Ground State Occupation Research Articles

On the spectral gap in the Kac–Luttinger model and Bose–Einstein condensation

We consider the Dirichlet eigenvalues of the Laplacian among a Poissonian cloud of hard spherical obstacles of fixed radius in large boxes of Rd, d≥2. In a large box of side-length 2ℓ centered at the origin, the lowest eigenvalue is known to be typically of order (logℓ)−2/d. We show here that with probability arbitrarily close to 1 as ℓ goes to infinity, the spectral gap stays bigger than σ(logℓ)−(1+2/d), where the small positive number σ depends on how close to 1 one wishes the probability. Incidentally, the scale (logℓ)−(1+2/d) is expected to capture the correct size of the gap. Our result involves the proof of new deconcentration estimates. Combining this lower bound on the spectral gap with the results of Kerner–Pechmann–Spitzer, we infer a type-I generalized Bose–Einstein condensation in probability for a Kac–Luttinger system of non-interacting bosons among Poissonian spherical impurities, with the sole macroscopic occupation of the one-particle ground state when the density exceeds the critical value.

Spectroscopic characterization of the quantum linear-zigzag transition in trapped ions

While engineered quantum systems are a general route to the manipulation of multipartite quantum states, access in a physical system to a continuous quantum phase transition under sufficient control offers the possibility of an intrinsic source of entangled states. To this end we realize the quantum version of the linear-zigzag structural transition for arrays of up to five ground state-cooled ions held in a linear Paul trap and we demonstrate several of the control requirements towards entangled-state interferometry near the critical point. Using in situ spectroscopy we probe the energy level structure and occupation of the soft mode associated with the structural transition, and show a stable critical point and majority ground state occupation crossing the transition. We resolve biases arising from trap electrode asymmetries that change the nature of the transition, show that they can be suppressed by varying the ion number, and demonstrate control of the transition bias using optical dipole forces.

Silicon anisotropy in a bi-dimensional optomechanical cavity

In this work, we study the effects of mechanical anisotropy in a 2D optomechanical crystal geometry. We fabricate and measure devices with different orientations, showing the dependence of the mechanical spectrum and the optomechanical coupling on the relative angle of the device to the crystallography directions of silicon. Our results show that the device orientation strongly affects its mechanical band structure, which makes the devices more susceptible to orientation fabrication imperfections. Finally, we show that our device is compatible with cryogenic measurements, reaching a ground state occupancy of 0.25 phonons at mK temperature.

The impact of spin-vibrational coupling on magnetic relaxation of a Co(II) single-molecule magnet.

Single-molecule magnets (SMMs) based on transition metals have appeared as enticing targets exploiting magnetic anisotropy in 3d elements. Among transition metals, Co based SMMs are very prominent as they often exhibit a high spin-reversal barrier (Ueff), owing to their large unquenched orbital angular momentum. Employing the wave function-based multireference CASSCF/NEVPT2 calculations, herein we substantiate the zero-field splitting parameters of four mononuclear Co complexes and one of them has shown potential as an SMM. The mechanism of magnetic relaxation has been studied to understand the molecular origin of the slow relaxation of magnetization. The combination of suppressed quantum tunneling of magnetization (QTM) at the ground state and the high negative D value usually manifests SMM behavior in a zero-applied magnetic field. However, mere fulfillment of these conditions ensures little about their SMM behavior, as spin-vibrational coupling often plays spoilsport by lowering the spin relaxation channels. A detailed study accounting for all the 46 vibrational modes below the first-excited state for the prospective Co(II) complex, reveals one of the vibrational modes providing a lower spin relaxation pathway. This results in the development of an SMM with a Ueff value of 239.30 cm-1, decreased by ∼81 cm-1 from the value without spin-vibrational coupling.

Beyond the Ground State: Predicting Electron Ionization Mass Spectra Using Excited-State Molecular Dynamics.

