Effects of retardation on many-body superradiance in chiral waveguide QED

The gist: Non-negligible photon propagation times in chiral waveguide QED can significantly alter collective decay dynamics by suppressing superradiant scaling, leading to an effective maximum cooperative system size and potentially inducing sustained oscillatory atomic dynamics <ref:2408.03390#pg12>.

Retardation Effects on Superradiance

The study investigates the superradiant decay of a chain of atoms coupled to a chiral waveguide, focusing on the regime where propagation times are non-negligible <ref:2408.03390#pg6>. Using an exact master equation description, the researchers obtained evidence suggesting that competition between collective decay and retardation results in an effective maximum number of atoms able to contribute to superradiant dynamics, leading to a plateau of the peak emission rate <ref:2408.03390#pg12>. This plateau signals the emergence of a maximum cooperative system size, resulting in local rather than global synchronisation among the atoms <ref:2408.03390#pg12>.

Emergent System Size and Scaling

The analysis characterizes this effect by studying atomic correlations and mean-field analyses <ref:2408.03390#pg12>. For a partially-inverted initial state, the peak emission rate scaling with system size plateaus to a maximum effective emission rate referred to as r eff <ref:2408.03390#pg12>. This suggests an effective maximum number of atoms Neff which can contribute to the superradiant decay <ref:2408.03390#pg12>. The asymptotic scaling for the peak emission rate r pk n instead plateaus to a maximum effective emission rate r eff <ref:2408.03390#pg12>. This plateau is related to the competition between retardation and cooperative emission, leading to an effective maximum number of atoms Neff which can contribute to the superradiant decay <ref:2408.03390#pg12>.

Oscillatory Dynamics and Domain Formation

Retardation can also result in persistent oscillatory atomic dynamics accompanied by a periodic sequence of emission bursts <ref:2408.03390#pg12>. The study further investigates inter-atomic correlations to find features consistent with the formation of individual superradiant domains <ref:2408.03390#pg12>. For sufficiently long chains, the dynamics can support sustained oscillatory dynamics, resulting in a periodic emission of intensity bursts <ref:2408.03390#pg12>. The multi-time correlator C(n, m, t) shows the emergence of a “light cone” structure for τ > 0 <ref:2408.03390#pg12>. Furthermore, correlations at peak emission decay with distance j over shorter length scales along the atomic array <ref:2408.03390#pg12>.

Theoretical Framework and Simulation Methods

The analysis employs an exact master equation description, which is derived in a time-shifted picture to account for delay effects <ref:2408.03390#pg12>. The dynamics are simulated using the Truncated Wigner Approximation (TWA) <ref:2408.03390#pg12>. Benchmarking against quantum trajectory simulations with Matrix Product States (MPS) confirms the accuracy of the TWA, especially when considering chiral interactions and time delays <ref:2408.03390#pg12>. The mean-field theory (MFT) is also employed, which shows excellent agreement with TWA numerics for sufficiently small delays <ref:2408.03390#pg12>.

Experimental Relevance and Limitations

The work establishes the setting of slow-light chiral WQED as a potential experimental platform for exploring this regime in a controlled manner <ref:2408.03390#pg12>. The results provide a glimpse into non-Markovian phenomena that have only been explored in the single-excitation regime <ref:2408.03390#pg12>. In realistic settings, independent decay into unguided free-space modes competes with waveguide emission, and a critical free-space decay rate gamma max depends on the retardation time τ <ref:2408.03390#pg12>. While sustained periodic dynamics in the thermodynamic limit raise questions about physical mechanisms, the work emphasizes that these dynamics represent a departure from the well-studied retardation-free superradiant dynamics <ref:2408.03390#pg12>.

Conclusion

In summary, nonnegligible photon propagation times lead to qualitative departures from standard theory by inhibiting global synchronisation, resulting in an effective maximum cooperative system size and sustained periodic atomic dynamics <ref:2408.03390#pg12>. The contribution of the work is providing a partial answer to superradiance in extended media and identifying slow-light chiral WQED as a platform for exploring non-Markovian collective decay dynamics <ref:2408.03390#pg12>.

