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When a quantum system undergoes slow changes, the evolution of its state depends only on the corresponding trajectory in Hilbert space. This phenomenon, known as quantum holonomy, brings to light the geometric aspects of quantum theory. Depending on the number of degrees of freedom involved, these purely geometric entities can be scalar or belong to a matrix-valued symmetry group. In their various forms, holonomies are vital elements in the description of the fundamental forces in particle physics as well as theories beyond the standard model such as loop quantum gravity or topological quantum field theory. Yet, implementing matrix-valued holonomies thus far has proven challenging, being further complicated by the difficulties involved in identifying suitable dark states for their construction in bosonic systems. Here we develop a representation of holonomic theory founded on the Heisenberg picture and leverage these insights for the experimental realization of a three-dimensional quantum holonomy. Its non-Abelian geometric phase is implemented via the judicious manipulation of bosonic modes constructed from indistinguishable photons and obeys the U(3) symmetry relevant to the strong interaction. Our findings could enable the experimental study of higher-dimensional non-Abelian gauge symmetries and the exploration of exotic physics on a photonic chip.

The generation of non-Abelian geometric phases from a system of evanescently coupled waveguides is extended towards the framework of nonorthogonal coupled-mode theory. Here, we study an experimentally feasible tripod arrangement of waveguides that contain dark states from which a nontrivial U(2)-mixing can be obtained by means of an adiabatic parameter variation. We investigate the influence of higher-order contributions beyond nearest-neighbour coupling as well as self-coupling on the stability of a U(3)-phase generated from an optical tetrapod setup. Our results indicate that, despite the mode nonorthogonality, the symmetry of dark states protects the geometric evolution of light from distortion., Comment: 8 pages, 11 figures

A proposal for the construction of geometrically robust linear optical quantum circuits by means of nonadiabatic geometric phases is presented. Our proof-of-principle designs of holonomic single-qubit gates enable high-fidelity unitaries.

We present the experimental realization of non-adiabatic holonomic quantum gates, paving the way towards noise-resilient quantum computing in integrated quantum optics. Their topologically protected functionalities are realized solely by means of non-Abelian geometric phases.

International audience; At nanometer separation, the dominant interaction between an atom and a material surface is the fluctuation-induced Casimir-Polder potential. We demonstrate that slow atoms crossing a silicon nitride transmission nanograting are a remarkably sensitive probe for that potential. A 15% difference between nonretarded (van der Waals) and retarded Casimir-Polder potentials is discernible at distances smaller than 51 nm. We discuss the relative influence of various theoretical and experimental parameters on the potential in detail. Our work paves the way to high-precision measurement of the Casimir-Polder potential as a prerequisite for understanding fundamental physics and its relevance to applications in quantum-enhanced sensing.

We study dipole interseries transitions between the yellow and green exciton series in cuprous oxide including the complex valence band structure. To this end, we extend previous studies of the spectrum of complex green exciton resonances [Phys. Rev. B 101, 075208 (2020)] to optical transitions between different exciton states in addition to transitions from the crystal ground state. This allows us to augment the calculations on interseries transitions using a hydrogen-like model [Phys. Rev. B 100, 085201 (2019)] by a more comprehensive treatment of the valence band structure., 12 pages, 5 figures

A common wisdom in quantum mechanics is that the Hamiltonian has to be Hermitian in order to ensure a real eigenvalue spectrum. Yet, parity–time (PT)-symmetric Hamiltonians are sufficient for real eigenvalues and therefore constitute a complex extension of quantum mechanics beyond the constraints of Hermiticity. However, as only single-particle or classical wave physics has been exploited so far, an experimental demonstration of the true quantum nature of PT symmetry has been elusive. In our work, we demonstrate two-particle quantum interference in a PT-symmetric system. We employ integrated photonic waveguides to reveal that the quantum dynamics of indistinguishable photons shows strongly counterintuitive features. To substantiate our experimental data, we analytically solve the quantum master equation using Lie algebra methods. The ideas and results presented here pave the way for non-local PT-symmetric quantum mechanics as a novel building block for future quantum devices. Parity–time symmetry in second quantization is demonstrated in an integrated non-Hermitian coupled waveguide structure. A counterintuitive shift of the position of the Hong–Ou–Mandel dip is observed in integrated lossy waveguide structures.

