Light has a way of surprising us just when we think we’ve cornered it. Spontaneous emission—the quiet habit of an excited quantum system shedding energy by emitting a photon—has long been a bedrock idea in quantum optics. Yet in many real materials the first excited state isn’t a tidy rung on a ladder. It’s a metastable resonance, a state with a finite lifetime that sits inside a sea of energies—the continuum. That subtle shift matters: it means the electron has nonradiative ways to dissipate energy, and the photon it emits comes from a more tangled set of possibilities than the textbook picture suggests. In such open systems, predicting how fast light bleeds out of an excited state becomes a delicate accounting problem between radiative and nonradiative channels.
A remarkable new study from Technion—Israel Institute of Technology in Haifa, conducted at the Andrew and Erna Viterbi Department of Electrical & Computer Engineering and the Helen Diller Quantum Center, with collaborators at the Guangdong Technion Israel Institute of Technology—asks a provocative question: when the initial state itself has a finite lifetime, should we still constrain the calculation to transitions to lower bound states? The answer, as the authors show, is both simple and surprising. You must include every possible final channel, including states with higher energy and the continuum, because the resonance you start from is really a spread across the continuum. And you don’t have to drown in that complexity forever—there is a powerful mathematical trick that makes it tractable: complex scaling, a non-Hermitian quantum-mechanics tool that recasts the continuum as a family of discrete complex poles that you can sum over without burning through computational time.
What follows is not a marginal tweak to the equations but a different way of thinking about how light and matter talk to each other when the system isn’t perfectly closed. The authors—led by Amir Sivan, with Milan Šindelka, Meir Orenstein, and Nimrod Moiseyev among the contributors—apply a unified framework in which the electron’s decay and the photon’s emission are treated on equal footing. By rotating the coordinates into the complex plane, they turn metastable resonances into square-integrable objects that behave like discrete states in a spectrum. The payoff is a practical route to compute spontaneous emission rates for metastable states in real, multi-electron systems—something that has been notoriously hard with traditional Hermitian quantum mechanics.
In short, the paper invites us to rethink spontaneous emission as a process connected to the entire landscape of final states, not a one-way street to a single, lower-energy destination. The implications ripple across engineered light sources—quantum dots, lasers, nanoscale emitters—and into the fundamental theory of open quantum systems. It’s a reminder that when a system leaks, our equations should leak too—in a controlled, computable way that reveals the full choreography of light and matter. The study stands as a concrete bridge between abstract non-Hermitian mathematics and the practical world of devices that glow in the wild.
A new lens on spontaneous emission
In the classic picture, spontaneous emission is captured by a version of Fermi’s golden rule that assumes the excited state is a true bound state. But metastable resonances live in the continuum; they have a finite lifetime and a width in energy. That means they aren’t simply waiting to drop to a lower bound level. They can decay via multiple channels, including nonradiative processes like autoionization or interaction with the environment. The paper makes clear that for such resonances, you can’t just tally decays to lower discrete levels. Instead, the total spontaneous emission rate must account for transitions to all discrete final states—both bound and resonant—as well as the continuum states that the system can reach after emitting a photon.
To handle this, the authors adopt a non-Hermitian quantum-mechanics viewpoint and derive a coherent expression for the spontaneous emission (SE) decay rate that explicitly links the rate to the finite lifetimes of the metastable states. The formalism treats electron dissociation and photon emission on the same footing and uses complex scaling to construct the spectrum of resonance states. This is more than a mathematical trick: it provides a way to organize the problem so that a computation that once looked hopeless becomes a sequence you can perform with a handful of well-chosen states.
One of the most striking conceptual shifts is the claim that the initial excitation of a resonance is not just a single energy, but a distribution across the real-energy continuum. Consequently, meaningful SE decay rates emerge only after you account for all accessible final states, including those with higher real energy than the nominal resonance energy. It’s a reminder that in the quantum world, what you start from can reach far beyond the most obvious destination once you open the door to the surrounding continuum. The paper’s careful treatment explicitly connects the measured decay rate to these multiple pathways, while the energy shifts (Lamb-like shifts) demand their own careful regularization.
From a practical standpoint, the team frames this as a necessary extension of the standard theory: when lifetimes become finite because the state is embedded in the continuum, the decay channels proliferate, and you must sum over them all to get a physically meaningful rate. This is the kind of result that sounds almost obvious in hindsight—open systems shouldn’t be modeled as perfectly closed—but it’s taken to a new level by showing how non-Hermitian machinery makes the calculation not just possible but convergent and transparent.
Turning the continuum into discrete poles
The heart of the approach is the complex-scaling (CS) transformation. Imagine you rotate the electronic coordinates by an angle θ into the complex plane. At the same time, the photonic coordinates are rotated in a coordinated way, so that the whole system’s continua rotate in a controlled fashion. The upshot is that metastable, embedded resonances turn into square-integrable states, and the continuum becomes a rotated set of complex poles in the energy plane. In plain terms: what used to be a dizzying, dense sea becomes a tractable scaffold of discrete points you can tally up.
From this rotated, non-Hermitian perspective, the total SE rate from a resonance state n splits into two kinds of contributions. The first is the sum of partial rates to all discrete final states, including bound states and the resonance poles themselves. The second is the integral over the rotated continuum, which captures the infinite family of continuum states that can participate in emission. While the split depends on the CS angle θ, the total rate is a physical observable and remains θ-independent—provided you account for how new poles emerge from the rotated continuum as you adjust θ. This is the key consistency check that keeps the math anchored to reality.
