Abstract
We present a series of quantum states that are characterized by dark solitons of the nonlinear Schrödinger equation (i.e. the Gross–Pitaevskii equation) for the one-dimensional Bose gas interacting through the repulsive delta-function potentials. The classical solutions satisfy the periodic boundary conditions and we simply call them classical dark solitons. Through exact solutions we show corresponding aspects between the states and the solitons in the weak coupling case: the quantum and classical density profiles completely overlap with each other not only at an initial time but also at later times over a long period of time, and they move together with the same speed in time; the matrix element of the bosonic field operator between the quantum states has exactly the same profiles of the square amplitude and the phase as the classical complex scalar field of a classical dark soliton not only at the initial time but also at later times, and the corresponding profiles move together for a long period of time. We suggest that the corresponding properties hold rigorously in the weak coupling limit. Furthermore, we argue that the lifetime of the dark soliton-like density profile in the quantum state becomes infinitely long as the coupling constant approaches zero, by comparing it with the quantum speed limit time. Thus, we call the quantum states quantum dark soliton states.
Highlights
The experimental technique of trapped one-dimensional atomic gases [1, 2] has become a fundamental tool for revealing nontrivial effects in quantum many-body systems [3, 4]
We suggest that it is due to the effect of manybody correlations in the 1D Bose gas such as observed in the study of the Bose-Einstein condensation (BEC) fraction [37] as mentioned in Introduction
We suggest that if the coupling constant c becomes much smaller, the square amplitude profiles |ψQ(x)|2 should be in much better agreement with the density profiles of classical dark soliton |ψC(x)|2
Summary
We suggest that if the coupling constant c becomes much smaller, the square amplitude profiles |ψQ(x)|2 should be in much better agreement with the density profiles of classical dark soliton |ψC(x)|2. (ii) Case of N = 20 for c = 1 With c = 1 and N = L = 20, for the profiles of square amplitude |ψQ(x, t)|2 and phase Arg[ψQ(x, t)]/π of the matrix element ψQ(x, t), the snapshots at times t are plotted by red solid lines in the third and fourth columns of panels of Fig. 9 from the left, respectively. (iii) Case of N = 500 for c = 0.01 With c = 0.01 and N = L = 500, for the profiles of square amplitude |ψQ(x, t)|2 and phase Arg[ψQ(x, t)]/π of the matrix element ψQ(x, t), the snapshots at times t are plotted by red solid lines in the first and second columns of panels of Fig. 10 from the left, respectively. The square-amplitude profile vanishes at the origin, and it has a density notch around at the origin and small ripples at the shoulders of the notch; the phase profiles change signs at the origin, etc
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