Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Indonesian Physical Review

TWO-PARTICLE THERMAL DENSITY MATRICES IN ONE DIMENSION USING FINITE DIFFERENCE TIME DOMAIN (FDTD) METHOD

subhan subhan, I Wayan Sudiarta, Lily Maysari Angraini

Published Oct 31, 2018• 1 citation
Publisher

Abstract

A quantum system in the thermal equilibrium state is a mixed state consisting of statistical ensembles of several different quantum systems can be represented by a thermal density matrix. In this research, the thermal density matrix is calculated for two-particle system case non-interaction in one-dimensional square well and one-dimensional harmonic oscillator using finite difference time domain (FDTD) method. In addition, thermal density matrix calculations are also performed for the case of two particle systems interacting in a one-dimensional harmonic oscillator. We present results of probability densities, partition functions, and internal energies for three cases: two distinguishable particles, two fermions and two bosons. Validation of numerical results of thermal density matrix and probability density is accurate with analytical solutions. Then, the result of partition function and internal energy the system is strongly effect by temperature. At low temperatures, internal energy the system will lead to the lowest energy or ground state.

DOI
10.29303/ipr.v1i1.13

Related papers

Penentuan Density Matrix Sistem Kuantum Satu Partikel Dimensi Satu dengan Metode Discretized Path Integral (DPI)

Iyan Islamiyati, I Wayan Sudiarta, Lily Maysari Angraini

A mixed quantum system can be represented by a density matrix. By knowing the density matrix of the system, other thermodynamic properties of the system such as Helmholtz free energy and entropy can be determined. Computation of the density matrix of a one-dimensional single-particle quantum system using the discretized path integral(DPI) method is presented. The results of the DPI method were validated using the finite difference time domain (FDTD) method for a particle in an infinite square well and harmonic oscillator potentials. The results of density matrix, Helmholtz’s free energy and entropy have shown that the DPI method produces correct numerical values compared to the FDTD results. Then, the DPI method is used in the double-well potential by giving variations in barrier potential and temperature of the system. The results show that the particle will be tunneling when the barrier potential energy is more than the kinetic energy of the particle. The particle also can tunnel easily if the system has the higher temperature.

Jurnal Teori dan Aplikasi Fisika•Feb 20, 2023•0 citations

The Finite Difference Time Domain Method for Computing Single-Particle Density Matrix

Sudiarta, I. Wayan, D. J. W. Geldart

A general method for numerical computation of the thermal density matrix of a single-particle quantum system is presented. The Schrodinger equation in imaginary time tau is solved numerically by the finite difference time domain (FDTD) method using a set of initial wave functions at tau=0 . By choosing this initial set appropriately, the set of wave functions generated by the FDTD method can be used to construct the thermal density matrix. The theoretical basis of the method, a numerical algorithm for its implementation, and illustrative examples in one, two and three dimensions are given in this paper. The numerical results show that the procedure is efficient and accurately determines the density matrix and thermodynamic properties of single-particle systems. Extensions of the method to more general cases are briefly indicated.

arXiv:0712.4317•Dec 28, 2007•0 citations

The finite difference time domain method for computing the single-particle density matrix

I Wayan Sudiarta, D. J. W. Geldart

A general method for numerical computation of the thermal density matrix of a single-particle confined quantum system is presented. The Schrodinger equation in imaginary time τ is solved numerically by the finite difference time domain (FDTD) method using a set of initial wavefunctions at τ = 0. By choosing this initial set appropriately, the set of wavefunctions generated by the FDTD method can be used to construct the thermal density matrix. The theoretical basis of the method, a numerical algorithm for its implementation and illustrative examples in one, two and three dimensions are given in this paper. The numerical results show that the procedure is efficient and accurately determines the density matrix and thermodynamic properties of single-particle systems. Extensions of the method to more general cases are briefly indicated.

