Correlated emission of electron-current waves

The gist The main result of this work is that subradiance requires emitters to be spaced at a distance closer than the Fermi wavelength, while the requirement for superradiance is generically looser

Correlated Emission into an Electronic Environment

Correlated emission occurs when a collection of localized quantum systems emit energy coherently into an environment First identified by R. H. Dicke [1], the phenomenon has been observed in platforms ranging from cold atomic gases [2–4], superconducting circuits [5], photonic waveguides [6, 7] and nitrogen-vacancy centers (NVs) in diamond [8–10]. It is important for quantum technologies, both as a potential hazard for error correction [11], and as a resource for entanglement generation [12] and quantum memories [13]. Correlated emission occurs only if the emitters are spaced closer than the wavelength of the emitted bosonic excitation. When that is the case, two types of correlated emission are generally identified: subradiance, when the rate of emission is suppressed due to cooperative effects, and superradiance, when the rate of emission is cooperatively enhanced. In contrast to correlated emission into a bosonic environment such as the electromagnetic vacuum, noise from an electronic gas occurs within a large range of length scales for a fixed emitter frequency. Electronic noise from conduction electrons is generically captured by the two-particle spectrum of a Fermi gas and is characterized by two length scales: the Fermi wavelength λF and a length vF /∆ depending on the frequency of the emitter ∆ and the Fermi velocity vF.

Conditions for Subradiance and Superradiance

The main result of this work is that subradiance requires emitters to be spaced closer than the Fermi wavelength, while the requirement for superradiance is generically looser. In particular, it depends on whether the emitters are initialized in a state with or without coherence between the two levels of the emitter. If the emitters are initialized without coherence, superradiance requires a spacing on a scale similar to the Fermi wavelength. While if the emitters are initialized with coherence, superradiance can occur with a spacing significantly further than the Fermi wavelength so long as a < pλF vF /∆. Subradiance is defined as the phenomenon in which the decay of the collective ensemble is slower than for independent emitters. For two dipoles, subradiance occurs when a <∼ λF, as shown in Fig. 2(b) for d = 0.

Modeling the Environment and Coupling

The dynamics of magnetic dipoles are described by a Lindblad master equation containing both coherent mediated interactions and dissipative processes such as excitation, dephasing, and decay. The magnetic-field noise at thermal equilibrium takes the form C−+ (ω, q) = Z d dk (2π) d−1 V − k,k+q V + k+q,k nk(1 − nk+q) × δ (ω + ωk − ωk+q), determined by the matrix elements V ± k,k+q = V x k,k+q ± iV y k,k+q, the Fermi-Dirac occupation function nk, and the kinetic constraint δ (ω + ωk − ωk+q) which restricts the particle at wave vector k to absorb only momentum ¯h q and energy ¯hω. For a homogeneous coupling, the magnetic fields at wave vector q are given by B(q) = √ V gqbq + g∗−q b −q B(q) = X k Vk,k+qf k f+q, where we have defined gq = gq(r = 0) and Vk,k+q = Vk,k+q(r = 0) for bosons and fermions respectively.

Superradiance Dynamics

For a homogeneous distribution of dipoles with mean spacing a > λF, there is a finite probability that two dipoles have a spacing r < λF. Thus, even at low densities, there exist subradiant states. Only when the densities are sufficiently large, n > min(λ−2 F, d−2), do subradiant states become typical and relevant for collective relaxation dynamics. For all in the excited state, the change in the total decay rate is given by ∂tR N γ2 0 = πλ2 SR − 1. The superradiance length scale is given by λSR ≈ λF ln(4EF /¯h∆), where we have assumed λF j vF /∆. A superradiant acceleration, ∂tR > 0, requires that the density of dipoles is larger than n > λ−2 SR/π ≈ λ−2 F/ ln(4EF /¯h∆)π.

