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Effects of finite-range Gaussian repulsion on trapped many-boson system

Mohd. Imran, M. A. H. Ahsan

Published Feb 11, 2016• 0 citations
OpenAlex

Abstract

We investigate a rotating system of $N$ spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive finite-range Gaussian interaction potential of large $s$-wave scattering length. Exact diagonalization of the Hamiltonian matrix in subspaces of quantized total angular momenta $L_{z}$ is carried out to obtain the $N$-body ground states in slow rotating regime of $0 \le L_{z} \le 2N$. We study the finite-range effects of the Gaussian repulsion potential on the many-body ground state properties in different subspaces of quantized $L_{z}$. The study ranges from the few-body ($N=2$) system to the many-body ($N=16$), system where quantum (Bose) statistics becomes perceptible. In particular, we analyze the effect of interaction range of the Gaussian potential on the ground state energy of different $L_{z}$ states as well on the critical angular velocity $\Omega_{\bf c1}$ of first vortex ($L_{z}=N$) state. For an optimal value of the range of the Gaussian interaction potential, the nucleation of the first centered vortex may begin at a lower value of the rotational angular velocity as compared to the zero-range interaction potential. Moreover, we explore the role of interaction potential range on the quantum correlation (measured in terms of the degree of condensation and the von Neumann entropy), for small systems ($N=2$ to $8$) where an understanding of few-body effects provide valuable insight into large systems ($N=8$ to $16$) which admit the possibility of many-body correlation. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential.

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Ground state properties of trapped boson system with finite-range Gaussian repulsion: exact diagonalization study

Mohd. Imran, M. A. H. Ahsan

Abstract We use exact diagonalization to study an interacting system of N spinless bosons with finite-range Gaussian repulsion, confined in a quasi-two-dimensional harmonic trap with and without an introduced rotation. The diagonalization of the Hamiltonian matrix using Davidson algorithm in subspaces of quantized total angular momentum L z is carried out to obtain the N -body lowest eigenenergy and eigenstate. To bring out the effect of quantum (Bose) statistics and consequent phase stiffness (rigidity) of the variationally obtained many-body wavefunction on various physical quantities, our study spans from few-body ( N = 2) to many-body ( N = 16) systems. Further, to examine the finite-range effect of the repulsive Gaussian potential on many-body ground state properties of the Bose-condensate, we obtain the lowest eigenstate, the critical angular velocity of single vortex state and the quantum correlation (measured) in terms of von Neumann entanglement entropy and degree of condensation. It is found that for small values of the range (measured by the parameter σ ) of Gaussian potential, the ground state energy increases for few-boson (2 ⩽ N ⩽ 8) systems but decreases for many-boson ( N > 8) systems. On the other hand for relatively large values of the range of Gaussian potential, the ground state energy exhibits a monotonic decrease, regardless of the number of bosons N . For a given N , there is found an optimal value of the range of Gaussian potential for which the first vortex (with L z = N ) nucleates at a lower value of the rotational angular velocity Ω c1 compared to the zero-range ( δ -function) potential. Further, we observe that the inter-particle interaction and the introduced rotation are competing effects with latter being dominant over the former. With increase in the range of Gaussian potential, the value of von Neumann entropy decreases and the degree of condensation increases implying an enhanced quantum correlation and phase rigidity.

arXiv:1602.03811•Journal of Physics B Atomic Molecular and Optical Physics•Mar 13, 2020•3 citations

Exact Diagonalization Study of Bose-Condensed Gas with Finite-Range Gaussian Interaction

Mohd. Imran, M. A. H. Ahsan

We investigate a system of $N$ spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive two-body finite-range Gaussian interaction potential of large $s$-wave scattering length. Exact diagonalization of the Hamiltonian matrix is carried out to obtain the $N$-body ground state as well as low-lying excited states, using Davidson algorithm in beyond lowest-Landau-level approximation. We examine the finite-range effects of the interaction potential on the many-body ground state energy as also the degree of condensation of the Bose-condensed gas. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential. We further analyze the effect of finite-range interaction potential on the breathing mode collective excitation. Our theoretical results may be relevant for experiments currently conducted on quasi-two-dimensional Bose gas with more realistic interaction potential.

