For ease of illustration, consider the interaction of two energy sub-levels of an atom with a quantized electromagnetic field. The behavior of any other two-state system coupled to a bosonic field will be
isomorphic to these dynamics. In that case, the
Hamiltonian for the atom-field system is: \hat{H} = \hat{H}_{A} + \hat{H}_F + \hat{H}_{AF} Where we have made the following definitions: • \hat{H}_A= E_g|g\rangle\langle g| +E_e|e\rangle\langle e| is the Hamiltonian of the atom, where the letters e, g are used to denote the excited and ground state respectively. Setting the zero of energy to the ground state energy of the atom simplifies this to \hat{H}_A= E_e|e\rangle\langle e|=\hbar \omega_{eg}|e\rangle \langle e| where \omega_{eg} is the resonance frequency of transitions between the sub-levels of the atom. • \hat{H}_F=\sum_{\mathbf{k},\lambda}\hbar\omega_{\mathbf{k}}\left(\hat{a}^{\dagger}_{\mathbf{k},\lambda}\hat{a}_{\mathbf{k},\lambda}+\frac{1}{2}\right) is the Hamiltonian of the quantized electromagnetic field. Note the infinite sum over all possible wave-vectors \mathbf{k} and two possible orthogonal polarization states \lambda. The operators \hat{a}^{\dagger}_{\mathbf{k},\lambda} and \hat{a}_{\mathbf{k},\lambda} are the photon creation and annihilation operators for each indexed mode of the field. The simplicity of the Jaynes–Cummings model comes from suppressing this general sum by considering only a
single mode of the field, allowing us to write \hat{H}_F = \hbar\omega_c\left(\hat{a}^{\dagger}_c \hat{a}_c + \frac{1}{2}\right) where the subscript c indicates that we are considering only the resonant mode of the cavity. • \hat{H}_{AF} =-\hat{\mathbf{d}}\cdot\hat{\mathbf{E}}(\mathbf{R}) is the dipole atom-field interaction Hamiltonian (here \mathbf{R} is the position of the atom).
Electric field operator of a quantized electromagnetic field is given by \hat{\mathbf{E}}(\mathbf{R})=i \sum_{\mathbf{k},\lambda}\sqrt{\frac{2\pi\hbar\omega_\mathbf{k}}{V}} \mathbf{u}_{\mathbf{k},\lambda} \left(\hat{a}_{\mathbf{k},\lambda}e^{i \mathbf{k}\cdot\mathbf{R}} -\hat{a}^\dagger_{\mathbf{k},\lambda}e^{-i \mathbf{k}\cdot\mathbf{R}}\right) and dipole operator is given by \hat{\mathbf{d}}=\hat{\sigma}_+\langle e| \hat{\mathbf{d}}|g\rangle +\hat{\sigma}_- \langle g| \hat{\mathbf{d}}|e\rangle. Setting \mathbf{R}=\mathbf{0} and making the definition \hbar g_{\mathbf{k},\lambda} = i\sqrt{\frac{2 \pi \hbar\omega_{\mathbf{k}}}{V}}\langle e| \hat{\mathbf{d}}|g\rangle\cdot\mathbf{u}_{\mathbf{k},\lambda}, where the \mathbf{u}_{\mathbf{k},\lambda} s are the orthonormal field modes, we may write \hat{H}_{AF} = -\sum_{\mathbf{k},\lambda}\hbar\left(g_{\mathbf{k},\lambda}\hat{\sigma}_+\hat{a}_{\mathbf{k},\lambda}-g^*_{\mathbf{k},\lambda}\hat{\sigma}_-\hat{a}^{\dagger}_{\mathbf{k},\lambda} -g_{\mathbf{k},\lambda}\hat{\sigma}_+\hat{a}^{\dagger}_{\mathbf{k},\lambda}+g^*_{\mathbf{k},\lambda}\hat{\sigma}_-\hat{a}_{\mathbf{k},\lambda}\right), where \hat{\sigma}_ +=|e\rangle\langle g| and \hat{\sigma}_-=|g\rangle\langle e| are the
raising and lowering operators acting in the \{|e\rangle,|g\rangle\} subspace of the atom. The application of the Jaynes–Cummings model allows suppression of this sum, and restrict the attention to a single mode of the field. Thus the atom-field Hamiltonian becomes: \hat{H}_{AF} = \hbar \left[\left(g_c \hat{\sigma}_+ \hat{a}_c - g_c^* \hat{\sigma}_- \hat{a}_c^{\dagger}\right) + \left(-g_c \hat{\sigma}_+ \hat{a}_c^{\dagger} + g_c^* \hat{\sigma}_- \hat{a}_c\right)\right].
