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arXiv:quant-ph/0403001 · Physical Review A

Large two-atom two-photon vacuum Rabi oscillations in a high-quality cavity

Pradyumna K. Pathak, G. S. Agarwal

Published Oct 11, 2004• 8 citations
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Abstract

We predict a large cooperative effect involving two-atom two-photon vacuum Rabi oscillations in a high-quality cavity. The two-photon emission occurs as a result of simultaneous deexcitation of both atoms with two-photon resonance condition ${\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}\ensuremath{\approx}{\ensuremath{\omega}}_{a}+{\ensuremath{\omega}}_{b}$, where ${\ensuremath{\omega}}_{1}$,${\ensuremath{\omega}}_{2}$ are the atomic transition frequencies and ${\ensuremath{\omega}}_{a}$,${\ensuremath{\omega}}_{b}$ are the frequencies of the emitted photons. The actual resonance condition depends on the vacuum Rabi couplings. The effect can be realized either with identical atoms in a bimodal cavity or with nonidentical atoms in a single-mode cavity.

DOI
10.1103/physreva.70.043807

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We predict a large cooperative effect involving two-atom two-photon vacuum Rabi oscillations in a high-quality cavity. The two-photon emission occurs as a result of simultaneous deexcitation of both atoms with two-photon resonance condition \ensuremathω1+\ensuremathω2\ensuremath≈\ensuremathωa+\ensuremathωb{\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}\ensuremath{\approx}{\ensuremath{\omega}}_{a}+{\ensuremath{\omega}}_{b}\ensuremathω1​+\ensuremathω2​\ensuremath≈\ensuremathωa​+\ensuremathωb​, where \ensuremathω1{\ensuremath{\omega}}_{1}\ensuremathω1​,\ensuremathω2{\ensuremath{\omega}}_{2}\ensuremathω2​ are the atomic transition frequencies and \ensuremathωa{\ensuremath{\omega}}_{a}\ensuremathωa​,\ensuremathωb{\ensuremath{\omega}}_{b}\ensuremathωb​ are the frequencies of the emitted photons. The actual resonance condition depends on the vacuum Rabi couplings. The effect can be realized either with identical atoms in a bimodal cavity or with nonidentical atoms in a single-mode cavity.
When an atom is strongly coupled to a cavity, the two systems can exchange a single photon through a coherent Rabi oscillation. This process enables precise quantum-state engineering and manipulation of atoms and photons in a cavity, which play a central role in quantum information and measurement. Recently, a new regime of cavity QED was reached experimentally where the strength of the interaction between light and artificial atoms (qubits) becomes comparable to the atomic transition frequency or the resonance frequency of the cavity mode. Here we show that this regime can strongly modify the concept of vacuum Rabi oscillations, enabling multiphoton exchanges between the qubit and the resonator. We find that experimental state-of-the-art circuit-QED systems can undergo two- and three-photon vacuum Rabi oscillations. These anomalous Rabi oscillations can be exploited for the realization of efficient Fock-state sources of light and complex entangled states of qubits.
We consider two separate atoms interacting with a single-mode optical resonator. When the frequency of the resonator field is twice the atomic transition frequency, we show that there exists a resonant coupling between one photon and two atoms, via intermediate virtual states connected by counter-rotating processes. If the resonator is prepared in its one-photon state, the photon can be jointly absorbed by the two atoms in their ground state which will both reach their excited state with probability close to one. Like ordinary quantum Rabi oscillations, this process is coherent and reversible, so that two atoms in their excited state will undergo a downward transition jointly emitting a single cavity photon. This joint absorption and emission processes can also occur with three atoms. The parameters used to investigate this process correspond to experimentally demonstrated values in circuit quantum electrodynamics systems.
If an excited atom is put in between high-reflectance mirrors, the light emitted by the atom can be bounced back and be re-absorbed and be re-emitted, again and again until the light leaks out of the mirrors or is re-emitted into a wrong direction. This coherent oscillation between atom and light, called vacuum Rabi oscillation, results from interference between two superposition states of the atom and the photon, which have different energies or frequencies, or vacuum Rabi splitting. The vacuum Rabi oscillation and splitting demonstrate the quantum nature of light. They also provide single-photon non-linearity for photonic applications. In recent years, these effects have been considered particularly useful for quantum information processing [1] since the oscillation presents a natural mechanism for information exchange between a stationary quantum bit