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Multi-vortex Bose-Einstein condensate: examining the role of interaction range using Gaussian potential

Md Hamid, M. A. H. Ahsan

We present exact diagonalization study on a system of $10 \leq N \leq 24$ spinless bosons interacting via repulsive Gaussian potential, harmonically confined in $xy$-plane with an externally impressed rotation about the $z$-axis. The two-body interaction strength in the Gaussian potential is taken in the strongly interacting regime with values of interaction range in the regime $0\leq σ\leq 1$. The diagonalization of the $N$-body Hamiltonian matrix, in subspaces of total angular momentum in the regime $0\le L_{z} \le 4N$ corresponds to the filling fraction $ν\lesssim 3.2$ is carried out to obtain the variationally exact ground-state wavefunction and the corresponding eigenvalue. It is found that an increase in interaction range $σ$ leads to (a) a systematic decrease in energy, (b) an increase in the critical angular velocity $Ω_{c_{i}}$ of the $i${th} vortex state and (c) an increase in the largest eigenvalue $λ_{1}$, (condensate fraction), of one-particle reduced density matrix (OPRDM). The von Neumann entropy $S_{1}\left(L_{z},σ\right)$, quantifying the quantum entanglement between the particles in the many-body ground state, is largely found to decrease with increase in $σ$. Crossings in von Neumann entropy for several of the angular momentum states are observed with variation in $σ$. The response of the Bose-Einstein condensate to rotation is examined through $L_{z}\left(σ\right)-Ω\left(σ\right)$ stability graph for several values of $σ$. A vortex state with larger plateau length on the $L_{z}-Ω$ graph is considered to be more stable. The internal structure of the condensate, as depicted by the conditional probability distribution (CPD) in the body-fixed frame, exhibits characteristic features with interaction range $σ$. One such feature is the merging of the cores of the two-vortex state.

arXiv:2205.12614•May 25, 2022•0 citations

Ground and Low-Lying Collective States of Rotating Three-Boson System

Mohd. Imran, M. A. H. Ahsan

Abstract The ground and low-lying collective states of a rotating system of N = 3 bosons harmonically confined in quasi-two-dimension and interacting via repulsive finite-range Gaussian potential is studied in weakly to moderately interacting regime. The N-body Hamiltonian matrix is diagonalized in subspaces of quantized total angular momenta 0 ≥ L ≥ 4N to obtain the ground and low-lying eigenstates. Our numerical results show that breathing modes with N-body eigenenergy spacing of 2ħω⊥, known to exist in strictly 2D system with zero-range (δ-function) interaction potential, may as well exist in quasi-2D system with finite-range Gaussian interaction potential. To gain an insight into the many-body states, the von Neumann entropy is calculated as a measure of quantum correlation and the conditional probability distribution is analyzed for the internal structure of the eigenstates. In the rapidly rotating regime the ground state in angular momentum subspaces L = (q/2)N (N − 1) with q = 2, 4 is found to exhibit the anticorrelation structure suggesting that it may variationally be described by a Bose-Laughlin like state. We further observe that the first breathing mode exhibits features similar to the Bose-Laughlin state in having eigenenergy, von Neumann entropy and internal structure independent of interaction for the three-boson system considered here. On the contrary, for eigenstates lying between the Bose-Laughlin like ground state and the first breathing mode, values of eigenenergy, von Neumann entropy and internal structure are found to vary with interaction.

arXiv:1508.05030•Communications in Theoretical Physics•Apr 1, 2016•5 citations

Two rotating particles interacting via two-body Gaussian potential harmonically confined in two spatial dimensions