Single-shot wide-angle diffraction imaging is a widely used method to investigate the structure of noncrystallizing objects such as nanoclusters, large proteins, or even viruses. Its main advantage is that information about the three-dimensional structure of the object is already contained in a single image. This makes it useful for the reconstruction of fragile and nonreproducible particles without the need for tomographic measurements. However, currently there is no efficient numerical inversion algorithm available that is capable of determining the object's structure in real time. Neural networks, on the other hand, excel in image processing tasks suited for such purpose. Here we show how a physics-informed deep neural network can be used to reconstruct complete three-dimensional object models of uniform, convex particles on a voxel grid from single two-dimensional wide-angle scattering patterns. We demonstrate its universal reconstruction capabilities for silver nanoclusters, where the network uncovers novel geometric structures that reproduce the experimental scattering data with very high precision.

We experimentally realize a U(3) holonomy. By adiabatically propagating quantum states in appropriately designed photonic waveguide systems, we evolve on closed loops within a degenerate subspace of dark states, resulting in a non-Abelian geometric phase.

The collective response of an atomic ensemble is shaped by its macroscopic environment. We demonstrate this effect in the near-resonant transmission of light through a thermal rubidium vapor confined in a planar nanocavity. Our model reveals density-dependent line shifts and broadenings beyond continuous electrodynamics models that oscillate with cavity width and have been observed in recent experiments. We predict that the amplitudes of these oscillations can be controlled by coatings that modify the cavity's Finesse., Comment: 12 pages, 5 figures

We investigate the influence of a static, uncorrelated distribution of charged impurities on the spectrum of bound excitons in the copper oxide Cu$_{\text{2}}$O. We show that the statistical distribution of Stark shifts and ionisation rates leads to the vanishing of Rydberg resonances into an apparent continuum. The appearance of additional absorption lines due to the broken rotational symmetry, together with spatially inhomogeneous Stark shifts, leads to a modification of the observed line shapes that agree qualitatively with the changes observed in the experiment., Comment: 6 pages, 4 figures

We investigate an all-out optical setup allowing for generation of non-Abelian geometric phases on its large degenerate eigenspaces. The proposal has the form of an M -pod system and can be implemented in terms of integrated photonic waveguide structures. We show that by injecting a larger number of photons into the optical setup, the degeneracy of eigenspaces scales rapidly. After studying the spectral properties of our system for the general case, we show how arbitrary U(4) transformations can be generated on the dark subspace of an optical tripod filled with two photons. Moreover, a degeneracy in the bright subspaces of the system, absent in any atomic analogue, allows for the generation of universal single-qubit manipulations. Finally, we address the complexity issue of holonomic computation. Particularly, we show how two-qubit and three-qubit states can be implemented on a photonic tripod, where a natural multi-partite structure is inherited from the spatial mode structure of the waveguides., Comment: 11 pages, 3 figures

We theoretically investigate Rydberg atom pair potentials of Rb atoms in front of a perfectly conducting plate. The pair potentials are perturbed by both the Casimir--Polder potential acting on a single atom and the scattering contribution to the interatomic interaction. In contrast to the pair potentials in free space, at atom-surface distances $d_s \lesssim 4\,\mu$m, avoided crossings appear. In the associated potential wells that are entirely due to dispersion interactions with the surface, there exist vibrational bound states, i.e. Rydberg macrodimers, with groundstate energies of up to $E = -68\,$MHz and radial expectation values of the order of several $\mu$m., Comment: 6 pages, 6 figures

Geometric phases, which are ubiquitous in quantum mechanics, are commonly more than only scalar quantities. Indeed, often they are matrix-valued objects that are connected with non-Abelian geometries. Here we show how generalized, non-Abelian geometric phases can be realized using electromagnetic waves travelling through coupled photonic waveguide structures. The waveguides implement an effective Hamiltonian possessing a degenerate dark subspace, in which an adiabatic evolution can occur. The associated quantum metric induces the notion of a geodesic that defines the optimal adiabatic evolution. We exemplify the non-Abelian evolution of an Abelian gauge field by a Wilson loop.