Here, we provide an algorithm that introduces excited states into the molecular dynamics prediction of the 70 eV electron ionization mass spectra. To decide the contributions of different electronic states, the ionization cross section associated with relevant molecular orbitals was calculated by the binary-encounter-Bethe (BEB) model. We used a fast orthogonalization model/single and double state configuration interaction (OM2/CISD) method to implement excited states calculations and combined this with the GFN1-xTB semiempirical model. Demonstrated by predicting the mass spectrum of urocanic acid, we showed better accuracies to experimental spectra using excited-state molecular dynamics than calculations that only used the ground-state occupation. For several histidine pathway intermediates, we found that excited-state corrections yielded an average of 73% more true positive ions compared to the OM2 method when matching to experimental spectra and 16% more true positive ions compared to the GFN method. Importantly, the exited state models also correctly predict several fragmentation reactions that were missing from both ground-state methods. Overall, for 48 calculated molecules, we found the best average mass spectral similarity scores for the mixed excited-state method compared to the ground-state methods using either cosine, weighted dot score, or entropy similarity calculations. Therefore, we recommend adding excited-state calculations for predicting the electron ionization mass spectra of small molecules in metabolomics.

Analytical framework for non-equilibrium phase transition to Bose–Einstein condensate

The theoretical description of non-equilibrium Bose–Einstein condensate (BEC) is one of the main challenges in modern statistical physics and kinetics. The non-equilibrium nature of BEC makes it impossible to employ the well-established formalism of statistical mechanics. We develop a framework for the analytical description of a non-equilibrium phase transition to BEC that, in contrast to previously developed approaches, takes into account the infinite number of continuously distributed states. We consider the limit of fast thermalization and obtain an analytical expression for the full density matrix of a non-equilibrium ideal BEC which also covers the equilibrium case. For the particular cases of 2D and 3D, we investigate the non-equilibrium formation of BEC by finding the temperature dependence of the ground state occupation and second-order coherence function. We show that for a given pumping rate, the macroscopic occupation of the ground state and buildup of coherence may occur at different temperatures. Moreover, the buildup of coherence strongly depends on the pumping scheme. We also investigate the condensate linewidth and show that the Schawlow–Townes law holds for BEC in 3D and does not hold for BEC in 2D.

Subgap states in superconducting islands

We study an interacting quantum dot in contact with a superconducting island described by the Richardson model with a Coulomb repulsion term controlling the number of electrons on the island. This Hamiltonian admits a compact matrix-product-operator representation and can be efficiently and accurately solved using the density-matrix renormalization group. We systematically explore the effects of the charging energy $E_c$. For $E_c$ comparable to the superconducting gap $\Delta$, the subgap states are stabilized by the combination of Kondo exchange coupling and charge redistribution driven by the Coulomb interaction. The subgap states exist for both even and odd superconductor ground-state occupancy, but with very distinctive excitation spectra in each case. The spectral peaks are not symmetric with respect to the chemical potential and may undergo discontinuous changes as a function of gate voltages.

Machine-learning semilocal density functional theory for many-body lattice models at zero and finite temperature

We introduce a machine-learning density-functional-theory formalism for the spinless Hubbard model in one dimension at both zero and finite temperature. In the zero-temperature case this establishes a one-to-one relation between the site occupation and the total energy, which is then minimised at the ground-state occupation. In contrast, at finite temperature the same relation is defined between the Helmholtz free energy and the equilibrium site occupation. Most importantly, both functionals are semi-local, so that they are independent from the size of the system under investigation and can be constructed over exact data for small systems. These 'exact' functionals are numerically defined by neural networks. We also define additional neural networks for finite-temperature thermodynamical quantities, such as the entropy and heat capacity. These can be either a functional of the ground-state site occupation or of the finite-temperature equilibrium site occupation. In the first case their equilibrium value does not correspond to an extremal point of the functional, while it does in the second case. Our work gives us access to finite-temperature properties of many-body systems in the thermodynamic limit.

Understanding radiative transitions and relaxation pathways in plexcitons

Summary Molecular aggregates on plasmonic nanoparticles have emerged as attractive systems for the studies of polaritonic light-matter states, called plexcitons. Such systems are tunable, scalable, easy to synthesize, and offer sub-wavelength confinement, all while giving access to the ultrastrong light-matter coupling regime, promising a plethora of applications. However, the complexity of these materials prevented the understanding of their excitation and relaxation phenomena. Here, we follow the relaxation pathways in plexcitons and conclude that while the metal destroys the optical coherence, the molecular aggregate coupled to surface processes significantly contributes to the energy dissipation. We use two-dimensional electronic spectroscopy with theoretical modeling to assign the different relaxation processes to either molecules or metal nanoparticle. We show that the dynamics beyond a few femtoseconds has to be considered in the language of hot electron distributions instead of the accepted lower and upper polariton branches and establish the framework for further understanding.