Md Hamid, M. A. H. Ahsan

We study two spinless bosons interacting via two-body Gaussian potential subjected to an externally impressed rotation about an axis confined in a harmonic trap in two-spatial dimensions. We obtain a transcendental equation for the relative angular momentum $|m|$ state with various values of the two-body interaction range $σ$ and the two-body interaction strength $g_{2}$ to study the resulting energy spectrum and analyze the role of Hilbert space dimensions $\widetilde{N}$. We compare results for both attractive and repulsive interaction for $δ$-function potential and Gaussian potential for various values of interaction range. We study the effects of interaction parameters and relative angular momentum on the ground state energy and its various components, namely, kinetic energy, trap potential and interaction potential. For a given $|m|$ and non-interacting case, we observe that the ground state energy becomes independent of interaction range. However, for a given relative angular momentum and interaction strength $g_{2}>0$, there is an increase in ground state energy with an increase in interaction range. Below the interaction strength $g_{2}V(r)\leq -1$, ground state energy diverges to physically unacceptable negative-infinity for $|m|=0$ state. Further, for $|m|=1$, the ground state energy becomes independent of the interaction strength. For a $|m|$, we present a comparative study between the Gaussian interaction potential and the $δ$-function potential. Further, we observe that for a given $g_{2}$ and $|m|$, for $δ$-function potential {\it i.e.} $σ\to 0$, to achieve the convergence of ground state energy, we require a considerably large critical Hilbert space. Whereas, in the case of Gaussian interaction potential with $σ\to 1$, the ground state energy converges for a considerably small critical Hilbert space.

arXiv:2205.11958•May 24, 2022•0 citations

Vortex patterns in moderately rotating Bose-condensed gas

Mohd. Imran, M A H Ahsan

Abstract Using exact diagonalization, we investigate the many-body ground state for regular vortex patterns in a rotating Bose-condensed gas of N spinless particles, confined in a quasi-two-dimensional harmonic trap and interacting repulsively via finite-range Gaussian potential. The N -body Hamiltonian matrix is diagonalized in given subspaces of quantized total angular momentum L z , to obtain the lowest-energy eigenstate. Further, the internal structure of these eigenstates is analyzed by calculating the corresponding conditional probability distribution. Specifically, the quantum mechanically stable as well as unstable states in a co-rotating frame are examined in the moderately rotating regime corresponding to angular momenta 4 N ≤ L z < 5 N for N = 16 bosons. In response to externally impressed rotation, the patterns of singly quantized vortices are formed, shaping into canonical polygons with a central vortex at the trap center. The internal structure of unstable states reveals the mechanism of entry, nucleation and pattern formation of vortices with structural phase transition, as the condensate goes from one stable vortical state to the other. The stable polygonal vortex patterns having discrete p -fold rotational symmetry with p = 5 and p = 6 are observed. The hexagonal vortex pattern with p = 6 symmetry is a precursor to the triangular vortex lattice of singly quantized vortices in the thermodynamic limit. For unstable states, quantum melting of vortex patterns due to uncertainty in positions of individual vortices, is also briefly discussed.