The time-dependent pseudo-Hermitian formulation of quantum mechanics allows to study open system dynamics in analogy to Hermitian quantum systems. In this setting, we show that the notion of holonomic quantum computation can equally be formulated for pseudo-Hermitian systems. Starting from a degenerate pseudo-Hermitian Hamiltonian we show that, in the adiabatic limit, a non-Abelian geometric phase emerges which realizes a pseudounitary quantum gate. We illustrate our findings by studying a pseudo-Hermitian gain/loss system which can be written in the form of a tripod Hamiltonian by using the biorthogonal representation. It is shown that this system allows for arbitrary pseudo-$\mathrm{U}(2)$ transformations acting on the dark subspace of the system., 8 pages, 1 figure

We investigate the infrared optical transitions between excitons of the yellow, green, and blue series in the cuprous oxide ${\mathrm{Cu}}_{2}\mathrm{O}$. We show that, in many cases, the dipole approximation is inadequate and, in particular, that it breaks down in yellow-blue transitions even for moderate principal quantum numbers of $n\ensuremath{\approx}10$. The interband matrix elements of the transition operator needed for the evaluation of the excitonic transition strengths are derived from known as well as from fitted band parameters.

Single-shot X-ray imaging of short-lived nanostructures such as clusters and nanoparticles near a phase transition or non-crystalizing objects such as large proteins and viruses is currently the most elegant method for characterizing their structure. Using hard X-ray radiation provides scattering images that encode two-dimensional projections, which can be combined to identify the full three-dimensional object structure from multiple identical samples. Wide-angle scattering using XUV or soft X-rays, despite yielding lower resolution, provides three-dimensional structural information in a single shot and has opened routes towards the characterization of non-reproducible objects in the gas phase. The retrieval of the structural information contained in wide-angle scattering images is highly non-trivial, and currently no efficient rigorous algorithm is known. Here we show that deep learning networks, trained with simulated scattering data, allow for fast and accurate reconstruction of shape and orientation of nanoparticles from experimental images. The gain in speed compared to conventional retrieval techniques opens the route for automated structure reconstruction algorithms capable of real-time discrimination and pre-identification of nanostructures in scattering experiments with high repetition rate -- thus representing the enabling technology for fast femtosecond nanocrystallography., 14 pages, 5 figures

Parity-Time (PT)-symmetric systems have been intensely studied since their first description by Bender & Boettcher in 1998 [1]. Especially in photonics, numerous PT-symmetric effects and systems were investigated [2]. However, all of these experiments used classical light or single photons. Here, we present the first observation of two-photon interference in a lossy directional coupler and its associated Hong-Ou-Mandel (HOM) dip.

When quantum systems evolve through Hilbert space, they acquire geometric phases [1]. For many applications, this can be expressed by a line integral over an (Abelian) vector potential, which leads to a scalar phase. However, systems with degenerate energy levels experience matrix-valued geometric phases [2], known as Non-Abelian holonomies. While Non-Abelian structures naturally arise in Yang Mills theories, they have also proven to be cruicial for topological quantum computation [3]. In our work we use an integrated photonic structure to realize a non-Abelian phase utilizing an Abelian gauge field, namely electro-magnetic wavepackets.

In this work we present the eigenspectra of a novel species of Wannier excitons when exposed to crossed electric and magnetic fields. In particular, we compute the eigenenergies of giant-dipole excitons in ${\mathrm{Cu}}_{2}\mathrm{O}$ in crossed fields. In our theoretical approach, we calculate the excitonic spectra within both an approximate as well as a numerically exact approach for arbitrary field configurations. We verify that stable bound excitonic giant-dipole states are only possible in the strong magnetic field limit, as this is the only regime providing sufficiently deep potential wells for their existence. Comparing both analytic as well as numerical calculations, we obtain excitonic giant-dipole spectra with level spacings in the range of 0.6--100 $\ensuremath{\mu}\mathrm{eV}$.

