Spectroscopic properties of inhomogeneously broadened spin ensembles in a cavity

The enhanced collective coupling to weak quantum fields may turn atomic or spin ensembles into an important component in quantum information processing architectures. Inhomogeneous broadening can, however, significantly reduce the coupling and the lifetime of the collective excitation that represent the quantum information. In this paper we show that the width and shape of the inhomogeneous broadening have a striking influence on the dynamics of the cavity-ensemble system and may lead to narrowing of the linewidth of the collective states. We underpin our findings with the examples of a Gaussian and a Lorentzian profile of the inhomogeneity.

Figure 1

The branch cut of the level-shift function $\mathrm{K̃}(z)$ displayed as the straight line $\omega -\frac{i}{2}{\gamma }_{\text{hom}}$ ($\omega \in \mathbb{R}$) in the complex plane. $\mathrm{K̃}(z)$ has a jump of ${\mathrm{K̃}}^{-}(z)-{\mathrm{K̃}}^{+}(z)=i{\Gamma }_c(z)$ when passing through the cut from above. Depending on whether $\kappa$ is smaller than, equal to, or larger than $\frac{1}{2}{\gamma }_{\text{hom}}$, the poles of the cavity-cavity propagator, i.e., the roots $E_{\pm }$ of Eq. (26), may lie above the cut, on the cut, or on the second Riemann sheet below the cut, respectively.

Figure 2

Numerical calculation for $N=25$ spins with an inhomogeneous width of $\Delta \omega$, sampled from a Gaussian distribution. The individual spins decay at a rate ${\gamma }_j=0.1\Delta \omega$ and are uniformly coupled to the cavity. The cavity decay rate is $\kappa =0.05\Delta \omega$. (a)–(c) The real part of the exact eigenenergies as a function of the cavity detuning. Color red (blue) indicates that the corresponding polariton is photon-like (spin-like). (d)–(f) Transmission spectrogram, the color hue (saturation) corresponds to the phase (logarithmic magnitude) of the susceptibility (25). (a), (d) In the strong-coupling regime $\Omega =4\Delta \omega$, only the two extremal eigenmodes have photonic attribute. (b), (e) In the intermediate-coupling regime $\Omega =\Delta \omega$, a band of intermediate eigenmodes has significant cavity content, and the cavity can “decay” into any of these modes. (c), (f) In the weak-coupling regime $\Omega =\Delta \omega /4$, there is no collective effect, and only a few resonant spins are dressed by the cavity.

Figure 3

(a), (c) Real and (b), (d) imaginary parts of the poles of the susceptibility (17) as a function of the collective coupling strength for (a), (b) a Lorentzian and (c), (d) a Gaussian coupling density profile. The quantities are directly related to the position and width of the peaks in the transmission spectrum. The cavity is tuned at the center of the ensemble (${\omega }_c={\omega }_a$), and $\kappa ={\gamma }_{\text{hom}}=0$ was assumed. The full width at half maximum of the coupling density is ${\gamma }_{\text{FWHM}}=\gamma$ for the Lorentzian and ${\gamma }_{\text{FWHM}}=\sqrt{8\ln 2}\hspace{0.5em}\sigma$ for the Gaussian. (a), (c) Dashed lines show the lowest order solutions of the asymptotic expansion given by Eq. (79), and correspond to the strong-coupling approximation described in Sec. 3a. (c), (d) Dotted curves show the asymptotic solutions in the next orders as given by Eqs. (64) and (65). (c) Dash-dotted curve shows Eq. (60) corresponding to the Weisskopf–Wigner approximation in the weak-coupling regime.

Figure 4

Level diagram showing the quasiclosed subspace consisting of the dressed polariton modes (superpositions of the cavity and the superradiant spin-wave mode) coupled to a chain of subradiant spin-wave modes. In the strong-coupling regime ${\Omega }_R\gg \Delta \omega$, the coupling between the dressed modes ${\Phi }_{\pm }$ and the subradiant mode $c$ is off-resonant, therefore, the inhomogeneity does not induce decoherence of the dressed modes: the dressed levels are not broadened.