Rotating frame and rotating-wave approximation Next, the analysis may be simplified by performing a
passive transformation into the so-called "co-rotating" frame. To do this, we use the
interaction picture. Take \hat{H}_0=\hat{H}_A+\hat{H}_F . Then the interaction Hamiltonian becomes: \hat{H}_{AF}(t)=e^{i\hat{H}_0t/\hbar}\hat{H}_{AF}e^{-i\hat{H}_0t/\hbar}=\hbar\left(g_c\hat{\sigma}_+\hat{a}_c^{\dagger}e^{i(\omega_c+\omega_{eg})t}+g_c^*\hat{\sigma}_-\hat{a}_ce^{-i(\omega_c+\omega_{eg})t}-g_c^*\hat{\sigma}_-\hat{a}_c^{\dagger}e^{-i(\omega_{eg}-\omega_c)t}-g_c\hat{\sigma}_+\hat{a}_ce^{i(\omega_{eg}-\omega_c)t}\right) We now assume that the resonance frequency of the cavity is near the transition frequency of the atom, that is, we assume |\omega_{eg}-\omega_c| \ll \omega_{eg}+\omega_c. Under this condition, the exponential terms oscillating at \omega_{eg} -\omega_c \simeq 0 are nearly resonant, while the other exponential terms oscillating at \omega_{eg}+\omega_c\simeq 2\omega_c are nearly anti-resonant. In the time \tau = \frac{2\pi}{\Delta}, \Delta \equiv \omega_{eg}-\omega_c that it takes for the resonant terms to complete one full oscillation, the anti-resonant terms will complete many full cycles. Since over each full cycle \frac{2 \pi}{2\omega_c} \ll \tau of anti-resonant oscillation, the net effect of the quickly oscillating anti-resonant terms tends to average to 0 for the timescales over which we wish to analyze resonant behavior. We may thus neglect the anti-resonant terms altogether, since their value is negligible compared to that of the nearly resonant terms. This approximation is known as the
rotating wave approximation, and it accords with the intuition that energy must be conserved. Then the interaction Hamiltonian (taking g_c to be real for simplicity) is: \hat{H}_{AF}(t)=-\hbar g_c \left(\hat{\sigma}_+\hat{a}_ce^{i(\omega_{eg}-\omega_c)t}+\hat{\sigma}_-\hat{a}_c^{\dagger}e^{-i(\omega_{eg}-\omega_c)t}\right) With this approximation in hand (and absorbing the negative sign into g_c ), we may transform back to the Schrödinger picture: \hat{H}_{AF}=e^{-i\hat{H}_0t/\hbar}\hat{H}_{AF}(t)e^{i\hat{H}_0t/\hbar} = \hbar g_c \left(\hat{\sigma}_+\hat{a}_c+\hat{\sigma}_-\hat{a}_c^{\dagger}\right)
Jaynes–Cummings Hamiltonian 2 Using the results gathered in the last two sections, we may now write down the full Jaynes–Cummings Hamiltonian: \begin{align} E_{n,\pm}&=\left(n\hbar\omega_c-\frac{1}{2}\hbar\Delta\right) \pm \frac{1}{2}\hbar\sqrt{\Delta^2+n\Omega^2}\\ \end{align} Where the angle \theta_n is defined by the relation \tan\theta_n=-\frac{\sqrt{n}\Omega}{\Delta} .