Based on a Liouville-space formulation of open systems, we present two methods to solve the quantum dynamics of coupled harmonic oscillators experiencing Markovian loss. Starting point is the quantum master equation in Liouville space which is generated by a Liouvillian that induces a Lie algebra. We show how this Lie algebra allows to define ladder operators that construct Fock-like eigenstates of the Liouvillian. These eigenstates are used to decompose the time-evolved density matrix and, together with the accompanying eigenvalues, provide insight into the transport properties of the lossy system. Additionally, a Wei-Norman expansion of the generated time evolution can be found by a structure analysis of the algebra. This structure analysis yields a construction principle to implement effective non-Hermitian Hamiltonians in lossy systems., Comment: 10 pages, 1 figure

We investigate the formation of waveguides for Rydberg excitons in ${\mathrm{Cu}}_{\text{2}}\mathrm{O}$ from cylindrical stressors as alternatives to optical traps. We show that the achievable potential depths can easily reach the meV and the trap frequencies the $\mathrm{GHz}$ regimes. For Rydberg excitons, we find that it is sufficient to consider only the shift of the band gap, whereas the excitonic binding energies remain almost unchanged.

Casimir–Polder forces—dispersion forces between point-like microscopic particles and macroscopic dielectric bodies caused by ground-state fluctuations of the electromagnetic field—have long been studied theoretically and experimentally for point-like particles. In order to describe the interaction of large molecules with typically more than 100 atoms, an extended theoretical concept beyond the point-dipole assumption is required. Here we develop a formalism to determine the rotationally averaged Casimir–Polder potential on extended rotating molecules.

We demonstrate under which conditions a violation of the charge-parity (CP) symmetry in molecules will manifest itself in the Casimir-Polder interaction of these with a magnetodielectric surface. Charge-parity violation induces a specific electric-magnetic cross-polarisability in a molecule that is not chiral, but time-reversal (T) symmetry violating. As we show, a detection of such an effect via the Casimir-Polder potential requires a material medium that is also sensitive to time-reversal, i.e., it must exhibit a non-reciprocal electromagnetic response. As simple examples of such media we consider a perfectly reflecting non-reciprocal mirror that is a special case of a perfect electromagnetic conductor as well as a Chern-Simons medium. In addition, we show that Chern-Simons and related media can induce unusual atom-surface interactions for anisotropic molecules with and without a chiral response., Comment: 5 pages

Van der Waals interactions, as a result of the exchange of photons between particles, can be altered by modifying the environment through which these photons propagate. As a consequence, phenomena such as the Rydberg blockade mechanism between highly excited atoms or excitons, can be controlled by adding reflecting surfaces. We provide the quantum electrodynamic framework for the van der Waals interaction in the nonretarded limit that is relevant for long-wavelength transitions such as those between Rydberg systems, and show its intimate connection with common static dipole-dipole interactions, thereby providing a generalization to include macroscopic bodies., 9 pages, 6 figures

In this article we extend the theory of ultra-long-range giant dipole molecules, formed by an atom in a giant dipole state and a ground-state alkali atom, by angular-momentum couplings known from recent works on Rydberg molecules. In addition to $s$-wave scattering, the next higher order of $p$-wave scattering in the Fermi-pseudopotential describing the binding mechanism is considered. Furthermore, the singlet and triplet channels of the scattering interaction as well as angular-momentum couplings such as hyperfine interaction and Zeeman interactions are included. Within the framework of Born--Oppenheimer theory, potential energy surfaces are calculated in both first-order perturbation theory and exact diagonalization. Besides the known pure triplet states, mixed-spin character states are obtained, opening up a whole new landscape of molecular potentials. We determine exact binding energies and wave functions of the nuclear rotational and vibrational motion numerically from the various potential energy surfaces., 8 pages, 7 figures

In this work we present the interaction potentials and eigenenergies of a novel species of Wannier excitons when exposed to crossed electric and magnetic fields. More precisely, we present the theory of giant-dipole excitons in Cu2O. We derive an exact formulation of the field-dressed excitonic Hamiltonian and exemplarily calculate the excitonic eigenenergies in an exact diagonalization approach for external field strengths of B = 4T and E = 1 kV/cm. For this particular field configuration, we obtain level spacings between 1.14 μV and 77.6 μeV.

