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High temperature collective spin-photon coupling in a microwave cavity

An ensemble of N identical noninteracting spins being in thermal equilibrium and coupled to the resonant mode of a lossless microwave cavity is studied at arbitrary temperature T. Near T = 0 the system is known to be in a coupled spin-photon state that manifests itself by the splitting of the cavity mode (vacuum Rabi splitting). The cavity emission spectrum is simulated for arbitrary T. It is shown that the spin-photon coherence can be partially preserved for T < w sqrt{N}/2, where w is the spin excitation energy, even in case when the spins are randomly directed. The calculations corroborate recent room-temperature observations of the collective coupling between the microwave cavity mode and the electron spin ensemble (NV centers in diamond, DPPH, Fe8 nanomagnets). At higher T, as a consequence of thermal excitations within the spin ensemble, the two lines of the emission spectrum merge into a narrow line with broad wings.