Vacuum Rabi oscillations Consider an atom entering the cavity initially in its excited state, while the cavity is initially in the
vacuum. Moreover, one assumes that the angular frequency of the mode can be approximated to the atomic transition frequency, involving \Delta \approx 0 . Then the state of the atom-field system as a function of time is: |\psi (t)\rangle = \cos\left(\frac{\Omega t}{2}\right)|e,0\rangle-i\sin\left(\frac{\Omega t}{2}\right)|g,1\rangle So the probabilities to find the system in the ground or excited states after interacting with the cavity for a time t are: \begin{align} P_e(t)&=|\langle e,0|\psi (t) \rangle |^2=\cos^2\left(\frac{\Omega t}{2}\right)\\ P_g(t)&=|\langle g,1|\psi (t) \rangle |^2=\sin^2\left(\frac{\Omega t}{2}\right)\\ \end{align} Thus the
probability amplitude to find the atom in either state oscillates. This is the quantum mechanical explanation for the phenomenon of
vacuum Rabi oscillation. In this case, there was only a single quantum in the atom-field system, carried in by the initially excited atom. In general, the Rabi oscillation associated with an atom-field system of n quanta will have frequency \Omega_n=\frac{\sqrt{n}\Omega}{2} . As explained below, this discrete spectrum of frequencies is the underlying reason for the collapses and subsequent revivals probabilities in the model.
Jaynes–Cummings ladder As shown in the previous subsection, if the initial state of the atom-cavity system is |e,n-1\rangle or |g,n\rangle , as is the case for an atom initially in a definite state (ground or excited) entering a cavity containing a known number of photons, then the state of the atom-cavity system at later times becomes a superposition of the
new eigenstates of the atom-cavity system: \begin{align} \end{align} This change in eigenstates due to the alteration of the Hamiltonian caused by the atom-field interaction is sometimes called "dressing" the atom, and the new eigenstates are referred to as the
dressed states. The energy difference between the dressed states is: \delta E=E_+-E_-=\hbar\sqrt{\Delta^2+n\Omega^2} Of particular interest is the case where the cavity frequency is perfectly resonant with the transition frequency of the atom, so \omega_{eg}=\omega_c\implies\Delta=0. In the resonant case, the dressed states are: |n,\pm \rangle = \frac{1}{\sqrt{2}}\left(|g,n \rangle\mp|e,n-1\rangle\right) With energy difference \delta E =\sqrt{n} \hbar\Omega . Thus the interaction of the atom with the field splits the
degeneracy of the states |e,n-1\rangle and |g,n\rangle by \sqrt{n} \hbar \Omega . This non-linear hierarchy of energy levels scaling as \sqrt{n} is known as the Jaynes–Cummings ladder. This non-linear splitting effect is purely quantum mechanical, and cannot be explained by any semi-classical model.
Collapse and revival of probabilities Consider an atom initially in the ground state interacting with a field mode initially prepared in a
coherent state, so the initial state of the atom-field system is: |\psi (0)\rangle = |g,\alpha \rangle = \sum_{n=0}^\infty e^{-|\alpha|^2/2}\frac{\alpha ^n}{\sqrt{n!}}|g,n\rangle For simplicity, take the resonant case ( \Delta = 0), then the Hamiltonian for the nth number subspace is: \hat{H}_n=\left(n+\frac{1}{2}\right)\hat{I}^{(n)}+\frac{\hbar\sqrt{n}\Omega}{2}\hat{\sigma}_x^{(n)} Using this, the time evolution of the atom-field system will be: \begin{align} &=e^{-|\alpha|^2/2}|g,0\rangle+\sum_{n=1}^\infty e^{-|\alpha|^2/2}\frac{\alpha^n}{\sqrt{n!}}e^{-in\omega_c t} \left(\cos{(\sqrt{n}\Omega t/2)}\hat{I}^{(n)}-i\sin{(\sqrt{n}\Omega t /2)}\hat{\sigma}_x^{(n)}\right)|g,n\rangle\\ &=e^{-|\alpha|^2/2}|g,0\rangle+\sum_{n=1}^\infty e^{-|\alpha|^2/2}\frac{\alpha^n}{\sqrt{n!