User:Sebhofer/sandbox

Jaynes-Cummings Hamiltonian
We consider a single optical (cavity) mode coupled to a two-level atom via electric dipole coupling in rotating wave approximation. The Hamiltonian for this system is given by


 * $$\hat{H}_{\text{JC}} = \hbar \omega \hat{a}^{\dagger}\hat{a}

+\hbar \omega_\mathrm{eg} \frac{\hat{\sigma}_z}{2} +\frac{\hbar \Omega}{2} \left(\hat{a}\hat{\sigma}_+ +\hat{a}^{\dagger}\hat{\sigma}_-\right).$$

where $$\omega_\mathrm{eg}$$ is the energy of the optical mode and $$\hat{a}^{\dagger}$$ and $$\hat{a} $$ are its creation and annihilation operators, fulfilling bosonic commutation relations $$[\hat{a},\hat{a}^\dagger]=1$$. The atom is described by the ground state $$|g\rangle$$ and the excited state $$|e\rangle$$ with a energy difference $$\omega_\mathrm{eg}$$. In terms of these states, the Pauli matrices are defined by $$\hat{\sigma}_\mathrm{z} = |e \rangle \langle e | - |g \rangle \langle g |$$, $$\hat{\sigma}_+ = |e \rangle \langle g |$$ and $$\hat{\sigma}_- = |g \rangle \langle e |$$, where $$\hat{\sigma}_+$$, $$\hat{\sigma}_-$$ are the atomic ladder operators. (Note that with the convention above, the ground state energy is $$-\omega_\mathrm{eg}$$ and the excited state energy is $$+\omega_\mathrm{eg}$$.)

The interaction term of the Hamiltonian has a straightforward interpretation: If the atom is in the ground state $$|g\rangle$$, it can absorb photons from the optical mode, thereby being excited to the excited state $$|e\rangle$$. If the photon is in the excited state, it may emit photons by going from the excited to the ground state.

We have set the zero field energy to zero for convenience.

For deriving the JCM interaction Hamiltonian the quantized radiation field is taken to consist of a single bosonic mode with the field operator $$\hat{E} = \hat{a} +\hat{a}^{\dagger}$$, where the operators $$\hat{a}^{\dagger}$$ and $$\hat{a} $$ are the bosonic creation and annihilation operators and $$\nu$$ is the angular frequency of the mode. On the other hand, the two-level atom is equivalent to a spin-half whose state can be described using a three-dimensional Bloch vector. (It should be understood that "two-level atom" here is not an actual atom with spin, but rather a generic two-level quantum system whose Hilbert space is isomorphic to a spin-half.) The atom is coupled to the field through its polarization operator $$\hat{S} = \hat{\sigma}_+ +\hat{\sigma}_-$$. The operators $$\hat{\sigma}_+ = |e \rangle \langle g |$$ and $$\hat{\sigma}_- = |g \rangle \langle e |$$ are the raising and lowering operators of the atom. The operator $$\hat{\sigma}_z = |e \rangle \langle e | - |g \rangle \langle g |$$ is the atomic inversion operator, and $$\omega$$ is the atomic transition frequency.

JCM Hamiltonian
Moving from the Schrödinger picture into the interaction picture (aka rotating frame) defined by the choice $$H_0 = \hat{H}_{\text{field}} +\hat{H}_{\text{atom}}$$, we obtain


 * $$\hat{H}_I(t) = \frac{\hbar \Omega}{2} \left(\hat{a}\hat{\sigma}_{-} e^{-i(\nu+\omega)t}

+\hat{a}^{\dagger}\hat{\sigma}_{+}e^{i(\nu+\omega)t} +\hat{a}\hat{\sigma}_{+}e^{i (-\nu+\omega) t} +\hat{a}^{\dagger}\hat{\sigma}_{-}e^{-i (-\nu+\omega) t}\right).$$

This Hamiltonian contains both quickly $$(\nu + \omega)$$ and slowly $$(\nu - \omega)$$ oscillating components. To get a solvable model, when $$|\nu - \omega| \ll \nu+\omega$$ the quickly oscillating "counter-rotating" terms can be ignored. This is referred to as the rotating wave approximation. Transforming back into the Schrödinger picture the JCM Hamiltonian is thus written as