Journal of Physics A Mathematical and Theoretical•Jun 25, 2009•11 citations

Reduced density matrix approach to one-dimensional ultracold bosonic systems

Mitchell J. Knight, Harry M. Quiney, Andy M. Martin

The variational determination of the two-boson reduced density matrix is described for a one-dimensional system of N N harmonically trapped bosons interacting via contact interaction, where N N ranges from 2 2 to 10^4 10 4 . The ground-state energies are calculated, and compared to existing methods, including the analytic case (for N=2) N = 2 ) and mean-field approaches such as the one-dimensional Gross-Pitaevskii equation and its variations. Structural properties such as the density and correlation functions are also derived, including the behaviour of the correlation function when boson coordinates coincide. This collectively demonstrates the capacity of the reduced density matrix method to accurately calculate collective ground-state properties of bosonic systems comprising few to many bosons, including the cross-over region between these extremes, across a large range of interaction strengths.

arXiv:2503.15811•SciPost Physics•Sep 28, 2026•0 citations

Quantum Joule expansion of one-dimensional systems

Jin Zhang, Yannick Meurice, Shan-Wen Tsai

We investigate the Joule expansion of nonintegrable quantum systems that contain bosons or spinless fermions in one-dimensional lattices. A barrier initially confines the particles to be in half of the system in a thermal state described by the canonical ensemble and is removed at time $t=0$. We investigate the properties of the time-evolved density matrix, the diagonal ensemble density matrix, and the corresponding canonical ensemble density matrix with an effective temperature determined by the total energy conservation using exact diagonalization. The weights for the diagonal ensemble and the canonical ensemble match well for high initial temperatures that correspond to negative effective final temperatures after the expansion. At long times after the barrier is removed, the time-evolved R\'enyi entropy of subsystems bigger than half can equilibrate to the thermal entropy with exponentially small fluctuations. The time-evolved reduced density matrix at long times can be approximated by a thermal density matrix for small subsystems. Few-body observables, like the momentum distribution function, can be approximated by a thermal expectation of the canonical ensemble with strongly suppressed fluctuations. The negative effective temperatures for finite systems go to non-negative temperatures in the thermodynamic limit for bosons, but there is a true thermodynamic effect for fermions, which is confirmed by finite-temperature density matrix renormalization group calculations. We propose the Joule expansion as a way to dynamically create negative temperature states for fermion systems with repulsive interactions.

arXiv:1903.01414•Physical Review A•Mar 10, 2020•3 citations

Two-component Fermi gas in a one-dimensional harmonic trap

Gao Xianlong, W. Wonneberger

A many-body theory for a two-component system of spin-polarized interacting fermions in a one-dimensional harmonic trap is developed. The model considers two different states of the same fermionic species and treats the dominant interactions between the two using the bosonization method for forward scattering. Asymptotically exact results for the one-particle matrix elements at zero temperature are given. Using them, occupation probabilities of oscillator states are discussed. Particle and momentum densities are calculated and displayed. It is demonstrated how interactions modify all these quantities. An asymptotic connection with Luttinger liquids is suggested. The relation of the coupling constant of the theory to the dipole-dipole interaction is also discussed.

