Subradiant Oscillating Correlations

• Subradiant oscillating correlations are phenomena where collective quantum states exhibit inhibited radiative decay with time-dependent oscillatory signatures due to destructive interference.
• Experimental investigations in photonic waveguides, circuit QED, and optical lattices demonstrate how tailored state preparation leads to measurable oscillatory correlations.
• Mathematical frameworks using Lindblad master equations and collective operators classify states as dark, subradiant, or superradiant, guiding applications in quantum sensing and decoherence-free subspaces.

Subradiant oscillating correlations describe the phenomenon in which collective oscillatory modes of correlated quantum systems exhibit inhibited radiative decay—i.e., subradiance—while manifesting temporal or spatially oscillating features in their correlation functions. Such phenomena have been rigorously analyzed in the context of both bosonic and spin (atomic) ensembles, including harmonic oscillator arrays, atomic chains, circuit QED, photonic lattices, and disordered Dicke models. The defining characteristics involve the formation of collective states (or correlations) that, due to destructive interference among radiative channels, suppress overall emission while supporting time-dependent (oscillatory) structure in their correlations.

1. Manifestation in Harmonic Oscillator Ensembles

Subradiant states in bosonic arrays, as shown for harmonic oscillator ensembles with collective damping, are eigenstates of a collective ladder operator $C_N = (\sum_j g_j b_j)/\mathcal{G}_N$ (with $\mathcal{G}_N = \sqrt{\sum_j g_j^2}$). States of the form $|{\boldsymbol{d}_L}, \Phi^0_L\rangle$ for which $C_N |{\boldsymbol{d}_L}, \Phi^0_L\rangle = 0$ are fully dark—i.e., all excitation energy remains trapped, corresponding to a decoherence-free subspace (Delanty et al., 2011).

Two-time correlations between modes, $$c_{i,j}(t,0) = \langle b_i^\dagger(t) b_j(0)\rangle - \langle b_i^\dagger(t)\rangle\langle b_j(0)\rangle$$, can display oscillatory or sign-changing behavior depending on the initial multimode quantum state. The correlated part of the intensity, $I_N^C(t)$, can be negative if initial states are anticorrelated, leading to inhibited emission and oscillatory features in measured correlation functions. This demonstrates that the presence of negative, temporally oscillating correlations is intimately connected to the subradiant suppression of radiative decay.

These phenomena are accessible in photonic waveguide arrays or multimode circuit QED setups, where state preparation and readout of multimode Fock or squeezed states permits direct observation of these oscillating subradiant correlations.

2. Dynamical and Oscillatory Features in Quantum Optical Platforms

Subradiant oscillating correlations are realized not only in bosonic platforms but also in driven atomic systems. In "Super/Subradiant Second Harmonic Generation" (Koganov et al., 2014), an N-atom ladder-type laser model exhibits a sharp transition between superradiant (intensity $\propto N^2$) and subradiant (intensity $\propto N^0$) regimes. In the subradiant regime, phase-space trajectories and stability analyses reveal transient nonstationary dynamics where the field’s phase and quadratures exhibit pronounced oscillatory behavior en route to their steady state—underscoring how subradiant states can host oscillating correlations even as the net emission is suppressed.

In subwavelength molecular systems, deeply subradiant excited states constructed via symmetry—forbidden transitions demonstrate extremely long lifetimes (McGuyer et al., 2014). There, the coherent superposition $$(|^1S_0\rangle|^3P_1\rangle + |^3P_1\rangle|^1S_0\rangle)/\sqrt{2}$$ produces destructive interference in radiative decay, with observed Rabi oscillations and time-resolved correlation measurements directly revealing oscillating subradiant correlations.

In one-dimensional optical lattices and atomic arrays, precisely imprinted spatial phase gradients lead to the creation of "De Moivre states"—engineered to be highly subradiant by design (Jen et al., 2016). The temporal dynamics of such states often project onto only a few eigenmodes, leading to Rabi-like decaying oscillations whose beating frequency reflects cooperative Lamb shifts in the subradiant subspace. Experimentally, this is manifested in time-dependent fluorescence showing clear oscillatory signatures tightly correlated with the degree of subradiance.

