Collective Lamb Shift Research Articles

Cooperative states and shift in resonant scattering of an atomic ensemble

We investigate the spectral shift known as the collective Lamb shift in forward scattering for a cold dense atomic cloud. The shift results from resonant dipole–dipole interaction mediated by real and virtual photon exchange, forming many-body states displaying various super- and subradiant behaviour. However, the scattering spectrum reflects the overall contributions from these states but also averages out the radiative details associated with the underlying spin orders, causing ambiguity in determination and raising controversy on the scaling property of this shift. We employ a Monte–Carlo simulation to study how the collective states contribute to emission. We thus distinguish two kinds of collective shift that follow different scaling laws. One results from dominant occupation of the near-resonant collective states. This shift is usually small and insensitive to the density or the number of participating atoms. The other comes from large spatial correlation of dipoles, associated with the states of higher degree of emission. This corresponds to larger collective shift that is approximately linearly dependent on the optical depth. We further demonstrate that the spatial spin order plays an essential role in superradiant emission. Our analysis provides a novel perspective for understanding collective scattering and cooperative effects.

Collective strong coupling in a plasmonic nanocavity.

Quantum plasmonics extends cavity quantum electrodynamics (cQED) concepts to the nanoscale, benefiting from the strongly subwavelength confinement of the plasmon modes supported by metal nanostructures. In this work, we describe in detail collective strong coupling to a plasmonic nanocavity. Similarities and differences to cQED are emphasized. We notably observe that the Rabi splitting can strongly deviate from the standard NeΔΩ1 law, where Ne is the number of emitters and ΔΩ1 is the Rabi splitting for a single emitter. In addition, we discuss the collective Lamb shift and the role of quantum corrections to the emission spectra.

Relative Entropy as a Measure of Difference between Hermitian and Non-Hermitian Systems.

We employ the relative entropy as a measure to quantify the difference of eigenmodes between Hermitian and non-Hermitian systems in elliptic optical microcavities. We have found that the average value of the relative entropy in the range of the collective Lamb shift is large, while that in the range of self-energy is small. Furthermore, the weak and strong interactions in the non-Hermitian system exhibit rather different behaviors in terms of the relative entropy, and thus it displays an obvious exchange of eigenmodes in the elliptic microcavity.

Large Collective Lamb Shift of Two Distant Superconducting Artificial Atoms.

Virtual photons can mediate interaction between atoms, resulting in an energy shift known as a collective Lamb shift. Observing the collective Lamb shift is challenging, since it can be obscured by radiative decay and direct atom-atom interactions. Here, we place two superconducting qubits in a transmission line terminated by a mirror, which suppresses decay. We measure a collective Lamb shift reaching 0.8% of the qubit transition frequency and twice the transition linewidth. We also show that the qubits can interact via the transmission line even if one of them does not decay into it.

Scalable collective Lamb shift of a 1D superconducting qubit array in front of a mirror

We theoretically investigate resonant dipole-dipole interaction (RDDI) between artificial atoms in a 1D geometry, implemented by N transmon qubits coupled through a transmission line. Similar to the atomic cases, RDDI comes from exchange of virtual photons of the continuous modes, and causes the so-called collective Lamb shift (CLS). To probe the shift, we effectively set one end of the transmission line as a mirror, and examine the reflection spectrum of the probe field from the other end. Our calculation shows that when a qubit is placed at the node of the standing wave formed by the incident and reflected waves, even though it is considered to be decoupled from the field, it results in large energy splitting in the spectral profile of a resonant qubit located at an antinode. This directly implies the interplay of virtual photon processes and explicitly signals the CLS. We further derive a master equation to describe the system, which can take into account mismatch of participating qubits and dephasing effects. Our calculation also demonstrates the superradiant and subradiant nature of the atomic states, and how the CLS scales when more qubits are involved.

Spectral Control of an X-Ray L-Edge Transition via a Thin-Film Cavity.

By embedding a thin layer of tantalum in an x-ray cavity, we observe a change in the spectral characteristics of an inner-shell transition of the metal. The interaction between the cavity mode vacuum and the L_{III}-edge transition is enhanced, permitting the observation of the collective Lamb shift, superradiance, and a Fano-like cavity-resonance interference effect. This experiment demonstrates the feasibility of cavity quantum electrodynamics with electronic resonances in the x-ray range with applications to manipulating and probing the electronic structure of condensed matter with high-resolution x-ray spectroscopy in an x-ray cavity setting.

Electro-optical effects in dense and cold atomic gases

On the basis of the general theoretical approach developed previously by Fofanov et al. [Ya. A. Fofanov et al., Phys. Rev. A 84, 053811 (2011)], we analyze the atomic polarization created by weak monochromatic light in an optically thick, dense, and cold atomic ensemble placed in an electrostatic field. On this basis, we determine complex refractive indexes and the permittivity tensor of the medium. We analyze the dispersion of the permittivity and its dependence on the strength of the field and on the density of the medium. We show that electro-optic effects in dilute and cold gases differ essentially from those in dense ones. In the latter case, electric field modifies the shape, amplitude, and typical linewidth of the atomic resonance. The electric field also modifies the collective Lamb shift. Observed peculiarities are explained as a result of the indirect influence of the field on optical properties of dense gases through modification of polyatomic collective effects caused by interatomic dipole-dipole interactions.