In this work we predict the existence of a novel species of Wannier excitons when exposed to crossed electric and magnetic fields. In particular, we present a theory of giant-dipole excitons in $\textrm{Cu}_2\rm O$ in crossed fields. Within our theoretical approach we perform a pseudoseparation of the center-of-mass motion for the field-dressed excitonic species, thereby obtaining an effective single-particle Hamiltonian for the relative motion. For arbitrary gauge fields we exactly separate the gauge-dependent kinetic energy terms from the effective single-particle interaction potential. Depending on the applied field strengths and the specific field orientation, the potential for the relative motion of electron and hole exhibits an outer well at spatial separations up to several micrometers and depths up to $380\, \mu \rm eV$, leading to possible permanent excitonic electric dipole moments of around three million Debye., Comment: 11 pages, 6 figures, 1 table, only minor changes compared to the first version, accepted for publication in Phys. Rev. B (in print)

In this work we perform a Green's function analysis of giant-dipole systems. First we derive the Green's functions of different magnetically field-dressed systems, in particular of electronically highly excited atomic species in crossed electric and magnetic fields, so-called giant-dipole states. We determine the dynamical polarizability of atomic giant-dipole states as well as the adiabatic potential energy surfaces of giant-dipole molecules in the framework of the Green's function approach. Furthermore, we perform an comparative analysis of the latter to and exact diagonalization scheme and show the general divergence behavior of the widely applied Fermi-pseudopotential approach. Finally, we derive the giant-dipole's regularized Green's function representation., Comment: 11 pages, 4 figures, few changes compared to first version, published in "Special issue on correlations in light-matter interactions" of J. Phys. B: At. Mol. Opt. Phys

We use two-color pump-probe spectroscopy to study Rydberg excitons in Cu$_2$O in the presence of free carriers injected by above-band-gap excitation. Already at plasma densities $\rho_\text{eh}$ below one hundredth electron-hole pair per \textmu m$^{3}$, the Rydberg exciton absorption lines are bleached while their energies remain constant, until they finally disappear, starting from the highest observed principal quantum number $n_\text{max}$. As confirmed by calculations, the band gap is reduced by many-particle effects caused by free carriers scaling as $\rho_\text{eh}^{1/2}$. An exciton line looses oscillator strength when the band edge approaches the exciton energy vanishing completely at the crossing point. We quantitatively describe this plasma blockade by introducing an effective Bohr radius that determines the energy distance to the shifted band edge. In combination with the negligible associated decoherence this opens the possibility to control the Rydberg exciton absorption through the plasma-induced band gap modulation.

Coherent optical control of individual particles has been demonstrated both for atoms and semiconductor quantum dots. Here we demonstrate the emergence of quantum coherent effects in semiconductor Rydberg excitons in bulk Cu$_2$O. Due to the spectral proximity between two adjacent Rydberg exciton states, a single-frequency laser may pump both resonances with little dissipation from the detuning. As a consequence, additional resonances appear in the absorption spectrum that correspond to dressed states consisting of two Rydberg exciton levels coupled to the excitonic vacuum, forming a V-type three-level system, but driven only by one laser light source. We show that the level of pure dephasing in this system is extremely low. These observations are a crucial step towards coherently controlled quantum technologies in a bulk semiconductor., Comment: 7 pages, 7 figures; substantially rewritten, close to published version including supplemental material

Dispersion forces such as Casimir, Casimir-Polder and van der Waals forces arise due to fluctuations of the quantised electromagnetic field. Any alteration of the environment in which the (virtual) photons propagate leads invariably to a modification of the respective interaction. Waveguide structures in particular are known to resonantly enhance Casimir-Polder interactions [1]. Here we focus on the van der Waals interaction between two isolated atoms inside a cylindrical waveguide and its applications to tunable interactions in hollow-core fibers. Based on the Green function formalism of macroscopic quantum electrodynamics, we have recently shown that the existence of a fundamental mode without cut-off [2] implies a far-field contribution to van der Waals interaction between excited atoms, thereby making it possible to observe the yet elusive retarded dispersion potential between two emitters. We show how to extend the fourth-order perturbation theory result [3] to van der Waals potentials at finite temperature and discuss the influence of guided modes on the resonantly enhanced interaction between (highly) excited atoms in a one-dimensional waveguide [4].

