arXiv subjects

Przemysław Kościk

Publications and source records attributed to Przemysław Kościk.

14 recordsLinked to original sources

arXiv2025

Exact collective occupancies of the Moshinsky model in two-dimensional geometry

In this paper, we investigate the ground state of $N$ bosonic atoms confined in a two-dimensional isotropic harmonic trap, where the atoms interact via a harmonic potential. We derive an exact diagonal representation of the first-order reduced density matrix in polar coordinates, in which the angular components of the natural orbitals are eigenstates of the angular momentum operator. Furthermore, we present an exact expression for the collective occupancy of the natural orbitals with angular momentum $l$, quantifying the fraction of particles carrying that angular momentum. The present study explores how the dependence of collective occupancy relies on angular momentum $l$ and the control parameters of the system. Building on these findings, we examine boson fragmentation into components with different $l$ and reveal a unique feature of the system: the natural orbitals contributing to the correlations are uniformly distributed across all significant $l$ components.

quant-ph↗

arXiv2023

Radial and Angular Correlations in a Confined System of Two Atoms in a Two-Dimensional Geometry

We study the ground-state correlations between two atoms in a two-dimensional isotropic harmonic trap. We consider a finite-range soft-core interaction that can be applied to simulate various atomic systems. We provide detailed results on the dependence of the correlations on the parameters of the system. Our investigations show that in the hardcore limit, the wave function can be approximated as the product of the radial and angular components regardless of the interaction range. This implies that the radial and angular correlations are independent of one another. However, correlations within the radial and angular components persist and are heavily influenced by the interaction range. The radial correlations are generally weaker than the angular correlations. When soft-core interactions are considered, the correlations exhibit more complex behavior.

quant-ph↗

arXiv2023

Universality of internal correlations of strongly interacting p-wave fermions in one-dimensional geometry

We consider the many-body ground state of polarized fermions interacting via zero-range $\mathfrak{p}$-wave forces in a one-dimensional geometry. We rigorously prove that in the limit of infinite attractions spectral properties of any-order reduced density matrix describing arbitrary subsystem are completely independent of the shape of an external potential. It means that quantum correlations between any two subsystems are in this limit insensitive to the confinement. In addition, we show that the purity of these matrices quantifying the amount of quantum correlations can be obtained analytically for any number of particles without diagonalizing them. This observation may serve as a rigorous benchmark for other models and methods describing strongly interacting p-wave fermions.

cond-mat.quant-gas↗

arXiv2023

On the Exponential Decay of Strongly Interacting Cold Atoms from a Double-Well Potential

In this article, we study an exponential decay for the gas of bosons with strong repulsive delta interactions from a double-well potential. We consider an exactly solvable model comprising an infinite wall and two Dirac delta barriers. We explore its features both within the exact method and with the resonance expansion approach. The study reveals the effect of the splitting barrier on the decay rate in dependence on the number of particles. Among other things, we find that the effect of the splitting barrier on the decay rate is most pronounced in systems with odd particle numbers. During exponential decay, the spatial correlations in an internal region are well captured by the "radiating state".

cond-mat.quant-gas↗

arXiv2021

Pair-correlation ansatz for the ground state of interacting bosons in an arbitrary one-dimensional potential

We derive and describe a very accurate variational scheme for the ground state of the system of a few ultra-cold bosons confined in one-dimensional traps of arbitrary shapes. It is based on assumption that all inter-particle correlations have two-body nature. By construction, the proposed ansatz is exact in the noninteracting limit, exactly encodes boundary conditions forced by contact interactions, and gives full control on accuracy in the limit of infinite repulsions. We show its efficiency in a whole range of intermediate interactions for different external potentials. Our results manifest that for generic non-parabolic potentials mutual correlations forced by interactions cannot be captured by distance-dependent functions.

cond-mat.quant-gas↗

arXiv2020

Decay properties of unstable Tonks-Girardeau gases from a split trap

We study the decay properties of Tonks-Girardeau gases escaping from a double-well potential. Initially, the gases are constrained between two infinite $δ$ barriers with an on-center $δ$-split potential. The strength of one of the obstacles is suddenly reduced, and the particles start to tunnel to the open space. Using the resonance expansion method, we derive the single-term approximate expression for the $N$-particle survival probability and demonstrate its effectiveness in both the exponential and nonexponential regimes. We also predict a parity effect and provide physical insights into its nature at different stages of the time evolution. We conclude that only the initial phase of the decay of the many-particle state may comply with an exponential law. The decay properties are dramatically affected by the presence of the split barrier. Our results reveal the overall decay mechanism of unstable Tonks-Girardeau gases from single and double quantum wells.

cond-mat.quant-gas↗

arXiv2020

Variational ansatz for $p$-wave fermions confined in a one-dimensional harmonic trap

We propose a very accurate and efficient variational scheme for the ground state of the system of $p$-wave attractively interacting fermions confined in a one-dimensional harmonic trap. By the construction, the method takes the non-analytical part of interactions exactly into account and thus it approximates the true ground-state wave function in a whole range of interactions very accurately. Within the method, we determine different properties of the system for a different number of particles and different interactions. In this way, we explore how the system and its features transit from the ideal non-interacting Fermi gas to the system of infinitely strong attractions. Additionally, we demonstrate that the ansatz may also be used on a repulsive branch of interactions where other numerical methods break down. The presented method of including zero-range interactions is very universal and may be easily generalized to other one-dimensional confinements.