arXiv:1601.06695•Journal of Physics B Atomic Molecular and Optical Physics•Jan 10, 2017•5 citations
We investigate a rotating system of NNN spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive finite-range Gaussian interaction potential of large sss-wave scattering length. Exact diagonalization of the Hamiltonian matrix in subspaces of quantized total angular momenta LzL_{z}Lz​ is carried out to obtain the NNN-body ground states in slow rotating regime of 0≤Lz≤2N0 \le L_{z} \le 2N0≤Lz​≤2N. We study the finite-range effects of the Gaussian repulsion potential on the many-body ground state properties in different subspaces of quantized LzL_{z}Lz​. The study ranges from the few-body (N=2N=2N=2) system to the many-body (N=16N=16N=16), system where quantum (Bose) statistics becomes perceptible. In particular, we analyze the effect of interaction range of the Gaussian potential on the ground state energy of different LzL_{z}Lz​ states as well on the critical angular velocity Ωc1\Omega_{\bf c1}Ωc1​ of first vortex (Lz=NL_{z}=NLz​=N) state. For an optimal value of the range of the Gaussian interaction potential, the nucleation of the first centered vortex may begin at a lower value of the rotational angular velocity as compared to the zero-range interaction potential. Moreover, we explore the role of interaction potential range on the quantum correlation (measured in terms of the degree of condensation and the von Neumann entropy), for small systems (N=2N=2N=2 to 888) where an understanding of few-body effects provide valuable insight into large systems (N=8N=8N=8 to 161616) which admit the possibility of many-body correlation. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential.
Abstract We use exact diagonalization to study an interacting system of N spinless bosons with finite-range Gaussian repulsion, confined in a quasi-two-dimensional harmonic trap with and without an introduced rotation. The diagonalization of the Hamiltonian matrix using Davidson algorithm in subspaces of quantized total angular momentum L z is carried out to obtain the N -body lowest eigenenergy and eigenstate. To bring out the effect of quantum (Bose) statistics and consequent phase stiffness (rigidity) of the variationally obtained many-body wavefunction on various physical quantities, our study spans from few-body ( N = 2) to many-body ( N = 16) systems. Further, to examine the finite-range effect of the repulsive Gaussian potential on many-body ground state properties of the Bose-condensate, we obtain the lowest eigenstate, the critical angular velocity of single vortex state and the quantum correlation (measured) in terms of von Neumann entanglement entropy and degree of condensation. It is found that for small values of the range (measured by the parameter σ ) of Gaussian potential, the ground state energy increases for few-boson (2 ⩽ N ⩽ 8) systems but decreases for many-boson ( N > 8) systems. On the other hand for relatively large values of the range of Gaussian potential, the ground state energy exhibits a monotonic decrease, regardless of the number of bosons N . For a given N , there is found an optimal value of the range of Gaussian potential for which the first vortex (with L z = N ) nucleates at a lower value of the rotational angular velocity Ω c1 compared to the zero-range ( δ -function) potential. Further, we observe that the inter-particle interaction and the introduced rotation are competing effects with latter being dominant over the former. With increase in the range of Gaussian potential, the value of von Neumann entropy decreases and the degree of condensation increases implying an enhanced quantum correlation and phase rigidity.
We investigate a system of NNN spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive two-body finite-range Gaussian interaction potential of large sss-wave scattering length. Exact diagonalization of the Hamiltonian matrix is carried out to obtain the NNN-body ground state as well as low-lying excited states, using Davidson algorithm in beyond lowest-Landau-level approximation. We examine the finite-range effects of the interaction potential on the many-body ground state energy as also the degree of condensation of the Bose-condensed gas. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential. We further analyze the effect of finite-range interaction potential on the breathing mode collective excitation. Our theoretical results may be relevant for experiments currently conducted on quasi-two-dimensional Bose gas with more realistic interaction potential.
We present exact diagonalization study on a system of 10≤N≤2410 \leq N \leq 2410≤N≤24 spinless bosons interacting via repulsive Gaussian potential, harmonically confined in xyxyxy-plane with an externally impressed rotation about the zzz-axis. The two-body interaction strength in the Gaussian potential is taken in the strongly interacting regime with values of interaction range in the regime 0≤σ≤10\leq σ\leq 10≤σ≤1. The diagonalization of the NNN-body Hamiltonian matrix, in subspaces of total angular momentum in the regime 0≤Lz≤4N0\le L_{z} \le 4N0≤Lz​≤4N corresponds to the filling fraction ν≲3.2ν\lesssim 3.2ν≲3.2 is carried out to obtain the variationally exact ground-state wavefunction and the corresponding eigenvalue. It is found that an increase in interaction range σσσ leads to (a) a systematic decrease in energy, (b) an increase in the critical angular velocity ΩciΩ_{c_{i}}Ωci​​ of the iii{th} vortex state and (c) an increase in the largest eigenvalue λ1λ_{1}λ1​, (condensate fraction), of one-particle reduced density matrix (OPRDM). The von Neumann entropy S1(Lz,σ)S_{1}\left(L_{z},σ\right)S1​(Lz​,σ), quantifying the quantum entanglement between the particles in the many-body ground state, is largely found to decrease with increase in σσσ. Crossings in von Neumann entropy for several of the angular momentum states are observed with variation in σσσ. The response of the Bose-Einstein condensate to rotation is examined through Lz(σ)−Ω(σ)L_{z}\left(σ\right)-Ω\left(σ\right)Lz​(σ)−Ω(σ) stability graph for several values of σσσ. A vortex state with larger plateau length on the Lz−ΩL_{z}-ΩLz​−Ω graph is considered to be more stable. The internal structure of the condensate, as depicted by the conditional probability distribution (CPD) in the body-fixed frame, exhibits characteristic features with interaction range σσσ. One such feature is the merging of the cores of the two-vortex state.
