if we place two atoms close together and excite them at the same time? Is the relative phase of the evolving dipole moments fixed or uncertain? If the relative phase is fixed, how close must the atoms be for the relative phase to be well defined? These questions about spatial coherence and Dicke superradiance will be covered later in the chapter.

Precession of a spin in a magnetic field

Precession of a spin can be viewed as an effect of coherence since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x\rangle=\frac{1}{\sqrt{2}}(|+z\rangle+|-z\rangle)} . In 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 |\psi(t)\rangle=\frac{1}{\sqrt{2}}(e^{-i\omega_L\frac{t}{2}}|+z\rangle+e^{i\omega_L\frac{t}{2}}|-z\rangle)} , so the precession is due to a coherence between 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 |\pm z\rangle} components of the spin. If no coherence existed, the spin would be in a statistical mixture 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 |+z\rangle} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |-z\rangle} . In the density matrix formalism,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho=\left( \begin{array}{cc} \frac{1}{2} & \frac{1}{2}e^{-i\omega_Lt} \\ \frac{1}{2}e^{i\omega_Lt} & \frac{1}{2} \\ \end{array} \right) }

in the z basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |+z\rangle=\left(\begin{array}{cc}1\\0\end{array}\right), |-z\rangle=\left(\begin{array}{cc}0\\1\end{array}\right)} .\\The expectation value 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 \hat\sigma_x=\left(\begin{array}{cc}0&1\\1&0\end{array}\right)} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\hat\sigma_x\rangle= Tr [\rho\hat\sigma_x]= Tr \left(\begin{array}{cc}\frac{1}{2}e^{-i\omega_Lt}&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}e^{i\omega_Lt}\end{array}\right)=\cos\omega_Lt. }

If the coherences (off-diagonal elements 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 \hat\rho} ) were smaller, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat\sigma_x} would be smaller. For a statistical mixture 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 |+z\rangle} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |-z\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 \rho=\left(\begin{array}{cc}\frac{1}{2}&0\\0&\frac{1}{2}\end{array}\right)} 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 \langle\hat\sigma_x\rangle=0} .

The Stern-Gerlach experiment and spatial loss of coherence

\begin{figure} \centering

\caption{Stern-Gerlach experiment. Where in the magnet (or outside) does the projection onto Failed to parse (MathML with SVG or PNG fallback (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 z\rangle} occur?} \end{figure} In the Stern-Gerlach experiment a particle initially spin-polarized 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 |x\rangle} has equal probability of following either 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 |+z\rangle} trajectory or 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 |-z\rangle} trajectory. So initially the particle is described by a density matrix for a pure 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 \rho_{in}=\left(\begin{array}{cc}\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}\end{array}\right), }

after passing the Stern-Gerlach apparatus (inhomogeneous magnetic field) the density matrix 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 \rho_{out}=\left(\begin{array}{cc}\frac{1}{2}&0\\0&\frac{1}{2}\end{array}\right) }

with no intereference possible between the two states. Why? Because describing the full quantum state of the particle also requires acocunting for its spatial wavefunction. The density matrix above does not contain all the relevant degrees of freedom. Correctly, the particle should initially be 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 |\psi\rangle=|x\rangle\otimes|\psi_{spat}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}{c}\psi_{in}( {\bf{r}} )\\\psi_{in}( {\bf{r}} )\end{array}\right) }