cond-mat.quant-gas↗

arXiv2020

Capturing non-exponential dynamics in the presence of two decay channels

The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the non-exponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through a specific quantum mechanical examples which can be accurately solved, we study general properties of a two-channel decay that apply for various unstable quantum states (among which also for unstable particles). Apart from small deviations at early times, the survival probability and the partial tunneling probability along the chosen direction are very well described by the exponential-decay model. In contrast, the ratios of the decay probabilities and probability currents are evidently not a simple constant (as they would be in the exponential limit) but display time-persisting oscillations. Hence, these ratios are optimal witnesses of deviations from the exponential decay law.

quant-ph↗

arXiv2019

Exactly solvable model of two interacting Rydberg-dressed atoms confined in a two-dimensional harmonic trap

Exactly solvable model of two Rydberg-dressed atoms moving in a quasi-two-dimensional harmonic trap is introduced and its properties are investigated. Depending on the strength of inter-particle interactions and the critical range of the potential, the two-particle eigenstates are classified with respect to the excitations of the center-of-mass motion, relative angular momentum, and relative distance variable. Having these solutions in hand, we discuss inter-particle correlations as functions of interaction parameters. We also present a straightforward prescription of how to generalize obtained solutions to higher dimensions.

cond-mat.quant-gas↗

arXiv2018

Variational approach for interacting ultra-cold atoms in arbitrary one-dimensional confinement

Standard analytical construction of the many-body wave function of interacting particles in one dimension, beyond mean-field theory, is based on the Jastrow approach. The many-body interacting ground state is build up from the ground state of the non-interacting system and the product of solutions of the corresponding interacting two-body problem. However, this is possible only if the center-of-mass motion is decoupled from the mutual interactions. In our work, based on the general constraints given by contact nature of the atom-atom interactions, we present an alternative approach to the standard construction of the \textit{pair-correlation} wave-function. Within the proposed ansatz, we study the many-body properties of trapped bosons as well as fermionic mixtures and we compare these predictions with the exact diagonalization approach in a wide range of particle numbers, interaction strengths, and different trapping potentials.

cond-mat.quant-gas↗

arXiv2018

Optimized Configuration Interaction Approach for Trapped Multiparticle Systems Interacting Via Contact Forces

For one-dimensional systems with delta-contact interactions, the convergence of the exact-diagonalization method is tested with a basis of harmonic oscillator eigenfunctions with frequency $Ω$ optimized through the minimization of the eigenenergy of the desired level. It is shown that within the framework of this approach the well-converged results can be achieved at much smaller dimensions of the Hamiltonian matrix than with the standard approach that uses $Ω=1$. We present calculations for model systems of identical bosons with harmonic and double-well potentials. Our results show promising potential for diminishing the computational cost of numerical simulations of various systems of trapped ultracold atoms.

cond-mat.quant-gas↗

arXiv2018

Exactly solvable model of two trapped quantum particles interacting via finite-range soft-core interactions

The exactly solvable model of two indistinguishable quantum particles (bosons or fermions) confined in a one-dimensional harmonic trap and interacting via finite-range soft-core interaction is presented and many properties of the system are examined. Particularly, it is shown that independently on the potential range, in the strong interaction limit bosonic and fermionic solutions become degenerate. For sufficiently large ranges a specific crystallization appears in the system. The results are compared to predictions of the celebrated Busch {\it et al.} model and those obtained in the Tonks-Girardeau limit. The assumed inter-particle potential is very similar to the potential between ultra-cold dressed Rydberg atoms. Therefore, the model can be examined experimentally.

cond-mat.quant-gas↗

arXiv2017

Quantum correlations in one-dimensional Wigner molecules

We studied one-dimensional systems formed by $N$ identical particles confined in a harmonic trap and subject to an inverse power law interaction potential $\sim|x|^{-d}$. The correlation properties of a Wigner molecule with the lowest energy are investigated in terms of their dependence on the number $N$ and the power $d$, including the limit as $d\rightarrow 0$. There are $N$-particle Wigner molecules with properties such that their correlations are mainly manifested in the $N$ lowest natural orbitals. The values of the control parameters of the system at which such states appear are identified. The properties of Wigner molecules formed in the limit as $d$ approaches infinity are also revealed.

quant-ph↗

arXiv2012

Determination of Resonances by the Optimized Spectral Approach

The Rayleigh-Ritz procedure for determining bound-states of the Schrödinger equation relies on spectral representation of the solution as a linear combination of the basis functions. Several possible extensions of the method to resonance states have been considered in the literature. Here we propose the application of the optimized Rayleigh-Ritz method to this end. The method uses a basis of the functions containing adjustable nonlinear parameters, the values of which are fixed so as to make the trace of the variational matrix stationary. Generalization to resonances proceeds by allowing the parameters to be complex numbers. Using various basis sets, we demonstrate that the optimized Rayleigh-Ritz scheme with complex parameters provides an effective algorithm for the determination of both the energy and lifetime of the resonant states for various one-dimensional and spherically symmetric potentials. The method is computationally inexpensive since it does not require iterations or predetermined initial values. The convergence rate compares favorably to other approaches.

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