Engineering a Bound State in the Continuum via Quantum Interference

TL;DR

The paper demonstrates a Friedrich–Wintgen bound state in the continuum (BIC) realized in genuine quantum matter by coherently coupling two Floquet-engineered Feshbach resonances in ultracold $^6$Li collisions. A minimal non-Hermitian two-level model and full coupled-channel calculations capture the interference that decouples a bound-like state from the scattering continuum at a critical detuning $\delta_{\mathrm{BIC}}$. Experimentally, both elastic and inelastic couplings to the continuum vanish at the BIC, as shown by loss spectroscopy, trap-quench dynamics, and rf photoassociation, with a narrow bound-state width of about $2.7$ mG and an accompanying broad resonance. These results establish quantum interference as a versatile tool to control openness in quantum systems and pave the way for engineered non-Hermitian dynamics and long-lived molecular states.

Abstract

Quantum mechanical interaction potentials typically support either localized bound states below the dissociation threshold or delocalized scattering states above it. While bound states are energetically isolated, scattering states embed a quantum system in a continuum of environmental modes, making dissipation and loss intrisic features of open quantum systems. A striking exception are bound states in the continuum (BICs), which remain localized despite lying within the scattering continuum due to destructive interference. It was predicted that such states can arise from the interference of two Feshbach resonances coupled to a common continuum, yet this mechanism has remained experimentally inaccessible in genuine quantum systems. Here we demonstrate the formation of such an interference-stabilized state in ultracold collisions of ${}^6$Li atoms by coherently coupling two tunable Feshbach resonances using Floquet engineering. At a critical parameter point, both elastic and inelastic coupling to the continuum vanish, yielding a molecular state above the dissociation threshold. Loss spectroscopy, quench dynamics, and rf-photoassociation directly reveal the resulting decoupling from scattering states. Our observations are quantitatively captured by full coupled-channel calculations and a minimal non-Hermitian model, identifying a Friedrich-Wintgen BIC. Our results establish quantum interference as a powerful mechanism for controlling openness in quantum matter and for engineering non-Hermitian Hamiltonians.

Figures (7)