 * $$\hat{H}_{\text{JC}} = \hbar \nu \hat{a}^{\dagger}\hat{a}

+\hbar \omega \frac{\hat{\sigma}_z}{2} +\frac{\hbar \Omega}{2} \left(\hat{a}\hat{\sigma}_+ +\hat{a}^{\dagger}\hat{\sigma}_-\right).$$

Eigenstates
It is possible, and often very helpful, to write the Hamiltonian of the full system as a sum of two commuting parts:


 * $$\hat{H}_\text{JC} = \hat{H}_I +\hat{H}_{II}$$

where



\begin{align} \hat{H}_{I} &= \hbar \nu \left(\hat{a}^{\dagger}\hat{a} +\frac{\hat{\sigma}_z}{2}\right)\\ \hat{H}_{II} &= \hbar \delta \frac{\hat{\sigma}_z}{2} +\frac{\hbar \Omega}{2} \left(\hat{a}\hat{\sigma}_+ +\hat{a}^{\dagger}\hat{\sigma}_-\right) \end{align} $$

with $$\delta = \omega - \nu$$ called the detuning (frequency) between the field and the two-level system.

The eigenstates of $$\hat{H}_{I}$$, being of tensor product form, are easily solved and denoted by $$|n,g\rangle, |n,e\rangle$$, where $$n \in \mathbb{N}$$ denotes the number of radiation quanta in the mode.

As the states $$|\psi_{1n}\rangle := |n,e\rangle$$ and $$|\psi_{2n}\rangle := |n+1,g\rangle$$ are degenerate with respect to $$\hat{H}_{I}$$ for all $$n$$, it is enough to diagonalize $$\hat{H}_{\text{JC}}$$ in the subspaces $$\text{span} \{ |\psi_{1n}\rangle ,|\psi_{2n}\rangle\}$$. The matrix elements of $$\hat{H}_{\text{JC}}$$ in this subspace, $${H}^{(n)}_{ij} := \langle\psi_{in}|\hat{H}_{\text{JC}}|\psi_{jn}\rangle,$$ read


 * $$H^{(n)} = \hbar

\begin{pmatrix} n \nu +\frac{\omega}{2} & \frac{\Omega}{2} \sqrt{n+1} \\[8pt] \frac{\Omega}{2} \sqrt{n+1} & (n+1)\nu -\frac{\omega}{2} \end{pmatrix} $$

For a given $$n$$, the energy eigenvalues of $$H^{(n)}$$ are


 * $$E_{\pm}(n) = \hbar\nu \left(n+\frac{1}{2}\right) \pm \frac{1}{2} \hbar\Omega_n(\delta),$$

where $$\Omega_n(\delta) = \sqrt{\delta^2 +\Omega^2(n+1)}$$ is the Rabi frequency for the specific detuning parameter. The eigenstates $$|n,\pm\rangle~$$ associated with the energy eigenvalues are given by


 * $$|n,+\rangle= \cos \left(\frac{\alpha_n}{2}\right)|\psi_{1n}\rangle+\sin \left(\frac{\alpha_n}{2}\right)|\psi_{2n}\rangle$$


 * $$|n,-\rangle= -\sin \left(\frac{\alpha_n}{2}\right)|\psi_{1n}\rangle+\cos \left(\frac{\alpha_n}{2}\right)|\psi_{2n}\rangle$$

where the angle $$\alpha_n$$ is defined through $$\alpha_n := \tan^{-1}\left(\frac{\Omega \sqrt{n+1}}{\delta}\right)$$

Schrödinger picture dynamics
It is now possible to obtain the dynamics of a general state by expanding it on to the noted eigenstates. We consider a superposition of number states as the initial state for the field, $$~|\psi_\text{field}(0)\rangle=\sum_n{C_n|n\rangle}~$$, and assume an atom in the excited state is injected into the field. The initial state of the system is