Microcanonical condensate fluctuations in one-dimensional weakly-interacting Bose gases

Weakly interacting Bose gases confined in a one-dimensional harmonic trap are studied using microcanonical ensemble approaches. Combining number theory methods, I present a new approach to calculate the particle number counting statistics of the ground state occupation. The results show that the repulsive interatomic interactions increase the ground state fraction and suppresses the fluctuation of ground state at low temperature.

Exact solution of spherical mean-field plus multi-pair interaction model with two non-degenerate j-orbits

The exact solution of spherical mean-field plus multi-pair interaction model with two non-degenerate j-orbits, which is an extension of the widely used standard (two-body) pairing model, is derived based on the Bethe–Richardson–Gaudin approach. The Bethe–Richardson–Gaudin equations in determining eigenstates and the corresponding eigen-energies of the model are provided and exemplified with up to three-pair interactions. With a suitable parameterization of the overall multi-pair interaction strengths, the model with one adjustable parameter and valence nucleons confined in the $$1d_{5/2}$$ and $$0g_{7/2}$$ orbits is applied to fit binding energies of $$^{102{\text {--}}112}$$Sn. It is shown that the ground-state occupation probabilities of nucleon pairs calculated from this model and those from the standard pairing model are almost the same with perfect ground-state overlap of the two models. A noticeable feature of the multi-pair interactions is that the even–odd staggering of pairing interaction strength appearing in the standard pairing model due to the Pauli-blocking is suppressed. As the result, the pairing interaction strength of the model only depends on the number of valence nucleon pairs in the system.

Engineering NonBinary Rydberg Interactions via Phonons in an Optical Lattice.

Coupling electronic and vibrational degrees of freedom of Rydberg atoms held in optical tweezer arrays offers a flexible mechanism for creating and controlling atom-atom interactions. We find that the state-dependent coupling between Rydberg atoms and local oscillator modes gives rise to two- and three-body interactions which are controllable through the strength of the local confinement. This approach even permits the cancellation of two-body terms such that three-body interactions become dominant. We analyze the structure of these interactions on two-dimensional bipartite lattice geometries and explore the impact of three-body interactions on system ground state on a square lattice. Focusing specifically on a system of ^{87}Rb atoms, we show that the effects of the multibody interactions can be maximized via a tailored dressed potential within a trapping frequency range of the order of a few hundred kilohertz and for temperatures corresponding to a >90% occupation of the atomic vibrational ground state. These parameters, as well as the multibody induced timescales, are compatible with state-of-the-art arrays of optical tweezers. Our work shows a highly versatile handle for engineering multibody interactions of quantum many-body systems in most recent manifestations on Rydberg lattice quantum simulators.

Floquet dynamics in the quantum measurement of mechanical motion

The radiation-pressure interaction between one or more laser fields and a mechanical oscillator gives rise to a wide range of phenomena: from sideband cooling and backaction-evading measurements to pondermotive and mechanical squeezing to entanglement and motional sideband asymmetry. In many protocols, such as dissipative mechanical squeezing, multiple lasers are utilized, giving rise to periodically driven optomechanical systems. Here we show that in this case, Floquet dynamics can arise due to presence of Kerr-type nonlinearities, which are ubiqitious in optomechanical systems. Specifically, employing multiple probe tones, we perform sideband asymmetry measurements, a macroscopic quantum effect, on a silicon optomechanical crystal sideband-cooled to 40% ground-state occupation. We show that the Floquet dynamics, resulting from the presence of multiple pump tones, gives rise to an artificially modified motional sideband asymmetry by redistributing thermal and quantum fluctuations among the initially independently scattered thermomechanical sidebands. For pump tones exhibiting large frequency separation, the dynamics is suppressed and accurate quantum noise thermometry demonstrated. We develop a theoretical model based on Floquet theory that accurately describes our observations. The resulting dynamics can be understood as resulting from a synthetic gauge field among the Fourier modes, which is created by the phase lag of the Kerr-type response. This novel phenomenon has wide-ranging implications for schemes utilizing several pumping tones, as commonly employed in backaction-evading measurements, dissipative optical squeezing, dissipative mechanical squeezing and quantum noise thermometry. Our observation may equally well be used for optomechanical Floquet engineering, e.g. generation of topological phases of sound by periodic time-modulation.

