Tmp Lecture 26

Lecture XXVI

Superradiance, continued

Now we can write the initial state as:

$|ge,0>={\frac {1}{2}}\underbrace {(|ge,0>+|eg,0>} _{{\sqrt {2}}|\mathrm {superradiant} >}+\underbrace {|ge,0>-|eg,0>)} _{{\sqrt {2}}|\mathrm {subradiant} >}$

where

$|\mathrm {superradiant} >={\frac {1}{\sqrt {2}}}(|ge,0>+|eg,0>)$

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 |\mathrm{subradiant}>=\frac{1}{\sqrt {2}}(|ge,0>-|eg,0>)}

and

$<gg,1|V|\mathrm {superradiant} >={\sqrt {2}}\hbar g;<gg,1|V|\mathrm {subradiant} >=0$

The initial state has a 50% probability to be the sub-radiant state, hence the system has a 50% probability of not decaying. The set of four 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 |gg,0>,|ge,0>,|eg,0>,|ee,0>} can be organized into a triplet and a singlet:

The state $|ge>-|eg>$ is "dark" in that it does not decay under the action of the Hamiltonian V. The matrix elements between the states, indicated by arrows, are expressed in units of the single atom coupling $g={\frac {1}{\hbar }}<e,0|V|g,1>$.

Just as we can identify the two-level system Failed to parse (MathML with SVG or PNG fallback (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>,|g>} , with a (pseudo)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 \frac{1}{2}} , we can identify the triplet and singlet 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 L=0,1} , and write

Failed to parse (MathML with SVG or PNG fallback (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=1,M=1>=|ee>}

Failed to parse (MathML with SVG or PNG fallback (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=1,M=0>=(|eg>+|ge>)/\sqrt{2}}

$|L=1,M=-1>=|gg>$

Failed to parse (MathML with SVG or PNG fallback (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=0,M=0>=(|ge>-|eg>)/\sqrt{2}}

Since V conserves parity (exchange of the two atoms), there is no coupling between singlet and triplet states. (This is no longer true when we consider spatially extended samples).

Supperradiance in N atoms

It is not difficult to generalize the formalism to more 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 V=-\vec{D}\cdot \vec{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 \vec{D}=\Sigma^N_{i=1}\vec{d}_ i}

The Dicke states, equivalent to the states obtained by summing N spin ${\frac {1}{2}}$ particles, are

Let us look at the leftmost (symmetric) ladder. Near the middle of the Dicke-ladder, Failed to parse (MathML with SVG or PNG fallback (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\simeq 0} , the matrix element 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 <V> \simeq \sqrt {\frac{N}{2}(\frac{N}{2}-1)}g\simeq N g. } The emission rate is proportional 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 |<V>|^2\simeq N^2g^2} , i.e. the rate is quadratic in atom number.

Classically, that is not too surprising: we have N dipoles oscillating in phase, which corresponds to a dipole $D=Nd$, the emission is proportional 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 D^2=N^2d^2} . However, the Dicke states 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 <D>=0} and nevertheless macroscopic emission. How do we see this? In the Bloch sphere, for the angular momentum representation the coherent 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 |L=N/2,M=-N/2>=|g...g>} corresponds to all atoms in the ground state.

A field that symmetrically couples to all atoms (e.g. ${\frac {\pi }{2}},\pi$ pulse) acts only within the completely symmetric Hilbert space Failed to parse (MathML with SVG or PNG fallback (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} . This space consists of states like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\cos \theta |e>+\sin \theta e^{i\phi }|g>)^ N} , corresponding to rotations of the 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...g>} around some axis on the Bloch sphere.

The states obtained by rotations of the state $|g...g>$ by symmetric operations that act on all individual atoms independently, i.e. of the form Failed to parse (MathML with SVG or PNG fallback (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>\rightarrow \cos \theta |e>+e^{i\phi }\sin \theta |g>} , are called coherent spin states (CSS). They are represented by a vector on the Bloch sphere with uncertainties Failed to parse (MathML with SVG or PNG fallback (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 {\frac{L}{2}}} in directions perpendicular to the Bloch vector.

If we prepare a system in the CSS corresponding to a slight angle away form Failed to parse (MathML with SVG or PNG fallback (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 =0} near $|e...e>=|L=N/2,M=N/2>$, then classically it will obey the eqs of motion of an inverted pendulum, and fall down along the Bloch sphere. (This can be shown using the classical analogy with a field.)

So what happens if we prepare the 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 |M=+L>=|e...e>} ? Does it:

a. evolve down along the Dicke ladder maintaining Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <D>=0} (but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <D^2>\neq 0} )?

b. fall like a Bloch vector along some angle chosen by vacuum fluctuations?

Answer: there is no way of telling unless you prepare a specific experiment. If we detect (with unity quantum efficiency) the emitted photons, then each detection projects the system one step down along the Dicke ladder, 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 <D>=0} .

