**SPECTROSCOPIC CHARACTERISTICS OF A MICROWAVE CAVITY AT **

**NONZERO TEMPERATURE **

**E.I. Baibekov **

*Kazan Federal University, 420008, Kazan, Russian Federation [email protected]*

We consider an ensemble of a large number

*N*

of non-interacting spins

*s*

1 2 coupled to a single mode of electromagnetic field in an ideal microwave cavity. The system is described by Tavis-Cummings Hamiltonian [1],

*H*

*p*

*s*

*S z*

*bS*

*b S*

, where

*b*

(

*b*

) is photon creation (annihilation) operator,

*S z*

and

*S*

are collective spin operators. Supposing that the photon number is <<

*N*

, one can obtain the corresponding energy spectra either numerically (for

*N*

~ 10

) or analytically. In the latter case, the spin ensemble must be very close to its highest polarization state, i.e.

*S z*

*N*

2 . If the spins are under thermal equilibrium, this corresponds to zero temperature. It is known [2] that emission spectrum of the cavity at resonance (

*p*

*s*

) consists of a single line at frequency

*p*

split by a gap of 2

*N*

(Rabi splitting).

If the spins belong to paramagnetic atoms distributed in a solid, the zero-temperature approximation is inadequate even for

*T*

of a few Kelvin. In the present work we develop a generalized calculation scheme and obtain the cavity emission spectrum for low polarization states. We calculate the lineshapes and obtain the generalized Rabi splitting of 2

2

*S z*

, where

*N*

*S z*

denotes temperature average. We show that the splitting remains under condition

*N*

2 , i.e. for a wide temperature range when

*N*

1

. At higher

*T*

, the lines gradually merge into one. The obtained results corroborate recent observations of Rabi splitting in magnetically coupled spin-photon system [3].

This work was supported by RFBR (Grant no. 12-02-31336) and by Dynasty

Foundation.

**References **

1. M. Tavis, F.W. Cummings, Phys. Rev.

**170**

, 379 (1968).

2. J.J. Sanchez-Mondragon, N.B. Narozhny, J.H. Eberly, Phys. Rev. Lett.

**51**

, 550 (1983).

3. I. Chiorescu, N. Groll, S. Bertaina, T. Mori, S. Miyashita, Phys. Rev. B

**82**

, 024413 (2010).