To illustrate the method without getting lost in high-dimensional thickets, the authors analyze a simple one-dimensional double-barrier potential. This toy model is enough to reveal the structure: as θ grows, more complex poles emerge from the rotated continuum, and the SE decay rate can be reconstructed by summing the contributions from these poles plus the rotated-continuum piece. The figures in the study show a striking pattern — the decay rate is already dominated by a relatively small handful of poles, and the remainder converges rapidly as more poles are included. In their example, the first twelve discrete poles account for more than 99.996% of the total rate. That’s not just elegant math; it’s a practical computational win.
Another subtle but important point is the way the method handles symmetry. In the model they analyze, certain discrete poles contribute with signs that reflect the spatial symmetry of the potential, so some poles can even contribute negatively to the total rate. This nuanced accounting would be nearly impossible to see or justify in a purely Hermitian treatment, but the non-Hermitian framework makes such structure natural and informative.
Rapid convergence makes it practical
The practical punchline is simple and powerful: you don’t need a mountain of continuum states to predict the SE decay rate of a metastable resonance. The trick is to pick a CS angle that reveals a compact set of dominant poles and then sum over those. The rest of the continuum’s influence is captured by the rotated-continuum integral, which becomes increasingly negligible as the pole set captures the essential physics. In their demonstration, including a dozen poles yields an almost exact rate, and adding more poles barely budges the result.
This isn’t just a numerical convenience. It reframes how we think about open systems at a fundamental level. In many-electron systems—think real atoms, quantum dots, and complex molecules—the spectrum is dense and the nonradiative channels multiply. The method shows a path to calculating SE rates from metastable resonances without sinking into a computational quagmire. It also clarifies how nonradiative lifetimes interplay with radiative decay, because the resonance width itself enters the SE rate through the complex energies, linking lifetime and emission in a single, coherent picture.
Of course, the energy shifts—the Lifshitz-like corrections that accompany emission—pose their own challenges. In a Hermitian world, these shifts are notorious for infrared divergences and require careful regularization. The authors extend the regularization logic to the non-Hermitian setting, showing how to define a finite, physically meaningful energy correction for metastable states. The upshot is a pair of tidy, physically interpretable quantities: a decay rate that incorporates all final channels and a finite energy correction that respects the open nature of the system. In short, the mathematics reads less like a maze and more like a robust, predictive framework that stays well-behaved as you scan across the spectrum.
Why this matters for real devices
Open quantum systems aren’t academic curiosities: they’re the rule in nanoscale devices. Quantum dots, photonic crystals, lasers, and other light-emitting technologies all operate in regimes where radiative decay competes with nonradiative channels, and where the environment—phonons, electromagnetic reservoirs, and imperfect isolation—plays a central role. The approach described in this paper offers a principled, first-principles way to predict SE lifetimes for metastable states in such systems, without resorting to ad hoc reservoir models or phenomenological damping terms. In other words, it’s a step toward models that can be trusted to guide the design of real-world emitters, not just idealized toy systems.
The study’s authors emphasize that predicting SE from metastable resonances is particularly relevant for devices where autoionization, predissociation, or strong environment coupling are at play. In those contexts, standard Hermitian single-branch analyses can suffice only so far. By incorporating the entire spectrum of possible final states in a unified, non-Hermitian framework, engineers and physicists gain a tool that can better reflect how a device actually behaves under operating conditions. This could, for example, influence how one tunes a quantum dot’s confinement landscape or engineers a photonic environment to suppress undesired nonradiative channels while preserving bright, efficient emission.
Crucially, the authors’ collaboration—rooted in Technion’s engineering and quantum-center ecosystems, with partners at GTIIT—highlights a broader push to bring advanced mathematical techniques into device-relevant physics. The lead author, Amir Sivan, together with Milan Šindelka, Meir Orenstein, and Nimrod Moiseyev, demonstrates that non-Hermitian methods aren’t exotic curiosities but practical instruments for contemporary quantum technologies. If you care about the lifetimes of quantum emitters, this work offers a new lens to predict, compare, and optimize what devices actually do when they glow in the presence of a real, messy environment.
A quiet revolution in how we model light and matter
The most enduring idea behind this work is not a single equation but a shift in perspective: bound-state intuition isn’t enough once metastable states and open channels enter the frame. Bound states in the continuum (BSCs) were long a theoretical curiosity, and metastable resonances are their practical cousins. The paper shows that the spontaneous emission rate from such resonances is determined by the entire spectrum—the discrete poles that emerge from the continuum and the continuum itself—yet remains computable with rapid convergence. It’s a unifying move that marries non-Hermitian quantum mechanics with concrete questions about how light is produced and lost in real systems.
In the end, the work is anchored in a real institution and a real team. The study comes from Technion’s Andrew and Erna Viterbi Department of Electrical & Computer Engineering and the Helen Diller Quantum Center, with significant contributions from the Guangdong Technion Israel Institute of Technology. The lead author, Amir Sivan, and collaborators Milan Šindelka, Meir Orenstein, and Nimrod Moiseyev chart a path that could illuminate a wide class of open-system problems—from quantum dots to lasers—by treating spontaneous emission as a spectrum-spanning, non-Hermitian journey rather than a single-step hop. If the future of photonics will be written in open-system language, this paper is a promising first paragraph.