• Figure 1: Emergence of a Friedrich–Wintgen BIC.(a) Energy spectrum and wavefunctions for a molecular potential $V_1$. Below the molecular dissociation threshold $V_1(r\to\infty)$, discrete bound states exist with wavefunctions localized in the potential well. Above the threshold, the spectrum becomes continuous and the solutions describe delocalized scattering states. (b) For alkali atoms, spin-dependent coupling between the open channel $V_1$ (triplet) and a closed channel $V_2$ (singlet) gives rise to Feshbach resonances, producing finite-lifetime resonant states at energy $E_1$ that mixes closed localized (blue) and open scattering (red) channels. Since the singlet and triplet channels possess different magnetic moments, an external magnetic field $B_0$ shifts their relative energies and thus allows the resonance position to be tuned. (c) In the presence of a second closed channel $V_3$ (yellow), destructive interference between two individual Feshbach resonances at energies $E_{1,2}$ can decouple one of them from the continuum if the detuning $\delta=E_1-E_2$ fulfills the BIC condition. In this case, the delocalized open channel contribution (red) vanishes completely and only spatially localized closed channel contributions (blue and yellow) remain, forming a Friedrich–Wintgen BIC above molecular dissociation threshold. Floquet engineering provides the modulation frequency $\nu$ as a continuous tuning parameter for $\delta$. (d) Effective non-Hermitian two level system modeling the Friedrich–Wintgen BIC mechanism. Two states with energies $E_{1,2}$ are coupled to each other with Rabi-frequency $\Omega$ and to a common continuum with coherent rates $\gamma_{1,2}$; rates $\gamma'_{1,2}$ take into account incoherent decay. Direct and continuum-mediated coupling pathways can interfere destructively canceling the effective decay of one resonance and yielding a non-decaying bound state embedded in the continuum.
• Figure 2: Appearance of a BIC at an avoided crossing of Feshbach resonances.(a) Modulation-frequency dependence of Floquet–Feshbach resonances. Floquet modulation dresses the molecular states and induces additional Feshbach resonances that tune differently with modulation frequency depending on the underlying molecular state causing the resonance. When two such resonances come close an avoided crossing forms. (b) Experimental atom-loss spectroscopy across the avoided crossing between the $(I=2,\Delta_N=0)$ and $(I=0,\Delta_N=+1)$ Floquet–Feshbach resonances. The lower loss branch vanishes in two distinct regions: I around a modulation frequency of $\qty{55.8}{MHz}$ and II near $\qty{58.5}{MHz}$. The BIC forms in region I. The faint horizontal feature between the two branches stems from the static $I=2$ resonance and is an artifact of the experimental sequence. (c) Coupled-channel calculations of the thermally averaged inelastic scattering cross section $\sigma^\text{avg}_\text{inel}$ reproduce the observed avoided crossing and identify the same regions I and II. (d) Dependence of the coupling strength $\Omega$ (upper panel) and numerically calculated position of the avoided crossing $\nu_0$ (lower panel) on the modulation amplitude $B_\text{rf}$. The coupling strength increases linearly with $B_\text{rf}$ (red dots: experiment; faint blue dots: coupled-channel calculations; blue line: linear fit, slope $\qty{37.65}{kHz/G}$. Strong modulation also induces an AC-Zeeman shift that quadratically shifts the crossing's center frequency $\nu_0$ (faint blue dots: numerical data; blue line: quadratic fit $\nu_0 = \alpha B_\text{rf}^2+\beta$, with: $\alpha=\qty{0.0149}{MHz/G^2}$ and $\beta=\qty{56.04}{MHz}$).
• Figure 3: Characterization of the BIC.(a), (b) Response of the gas to a sudden trap quench recorded at points I (a) and II (b). At point I, the pole in $\text{Re}(a_s)$ vanishes, and the cloud exhibits no collective motion, demonstrating that the state is decoupled from the continuum forming a BIC. In contrast at II, the resonance pole in $\text{Re}(a_s)$ drives pronounced collective breathing oscillations, which can be fitted with an exponentially decaying sine with amplitude $A$ and decay coefficient $\gamma$ (see Methods). (c) Cuts through the avoided crossing at selected modulation frequencies. The upper panels show the relative remaining atom number after modulation; the lower panels show the real (blue) and imaginary (ochre) parts of the $s$-wave scattering length $a_s$, associated with elastic and inelastic processes, respectively. At point I both the pole in $\text{Re}(a_s)$ and the extrema in $\text{Im}(a_s)$ vanish, indicating decoupling from the continuum and formation of a BIC. At point II only the imaginary part is suppressed, yielding resonant elastic scattering with suppressed losses.
• Figure 4: Comparison with the non-Hermitian two-level model.(a) Resonance positions extracted from the real parts of the eigenvalues of the effective non-Hermitian two-level Hamiltonian (solid line), compared with full coupled-channel calculations (crosses) and experimental data (dots). (b) Corresponding resonance widths derived from the imaginary parts of the eigenvalues (solid line) and from coupled-channel calculations (crosses). The BIC appears at the minimum of the resonance width. The small discrepancy between the predicted and experimentally observed BIC frequencies likely originates from uncertainties in the calibrated modulation amplitude and the underlying interaction potentials. The parameters of Eq. \ref{['eq:two_level_hamiltonian']} were iteratively optimized to reproduce the coupled-channel results; details are provided in the Methods section. Colors indicate the admixture of the uncoupled bare states to the resulting eigenstates.
• Figure 5: RF photoassociation spectroscopy.(a) Energy-level diagram and working principle of RF photoassociation. Dashed lines denote atomic thresholds (thrs.), while solid lines indicate molecular bound states below threshold and scattering states above. The gas is prepared in a $\ket{a}$-$\ket{c}$ mixture, and the RF field drives the $\ket{c}\rightarrow\ket{b}$ transition (arrow (1)). Besides single-atom spin flips, two colliding atoms in the $\ket{a}$-$\ket{c}$ channel can be photoassociated into $\ket{a}$-$\ket{b}$ molecules (arrow (2)), with a rate set by the Franck–Condon overlap of the scattering states. (b) Atom-loss spectrum by rf photoassociation (lower panel) and imaginary part of $a_s$ (upper panel) without Floquet modulation. The RF detuning $\nu_\delta$ defines the probed kinetic energy $E_\text{kin}$. Two loss features appear: (1) the single-atom spin-flip transition and (2) a two-body rf photoassociation resonance linked to the static $I=2$ Feshbach resonance. (c) With Floquet modulation of $\qty{55.74}{MHz}$, the dressed $(I=0,\Delta_N=+1)$ resonance intersects the static $(I=2,\Delta_N=0)$ resonance at the $\ket{a}$-$\ket{b}$ threshold. (d) Corresponding spectrum showing the atomic spin-flip resonance (1) and two photoassociation features: the original narrow (2) and a broad (3) resonance. (3) acquires the full width, while (2) becomes narrow, marking the formation of a BIC. The deviation of the fit from the data for (3) might be caused by a mixture of energy broadening and a varying rabi frequency with spectroscopy frequency. A slight position shift can be observed due to the coupling between the two resonances.