Atoms and cavities

This section introduces methods for studying one two-level atom, interacting with a single mode of light. We begin with a brief derivation of the interaction Hamiltonian needed, known as the Jaynes-Cummings Hamiltonian, starting from quantum electrodynamics (QED). We then review the physics of a classically controlled spin. Studying the same scenario, but with a full quantum treatment based on the Jaynes-Cummings Hamiltonian then allows us to appreciate some of the richness of atom-photon interactions, and the limitations of semiclassical approximations, particularly in the context of cavity QED.

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 p} is the electron's 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 m} its mass, Failed to parse (MathML with SVG or PNG fallback (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} its charge; Failed to parse (MathML with SVG or PNG fallback (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{A}(\vec{r},t)} is the vector potential of the electromagnetic field at the position Failed to parse (MathML with SVG or PNG fallback (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{r}} of the electron; Failed to parse (MathML with SVG or PNG fallback (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(\vec{r},t)} is the scalar potential; Failed to parse (MathML with SVG or PNG fallback (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(r)} is the potential binding the electron to a certain position (eg as in an atom), 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 H_{\rm field}} is the free field Hamiltonian which we have previously modeled as being $\hbar \omega a^{\dagger }a$ . Recall that the electric and magnetic fields are related to the vector and scalar potentials through Failed to parse (MathML with SVG or PNG fallback (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{E}=-\nabla\phi - \partial_t \vec{A}} 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 \vec{B}=\nabla\times \vec{A}} , and that we may choose a gauge 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 \phi=0} and $\nabla \cdot {\vec {A}}=0$ (the Coulomb, or "radiation" gauge). Suppose the field is a plane wave, interacting with the atom binding the charge. Because the atom is typically much smaller than the wavelength Failed to parse (MathML with SVG or PNG fallback (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/k} of the field, we may approximate Failed to parse (MathML with SVG or PNG fallback (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{k}\cdot\vec{r}\ll 1} , 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 \vec{A}(\vec{r},t) \sim \vec{A}(\vec{r}_0,t)} , 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 \vec{r}_0} is the position of the atom. The Schr\"odinger equation for this 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 H |\psi \rangle = i\hbar \partial_t |\psi{\rangle} \,, }

is not immediately solvable, through direct exponentiation 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 H} , because Failed to parse (MathML with SVG or PNG fallback (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} is time varying (due to the field). Solution of this equation of motion may be accomplished by transforming into a moving frame of reference, in a manner which is useful for later reference. Specifically, we may define the moving frame 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 \rangle = \exp \left[ {\frac{ie}{\hbar}\left( \vec{A}(\vec{r}_0,t)\cdot \vec{r}\right)} \right] |\phi{\rangle} \,, }

motivated by the fact that $\exp \left[{{\frac {ie}{\hbar }}\left({\vec {A}}({\vec {r}}_{0},t)\cdot {\vec {r}}\right)}\right]$ is a unitary operator which shifts the momentum by amount Failed to parse (MathML with SVG or PNG fallback (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\vec{A}(\vec{r}_0,t)} ; this is precisely what is needed to remove the time varying field from Failed to parse (MathML with SVG or PNG fallback (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} .

In particular, after substitution and simplification, we find that the equation of motion 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 |\phi{\rangle}} is

i\hbar \partial _{t}|\phi \rangle =\left[{\left({\frac {p^{2}}{2m}}+V(r)+H_{\rm {field}}\right)-e{\vec {r}}\cdot {\vec {E}}}\right]|\phi {\rangle }\,,

where the first term in parentheses on the right is the free system 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} , 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\vec{r}\cdot\vec{E}} is interpreted as the dipole interaction Hamiltonian.

Let us focus on Failed to parse (MathML with SVG or PNG fallback (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_I} , in the case of a two-level atom. Note that ${\vec {r}}$ is an operator. For a two-level system, with energy 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{\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}} , it is usually the case 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 \langle r{\rangle}=0} for both of these eigenstates. $\langle r{\rangle }$ is nonzero for superpositions, such 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 (|g{\rangle}+|e \rangle )/\sqrt{2}} . Without loss of generality, we may thus let Failed to parse (MathML with SVG or PNG fallback (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{r} = d\hat{x} (|g \rangle \langle e| + |e \rangle \langle g|)} , or in terms of the Pauli 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 X} , $Y$ , 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} , we may 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 \vec{r}=\hat{x} dX} . Assuming the electric field is also 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 \hat{x}} direction, 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 \vec{E} = E \hat{x} (a+ a^\dagger )} , 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 H_I = dE (|g \rangle \langle e| + |e \rangle \langle g|)(a+ a^\dagger ) }

