Cooperative Light Emission in the Presence of Strong Inhomogeneous Broadening

We study photon emission by an ensemble of two-level systems, with strong inhomogeneous broadening and coupled to a cavity mode whose frequency has linear time dependence. The analysis shows that, regardless of the distribution of energy level splittings, a sharp phase transition occurs between the weak and strong cooperative emission phases near a critical photonic frequency sweeping rate. The associated scaling exponent is determined. We suggest that this phase transition can be observed in an ensemble of negatively charged ${\mathrm{NV}}^{-}$ centers in diamond interacting with a microwave half-wavelength cavity mode even in the regime of weak coupling and at strong disorder of two-level splittings.

Figure 1

The adiabatic energy levels of the Hamiltonian (1) in the sector with five up-polarized spins ($2^5=32$ states in the phase space) and zero bosons in the initial state; $\omega =-\beta t$. The red dotted curve marks the ground energy level for this sector. Here, $\beta =1$, $g_k=0.25$, $\forall k$, and ${ϵ}_k=0.8(k-N/2+r_k)$, where $r_k$ are random numbers from the range of (0,0.5).

Figure 2

Comparison of the normalized average number of photons $n_{\mathrm{ph}}\equiv N_{\mathrm{ph}}/N$ with $N_{\mathrm{ph}}\equiv ⟨{\widehat{a}}^{†}\widehat{a}⟩$ emitted by $N=8$ initially up-polarized spins with the Hamiltonian (1) after a sudden quench (blue curve) that places $\omega =⟨{ϵ}_k⟩$ at $t=-200$ and keeps $\omega$ constant afterwards; and during a linear sweep of the frequency (black curve) according to Eq. (2) with $\beta =0.03$. Here, ${ϵ}_k$ and $g_k$ are random Gaussian numbers with $⟨{ϵ}_k⟩=0$, $\sqrt{\mathrm{var}({ϵ}_k)}=1$, $⟨g_k⟩=0.1$, and $\sqrt{\mathrm{var}(g_k)}=0.02$. Both curves are averaged over 10 different realizations of random parameters.

Figure 3

Distribution $P_n$ for $N=30$ spins and the ratio $\beta /g^2$ being $4N$ (solid black), $3N$ (dotted black), $2.42N$ (solid blue), and $1.5N$ (dashed black). Lines connect discrete points for better visibility. The blue curve corresponds to the critical $\beta$ at which the distribution maximum starts moving from $N^{*}=0$ to the right. The inset shows the normalized position of the maximum, $n^{*}=N^{*}/N$, as a function of $f=g^{2}N/(\beta {\log }_{e}N)$ in the limit $N\to \infty$. The discontinuity of $n^{*}(f)$ marks the position of a phase transition.

Figure 4

Left: (discrete points) $n_{\mathrm{ph}}(f)$ and (solid curves) $n^{*}(f)$ given by (4), for $N=50$ and $N={10}^8$ and the distribution (3). The discontinuity at $f=f_c$ emerges for $n_{\mathrm{ph}}(f)$ as $N\to \infty$. Right: the derivative of the number of produced photons, $d{N}_{\mathrm{ph}}/d(1/\beta )$, vs the inverse sweeping rate. The sharply rising piece of the curve before the critical $1/{\beta }_c$ [from Eq. (9)] becomes vertical in the limit $N\to \infty$.