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S1 Exact master equation in time-shifted picture<ref:2408.03390#pg10>

The goal of this section is to derive the master equation in the main text <ref:2408.03390#pg10>. The starting point for this derivation is the Heisenberg equation of motion under the atom-waveguide Hamiltonian for an arbitrary emitter operator O(t), O˙(t) = i√ΓvΘ(t)Xn[σ†n(t), O(t)]b(xn, t) + b† (xn, t)[σn(t), O(t)] <ref:2408.03390#pg10>. Note that the Hamiltonian can also be expressed in terms of waveguide modes b(k) = R ∞−∞ dx e−ikxb(x)/√2π with well-defined momenta k as H(t) = Hwg + Θ(t)PN n=1 Hn, where Hwg = vZ ∞−∞ dk k b† (k)b(k) and Hn = rΓv2π Z ∞−∞ dk eikxn σ†n b(k) + H.c. <ref:2408.03390#pg10>. From this expression, we can derive the Heisenberg equations of motion for the waveguide modes, ∂tb(k, t) = −ivk b(k, t) − i rΓv2π Θ(t)Xn e−ikxn σn(t), which can be integrated with respect to time to obtain b(k, t) = e−ivkt b(k, 0) − i rΓv2π Xn e−ikxn Z t0 ds e−ivk(t-s)σn(t). <ref:2408.03390#pg10>. The time-evolved real-space modes can be obtained by means of a Fourier transform, b(x, t) = R ∞−∞ dk eikxb(k, t)/√2π. <ref:2408.03390#pg10>. Evaluated at the emitter positions x = xn, they are given by b(xn, t) = b(xn − vt, 0) − i 2rΓvΘ(t)σn(t) − i rΓv Xm. The first term corresponds to the free evolution of the waveguide modes, i.e., in the absence of emitters <ref:2408.03390#pg10>. For a nonzero coupling Γ > 0, the field at position xn starts to depend on the state of the atoms further upstream after the corresponding delay times, tmn = tm − tn = (xn − xm)/v, for 1 ≤ m ≤ n <ref:2408.03390#pg10>. Substituting Eq. (5) into Eq. (1), we realize that the evolution of an operator on the nth emitter, On(t), depends on operators of other emitters further upstream, evaluated at earlier times, ∂tOn(t) = i√ΓvΘ(t)[σ†n(t), On(t)] (b(xn − vt, 0) − i 2rΓvσn(t) − i rΓv Xm. The different time-dependencies can be neatly accommodated by defining time-shifted local operators, Oˆn(t) = On(t − tn). These operators in the time-shifted picture obey ∂tOˆn(t) = i√ΓvΘ(t - tn)[ˆσ†n(t), Oˆn(t)] (b(xN − vt, 0) − i 2rΓvσ̂n(t) − i rΓv Xm.

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S1.

1. The AI system can model collective decay dynamics in chiral waveguide QED by incorporating time-shifted operators, allowing it to account for delay effects which are crucial for non-Markovian dynamics. This enables the AI to predict how retardation suppresses the characteristic superradiant scaling of the peak emission rate, leading instead to a plateau in peak emission along the chain of waveguide-coupled atoms.

2. The system can simulate domain structure formation by analyzing atomic correlations and identifying features consistent with the formation of individual superradiant domains. This allows for the prediction that the competition between retardation and cooperative emission leads to an effective maximum number of atoms Neff which can contribute to the superradiant decay, resulting in local rather than global synchronisation among the atoms.

3. The AI can generate time-dependent emission predictions by identifying regimes where retardation induces sustained oscillatory dynamics, leading to a periodic sequence of emission bursts. This capability allows for characterizing phenomena such as the evolution of all atoms sufficiently far down the chain will be determined by an identical environment of a fixed number of atoms further upstream, manifesting as oscillations in the local emission rate.

4. The system can characterize the spatial extent of collective effects by analyzing the asymptotic correlation length ξeff and predicting its scaling with retardation time, as shown by ξeff/d ∼ (Γτ)−1/2 at sufficiently small τ. This provides a quantitative measure of the emergent cooperative system size.

5. The AI can distinguish between different physical regimes by analyzing the behavior of the peak emission rate scaling, noting that for fully-inverted initial states, a better fit is given by the more general compressed exponential function C pk n(j) = C0e−(jd/ξn(t))α introduced in the main text. This allows for accurate prediction of correlations based on the initial state.

We study the superradiant decay of a chain of atoms coupled to a chiral waveguide, focusing on the regime of non-negligible photon propagation time. Using an exact master equation description which accounts for delay effects, we obtain evidence to suggest that competition between collective decay and retardation leads to the emergence of an effective maximum number of atoms able to contribute to the superradiant dynamics, resulting in a plateau of the peak emission rate. To develop this analysis further, we investigate the inter-atomic correlations to find features consistent with the formation of individual superradiant domains. Moreover, we find that retardation can also result in persistent oscillatory atomic dynamics accompanied by a periodic sequence of emission bursts.