Of the four terms in this expression, 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 a|g \rangle \langle 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 a^\dagger |e \rangle \langle g|} terms involve removing and adding two quanta of energy (one photon and one atomic transition). When those two energies are nearly equal, those two interactions are much more unlikely to occur than 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 a|e \rangle \langle g|} and $a^{\dagger }|g\rangle \langle e|$ terms, which move quanta of energy between the field and atom, conserving energy. It is thus a reasonable to drop the two-quanta terms (the "rotating wave approximation"), leaving us with the interaction 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_I = g \left[ { a^\dagger \sigma^- + a \sigma^+ } \right] \,, }

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 \sigma^- = |g \rangle \langle e|} and $\sigma ^{+}=|e\rangle \langle g|$ . This is the Jaynes-Cummings interaction Hamiltonian, and will be the basis for all the following discussion, as well as much of the fields of quantum optics and atomic physics. It describes the interaction of one atom with a single mode of the electromagnetic field, with no decay mechanisms (in particular, no spontaneous emission), and no photon loss. Physically, you can think of the scenario governed as being an infinitely massive atom held fixed in the middle of a single mode optical cavity with perfect mirrors.

Classical control of a spin

We would now like to consider some of the physics of the Jaynes-Cummings interaction Hamiltonian, in the limit of a classical electromagnetic field. This will provide us with some intuition about how a two-level system behaves, in the absence of complication about the quantum nature of the field. It will also let us review some basic atomic physics, specifically the physics of resonance and two-state spins, using the language which will later be employed in our study of the optical Bloch equations.

When the electromagnetic field is a strong coherent state $|\alpha {\rangle }$ with $\alpha \gg 1$ , we may approximate 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 a|\alpha \rangle \sim \alpha|\alpha{\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 a^\dagger |\alpha \rangle \sim \alpha^*|\alpha{\rangle}} , so 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 \alpha = \alpha_0 e^{i\omega t}} . This gives us an atom-field Hamiltonian (letting Failed to parse (MathML with SVG or PNG fallback (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=1} ):

Failed to parse (MathML with SVG or PNG fallback (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 = \frac{\omega_0}{2} Z + g_0\alpha_0 \left[ { e^{-i\omega t} \sigma^- + e^{i\omega t} \sigma^+ } \right] \,, }

where the first term is the free Hamiltonian of the atom, with transition 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 2\pi\times \omega_0} , and the field has 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 2\pi\times \omega} . Letting Failed to parse (MathML with SVG or PNG fallback (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_0\alpha_0} (this turns out to be half the Rabi frequency), and rewriting the atomic raising and lowering operators with Pauli operators, we find that

H={\frac {\omega _{0}}{2}}Z+g(X\cos \omega t+Y\sin \omega t)\,,

Define Failed to parse (MathML with SVG or PNG fallback (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(t) \rangle = e^{i\omega t Z/2} |\chi(t){\rangle}} , such that the Schr\"odinger 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 \partial_t |\chi(t) \rangle = H |\chi(t){\rangle} }

can be re-expressed 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 i \partial_t |\phi(t){\rangle} = \left[ { e^{i\omega Z t/2} H e^{-i\omega Z t/2} - \frac{\omega}{2} Z } \right] |\phi(t){\rangle} \,. }

(7.1.1)

Since

e^{i\omega Zt/2}Xe^{-i\omega Zt/2}=(X\cos \omega t-Y\sin \omega t)\,,

Eq.(7.1.1) simplifies to become

Failed to parse (MathML with SVG or PNG fallback (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 \partial_t |\phi(t){\rangle} = \left[ { \frac{\omega_0 - \omega}{2} Z + g X } \right] |\phi(t){\rangle} \,, }

where the terms on the right multiplying the state can be identified as the effective `rotating frame' Hamiltonian. The solution to this equation 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 |\phi(t) \rangle = e^{i \left[ { \frac{\omega_0 - \omega}{2} Z + g X } \right] t} |\phi(0){\rangle} \,. }

The concept of resonance arises from the behavior of this time evolution, which can be understood as being a single qubit rotation about the axis

Failed to parse (MathML with SVG or PNG fallback (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 n = \frac{\hat z + \frac{2g}{\omega_0 - \omega} \,\hat x} {\sqrt{1+ \left( {\frac{2g}{\omega_0-\omega}} \right) ^2}} }

by an angle

|{\vec {n}}|=t{\sqrt {\left({\frac {\omega _{0}-\omega }{2}}\right)^{2}+g^{2}}}\,.

