Quantum many-body dynamics of dark solitons in optical lattices

Mishmash, Ryan V.; Danshita, Ippei; Clark, Charles W.; Carr, Lincoln D.

doi:10.1103/physreva.80.053612

Cited by 61 publications

(81 citation statements)

References 68 publications

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“…2,5,14,[54][55][56][57][58][59][60][61][62] Symmetries have also been used in more recent proposals to simulate time evolution with MPSs. [10][11][12][13][14][63][64][65][66][67][68] When considering symmetries, it is important to notice that an MPS is a trivalent tensor network. That is, in an MPS each tensor has at most three indices.…”

Section: Introductionmentioning

confidence: 99%

Tensor network states and algorithms in the presence of a global U(1) symmetry

2011

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Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. In a recent paper [Phys. Rev. A 82, 050301 (2010)] we discussed how to incorporate a global internal symmetry, given by a compact, completely reducible group G, into tensor network decompositions and algorithms. Here we specialize to the case of Abelian groups and, for concreteness, to a U(1) symmetry, associated, e.g., with particle number conservation. We consider tensor networks made of tensors that are invariant (or covariant) under the symmetry, and explain how to decompose and manipulate such tensors in order to exploit their symmetry. In numerical calculations, the use of U(1)-symmetric tensors allows selection of a specific number of particles, ensures the exact preservation of particle number, and significantly reduces computational costs. We illustrate all these points in the context of the multiscale entanglement renormalization Ansatz.

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Section: Introductionmentioning

confidence: 99%

Tensor network states and algorithms in the presence of a global U(1) symmetry

2011

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“…The latter situation has been considered in Refs. [19,20] within many-body numerical calculations. Analysis of the second order correlation function leads the authors to the conclusion that a dark soliton cannot be observed when the single particle density becomes uniform.…”

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“…In one-dimensional (1D) space, this non-linear wave equation possesses a dark (bright) soliton solution if particle interactions are effectively repulsive (attractive). The GPE assumes all particles in the same single particle state and neglects that the interactions can populate other modes.There is a debate in the literature concerning manybody effects in a dark soliton [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. The GPE predicts a stable dark soliton state in 1D.…”

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confidence: 99%

“…Following Refs. [19,20,34] we choose as initial state the ground state of the system in the presence of a Gaussian barrier V (x) = V 0 e −x 2 /σ 2 0 located at the center of the box. For V 0 = 18 and σ 0 = 0.2 the one-body density profile is nearly identical to the density profile of the dark soliton described by eq.…”

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confidence: 99%

“…The corresponding parameter for the Lieb-Liniger model [24] is γ = 0.04, indicating the weakly interacting regime. Space is discretized on a grid with ∆x = 0.2, so that the many-body Hamiltonian in the second-quantization formalism is a Bose-Hubbard Hamiltonian with tunneling amplitude 1/(2∆x 2 ) and interaction energy g 0 /∆x [19,20,29,30] [37]. An N -body state of the system can be represented by a matrix product state (MPS) α l =1,2... are rapidly decaying numbers and a cutoff χ can be introduced in the sum over Greek indices.…”

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Many-Body Matter-Wave Dark Soliton

Delande

Sacha

2014

Phys. Rev. Lett.

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The Gross-Pitaevskii equation -which describes interacting bosons in the mean-field approximation -possesses solitonic solutions in dimension one. For repulsively interacting particles, the stationary soliton is dark, i.e. is represented by a local density minimum. Many-body effects may lead to filling of the dark soliton. Using quasi-exact many-body simulations, we show that, in single realizations, the soliton appears totally dark although the single particle density tends to be uniform. [4][5][6][7][8][9]. Within the mean field approach and at zero temperature, an atomic Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation (GPE). In one-dimensional (1D) space, this non-linear wave equation possesses a dark (bright) soliton solution if particle interactions are effectively repulsive (attractive). The GPE assumes all particles in the same single particle state and neglects that the interactions can populate other modes.There is a debate in the literature concerning manybody effects in a dark soliton [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. The GPE predicts a stable dark soliton state in 1D. The Bogoliubov corrections show that signatures of the soliton tend to disappear in the single particle density because particles depleted from the condensate fill the soliton notch [12][13][14][15]. However, the single particle density refers to results averaged over many experimental realizations and is not able to predict a single experimental outcome. Within the Bogoliubov approach, it is possible to simulate single experiments: in a single photo, the soliton is completely dark but its position varies randomly from realization to realization [14,15]. This is a signature of the quantum character of the soliton, i.e. the soliton position is not described by a classical variable but by a probability density. The Bogoliubov analysis is limited to quantum fluctuations of the soliton position on a scale smaller than the healing length and cannot predict what happens when many-body effects become stronger. The latter situation has been considered in Refs. [19,20] within many-body numerical calculations. Analysis of the second order correlation function leads the authors to the conclusion that a dark soliton cannot be observed when the single particle density becomes uniform. However, full simulations of single experiments involve much higher order correlation functions [21,22]. It is the goal of the present work to perform many-body numerical simulations of the entire experiment starting from soliton preparation till the destructive measurement of all particle positions. We show that a fully dark soliton can be observed when manybody effects are strong, although its measured position is random and varies among experimental realizations. In the following we assume the zero temperature limit and do not consider thermodynamical instability of the dark soliton [26,27].The stationary solution of the 1D GPE for particles of mass m and interaction coefficient g 0 , associated with a da...

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Open Source Code Development

Wall

2015

Quantum Many-Body Physics of Ultracold Molecules in Optical Lattices

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Quantum many-body dynamics of dark solitons in optical lattices

Cited by 61 publications

References 68 publications

Tensor network states and algorithms in the presence of a global U(1) symmetry

Tensor network states and algorithms in the presence of a global U(1) symmetry

Many-Body Matter-Wave Dark Soliton

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