Ground-state shape evolution in Er and Yb isotopes

The ground-state shape (phase) evolution in Er and Yb isotopes is manifested in the axially deformed Nilsson mean-field plus extended pairing model. The energy ratio R02+/21+, the odd-even mass differences, the ground-state occupation probabilities of valence nucleon pairs with different angular momenta and the information entropy are calculated. It is shown from these quantities as functions of the quadrupole deformation parameter and the overall pairing interaction strength that the ground-state shape (phase) evolution is mainly driven by the pairing interaction and less affected by the quadrupole deformation, which thus provides a possible origin of the ground-state shape (phase) evolution of these isotopes.

Modeling near ground-state cooling of two-dimensional ion crystals in a Penning trap using electromagnetically induced transparency

Penning traps, with their ability to control planar crystals of tens to hundreds of ions, are versatile quantum simulators. Thermal occupations of the motional drumhead modes, transverse to the plane of the ion crystal, degrade the quality of quantum simulations. Laser cooling using electromagnetically induced transparency (EIT cooling) is attractive as an efficient way to quickly initialize the drumhead modes to near ground-state occupations. We numerically investigate the efficiency of EIT cooling of planar ion crystals in a Penning trap, accounting for complications arising from the nature of the trap and from the simultaneous cooling of multiple ions. We show that, in spite of challenges, the large bandwidth of drumhead modes (hundreds of kilohertz) can be rapidly cooled to near ground-state occupations within a few hundred microseconds. Our predictions for the center-of-mass mode include a cooling time constant of tens of microseconds and an enhancement of the cooling rate with increasing number of ions. Successful experimental demonstrations of EIT cooling in the NIST Penning trap [E. Jordan, K. A. Gilmore, A. Shankar, A. Safavi-Naini, M. J. Holland, and J. J. Bollinger, "Near ground-state cooling of two-dimensional trapped-ion crystals with more than 100 ions", (2018), submitted.] validate our predictions.

Optimizing the orbital occupation in the multiple minima problem of magnetic materials from the metaheuristic firefly algorithm.

We present the use and implementation of the firefly algorithm to help in scanning the multiple metastable minima of orbital occupations in density functional theory (DFT) plus Hubbard U correction and to identify the ground state occupations in strongly correlated materials. We show the application of this implementation with the Abinit code on KCoF3 and UO2 crystals, which are typical d and f electron systems with numerous occupation minima. We demonstrate the validity and performance of the method by comparing with previous methodologies. The method is general and can be applied to any code using constrained occupation matrices.

Asymmetric Oxygen Bridged Copper(II) Carboxylate: Synthesis, Complete Characterization, and Crystal Structure

Binuclear centrosymmetric copper(II) complex of the formula bipyCu(L)4Cubipy, where bipy = 2,2′- bipyridine and L = 4-methoxy-2-phenyl acetate, is synthesized and characterized by FT-IR, UV-Visible, ESR and mass spectroscopy, electrochemical, thermogravimetric, and single crystal XRD techniques. The complex contains two differently oriented molecules per unit cell stabilized via intermolecular interactions. Geometry around each Cu(II) was found to be square pyramidal affected by asymmetrically bridging oxygen atoms occupying the apical position of one square pyramid and the axial position of another in the same binuclear molecule. The square base is formed by two oxygen atoms from two carboxylate ligands and two nitrogen atoms from the bipyridine moiety. TGA shows that the complex is stable up to 170 °C followed by stepwise loss of coordinated ligands, which was supported by GC-MS data as well. A broad absorption spectrum indicated 2B1g as the ground state and single electron occupancy of the dx2–y2 orbital, which was confirmed by the ESR spectrum. The electrochemical study gives two oxidation curves corresponding to Cu(II)/Cu(III) and Cu(I)/Cu(II) and a reduction signal corresponding to the Cu(II)/Cu(I) process. The robust complex represents an interesting contribution to the understanding of coordination chemistry.

Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose-Einstein Gas in a Box.

