Stabilization of an optical transition energy via nuclear Zeno dynamics in quantum-dot–cavity systems

We investigate the effect of nuclear spins on the phase shift and polarization rotation of photons scattered off a quantum-dot–cavity system. We show that as the phase shift depends strongly on the resonance energy of an electronic transition in the quantum dot, it can provide a sensitive probe of the quantum state of nuclear spins that broaden this transition energy. By including the electron-nuclear spin coupling at a Hamiltonian level within an extended input-output formalism, we show how a photon-scattering event acts as a nuclear spin measurement, which when rapidly applied leads to an inhibition of the nuclear dynamics via the quantum Zeno effect, and a corresponding stabilization of the optical resonance. We show how such an effect manifests in the intensity autocorrelation $g^{\left(2\right)}\left(\tau \right)$ of scattered photons, whose long-time bunching behavior changes from quadratic decay for low photon-scattering rates (weak laser intensities) to ever slower exponential decay for increasing laser intensities as optical measurements impede the nuclear spin evolution.

Figure 1

(a) Experimental setup used to measure photon phase shifts due to a quantum dot inside a micropillar cavity. The states $|H\rangle$ and $|V\rangle$ denote horizontally and vertically polarized light, while (P)BS labels a (polarizing) 50:50 beam splitter. A variable delay $\tau$ between detectors $D_1$ and $D_2$ can be used to measure the intensity autocorrelation function of light scattered into a cross-polarized (vertical) channel, as shown in (c). (b) Right $|R\rangle$ and left $|L\rangle$ circularly polarized photons couple to the spin ground states $|\uparrow \rangle$ and $|\downarrow \rangle$, respectively. In a large magnetic field, the $|\downarrow \rangle$ is far detuned, and $|L\rangle$ reflects off an effectively empty cavity, while $|R\rangle$ experiences a phase shift ${\varphi }_g\left(\delta \right)$ that depends on the nuclear spin Overhauser field $\delta$. (c) Cross-polarized intensity correlation function for different average photon count rates obtained from a Monte Carlo simulation including eight nuclear spins. While the quadratic behavior characteristic of unitary evolution is observed for low count rates, the evolution of the nuclear spins is impeded by more frequent photon scattering, which constitutes a quantum Zeno effect. Note that the antibunching at subnanosecond timescales is neglected and that the weak-driving assumption is satisfied. Parameters $A_k$ and ${\omega }_k$ were randomly drawn from a Gaussian distribution with $\langle A_k\rangle =\langle {\omega }_k\rangle =0.5\mu \mathrm{eV}$, $\sigma \left(A_k\right)=0.25\mu \mathrm{eV}$, and $\sigma \left({\omega }_k\right)=10\mathrm{neV}$.

Figure 2

Probability for a photon-scattering event into the copolarized $H\to H$ (blue) or cross-polarized $H\to V$ channel $\langle \delta |M_c^{†}{M}_c|\delta \rangle$, shown for a nuclear system in an Overhauser shift eigenstate $|\delta \rangle$, and as a function of the shift $\delta$. For zero Overhauser shift, cross-polarized photons are preferred, with this bias reversing as the Overhauser shift becomes greater than the linewidth of the electronic transition in the cavity $2g^2/\kappa$. Parameters ($\mu \mathrm{eV}$): $\kappa =4000$, $g=30$, ${\omega }_c={\omega }_0=\omega$.

Figure 3

Left: Representative time evolution of the nuclear spin system, here taken to be two spins spanned by the phase shift eigenstates $|\delta \rangle$ and $|{\delta }^{\prime }\rangle$, with the dynamics generated following Eqs. (35a) and (35b). The plot shows the path of a Bloch vector representation of the nuclear spin state, with Overhauser amplitude and coherence operators respectively ${\sigma }_z=|\delta \rangle \langle \delta |-|{\delta }^{\prime }\rangle \langle {\delta }^{\prime }|$, ${\sigma }_y=|+\rangle \langle +|-|-\rangle \langle -|$, with $|\pm \rangle =|\delta \rangle \pm i|{\delta }^{\prime }\rangle$. State trajectories are shown for no photon-scattering events (blue), photon scattering every $\mathrm{\Delta }=5$ ns (orange) and $\mathrm{\Delta }=2$ ns (green). The dotted red line shows the surface of the Bloch sphere, a cross section of which is shown in the inset (red section near the pole corresponds to the outer plot). Right: Exclusive joint probability ${\mathcal{P}}_n$ following Eq. (36) of scattering into cross polarization at times 0 and $\tau$ corresponding to the state evolution shown to the left, where ${\mathcal{P}}_n\propto \langle {\sigma }_z\rangle$. Parameters: $A_1=1$, $A_2=3$, ${\omega }_1=2.5$, ${\omega }_1=0.5$, $\omega =40$.

Figure 4

Cross-polarized optical intensity autocorrelation $g_V^{\left(2\right)}\left(\tau \right)$ including Markovian resonance fluctuations noise. The transparent lines show the data from Fig. 1 without Markovian noise for comparison. A random, uncorrelated resonance shift $s$ is sampled from a Gaussian distribution with zero mean and 250 MHz variance at each photon-scattering event of the Monte Carlo simulation, with all other parameters as in Fig. 1.