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 \psi_{+z}( {\bf{r}} )=\psi_{-z}( {\bf{r}} )=\psi_{in}( {\bf{r}} )} , i.e. a spatial wavefunction independent of internal 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 \rho_{in}=\rho_{in,spin}\otimes \rho_{in, spatial}} . In the inhomogeneous magnetic field, the wavefuntion components Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{+z},\psi_{-z}} evolve differently because there is a different potential energy seen by the two spin states Failed to parse (MathML with SVG or PNG fallback (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 z\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 |\psi(t)\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}{c}\psi_{+z}( {\bf{r}} ,t)\\\psi_{-z}( {\bf{r}} ,t)\end{array}\right). }

\begin{figure} \centering

\caption{Spatial wave function and corresponding spin density matrix in the Stern-Gerlach experiment.} \end{figure} The coherence (interference) between $|+z\rangle$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |-z\rangle} components to form $|x\rangle$ exists only in the region where there is at least partial overlap between the two wavefunctions $\psi _{+z}({\bf {r}}),\psi _{-z}({\bf {r}})$. When the wavefunctions do not overlap, there is no significance to a relative phase between $|+z\rangle$ and $|-z\rangle$, i.e. no interference term. (Of course, if the wavefunctions are steered back to overlap, we can ask if there was a well-defined relative phase between them while they were separated.) In a more complete description, the inhomogeneous magnetic field entangles the spatial and spin degrees of freedom. When the spatial overlap disappears, or equivalently, when we trace over the spatial wavefunction (by measuring the particle either at location 1 or at location 2), the interference between $|+z\rangle$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |-z\rangle} giving rise 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 |x\rangle} disappears. In a measurement language, the inhomogeneous magnetic field entangles the "variable" (Failed to parse (MathML with SVG or PNG fallback (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 z\rangle} ) with the "meter" (the spatial wavefunctions of the particle). Once the spatial wavefunctions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{+z}( {\bf{r}} ),\psi_{-z}( {\bf{r}} )} cease to overlap, the particle's position can serve as a "meter" for the variable to be measured, the spin along $\pm z$. However, until the particle hits the screen, or is subjected to uncontrolled or unknown magnetic fields, the meter-variable entanglement is still reversible, and a "measurement" has not been made.

Quantum Beats

Quantum beats can be thought of as a two-level effect, though they are observed in multilevel atoms. They allow one to measure level spacings with high resolution when a narrowband excitation source (narrowband laser) is not available. \begin{figure} \centering

\caption{Multiple levels within energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar\Delta} from 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 \Delta>\Gamma,\Gamma'} ,\ldots}

\end{figure} Consider the scenario of Figure \ref{fig:qb-levels}, where we have multiple excited levels in a narrow energy interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar\Delta} , all decaying to a common ground state. If we excite with a pulse of duration Failed to parse (MathML with SVG or PNG fallback (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\ll\frac{1}{\Delta}} , we cannot resolve the levels, and they will be populated according to the coupling strength to the ground state $|g\rangle$ for the given excitation method:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi(0)\rangle=\Sigma c_i |e^{(i)}\rangle }

and for times Failed to parse (MathML with SVG or PNG fallback (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 \ll \frac{1}{\Gamma}, \frac{1}{\Gamma'}, \frac{1}{\Gamma''}, \ldots} , the state vector 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 |\psi(t)\rangle=\Sigma c_i e^{-i\omega_it}|e^{(i)}\rangle. }

It follows that, in directions where the radiation from the levels Failed to parse (MathML with SVG or PNG fallback (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^{(i)}\rangle} interferes, there will be oscillating terms at frequencies $\omega _{i}-\omega _{j}$ on top of the excited state decay. \begin{figure} \centering

\caption{Quantum beats}

\end{figure} This allows one to measure excited-state splittings in spite of the lack of a sufficiently narrow excitation source. Compared to our initial example of a two-level atom, here the coherence is initially purely between the excited states (definite excitation phase between them), and no coherence between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e^{(i)}\rangle} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |g\rangle} exists initially. Of course, as the atom decays, coherence between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e^{(i)}\rangle} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |g\rangle} (e.g. dipole moments) build up, and the coherence between the emitted fields of the different dipoles gives rise to the observed effect. Figures \ref{fig:qb-ideal} and \ref{fig:qb-data} show respectively an idealized quantum beat signal and real data from an experimental demonstration of the technique. \begin{figure} \centering