Abstract The ground and low-lying collective states of a rotating system of N = 3 bosons harmonically confined in quasi-two-dimension and interacting via repulsive finite-range Gaussian potential is studied in weakly to moderately interacting regime. The N-body Hamiltonian matrix is diagonalized in subspaces of quantized total angular momenta 0 ≥ L ≥ 4N to obtain the ground and low-lying eigenstates. Our numerical results show that breathing modes with N-body eigenenergy spacing of 2ħω⊥, known to exist in strictly 2D system with zero-range (δ-function) interaction potential, may as well exist in quasi-2D system with finite-range Gaussian interaction potential. To gain an insight into the many-body states, the von Neumann entropy is calculated as a measure of quantum correlation and the conditional probability distribution is analyzed for the internal structure of the eigenstates. In the rapidly rotating regime the ground state in angular momentum subspaces L = (q/2)N (N − 1) with q = 2, 4 is found to exhibit the anticorrelation structure suggesting that it may variationally be described by a Bose-Laughlin like state. We further observe that the first breathing mode exhibits features similar to the Bose-Laughlin state in having eigenenergy, von Neumann entropy and internal structure independent of interaction for the three-boson system considered here. On the contrary, for eigenstates lying between the Bose-Laughlin like ground state and the first breathing mode, values of eigenenergy, von Neumann entropy and internal structure are found to vary with interaction.
We study two spinless bosons interacting via two-body Gaussian potential subjected to an externally impressed rotation about an axis confined in a harmonic trap in two-spatial dimensions. We obtain a transcendental equation for the relative angular momentum ∣m∣|m|∣m∣ state with various values of the two-body interaction range σσσ and the two-body interaction strength g2g_{2}g2​ to study the resulting energy spectrum and analyze the role of Hilbert space dimensions N~\widetilde{N}N. We compare results for both attractive and repulsive interaction for δδδ-function potential and Gaussian potential for various values of interaction range. We study the effects of interaction parameters and relative angular momentum on the ground state energy and its various components, namely, kinetic energy, trap potential and interaction potential. For a given ∣m∣|m|∣m∣ and non-interacting case, we observe that the ground state energy becomes independent of interaction range. However, for a given relative angular momentum and interaction strength g2>0g_{2}>0g2​>0, there is an increase in ground state energy with an increase in interaction range. Below the interaction strength g2V(r)≤−1g_{2}V(r)\leq -1g2​V(r)≤−1, ground state energy diverges to physically unacceptable negative-infinity for ∣m∣=0|m|=0∣m∣=0 state. Further, for ∣m∣=1|m|=1∣m∣=1, the ground state energy becomes independent of the interaction strength. For a ∣m∣|m|∣m∣, we present a comparative study between the Gaussian interaction potential and the δδδ-function potential. Further, we observe that for a given g2g_{2}g2​ and ∣m∣|m|∣m∣, for δδδ-function potential {\it i.e.} σ→0σ\to 0σ→0, to achieve the convergence of ground state energy, we require a considerably large critical Hilbert space. Whereas, in the case of Gaussian interaction potential with σ→1σ\to 1σ→1, the ground state energy converges for a considerably small critical Hilbert space.
Abstract Using exact diagonalization, we investigate the many-body ground state for regular vortex patterns in a rotating Bose-condensed gas of N spinless particles, confined in a quasi-two-dimensional harmonic trap and interacting repulsively via finite-range Gaussian potential. The N -body Hamiltonian matrix is diagonalized in given subspaces of quantized total angular momentum L z , to obtain the lowest-energy eigenstate. Further, the internal structure of these eigenstates is analyzed by calculating the corresponding conditional probability distribution. Specifically, the quantum mechanically stable as well as unstable states in a co-rotating frame are examined in the moderately rotating regime corresponding to angular momenta 4 N ≤ L z < 5 N for N = 16 bosons. In response to externally impressed rotation, the patterns of singly quantized vortices are formed, shaping into canonical polygons with a central vortex at the trap center. The internal structure of unstable states reveals the mechanism of entry, nucleation and pattern formation of vortices with structural phase transition, as the condensate goes from one stable vortical state to the other. The stable polygonal vortex patterns having discrete p -fold rotational symmetry with p = 5 and p = 6 are observed. The hexagonal vortex pattern with p = 6 symmetry is a precursor to the triangular vortex lattice of singly quantized vortices in the thermodynamic limit. For unstable states, quantum melting of vortex patterns due to uncertainty in positions of individual vortices, is also briefly discussed.