Shock waves in a 1D Bose gas: from a BEC to a Tonks gas

We derive and analyze shock-wave solutions of hydrodynamic equations describing repulsively interacting one-dimensional Bose gas. We also use the number-conserving Bogolubov approach to verify accuracy of the Gross-Pitaevskii equation in shock wave problems. We show that quantum corrections to dynamics of shocks (dark-shock-originated solitons) in a Bose-Einstein condensate are negligible (important) for a realistic set of system parameters. We point out possible signatures of a Bose-Einstein condensate---Tonks crossover in shock dynamics. Our findings can be directly verified in different experimental setups.We derive and analyze shock-wave solutions of hydrodynamic equations describing repulsively interacting one-dimensional Bose gas. We also use the number-conserving Bogolubov approach to verify accuracy of the Gross-Pitaevskii equation in shock wave problems. We show that quantum corrections to dynamics of shocks (dark-shock-originated solitons) in a Bose-Einstein condensate are negligible (important) for a realistic set of system parameters. We point out possible signatures of a Bose-Einstein condensate---Tonks crossover in shock dynamics. Our findings can be directly verified in different experimental setups.

arXiv:cond-mat/0506210Physical Review AApr 3, 200640 citations

Shock waves in quasi one-dimensional Bose-Einstein condensate

Luca Salasnich

arXiv:1504.07517The European Physical Journal PlusMar 1, 20167 citations

Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics

Mark A. Hoefer, Mark J. Ablowitz, Ian R. Coddington +3 more

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock-wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, e.g., traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation, hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical one-dimensional (1D) dissipative and dispersive shock problem shows significant differences in shock structure and shock-front speed. Numerical results associated with the three-dimensional experiment show that three- and two-dimensional approximations are in excellent agreement and 1D approximations are in good qualitative agreement. Using 1D DSW theory, it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock-wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, e.g., traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation, hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical one-dimensional (1D) dissipative and dispersive shock problem shows significant differences in shock structure and shock-front speed. Numerical results associated with the three-dimensional experiment show that three- and two-dimensional approximations are in excellent agreement and 1D approximations are in good qualitative agreement. Using 1D DSW theory, it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.

arXiv:cond-mat/0603389Physical Review AAug 29, 2006318 citations

Wave functions of the super-Tonks-Girardeau gas and the trapped one-dimensional hard-sphere Bose gas

Marvin D. Girardeau, G. E. Astrakharchik

Recent theoretical and experimental results demonstrate a close connection between the super-Tonks-Girardeau (STG) gas and a one-dimensional (1D) hard-sphere Bose (HSB) gas with hard-sphere diameter nearly equal to the 1D scattering length ${a}_{1\mathrm{D}}$ of the STG gas, a highly excited gaslike state with nodes only at interparticle separations $|{x}_{j\ensuremath{\ell}}|={x}_{\mathrm{node}}\ensuremath{\approx}{a}_{1\mathrm{D}}$. It is shown herein that when the coupling constant ${g}_{B}$ in the Lieb-Liniger interaction ${g}_{B}\ensuremath{\delta}({x}_{j\ensuremath{\ell}})$ is negative and $|{x}_{12}|\ensuremath{\geqslant}{x}_{\mathrm{node}}$, the STG and HSB wave functions for $N=2$ particles are not merely similar, but identical; the only difference between the STG and HSB wave functions is that the STG wave function allows a small penetration into the region $|{x}_{12}|<{x}_{\mathrm{node}}$ , whereas for a HSB gas with hard-sphere diameter ${a}_{\mathrm{hs}}={x}_{\mathrm{node}}$ , the HSB wave function vanishes when $|{x}_{12}|<{a}_{\mathrm{hs}}$ . Arguments are given suggesting that the same theorem holds also for $N>2$ . The STG and HSB wave functions for $N=2$ are given exactly in terms of a parabolic cylinder function, and for $N\ensuremath{\geqslant}2$, ${x}_{\mathrm{node}}$ is given accurately by a simple parabola. The metastability of the STG phase generated by a sudden change of the coupling constant from large positive to large negative values is explained in terms of the very small overlap between the ground state of the Tonks-Girardeau gas and collapsed cluster states.Recent theoretical and experimental results demonstrate a close connection between the super-Tonks-Girardeau (STG) gas and a one-dimensional (1D) hard-sphere Bose (HSB) gas with hard-sphere diameter nearly equal to the 1D scattering length ${a}_{1\mathrm{D}}$ of the STG gas, a highly excited gaslike state with nodes only at interparticle separations $|{x}_{j\ensuremath{\ell}}|={x}_{\mathrm{node}}\ensuremath{\approx}{a}_{1\mathrm{D}}$. It is shown herein that when the coupling constant ${g}_{B}$ in the Lieb-Liniger interaction ${g}_{B}\ensuremath{\delta}({x}_{j\ensuremath{\ell}})$ is negative and $|{x}_{12}|\ensuremath{\geqslant}{x}_{\mathrm{node}}$, the STG and HSB wave functions for $N=2$ particles are not merely similar, but identical; the only difference between the STG and HSB wave functions is that the STG wave function allows a small penetration into the region $|{x}_{12}|<{x}_{\mathrm{node}}$, whereas for a HSB gas with hard-sphere diameter ${a}_{\mathrm{hs}}={x}_{\mathrm{node}}$, the HSB wave function vanishes when $|{x}_{12}|<{a}_{\mathrm{hs}}$. Arguments are given suggesting that the same theorem holds also for $N>2$. The STG and HSB wave functions for $N=2$ are given exactly in terms of a parabolic cylinder function, and for $N\ensuremath{\geqslant}2$, ${x}_{\mathrm{node}}$ is given accurately by a simple parabola. The metastability of the STG phase generated by a sudden change of the coupling constant from large positive to large negative values is explained in terms of the very small overlap between the ground state of the Tonks-Girardeau gas and collapsed cluster states.

arXiv:0912.1633Physical Review AJun 21, 201035 citations

Supersonic effects and shock waves in a Bose–Einstein condensate

Michail A. Zak, Igor K. Kulikov

Physics Letters AJan 1, 200310 citations

{a}_{1\mathrm{D}}

of the STG gas, a highly excited gaslike state with nodes only at interparticle separations $|{x}_{j\ensuremath{\ell}}|={x}_{\mathrm{node}}\ensuremath{\approx}{a}_{1\mathrm{D}}$. It is shown herein that when the coupling constant

{g}_{B}

in the Lieb-Liniger interaction ${g}_{B}\ensuremath{\delta}({x}_{j\ensuremath{\ell}})$ is negative and $|{x}_{12}|\ensuremath{\geqslant}{x}_{\mathrm{node}}$, the STG and HSB wave functions for

N=2

particles are not merely similar, but identical; the only difference between the STG and HSB wave functions is that the STG wave function allows a small penetration into the region

|{x}_{12}|<{x}_{\mathrm{node}}

, whereas for a HSB gas with hard-sphere diameter

{a}_{\mathrm{hs}}={x}_{\mathrm{node}}

, the HSB wave function vanishes when

|{x}_{12}|<{a}_{\mathrm{hs}}

. Arguments are given suggesting that the same theorem holds also for

N>2

. The STG and HSB wave functions for

N=2

are given exactly in terms of a parabolic cylinder function, and for $N\ensuremath{\geqslant}2$,

{x}_{\mathrm{node}}

is given accurately by a simple parabola. The metastability of the STG phase generated by a sudden change of the coupling constant from large positive to large negative values is explained in terms of the very small overlap between the ground state of the Tonks-Girardeau gas and collapsed cluster states.