carried by the atom and a flying quantum bit carried by the photon, making scalable and distributed quantum computing possible. Realization of the vacuum Rabi oscillation and splitting, however, is highly non-trivial. It requires the so-called strong coupling condition, that is, the exchange between the atom and the photon must be faster than the photon leaks out of the space enclosed by the mirrors, i.e. the cavity, and faster than the atom emits the light into a wrong direction. The technique to realize the strong coupling condition is to make the cavity small so that the photon is bounced back more frequently to the atom and to make the mirrors more reflective so that the photon can be bounced more times. That means to decrease the cavity volume V and to increase the quality factor Q. Remarkable progresses have been made to make optical microcavities with small V and high Q, facilitating observation of vacuum Rabi oscillation and splitting in highly dissipative solid-state systems [2,3]. However, there is no systematic method to reduce V/Q—the key factor for realizing strong coupling (see Fig. 1a). The reason appears to be fundamental. When the size of a cavity is made as small as comparable to the light wavelength, small deformation of the cavity would result in large evanescent wave and hence photon leakage [4]. (a) An atom and a small cavity have strong coupling but the cavity leakage is also strong. A large cavity has small photon leakage but its coupling to the atom is small (so emission goes to free space). The atom and the large cavity can have strong coupling, with photons exchanged through but not occupying the small cavity—using the quantum tunneling effect. (b) In the classical world, two lower players have to provide sufficient energy to pass the ball through the higher player. (c) In the quantum world, the two lower players can pass balls through the wall via the higher players but with energy of the balls insufficient to reach the higher player. A recent proposal by [5] provides a general solution, using a divide-and-conquer strategy. A small cavity with relatively low Q is used to enhance the interaction between the emitter and the photon, while the photon stays in a large cavity with high Q. [5] demonstrate the strong coupling by showing Rabi splitting, Rabi oscillation, and single-photon non-linearity of the system. The scheme at the first sight does not seem to work since the emitter has no contact with the high-Q large cavity and if the photon enters the small cavity, it would leak out. The clever point of the scheme is: in the quantum mechanical world, the photon can interact with the emitter without occupying the small cavity! Similar to the ball game in Fig. 1b, a classical particle needs to have sufficient energy to pass a barrier. The small cavity is designed to have a frequency higher than those of the emitter and the large cavity, which would prevent the emitted photons entering the large cavity. However, quantum ball games allow tunneling of a barrier (Fig. 1c). So the emitter and the large cavity can indeed exchange photons. The small cavity would not be occupied since its frequency is not right, which prevents photon leakage. It looks the scheme applies generally to all types of photonic systems—photonic crystals, microspheres, microtoroids, etc. Considering the simplicity of the scheme, we expect applications be found soon.
We consider two separate atoms interacting with a single-mode optical or microwave resonator. When the frequency of the resonator field is twice the atomic transition frequency, we show that there exists a resonant coupling between one photon and two atoms, via intermediate virtual states connected by counterrotating processes. If the resonator is prepared in its one-photon state, the photon can be jointly absorbed by the two atoms in their ground state which will both reach their excited state with a probability close to one. Like ordinary quantum Rabi oscillations, this process is coherent and reversible, so that two atoms in their excited state will undergo a downward transition jointly emitting a single cavity photon. This joint absorption and emission process can also occur with three atoms. The parameters used to investigate this process correspond to experimentally demonstrated values in circuit quantum electrodynamics systems.
The quantum electrodynamics of an atom in a cavity is reexamined from the view point of the vacuum-field Rabi oscillations. Many features of such oscillations in a single atom as well as in a cooperative system are discussed. Various physical quantities that carry the signature of these vacuum-field oscillations are calculated. Even oscillator systems are shown to exhibit such vacuum-field Rabi oscillations, which is in contrast to the external-field-induced Rabi oscillations. Such oscillations are found to occur in situations involving both single-photon and multiphoton transitions.