 * $$|\psi_\text{tot}(0)\rangle= \sum_n C_n \left[ \cos \left(\frac{\alpha_n}{2}\right)|n,+\rangle-\sin \left(\frac{\alpha_n}{2}\right)|n,-\rangle\right].$$

Since the $$~|n,\pm\rangle~$$ are stationary states of the field-atom system, then the state vector for times $$~t>0~$$ is just given by


 * $$|\psi_\text{tot}(t)\rangle = e^{i\hat{H}_{\text{JC}}t/\hbar}|\psi_\text{tot}(0)\rangle = \sum_n C_n \left[ \cos \left(\frac{\alpha_n}{2}\right)|n,+\rangle e^{iE_+(n)t/\hbar}- \sin \left(\frac{\alpha_n}{2}\right)|n,-\rangle e^{iE_-(n)t/\hbar}\right].$$

The Rabi oscillations can readily be seen in the sin and cos functions in the state vector. Different periods occur for different number states of photons. What is observed in experiment is the sum of many periodic functions that can be very widely oscillating and destructively sum to zero at some moment of time, but will be non-zero again at later moments. Finiteness of this moment results just from discreteness of the periodicity arguments. If the field amplitude were continuous, the revival would have never happened at finite time.

Heisenberg picture dynamics
It is possible in the Heisenberg notation to directly determine the unitary evolution operator from the Hamiltonian:
 * $$\begin{matrix}\begin{align}

\hat{U}(t) &= e^{-i\hat{H}_{\text{JC}}t/\hbar}\\ &= \begin{pmatrix} e^{- i \nu t (\hat{a}^{\dagger} \hat{a} + \frac{1}{2})}\left( \cos t \sqrt{\hat{\varphi} + g^2} - i \delta/2 \frac{\sin t \sqrt{\hat{\varphi} + g^2}}{\sqrt{\hat{\varphi} + g^2}}\right) & - i g e^{- i \nu t (\hat{a}^{\dagger} \hat{a} + \frac{1}{2})} \frac{\sin t \sqrt{\hat{\varphi} + g^2}}{\sqrt{\hat{\varphi} + g^2}} \,\hat{a} \\

-i g e^{- i \nu t (\hat{a}^{\dagger} \hat{a} - \frac{1}{2})}\frac{\sin t \sqrt{\hat{\varphi}}} {\sqrt{\hat{\varphi}}}\hat{a}^{\dagger} & e^{- i \nu t (\hat{a}^{\dagger} \hat{a} - \frac{1}{2})} \left( \cos t \sqrt{\hat{\varphi}} + i \delta/2 \frac{\sin t \sqrt{\hat{\varphi}}}{\sqrt{\hat{\varphi} }}\right) \end{pmatrix} \end{align}\end{matrix}$$

where the operator $$~\hat{\varphi}~$$ is defined as


 * $$ \hat{\varphi} = g^2 \hat{a}^{\dagger} \hat{a} + \delta^2/4 $$

The unitarity of $$~\hat{U}~$$ is guaranteed by the identities


 * $$\frac{\sin t\,\sqrt{\hat{\varphi} + g^2}}{\sqrt{\hat{\varphi} + g^2}}\; \hat{a} = \hat{a}\; \frac{\sin t\,\sqrt{\hat{\varphi}}}{\sqrt{\hat{\varphi}}} ,$$


 * $$\cos t\, \sqrt{\hat{\varphi} + g^2}\; \hat{a} = \hat{a}\; \cos t

\sqrt{\hat{\varphi}},$$

and their Hermitian conjugates.

By the unitary evolution operator one can calculate the time evolution of the state of the system described by its density matrix $$~\hat{\rho}(t)~$$, and from there the expectation value of any observable, given the initial state:


 * $$\hat{\rho}(t)=\hat{U}^{\dagger}(t)\hat{\rho}(0)\hat{U}(t)$$


 * $$\langle\hat{\Theta}\rangle_{t}=\text{Tr}[\hat{\rho}(t)\hat{\Theta}]$$

The initial state of the system is denoted by $$~\hat{\rho}(0)~$$ and $$~\hat{\Theta}~$$ is an operator denoting the observable.