User:Sivaprasad0811/sandbox/Vacuum Rabi Splitting

Vacuum Rabi splitting, is the splitting of the energy levels of the first excited state of an atom-cavity coupled system in the presence of the atoms when the system is in the strong coupling regime such that the interaction strength dominates the spontaneous decay. This has very interesting implications as it can be considered as a consequence of the manifestation of the quantum nature of the electromagnetic field, and can be considered as a simple consequence of the linear absorption and dispersion of the intracavity atoms.

Theory of Vacuum Rabi Splitting


The Jaynes-Cummings Hamiltonian is given as,
 * $$\hat{H}_{JC} = \hbar \omega_c \hat{a}^{\dagger}\hat{a}

+\hbar \omega_{eg} | e\rangle \langle e| +\hbar g \left(\hat{a}\hat{\sigma}_+ +\hat{a}^{\dagger}\hat{\sigma}_-\right),$$ where the operators $$\hat{a}^{\dagger}$$ and $$\hat{a} $$ are the creation and annihilation operators, and $$\hat{\sigma}_+ = |e \rangle \langle g |$$ and $$\hat{\sigma}_- = |g \rangle \langle e |$$ are the raising and lowering operators of the atom. $$\omega_c$$ is the angular frequency of the mode and $$g$$ gives the interaction strength between the atom and the cavity.

$$N_T=\hat{a}^{\dagger}\hat{a}+\hat{\sigma}_+\hat{\sigma}_-$$, which represents the total excitation in the system, is conserved and hence $$\hat{H}_{JC}$$ is block diagonlizable in the basis given by $$\{|e,n-1\rangle |g,n\rangle\}$$. The matrix elements of $$\hat{H}_{\text{JC}}$$ in this subspace read


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

\begin{pmatrix} n \omega_c -\Delta & g \sqrt{n} \\[8pt] g \sqrt{n} & n\omega_c \end{pmatrix}, $$

where $$\Delta=\omega_c-\omega_{eg}$$ and for a given $$n$$, the energy eigenvalues of $$H^{(n)}$$ are


 * $$E_{\pm}(n) = n\hbar\omega_c -\hbar \frac{\Delta}{2} \pm\frac{\hbar}{2}\sqrt{\Delta^2+n(2g)^2}.$$

For the case when $$\Delta=0$$ and $$n=1$$, which represents the first exited states of the system, the energy eigenvalues are given as,


 * $$E_{\pm}(1) = \hbar \omega_c\pm 2\hbar g,$$,

the eigenstates of which are given as $$|\pm\rangle=\frac{|e,0\rangle\pm|g,1\rangle}{\sqrt{2}}$$ and these levels are separated by gap of $$2g$$. This splitting in the energy of the eigenstates in the presence of interaction is the Vaccum Rabi Splitting. So if a laser photon is sent to the cavity with no atom the resonance is at frequency $$ \omega_c$$: however, if there is an atom present inside the cavity the spectrum is split by $$2g$$.

First Experimental observation of single-atom Vacuum Rabi Splitting
In the optical regime, the experimental confirmation of the single-atom vacuum Rabi splitting has been precluded by the smallish size of the coupling between the atom and the cavity; however, the single-atom vacuum Rabi splitting was first observed in the seminal paper. This was an experiment done in the strong coupling regime of the interaction such that the rate $$ \beta$$ for irreversible decay into the continuum degrees of freedom is much less than the frequency $$g$$ associated with the reversible evolution. In this regime, a single photon emitted by an atom into a cavity mode is likely to be repeatedly absorbed and reemitted before irreversibly escaping into the environment, which is the case in the weak coupling regime. Under these conditions instead of a single Lorentzian resonance, a double peak was observed. However, this simple analysis requires a lot of complicated experimental requirements, each atom probed a spatially varying standing wave pattern in the mode, resulting in a dispersion of vacuum Rabi frequencies also, the number of atoms coupled to the mode at a given time fluctuates around$$~1,~$$ leading to a partial blurring of the splitting. These effects were well-understood and a good agreement between the theoretical data in this work and hence the vacuum Rabi splitting was experimentally confirmed. The experiment provided even clearer evidence of the vacuum Rabi splitting using single atom in optical cavity.

Multi Atom vacuum Rabi Splitting
Even though single-atom vacuum Rabi splitting is a very tough phenomenon, it has been shown that cavity resonance splitting also occurs when many atoms are inserted into a cavity, and that the magnitude of the splitting increases with the square root of the number of atoms present. Multi-atom vacuum Rabi splitting was observed in Microwave Absorptionby Rydberg Atoms in a Cavity. The atoms interact with a single mode of the quantized radiation field in the vacuum state and the absorption spectra show a doublet structure. Such a doublet structure, in the case of exact resonance, corresponds to the vacuum-field Rabi oscillations.

Also, multi-atom vacuum Rabi splitting of a composite cavity and atom system was observed in the paper. They observe vacuum Rabi splitting for a composite cavity and atom system, this happens when the coupling field applied to the atoms which induce the resonant two-photon Raman transition with the cavity field in a $$\Lambda$$ type system. A cavity transmission spectrum is seen with two vacuum Rabi sidebands and a central peak representing the intracavity dark state. The central peak linewidth is significantly narrowed by the dark-state resonance and its position is not sensitive to the frequency change in the empty cavity.

Vacuum Rabi Splitting in Semiconductors as an indicator of Strong coupling Regime
Since the Vacuum Rabi splitting happens in the strong coupling regime, the ability to control the spontaneous emission of a quantum dot in a cavity makes it a good candidate to study Vacuum Rabi splitting. The observation of vacuum Rabi splitting indicates a strong coupling between a quantum dot and a superconducting coplanar waveguide (CPW) resonator. The decoherence of the system is dominated by the intrinsic piezoelectric effect and thus if a semiconductor system exhibits vacuum Rabi splitting it gives as a guideline to use semiconductor devices for the construction of solid-state hybrid devices for quantum information processing and quantum communication. For a detailed review of vacuum Rabi splitting in semiconductors and it's applications see

Cavity-free vacuum-Rabi splitting in circuit quantum acoustodynamics
Artificial atoms coupled to surface acoustic waves (SAWs) have played a crucial role in the recent development of circuit quantum acoustodynamics. The existence of a cavity regime circuit quantum acoustodynamics is expected due to the spatial extension of the interdigital transducer (IDT). For a good enough coupling strength, the atom has time to interchange an excitation with the part of the substrate covered by the IDT multiple times before it leaves the system at either ends. This coherent exchange leads to vacuum-Rabi splitting as the atom hybridizes with the phononic vacuum inside the IDT. Thus it is found that vacuum-Rabi splitting occurs for a surface acoustic waves (SAWs)-coupled atom as long as its decay rate exceeds the inverse phononic traveling time across the IDT, $$\gamma t\leq 1$$.