From basic principles, we review some fundamentals of entropy calculations, some of which are implicit in the literature. We mainly deal with microcanonical ensembles to effectively compare the counting of states in continuous and discrete settings. When dealing with non-interacting elements, this effectively reduces the calculation of the microcanonical entropy to counting the number of certain partitions, or compositions of a number. This is true in the literal sense, when quantization is assumed, even in the classical limit. Thus, we build on a moderately dated, ingenuous mathematical work of Haselgrove and Temperley on counting the partitions of an arbitrarily large positive integer into a fixed (but still large) number of summands, and show that it allows us to exactly calculate the low energy/temperature entropy of a one-dimensional Bose–Einstein gas in a box. Next, aided by the asymptotic analysis of the number of compositions of an integer as the sum of three squares, we estimate the entropy of the three-dimensional problem. For each selection of the total energy, there is a very sharp optimal number of particles to realize that energy. Therefore, the entropy is ‘large’ and almost independent of the particles, when the particles exceed that number. This number scales as the energy to the power of -rds in one dimension, and -ths in three dimensions. In the one-dimensional case, the threshold approaches zero temperature in the thermodynamic limit, but it is finite for mesoscopic systems. Below that value, we studied the intermediate stage, before the number of particles becomes a strong limiting factor for entropy optimization. We apply the results of moments of partitions of Coons and Kirsten to calculate the relative fluctuations of the ground state and excited states occupation numbers. At much lower temperatures than threshold, they vanish in all dimensions. We briefly review some of the same results in the grand canonical ensemble to show to what extents they differ.

At which magnetic field, exactly, does the Kondo resonance begin to split? A Fermi liquid description of the low-energy properties of the Anderson model

This paper is a corrected version of Phys. Rev. B 95, 165404 (2017), which we have retracted because it contained a trivial but fatal sign error that lead to incorrect conclusions. --- We extend a recently-eveloped Fermi-liquid (FL) theory for the asymmetric single-impurity Anderson model [C. Mora $et al.$, Phys. Rev. B, 92, 075120 (2015)] to the case of an arbitrary local magnetic field. To describe the system's low-lying quasiparticle excitations for arbitrary values of the bare Hamiltonian's model parameters, we construct an effective low-energy FL Hamiltonian whose FL parameters are expressed in terms of the local level's spin-dependent ground-state occupations and their derivatives with respect to level energy and local magnetic field. These quantities are calculable with excellent accuracy from the Bethe Ansatz solution of the Anderson model. Applying this effective model to a quantum dot in a nonequilibrium setting, we obtain exact results for the curvature of the spectral function, $c_A$, describing its leading $\sim\varepsilon^2$ term, and the transport coefficients $c_V$ and $c_T$, describing the leading $\sim V^2$ and $\sim T^2$ terms in the nonlinear differential conductance. A sign change in $c_A$ or $c_V$ is indicative of a change from a local maximum to a local minimum in the spectral function or nonlinear conductance, respectively, as is expected to occur when an increasing magnetic field causes the Kondo resonance to split into two subpeaks. We find that the fields $B_A$, $B_T$ and $B_V$ at which $c_A$, $c_T$ and $c_V$ change sign, respectively, are all of order $T_K$, as expected, with $B_A = B_T = B_V = 0.75073\,T_K$ in the Kondo limit.

How sharply does the Anderson model depict a single-electron transistor?

The single-impurity Anderson model has been the focus of theoretical studies of molecular junctions and the single-electron transistor, a nanostructured device comprising a quantum dot that bridges two otherwise decoupled metallic leads. The low-temperature transport properties of the model are controlled by the ground-state occupation of the quantum dot, a circumstance that recent density-functional approaches have explored. Here we show that the ground-state dot occupation also parametrizes a linear mapping between the thermal dependence of the zero-bias conductance and a universal function of the temperature scaled by the Kondo temperature. Careful measurements by Grobis and co-workers are very accurately fitted by the universal mapping. Nonetheless, the dot occupation and an asymmetry parameter extracted from the same mapping are relatively distant from the expected values. We conclude that mathematical results derived from the model Hamiltonian reproduce accurately the universal physical properties of the device. In contrast, non-universal features cannot be reproduced quantitatively. To circumvent this limitation, \emph{ab initio} studies of the device at high energies seem necessary, to accurately define the model Hamiltonian. Our conclusion reinforces findings by Gross and coworkers, who applied time-dependent density-functional theory to show that, to describe the low-energy properties of molecular junctions, one must be able to describe the high-energy regime.