Non-Hermiticity and conservation of orthogonal relation in dielectric microcavity

Non-Hermitian properties of open quantum systems and their applications have attracted much attention in recent years. While most of the studies focus on the characteristic nature of non-Hermitian systems, here we focus on the following issue: A non-Hermitian system can be a subsystem of a Hermitian system as one can clearly see in Feshbach projective operator (FPO) formalism. In this case, the orthogonality of the eigenvectors of the total (Hermitian) system must be sustained, despite the eigenvectors of the subsystem (non-Hermitian) satisfy the bi-orthogonal condition. Therefore, one can predict that there must exist some remarkable processes that relate the non-Hermitian subsystem and the rest part, and ultimately preserve the Hermiticity of the total system. In this paper, we study such processes in open elliptical microcavities. The inner part of the cavity is a non-Hermitian system, and the outer part is the coupled bath in FPO formalism. We investigate the correlation between the inner- and the outer-part behaviors associated with the avoided resonance crossings (ARCs), and analyze the results in terms of a trade-off between the relative difference of self-energies and collective Lamb shifts. These results come from the conservation of the orthogonality in the total Hermitian quantum system.

Collective Lamb Shift of a Nanoscale Atomic Vapor Layer within a Sapphire Cavity.

We measure the near-resonant transmission of light through a dense medium of potassium vapor confined in a cell with nanometer thickness in order to investigate the origin and validity of the collective Lamb shift. A complete model including the multiple reflections in the nanocell reproduces accurately the observed line shape. It allows the extraction of a density-dependent shift and width of the bulk atomic medium resonance, deconvolved from the cavity effect. We observe an additional, unexpected dependence of the shift with the thickness of the medium. This extra dependence demands further experimental and theoretical investigations.

Shannon entropy and avoided crossings in closed and open quantum billiards.

The relation between Shannon entropy and avoided crossings is investigated in dielectric microcavities. The Shannon entropy of the probability density for eigenfunctions in an open elliptic billiard as well as a closed quadrupole billiard increases as the center of the avoided crossing is approached. These results are opposite to those of atomic physics for electrons. It is found that the collective Lamb shift of the open quantum system and the symmetry breaking in the closed chaotic quantum system have equivalent effects on the Shannon entropy.

Experimental Observation of One-Dimensional Superradiance Lattices in Ultracold Atoms.

We measure the superradiant emission in a one-dimensional (1D) superradiance lattice (SL) in ultracold atoms. Resonantly excited to a superradiant state, the atoms are further coupled to other collectively excited states, which form a 1D SL. The directional emission of one of the superradiant excited states in the 1D SL is measured. The emission spectra depend on the band structure, which can be controlled by the frequency and intensity of the coupling laser fields. This work provides a platform for investigating the collective Lamb shift of resonantly excited superradiant states in Bose-Einstein condensates and paves the way for realizing higher dimensional superradiance lattices.

Collective radiation spectrum for ensembles with Zeeman splitting in single-photon superradiance

In an ensemble of identical atoms, cooperative effects like sub- or superradiance may alter the decay rates and the energy of specific transitions may be shifted from the single-atom value by the so-called collective Lamb shift. While such effects in ensembles of two-level systems are by now well understood, realistic multi-level systems are more difficult to handle. In this work we show that in a system of atoms or nuclei under the action of an external magnetic field, the collective contribution to the level shifts can amount to seizable deviations from the single-atom Zeeman or magnetic hyperfine splitting picture. We develop a formalism to describe single-photon superradiance in multi-level systems in the small sample limit and quantify the parameter regime for which the collective Lamb shift leads to measurable deviations in the magnetic-field-induced splitting. In particular, we show that this effect should be observable in the nuclear magnetic hyperfine splitting in M\"ossbauer nuclei embedded in thin-film x-ray cavities.

Collective dipole-dipole interactions in an atomic array

The coherent dipole-dipole interactions of atoms in an atomic array are studied. It is found that the excitation probability of an atom in an array parallel to the direction of laser propagation ($\boldsymbol{\hat{k}}$) will either grow or decay logarithmically along $\boldsymbol{\hat{k}}$, depending on the detuning of the laser. The symmetry of the system for atomic separations of $\delta r = j\lambda/2$, where $j$ is an integer, causes the excitation distribution and scattered radiation to abruptly become symmetric about the center of the array. For atomic separations of $\delta r < \lambda/2$, the appearance of a collection of extremely subradiant states ($\Gamma\sim 0$), disrupts the described trend. In order to interpret the results from a finite array of atoms, a band structure calculation in the $N\rightarrow \infty$ limit is conducted where the decay rates and the Collective Lamb Shifts of the eigenmodes along the Brillouin zone are shown. Finally, the band structure of an array strongly affects its scattered radiation, allowing one to manipulate the Collective Lamb Shift as well as the decay rate (from superradiant to subradiant) by changing the angle of the driving laser.