\caption{Schematic level diagrams and observed quantum beats of $^{4}He3^{3}P$ at 475 keV/atom; H, n=3, and H, n=4 at 133 keV/atom. \cite{Andra1970}}

Clarification on coherence and dipole moment

Consider the coherence of the atom after coherent excitation with a short pulse (shorter than emission rate). Let the state of the atom be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi> = C_g |g> + C_e |e>} . Then, the coherence between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e>} 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 |g>} is maximum 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 C_ e=C_ g=\frac{1}{\sqrt {2}}} , i.e. with ${\frac {\pi }{2}}$ pulse. (Coherence 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 \rho_{ab}} in the density matrix. For a pure state, it 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 C_g C_e} ). Now, let us consider a system with the atom and (external) EM mode. We consider the case where there is only a single EM mode coupled to the atom (ex, an atom strongly coupled to a cavity). Then, emission couples atomic states with photon number states: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |g,1>} 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 |e,0>} . Thus, a ${\frac {\pi }{2}}$ pulse also maximizes the coherence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |g,1>} 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 |e,0>} .

On the other hand, for continuous excitation (not a short pulse), saturation of the atom leads to emission of increasingly incoherent light (Mollow triplet). For Mollow triplet, see Cohen-Tannoudji p:424.

Next, consider a coherent light which is very weak. Monochromatic, coherent light is represented by a coherent stateFailed to parse (MathML with SVG or PNG fallback (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 >} that has a Poissonian distribution of photon numbers: Failed to parse (MathML with SVG or PNG fallback (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 >=e^{-\frac{1}{2}|\alpha|^{2}} \sum_{n = 0}^{\infty } \frac{ \alpha^{n}}{\sqrt{n+1}} |n>} . For $|\alpha |\ll 1$, the population of the states 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 n > 1} is negligible, and the atom prepared in a 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 |g>+\epsilon |e>} 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 |\epsilon | \ll 1} emits a coherent state of light, in agreement with what is expected for small saturation.

First observation of coherent population trapping CPT

Prepare a multimode laser with regular frequency spacing (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1, \cdots} , in the figure).

Prepare a gas in a cylindrical volume with gradient of magnetic field in z direction, and observe fluorescence.

Dark regions show the phaces where Zeeman shift between magnetic sublevels equals frequency difference between laser modes.

Absorption calculation by interference, goin without inversion

(Steve Harris, PRL 62, 1033 (1989)) http://prl.aps.org/abstract/PRL/v62/i9/p1033_1

It is commonly believed that we need $N_{e}>N_{g}$ for optical gain. But: Consider a V system with two unstable states that decay by coupling to the same continuum (This is a fairly special situation, e.g. different m-levels do not qualify, since they emit photons of different polarizations, thus the continue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |k_1>} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |k_2>} are distinguishable.)

Consider three level systems as in the figure 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 |e_1>} 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 |e_2>} decay to the continuum. A surprising feature of this system is the fact that there is a frequency at which the absorption rate becomes zero. To formally confirm this, we need to compute the second-order matrix element $M=\sum _{i=1,2}{\frac {<continuum|V|e_{i}><e_{i}|V|g>}{\Delta _{i}-i{\frac {\Gamma _{i}}{2}}}}$. Then, we know that there is a frequency at which this matrix element almost cancels. Let the frequency be $\omega _{o}$ that corresponds to an energy between the two levels. Note $\omega _{o}$ depends on the two matrix elements and we assume $(|\Delta _{1}|,|\Delta _{2}|\gg r_{1},r_{2}$.

One may understand this by considering the fact that the two-photon scattering process $|g>\rightarrow |continuum>$ can proceed via two pathways that are fundamentally indistinguishable. In other words, it is impossible to tell whether it came from $|e_{1}>$ or Failed to parse (MathML with SVG or PNG fallback (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_2>} . Thus, we need to consider quantum interference between them.