We construct the dyadic Green function of the electromagnetic field in the presence of a spatially dispersive spherical object. It contains all the information on light-scattering processes off a spatially dispersive sphere, and, in particular, its imaginary part gives the local density of states. Our construction of the Green function automatically yields a dispersion relation for surface polaritons in terms of surface impedances. From a methodological point of view, our approach is based on the extinction theorem and Huygens' principle. This allows for the straightforward application of Maxwell boundary conditions, without the need to resort to so-called additional boundary conditions. With this, we investigate the influence of spatial dispersion on the local density of states in the proximity of a nanosphere.

Rydberg excitons are semiconductor analogues to Rydberg atoms, where one electron is promoted to an energy level of large principal quantum number η and which behave in a manner similar to hydrogen. Their huge spatial extent results in giant dipole moments and interaction effects, which can be used to create nonlinearities at the single excitation level. In contrast to hydrogen, the effective masses and Rydberg energies involved are moderately small, so that in contrast to Rydberg atoms the high field limit of Rydberg physics can be studied using fields strengths that can be realized in the lab. Here we investigate the effects of external magnetic fields of up to 7T on Rydberg excitons both in Faraday and Voigt geometry. In both cases complicated splitting patterns emerge. We investigate the differences between the two geometries and highlight spectroscopic features that are especially easy to access using them. We show that the large number of resonances in the spectrum renders a microscopic treatment of each individual resonance implausible. We instead demonstrate general effects introduced by the field like avoided crossings and discuss alternative approaches to the level structure in terms of collective descriptions.

Recent high-resolution absorption spectroscopy on excited excitons in cuprous oxide [Nature (London) 514, 343 (2014)] has revealed significant deviations of their spectrum from that of the ideal hydrogen-like series. Here we show that the complex band dispersion of the crystal, which determines the kinetic energy of electrons and holes, strongly affects the exciton binding energy. Specifically, we show that the nonparabolicity of the band dispersion is the main cause of the deviation from the hydrogen series. Experimental data collected from high-resolution absorption spectroscopy in electric fields validate the assignment of the deviation to the nonparabolicity of the band dispersion.

We discuss the impact of gain and loss on the evolution of photonic quantum states and find that -symmetric quantum optics in gain/loss systems is not possible. Within the framework of macroscopic quantum electrodynamics we show that gain and loss are associated with non-compact and compact operator transformations, respectively. This implies a fundamentally different way in which quantum correlations between a quantum system and a reservoir are built up and destroyed.

We formulate a symmetry principle on the basis of the duality of electric and magnetic fields and apply it to dispersion forces. Within the context of macroscopic quantum electrodynamics, we rigorously establish duality invariance for the free electromagnetic field in the presence of causal magnetoelectrics. Dispersion forces are given in terms of the Green tensor for the electromagnetic field and the atomic response functions. After discussing the behavior of the Green tensor under a duality transformation, we are able to show that Casimir forces on bodies in free space as well as local-field corrected Casimir-Polder and van der Waals forces are duality invariant., 8 pages

This review surveys the physical principles and recent developments in manufacturing single-photon sources. Special emphasis is placed on important potential applications such as linear optical quantum computing (LOQC), quantum key distribution (QKD) and quantum metrology that drive the development of these sources of single photons. We discuss the quantum-mechanical properties of light prepared in a quantum state of definite photon number and compare it with coherent light that shows a Poissonian distribution of photon numbers. We examine how the single-photon fidelity directly influences the ability to transmit secure quantum bits over a predefined distance. The theoretical description of modified spontaneous decay, the main principle behind single-photon generation, provides the background for many experimental implementations such as those using microresonators or pillar microcavities. The main alternative way to generate single photons using postselection of entangled photon pairs from parametric dow...