When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} is far from Failed to parse (MathML with SVG or PNG fallback (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 qubit is negligibly affected by the laser field; the axis of its rotation is nearly parallel with ${\hat {z}}$ , and its time evolution is nearly exactly that of the free atom Hamiltonian. On the other hand, when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0\approx \omega} , the free atom contribution becomes negligible, and a small laser field can cause large changes in the state, corresponding to rotations about 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 \hat x} axis. The enormous effect a small field can have on the atom, when tuned to the appropriate frequency, is responsible for the concept of atomic `resonance,' as well as nuclear magnetic resonance. Let Failed to parse (MathML with SVG or PNG fallback (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-\omega} be the detuning between atom and field. 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} , the on-resonance case, the coherent field causes a rotation of the atomic state 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 \exp(igt X)} , such that for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gt=\pi/4} we have 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 90^\circ} rotation of the spin about 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 \hat{x}} axis. In the limit of large $\delta$ , the far off-resonance case, the spin is rotated 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 \exp(i\delta t Z)} . The leading correction to it, 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 g^2/\delta } , is the AC Stark shift.

These spin dynamics are widely observed, but nevertheless, still just an approximation. When the 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 |\alpha{\rangle}} is weak, then the original assumptions made, specifically that $a^{\dagger }|\alpha \rangle \sim a^{*}|\alpha {\rangle }$ , are no longer good. For example, when the mean photon number in the 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 |\alpha|^2} is, 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 16} , the true dynamics of the system are far from the semiclassical NMR-like picture given here.

Jaynes-Cummings Hamiltonian

The full Jaynes-Cummings Hamiltonian, describing the quantum evolution of a single two-level atom with a single mode electromagnetic field, 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 H = \frac{\omega_0}{2} Z + \omega a^\dagger a + g \left[ { a^\dagger \sigma^- + a \sigma^+ } \right] \,, }

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 2\pi\times \omega_0} is the transition frequency of the atom, and the field has frequency $2\pi \times \omega$ . One of the most important facts about this Hamiltonian is that it is fully solvable.

Here, we provide a solution in the interaction picture, obtained at zero 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_0-\omega = 0} , in the frame of reference of bare Hamiltonians of the atom and field. The Hamiltonian in this frame is simply the Jaynes-Cummings interaction 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_I = g \left[ { a^\dagger \sigma^- + a \sigma^+ } \right] \,, }

which is easily exponentiated using the fact that for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^+ = |e \rangle \langle g|} 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 \sigma^- = |g \rangle \langle e|} ,

{\begin{array}{rcl}\sigma ^{+}\sigma ^{-}&=&|e\rangle \langle e|\\\sigma ^{-}\sigma ^{+}&=&|g\rangle \langle g|\,.\end{array}}

From this, it follows 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 \begin{array}{rcl} \left[ { a^\dagger \sigma^- + a \sigma^+ } \right] ^{2k} &=& ( a^\dagger a)^k |e \rangle \langle e| + ( a^\dagger a)^k |g \rangle \langle g| \\ \left[ { a^\dagger \sigma^- + a \sigma^+ } \right] ^{2k+1} &=& (a a^\dagger )^k a |e \rangle \langle g| + a^\dagger (a a^\dagger )^k |g \rangle \langle e| \,. \end{array}}

Thus, letting Failed to parse (MathML with SVG or PNG fallback (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= a^\dagger a} , we find for the time evolution 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 U}

{\begin{array}{rcl}U&=&e^{-iHt}\\&=&\sum _{k}{\frac {(-iHt)^{k}}{k!}}\\&=&\cos(gt{\sqrt {n+1}})|e\rangle \langle e|+\cos(gt{\sqrt {n}})|g\rangle \langle g|-i{\frac {\sin(gt{\sqrt {n+1}})}{\sqrt {n+1}}}a|e\rangle \langle g|-ia^{\dagger }{\frac {\sin(gt{\sqrt {n+1}})}{\sqrt {n+1}}}|g\rangle \langle e|\,.\end{array}}

An arbitrary state of the atom and field can be written as

|\psi \rangle =\sum _{n}\alpha _{n}|e,n\rangle +\beta _{n}|g,n{\rangle }\,.

so that the state at time $t$ is given by $|\psi (t)\rangle =U(t)|\psi {\rangle }$ .