3. Mathematical Formulation and Decay Classification

The mathematical delineation of subradiant and superradiant regimes follows from the structure of the collective operators and the Lindblad equation governing the ensemble:

$$\dot\rho_b = (N\Gamma/2)\,\mathcal{D}[C_N]\rho_b, \quad \mathcal{D}[A]\rho = 2A\rho A^\dagger - A^\dagger A \rho - \rho A^\dagger A,$$

where subradiant eigenstates satisfy $C_N|\text{state}\rangle=0$ and thus do not couple to the bath. The "dark" fraction,

$F = \langle L \rangle / \langle M \rangle,$

classifies the degree of subradiance: $F > F_N=1-1/N$ indicates subradiance, $F = F_N$ normal radiance, and $F < F_N$ superradiance (Delanty et al., 2011).

For oscillator ensembles and arrays, the time-dependent two-mode correlations,

$$c_{i,j}(t,0) = \langle b_i^\dagger(t)b_j(0)\rangle - \langle b_i^\dagger(t)\rangle\langle b_j(0)\rangle,$$

directly resolve the build-up, sign, and oscillatory dynamics of quantum correlations in both subradiant and superradiant sectors.

Analogous classification arises for atomic and spin systems: collective eigenmodes can be superradiant (enhanced decay), subradiant (suppressed decay), or dark (decayless). State preparation protocols often rely on collective excitation of symmetric or antisymmetric superpositions, manipulation of spatial or polarization phase, and controlled driving/hybridization to access these regimes.

4. Contrasts with Superradiance and Interplay with Oscillating Dynamics

Superradiant and subradiant regimes represent opposing extremes in collective radiative behavior. Superradiant states (large $R$, $F < F_N$) exhibit constructive build-up of correlations, with intensity scaling as $N^2$ and rapid decay—typically manifested in correlated emission bursts. Subradiant states (small $R$, ideally $R=0$, $F=1$) have destructive interference suppressing emission; their correlation functions, rather than building up constructively, may display temporal oscillations or even negative values—a clear signature of coherent mixing of dark and residual bright sectors.

Measured two-time and spatial correlations thus provide a transparent diagnostic: constructive, in-phase oscillations denote superradiance, while anticorrelated or out-of-phase oscillations reflect the inhibited emission channels characteristic of subradiance.

5. Experimental Platforms and Applications

Subradiant oscillating correlations are accessible in multiple physical platforms:

• Integrated waveguide photonics: Engineered arrays with tailored couplings enable realization and measurement of bosonic Dicke states with controllable collective decay rates.
• Circuit QED: Superconducting resonator arrays hosting bosonic modes and qubits achieve strong coupling to bath modes (damped resonator), supporting state preparation and time-resolved correlation measurements.
• Ultracold molecules: Optical lattice clock platforms allow direct observation of long-lived subradiant states and their oscillatory population dynamics via optical spectroscopy (McGuyer et al., 2014).
• Exciton-polariton microcavities: Second-order photon correlation measurements reveal counterintuitive subradiant dips and nonmonotonic bunching due to stochastic matter–light conversion (Rubo et al., 2015).

Practical significance includes the realization of decoherence-free subspaces for quantum memories, entanglement-enabled sensing, and protocols for continuous variable quantum information processing.

6. Broader Context: Role in Many-Body Physics and Quantum Technologies

Subradiant oscillating correlations form the basis for protected many-body quantum phases with long-lived coherence. The ability to engineer and control the collective radiative properties—by selecting spatial phase patterns, exploiting many-body interactions, and manipulating system–bath coupling—enables precision metrology, robust quantum information storage, and the fundamental exploration of symmetry-induced decoherence suppression.

In the bosonic and spin cases, the theoretical frameworks—rooted in the bosonic Dicke basis and Lindblad master equations—enable quantitative predictions of decay rates, correlation dynamics, and parameter regimes for observable subradiant effects. The distinction between normal, super-, and subradiant regimes, underpinned by the structural form of the collective operators and state decomposition, provides a unifying perspective across quantum optics, open quantum systems, and wave physics.

The interplay between oscillatory correlation dynamics and radiative suppression is a central theme in the engineering of quantum devices that must operate robustly in the presence of dissipation, making subradiant oscillating correlations both of fundamental and practical interest.