}}e^{-in\omega_c t} \left(\cos{(\sqrt{n}\Omega t/2)}|g,n\rangle-i\sin{(\sqrt{n}\Omega t /2)}|e,n-1\rangle\right) \end{align} Note neither of the constant factors \frac{\hbar\omega_c}{2}\hat{I}^{(n)} nor \hat{H}_0 contribute to the dynamics beyond an overall phase, since they represent the zero-point energy. In this case, the probability to find the atom having flipped to the excited state at a later time t is: \begin{align} P_e(t) = \left|\langle e|\psi (t)\rangle \right|^2 &= \sum_{n=1}^\infty\frac{e^{-|\alpha|^2}}{n!}|\alpha|^{2n} \sin^2\left(\tfrac{1}{2} \sqrt{n} \Omega t\right) \\[2ex] &= \sum_{n=1}^\infty\frac{e^{-\langle n \rangle}\langle n \rangle^n}{n!} \sin^2\left(\tfrac{1}{2} \sqrt{n}\Omega t \right) \\[2ex] &= \sum_{n=1}^\infty\frac{e^{-\langle n \rangle}\langle n \rangle^n}{n!} \sin^2(\Omega_n t) \\{} \end{align} Where we have identified \langle n \rangle = |\alpha|^2 to be the mean photon number in a coherent state. If the mean photon number is large, then since the statistics of the coherent state are
Poissonian we have that the variance-to-mean ratio is \langle (\Delta n)^2\rangle /\langle n \rangle ^2 \simeq 1/\langle n \rangle . Using this result and expanding \Omega_n around \langle n \rangle to lowest non-vanishing order in n gives: \Omega_n\simeq\frac{\Omega}{2}\sqrt{\langle n \rangle}\left(1+\frac{1}{2}\frac{n-\langle n \rangle}{\langle n \rangle}\right) Inserting this into the sum yields a complicated product of exponentials: P_e(t)\simeq \frac{1}{2}-\frac{e^{-\langle n\rangle}}{4}\cdot\left(e^{-i\sqrt{\langle n \rangle }\Omega t/2} \exp\left[\langle n \rangle \exp\left(-\frac{i\Omega t}{2 \sqrt{\langle n \rangle}}\right)\right]+e^{i\sqrt{\langle n \rangle }\Omega t/2} \exp\left[\langle n \rangle \exp\left(\frac{i\Omega t}{2 \sqrt{\langle n \rangle}}\right)\right]\right) For "small" times such that \frac{\Omega t}{2} \ll \sqrt{\langle n \rangle} , the inner exponential inside the double exponential in the last term can be expanded up second order to obtain: P_e(t)\simeq \frac{1}{2}-\frac{1}{2}\cdot \cos\left[\sqrt{\langle n \rangle}\Omega t\right]e^{-\Omega^2 t^2/8} This result shows that the probability of occupation of the excited state
oscillates with effective frequency \Omega_{\text{eff}} = \sqrt{\langle n \rangle}\Omega . It also shows that it should decay over characteristic time: \tau_c=\frac{\sqrt{2}}{\Omega} The collapse can be easily understood as a consequence of destructive interference between the different frequency components as they de-phase and begin to destructively interfere over time. However, the fact that the frequencies have a discrete spectrum leads to another interesting result in the longer time regime; in that case, the periodic nature of the slowly varying double exponential predicts that there should also be a
revival of probability at time: \tau_r=\frac{4\pi}{\Omega}\sqrt{\langle n \rangle} . The revival of probability is due to the re-phasing of the various discrete frequencies. If the field were classical, the frequencies would have a continuous spectrum, and such re-phasing could never occur within a finite time. A plot of the probability to find an atom initially in the ground state to have transitioned to the excited state after interacting with a cavity prepared a in a coherent state vs. the unit-less parameter gt = \Omega t /2 is shown to the right. Note the initial collapse followed by the clear revival at longer times. == Collapses and revivals of quantum oscillations ==