There are many other ways to solve the Jaynes-Cummings interaction, with $\delta =0$ or even otherwise. The approach given here is sufficient for our goal, to explore some of the non-classical behavior of a single atom with a single mode field.

Cavity QED

Two of the most important features of a single atom interacting with a single mode electromagnetic field, in the absence of decay and loss, may be obtained from the above solution of the Jaynes-Cummings Hamiltonian. In particular, we find that an initial state with the atom being in $|e{\rangle }$ , and the field being arbitrary evolves to become

{\begin{array}{rcl}U(t)\sum _{n}\alpha _{n}^{0}|e,n{\rangle }&=&\sum _{n}\alpha _{n}^{0}\left[{\cos(gt{\sqrt {n+1}})|e,n{\rangle }-i\sin(gt{\sqrt {n+1}})|g,n+1\rangle }\right]\\&=&\sum _{n}\left[{\alpha _{n}(t)|e,n{\rangle }+\beta _{n}(t)|g,n+1\rangle }\right]\,,\end{array}}

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 \begin{array}{rcl} \alpha_n(t) &=& \alpha^0_n \cos(gt\sqrt{n+1}) \\ \beta_{n+1}(t) &=& - i \alpha^0_n\sin(gt\sqrt{n+1}) \,. \end{array}}

Let Failed to parse (MathML with SVG or PNG fallback (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(t) = \sum_n |\alpha_n(t)|^2 - |\beta_n(t)|^2} be the polarization of the atom. Defining $\Omega _{n}^{2}=\delta ^{2}+4g^{2}(n+1)$ , one can show that at finite 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_0-\omega} , this polarization 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(t) = \sum_n \alpha_n^2 \left[ { \frac{\delta^2}{\Omega_n^2} + \frac{4g^2(n+1)}{\Omega_n^2} \cos(\Omega_n t) } \right] \,. }

Vacuum Rabi Oscillations

Suppose initially there are no photons, so only Failed to parse (MathML with SVG or PNG fallback (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_0=1} . Then

P(t)={\frac {\delta ^{2}+4g^{2}\cos(\Omega _{0}t)}{\Omega _{0}^{2}}}\,,

meaning that the atom in its ground or excited states is not in a stationary state. Specifically, the state of the system oscillates 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,0{\rangle}} , an excited atom with no photon in the cavity, 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,1{\rangle}} , a ground state atom with a single photon in the cavity. The frequency of this oscillation 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} is $2g$ , a quantity known as the vacuum Rabi splitting, and the oscillations are known as vacuum Rabi oscillations. Such oscillations have been observed in a wide variety of experimental systems, including solid state devices.

Collapse and Revival

Finally, let us return to the approximation made in studying the classical control of the two-level atom. Our solution of the Jaynes-Cummings Hamiltonian allows us to now compute what happens when the control field is a coherent state, but instead of being a strong, it has few photons. At zero 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 P(t) = \sum_n \alpha_n^2 \cos(\Omega_n t) \,. }

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_n = 2g\sqrt{n+1}} may be interpreted as being the Rabi frequency induced 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} photons.

For a strong coherent state, the photon number distribution Failed to parse (MathML with SVG or PNG fallback (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_n^2} is strongly peaked about $|\alpha |^{2}$ , with a width 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 |\alpha|} , so that the width is much smaller than the mean for 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 \alpha} .

For 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 \alpha} , however, the fields oscillating at different frequencies can interfere with each other, causing the net atomic polarization to decay, in sharp contrast to the continuous rotations expected in the semiclassical picture. Moreover, because of the discreteness of the number of oscillating frequencies, there can be Poincare recurrences in the polarization. Here is a plot of the case when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_n} is the photon distribution for a $|\alpha |^{2}=16$ coherent state:

<jwplayer width="560" height="440" repeat="true" displayheight="420" image="http://cua.mit.edu/8.422/HANDOUTS/jcr1.png" autostart="false">http://feynman.mit.edu/8.422/jcrevivals1b.flv</jwplayer>

References

<refbase>5242</refbase>
<refbase>5241</refbase>
<refbase>5238</refbase>

Atoms and cavities

Contents

The QED Hamiltonian

Classical control of a spin

Jaynes-Cummings Hamiltonian

Cavity QED

Vacuum Rabi Oscillations

Collapse and Revival

References

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