Now assume that with some mechanism we populate, say, Failed to parse (MathML with SVG or PNG fallback (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_2>} with a small number of atoms Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_2<N_ g} . These atoms have maximum stimulated emission probability on 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 |e_2>\rightarrow |g>} , but there is also even larger absorption, since $N_{g}>N_{2}$. However, because of the finite 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 r_2} of level Failed to parse (MathML with SVG or PNG fallback (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_2>} , there is also stimulated emission gain at the "magical" (absorption-free) 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 _ o} . Since 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_ g} atoms do not absorb here, there is net gain at this frequency in spite of $N_{2}<N_{g}$, which can lead to "lasing without inversion." Note: this only works if the two excited states decay to the same continuum, such that the paths are indistinguishable. How can a system for lasing without inversion be realized?

Possibility 1: hydrogen and dc electric field

Possibility 2: use ac electric field to mix non-degenerate s state with p state.

Electromagnetically induced transparency

"Is it possible to send a laser beam through a brick wall?"

Radio Yerevan: "In principle yes, but you need another very powerful laser..."

Steve Harris thought initially of special, ionizing excited states. However it is possible to realize the requirement of identical decay paths in a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} -system with a a(strong) coupling laser. The phenomenon is closely related to coherent population trapping.

For resonant fields Failed to parse (MathML with SVG or PNG fallback (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 _{ge}, \omega_2=\omega _{fe}} , we have

As we turn up the power of the coupling laser the transmission improves and then broadens (in the realistic case of a finite decoherence rate Failed to parse (MathML with SVG or PNG fallback (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_{gf}=0} , an infinitesimally small coupling Rabi frequency, but the frequency window over which transmission occurs is very narrow and 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 \Delta \omega =\Omega_c} .

Two-phase absorption, Fano profiles

Let us assume large one-photon detuning, $\Delta \gg \Gamma$, weak probe Failed to parse (MathML with SVG or PNG fallback (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} and strong control 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 _2} (we also define the two-photon 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 =\Delta _1-\Delta _2} ).

In this limit analytic expressions for the absorption cross section for beam Failed to parse (MathML with SVG or PNG fallback (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} and the refractive index seen by beam $\omega _{1}$ exist, e.g. [Muller et al., PRA 56, 2385 (1997)]

The refractive index 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 n(d)=1+\frac{3}{8\pi }\tilde{n}\lambda ^3f(\Delta )}

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 \tilde{n}} is the atomic density, Failed to parse (MathML with SVG or PNG fallback (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(\delta )=\frac{\Gamma}{2}\delta \frac{A}{A^2+B^2},A=\omega _2^2-\delta \Delta _2, B=\delta\Gamma} .

For zero ground-state linewidth (decoherence between the ground-states) Failed to parse (MathML with SVG or PNG fallback (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(\delta)=\sigma_{0}.g\left(\delta\right)} where $\sigma _{0}={\frac {3}{2\pi }}\lambda ^{2}$ is the resonant cross-section, 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 g(\Delta)=\frac{\Gamma}{2}\frac{\delta B}{A^2+B^2}} .

The absorption cross section 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 _2>0} :

This is like a ground state coupling to one narrow and one wide excited state, except that there is EIT in between because both states decay to the same continuum.

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=\Delta _{2}} , we have one-photon absorption, which is a two-photon scattering process:

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=-\frac{\omega _2^2}{\Delta _2}} , two-photon absorption, which is (at least) four-photon scattering process:

For the EIT condition $\delta =0$, there is no coupling to the excited state, and the refractive index is zero. In the vicinity of EIT, there is steep dispersion, resulting in a strong alteration of the group velocity of lightFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rightarrow } slowing and stopping light.

Slow light, adiabatic changes of velocity of light

The group velocity of light in the presence of linear dispersion Failed to parse (MathML with SVG or PNG fallback (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{dn}{d\omega }} is given by [Harris and Hau, PRL 82, 4611 (1999)]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_ g=\frac{c}{1+\omega \frac{dn}{d\omega }}} for light at frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega } .

A strong linear dispersion with positive slope near EIT then corresponds to very slow light.

As $n=1$, the electric field is unchanged so the power per area Failed to parse (MathML with SVG or PNG fallback (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{P}{A}=\frac{1}{2}\epsilon _ oC|E|^2} also remains unchanged.

Due to slowed group velocity, the pulse is compressed in the medium. As a consequence, the energy density is increased, and the light is partly in the form of an atomic excitation (coherent superposition of the ground states, although most of the energy is exchanged to the control field).

For sufficiently 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 \omega _2} , the velocity of light may be very small, down to a few m/s [L.V. Hau, S.E. Harris et al., Nature 397, 594 (1999)], as observed in a BEC. Reduced group velocity can also be observed in room-temperature experiment if the setup is Doppler free (co-propagating probe and control fields Failed to parse (MathML with SVG or PNG fallback (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} 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 _2} ) (otherwise only a small velocity class satisfies the two-photon resonance condition).

If we change control field adiabatically while the pulse is inside the medium, we can coherently stop light, i.e. convert it into an atomic excitation or spin wave. With the reverse process we can then convert the stored spin-wave back into the original light field. The adiabatic conversion is made possible by the finite splitting between bright and dark states. In principle, all coherence properties and other (qm) features of the light are maintained, and it is possible to store non-classical states of light by mapping photon properties one-to-one onto quantized spin waves. We will learn more about these quanta called "dark-state polaritons" once we have introduced Dicke states. Is it possible to make use of EIT for, e.g. atom detection without absorption? Answer: no improvement for such linear processes. However: improvement for non-linear processes is possible.

Superradiance

Assume that two identical atoms, one in its ground and the other in its excited state, are placed within a distance $d\gg \lambda _{eg}$ of each other. What happens?

For a single atom we have, for the emission rate Failed to parse (MathML with SVG or PNG fallback (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\left(t\right) } at time t 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 R\left(t\right)=\Gamma e^{-\Gamma t}} (product of the spontaneous emission rate and occupation probability). Thus, the emission probability to have emitted a photon by time t 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 p_{1}(t)=\int _0^ t R(t') dt'=1-e^{-\Gamma t}}

(eqtn:superradiance1)

What about two atoms? It turns out that the correct answer 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 R_2\left(t\right)=\Gamma e^{-2 \Gamma t}}

(eqtn:superradiance2)

Failed to parse (MathML with SVG or PNG fallback (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_2(t)=\int ^ t_0 R_2(t')dt'=\frac{1}{2}(1-e^{-2rt})}

(eqtn:superradiance3)

The photon is emitted with the same initial rate, but has only probability ${\frac {1}{2}}$ of being emitted at all! How can we understand this? The interaction Hamiltonian is (classically):

$V=-{\vec {d}}_{1}\cdot {\vec {E}}({\vec {\gamma }},t)-{\vec {d}}_{2}\cdot {\vec {E}}({\vec {\gamma }},t)=-{\vec {D}}\cdot {\vec {E}}({\vec {\gamma }},t)$

(eqtn:superradiance5)

In QED:

$V=\hbar g(\sigma _{1}^{+}+\sigma _{1}^{-})(a+a^{+})+\hbar g(\sigma _{2}^{+}+\sigma _{2}^{-})(a+a^{+})$ $=\hbar g(\sigma _{1}^{+}+\sigma _{2}^{+}+\sigma _{1}^{-}+\sigma _{2}^{-})(a+a^{+})=\hbar g(\Sigma ^{+}+\Sigma ^{-})(a+a^{+})$

(eqtn:superradiance6)

with $\Sigma ^{\pm }=\Sigma _{i=1}^{2}\sigma _{i}^{\pm },{\vec {D}}=\Sigma _{i=1}^{2}{\vec {d}}_{i}$