If we measure the phase of the emitted light, say with some heterodyne technique, then we find that the system evolves as a Bloch state.

Dicke states of extended samples

Consider an elongated atomic sample

such that a preferential mode (along x) is defined. Then we can define Dicke states with respect to that mode 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 |L,M=-L>=|g...g>}

Failed to parse (MathML with SVG or PNG fallback (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,M=-L+1>=\frac{1}{\sqrt {N}}(e^{i\kappa x_1}|eg...g>+e^{ihx_2}|geg...>+...+e^{ihx_N}|g...ge>)}

Failed to parse (MathML with SVG or PNG fallback (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,M=-L+2>=\frac{\sqrt {2}}{\sqrt {N(N-1)}}|e^{i\kappa x_1}e^{i\kappa x_ 2}|eeg...g>+e^{i\kappa x_1}e^{i\kappa x_ 2}|egeg...>+...}

etc.

Then one can easily see that the phase factors are such that the interaction Hamiltonian

$V=\Sigma _{i}{\vec {d_{i}}}\cdot {\vec {E}}({\vec {x_{i}}})=\Sigma _{i}{\vec {d}}\cdot {\vec {E}}_{0}e^{i\kappa x_{i}}$

is such that the Dicke ladder has the same couplings as before, i.e. superradiance occurs. However, emission along a direction other than the preferred mode now leads to diagonal couplings between the Dicke ladders , since emission along some other direction with 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 e^{i\vec{k}\cdot \vec{r}}} does not preserve the symmetry of the state with respect to permutations of the atoms. However, if the atom number along the preferred direction is large enough, superradiance still occurs. The condition for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\ll\lambda^2} 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 N\geq 2} , but for $A>\lambda ^{2}$ the condition is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{N\lambda ^2}{A}>1} . This is exactly the condition for sufficient optical gain in an inverted system for optical amplification (lasing) to occur, 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 \lambda ^2} is the stimulated emission cross section for an atom in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e>} .

Observation in a BEC, in multimode optical cavities.

Oscillating and overdamped regimes of superradiance

The photon leaves the sample 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 \frac{L}{c}} . 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 \sqrt{N}g<\frac{c}{L}} , then the damping is faster than Rabi flopping, and we are in the rate equation limit where the emission proceeds 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{(\sqrt {N}g)^2}{c/L}=\frac{Ng^2L}{c}} , rather than as emission by independent atoms that would decay 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{g^2}{c/L}} . 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 \sqrt {N}g>\frac{c}{L}} , then Rabi flopping occurs during the decay.

Note:Failed to parse (MathML with SVG or PNG fallback (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 = g^2L/C} .

Raman Superradiance

Image

In the limit of large Failed to parse (MathML with SVG or PNG fallback (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 } and low saturation Failed to parse (MathML with SVG or PNG fallback (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\ll \Delta } , we can eliminate the excited state and have an effective system.

Image

$r_{sc}={\frac {\omega _{1}^{2}}{\Delta ^{2}}}{\frac {r}{2}}$

We can now adjust the 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_ sc} via Failed to parse (MathML with SVG or PNG fallback (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 also make the 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 .|\tilde{e}>} suddenly stable by turning off Failed to parse (MathML with SVG or PNG fallback (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} . In fact we can switch ground and excited states by applying a laser beam on the other Raman leg instead.

Storing light, catching photons

Image

When we consider a quantized field on the $|g>\rightarrow |e>$ transition, there is a family of dark states, corresponding 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 n=0,1,2,...} excitations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |D,n>=\Sigma ^ n_{n=0}\sqrt {\frac{n!}{n!(n-\kappa )!}}\frac{(-g)^{\kappa }N^{\frac{\kappa }{2}}\Omega ^{n-\kappa }}{(Ng^2+\Omega ^2)^{\frac{n}{2}}}|n-\kappa >_{photons}|C=\frac{N}{2},M=-L+\kappa >}

(Lukin, Yelin, and Fleischhauer PRL <underline> <attributes> </attributes> 84 </underline>, 4233 (2000)) As $\Omega \gg Ng^{2}$, these states are purely photonic, 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 \omega \ll Ng^2} , these states are purely atomic excitations.

|D,n> Failed to parse (MathML with SVG or PNG fallback (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 \gg Ng^2|n>_{photons}|L,M=-L>_{atoms}} $\Omega \ll Ng^{2}|0>_{photons}|L,M=-L+n>_{atoms}$

In general, these excitations n=0,1,2,... are called dark state polarizations, they are a mixture of photonic excitations and spin-wave excitations.

By adiabatically changing Failed to parse (MathML with SVG or PNG fallback (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 \rightarrow 0} after the pulse has entered the medium, we can map any photonic 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 |\psi >_{photon}=\Sigma ci|i>_{photon}} onto a spin wave, store it and map it back onto a light-field by turning on the coupling laser $\Omega$ again.