arXiv:quant-ph/0111104•Physical Review A•Feb 8, 2002•18 citations
A quantum system in the thermal equilibrium state is a mixed state consisting of statistical ensembles of several different quantum systems can be represented by a thermal density matrix. In this research, the thermal density matrix is calculated for two-particle system case non-interaction in one-dimensional square well and one-dimensional harmonic oscillator using finite difference time domain (FDTD) method. In addition, thermal density matrix calculations are also performed for the case of two particle systems interacting in a one-dimensional harmonic oscillator. We present results of probability densities, partition functions, and internal energies for three cases: two distinguishable particles, two fermions and two bosons. Validation of numerical results of thermal density matrix and probability density is accurate with analytical solutions. Then, the result of partition function and internal energy the system is strongly effect by temperature. At low temperatures, internal energy the system will lead to the lowest energy or ground state.
A mixed quantum system can be represented by a density matrix. By knowing the density matrix of the system, other thermodynamic properties of the system such as Helmholtz free energy and entropy can be determined. Computation of the density matrix of a one-dimensional single-particle quantum system using the discretized path integral(DPI) method is presented. The results of the DPI method were validated using the finite difference time domain (FDTD) method for a particle in an infinite square well and harmonic oscillator potentials. The results of density matrix, Helmholtz’s free energy and entropy have shown that the DPI method produces correct numerical values compared to the FDTD results. Then, the DPI method is used in the double-well potential by giving variations in barrier potential and temperature of the system. The results show that the particle will be tunneling when the barrier potential energy is more than the kinetic energy of the particle. The particle also can tunnel easily if the system has the higher temperature.
A general method for numerical computation of the thermal density matrix of a single-particle quantum system is presented. The Schrodinger equation in imaginary time tau is solved numerically by the finite difference time domain (FDTD) method using a set of initial wave functions at tau=0 . By choosing this initial set appropriately, the set of wave functions generated by the FDTD method can be used to construct the thermal density matrix. The theoretical basis of the method, a numerical algorithm for its implementation, and illustrative examples in one, two and three dimensions are given in this paper. The numerical results show that the procedure is efficient and accurately determines the density matrix and thermodynamic properties of single-particle systems. Extensions of the method to more general cases are briefly indicated.
A general method for numerical computation of the thermal density matrix of a single-particle confined quantum system is presented. The Schrodinger equation in imaginary time τ is solved numerically by the finite difference time domain (FDTD) method using a set of initial wavefunctions at τ = 0. By choosing this initial set appropriately, the set of wavefunctions generated by the FDTD method can be used to construct the thermal density matrix. The theoretical basis of the method, a numerical algorithm for its implementation and illustrative examples in one, two and three dimensions are given in this paper. The numerical results show that the procedure is efficient and accurately determines the density matrix and thermodynamic properties of single-particle systems. Extensions of the method to more general cases are briefly indicated.
The variational determination of the two-boson reduced density matrix is described for a one-dimensional system of N N harmonically trapped bosons interacting via contact interaction, where N N ranges from 2 2 to 10^4 10 4 . The ground-state energies are calculated, and compared to existing methods, including the analytic case (for N=2) N = 2 ) and mean-field approaches such as the one-dimensional Gross-Pitaevskii equation and its variations. Structural properties such as the density and correlation functions are also derived, including the behaviour of the correlation function when boson coordinates coincide. This collectively demonstrates the capacity of the reduced density matrix method to accurately calculate collective ground-state properties of bosonic systems comprising few to many bosons, including the cross-over region between these extremes, across a large range of interaction strengths.
We investigate the Joule expansion of nonintegrable quantum systems that contain bosons or spinless fermions in one-dimensional lattices. A barrier initially confines the particles to be in half of the system in a thermal state described by the canonical ensemble and is removed at time t=0t=0t=0. We investigate the properties of the time-evolved density matrix, the diagonal ensemble density matrix, and the corresponding canonical ensemble density matrix with an effective temperature determined by the total energy conservation using exact diagonalization. The weights for the diagonal ensemble and the canonical ensemble match well for high initial temperatures that correspond to negative effective final temperatures after the expansion. At long times after the barrier is removed, the time-evolved R\'enyi entropy of subsystems bigger than half can equilibrate to the thermal entropy with exponentially small fluctuations. The time-evolved reduced density matrix at long times can be approximated by a thermal density matrix for small subsystems. Few-body observables, like the momentum distribution function, can be approximated by a thermal expectation of the canonical ensemble with strongly suppressed fluctuations. The negative effective temperatures for finite systems go to non-negative temperatures in the thermodynamic limit for bosons, but there is a true thermodynamic effect for fermions, which is confirmed by finite-temperature density matrix renormalization group calculations. We propose the Joule expansion as a way to dynamically create negative temperature states for fermion systems with repulsive interactions.
A many-body theory for a two-component system of spin-polarized interacting fermions in a one-dimensional harmonic trap is developed. The model considers two different states of the same fermionic species and treats the dominant interactions between the two using the bosonization method for forward scattering. Asymptotically exact results for the one-particle matrix elements at zero temperature are given. Using them, occupation probabilities of oscillator states are discussed. Particle and momentum densities are calculated and displayed. It is demonstrated how interactions modify all these quantities. An asymptotic connection with Luttinger liquids is suggested. The relation of the coupling constant of the theory to the dipole-dipole interaction is also discussed.