Current Response and Dissipation

The imaginary part of the current response Im GR0 from Eq. (56) (capturing dissipation) and the magnetic noise C−+ from Eq. (20) (originating from current fluctuations) are related to each other via the fluctuation-dissipation theorem. The static response agrees with the well known result of Landau diamagnetism. The total response current shown in Fig. 3 also contains the diamagnetic contribution Jdia given in Eq. (47) whose dynamics is directly determined by the vector potential.

Current Wave Features

The paramagnetic current distribution forms the pattern of Archimedean spirals with two arms. This two spiral arms merge at the origin (the position of the macrospin) in a way that ensures ∇ · J = 0. The parametric form for the paramagnetic current response is given by Jpara ρ(ρ, ϕ, t) ∼ 1/ρ cubed − hpara ρx t − ρ/vF sin ϕ + hpara ρy t − ρ/vF cos ϕ.

Discussion on Material Effects

Materials introduce both disorder and interaction effects, which might modify length scales at which correlated emission can occur. For example, the mean free path l in many common materials is shorter than vF /∆, and the condition for superradiance will be modified: likely requiring a < l. The exception is if there are strong enough interactions that a transverse-current zero-sound mode appears in the spectrum. Temperature will only induce thermal noise in the collective relaxation process, and could be relatively small when the number of dipoles is large N k kBT /¯h∆.

Supplemental Material Details

The Supplemental material first introduces a general model for magnetic dipoles interacting with bosonic and fermionic environments, discusses the projection of NV center spin-1 triplets to effective two-level systems, and analyzes the coupling to current fluctuations in two-dimensional conductors. It presents the response of electron currents to a decaying macrospin, including analytic and numerical results, and connects the dynamic and static current responses to established physical concepts like Landau diamagnetism. The document includes a video of the current emitted by the superradiant decay of a macrospin.

References

[1] R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954)

[2] N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, Observation of dicke superradiance in optically pumped hf gas, Phys. Rev. Lett. 30, 309 (1973)

[3] M. Gross, P. Goy, C. Fabre, S. Haroche, and J. M. Raimond, Maser oscillation and microwave superradiance in small systems of rydberg atoms, Phys. Rev. Lett. 43, 343 (1979)

[4] M. A. Norcia, M. N. Winchester, J. R. K. Cline, and J. K Thompson, Superradiance on the millihertz linewidth strontium clock transition, Science Advances 2, e1601231 (2016)

[5] M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, Superradiance of quantum dots, Nature Physics 3, 106 (2007)

[6] C. Liedl, F. Tebbenjohanns, C. Bach, S. Pucher, A. Rauschenbeutel, and P. Schneeweiss, Observation of superradiant bursts in a cascaded quantum system, Phys Rev X 14 011020 (2024)

1. Bold header: Collective Emission Modeling for Quantum Memory Fidelity

This system can predict subradiance and superradiance effects by calculating the decay rate matrix elements γnm, directly addressing how the emitted current burst has a spiral form. It can assess whether emitters should be spaced closer than the Fermi wavelength for subradiance or less than pλF vF/∆ for superradiance.

2. Bold header: Entanglement Generation via Correlated Emission

The AI can simulate the conditions required for entanglement generation between atomic spins by analyzing the nonlocal decay rates γnm. It can determine if a state initialized with coherence will result in a "superradiant acceleration, ∂tR > 0," which is crucial for generating entangled states.

3. Bold header: Sensing and Characterizing Condensed Matter Environments

The system can be used to probe the environment by analyzing Oersted magnetic-field noise produced by the transverse-current fluctuations and its spectral density C−+(ω, q). It can identify the relevant length scales—one set by the Fermi velocity and the frequency of the color centers vF /∆, and the other set by the Fermi wavelength λF j vF /∆—to characterize a material's noise signature.

4. Bold header: Correlated Dissipation Analysis

The AI can model correlated dissipation by color-center qubits coupled to electronic environments by solving the adjoint master equation for correlated loss, allowing for the prediction of how collective effects influence relaxation dynamics compared to independent emission.