Symmetry-protected single-photon subradiance

We study the protection of subradiant states by the symmetry of the atomic distributions in the Dicke limit, in which collective Lamb shifts cannot be neglected. We find that anti-symmetric states are subradiant states for distributions with reflection symmetry. Continuous symmetry can also be used to achieve subradiance. This study is relevant to the problem of robust quantum memory with long storage time and fast readout.

Single-photon superradiance and radiation trapping by atomic shells

The collective nature of light emission by atomic ensembles yields fascinating effects such as superradiance and radiation trapping even at the single-photon level. Light emission is influenced by virtual transitions and the collective Lamb shift which yields peculiar features in temporal evolution of the atomic system. We study how two-dimensional atomic structures collectively emit a single photon. Namely, we consider spherical, cylindrical, and spheroidal shells with two-level atoms continuously distributed on the shell surface and find exact analytical solutions for eigenstates of such systems and their collective decay rates and frequency shifts. We identify states which undergo superradiant decay and states which are trapped and investigate how size and shape of the shell affects collective light emission. Our findings could be useful for quantum information storage and the design of optical switches.

Properties of far-field fluorescence from an ensemble of interacting Sr atoms

We systematically investigate the far-zone fluorescence from a number of laser-driven two-level atoms with the cooperative effects stemming from the virtual exchange of photons between identical atoms. Due to the large interaction-induced energy level shifts, i.e. the collective Lamb shifts, both atomic excitation and photon emission do not maximize at the resonant light-atom coupling. In addition, the long-range dissipative interactions among atoms lead to the disproportion between the atomic excitation and the photon emission. Moreover, we discuss the temporal properties of the far-field fluorescence light from the atomic ensemble. The first-order correlation function indicates a damped light-atom dynamics combined with a strong background of the elastic photon scattering. The second-order correlation function illustrates that in the weak-excitation limit the scattered light exhibits the nonclassical photon antibunching behavior, while the bunched photons arrive at the photodetector in the saturated-excitation limit.

Photon-mediated qubit interactions in one-dimensional discrete and continuous models

In this work we study numerically and analytically the interaction of two qubits in a one-dimensional waveguide, as mediated by the photons that propagate through the guide. We develop strategies to assert the Markovianity of the problem, the effective qubit-qubit interactions and their individual and collective spontaneous emission. We prove the existence of collective Lamb-shifts that affect the qubit-qubit interactions and the dependency of coherent and incoherent interactions on the qubit separation. We also develop the scattering theory associated to these models and prove single photon spectroscopy does probes the renormalized resonances of the single- and multi-qubit models, in sharp contrast with earlier toy models where individual and collective Lamb shifts cancel.

Shifts of a Resonance Line in a Dense Atomic Sample

We study the collective response of a dense atomic sample to light essentially exactly using classical-electrodynamics simulations. In a homogeneously broadened atomic sample there is no overt Lorentz-Lorenz local field shift of the resonance, nor a collective Lamb shift. However, the addition of inhomogeneous broadening restores the usual mean-field phenomenology.

Cooperative emission from nuclei: The collective Lamb shift and electromagnetically induced transparency

AbstractThe narrow nuclear resonances of Mössbauer isotopes constitute almost ideal two‐level systems to study cooperative effects in the interaction of x‐rays with matter. Embedding an ensemble of resonant Mössbauer nuclei into a planar cavity facilitates the excitation of superradiant oscillatory eigenmodes via pulses of synchrotron radiation. These modes exhibit a large collective Lamb shift that can be controlled via the excitation conditions and the cavity geometry. Cooperative emission in connection with the radiative coupling of two resonant ensembles in a cavity leads to electromagnetically induced transparency (EIT). This contribution gives a review on cooperative effects in nuclear resonant scattering of synchrotron radiation. Special emphasis is given to the basic principles of recent experiments that led to the observation of the collective Lamb shift and EIT with nuclear resonances in the regime of hard x‐rays.

The collective Lamb shift in nuclear γ-ray superradiance

The electromagnetic transitions of Mossbauer nuclei provide almost ideal two-level systems to transfer quantum optical concepts into the regime of hard x-rays. If many identical atoms collectively interact with a resonant radiation field, one observes (quantum) optical properties that are strongly different from those of a single atom. The most prominent effect is the broadening of the resonance line known as collective enhancement, resulting from multiple scattering of real photons within the atomic ensemble. On the other hand, the exchange of virtual photons within the ensemble leads to a tiny energy shift of the resonance line, the collective Lamb shift, that remained experimentally elusive for a long time after its prediction. Here we illustrate how highly brilliant synchrotron radiation allows one to prepare superradiant states of excited Mossbauer nuclei, an important condition for observation of the collective Lamb shift.