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Introduction and Theoretical Basics

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Non-Equilibrium Dynamics Beyond Dephasing

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

This chapter discusses the theoretical foundations of \(\mathrm {1d}\) Bose gases, geared towards the systems investigated in our setup. After a short introduction of interacting \(\mathrm {1d}\) Bose gases we briefly introduce the seminal Lieb-Liniger model, discussing its distinct phases before giving the quasi-condensates regime a more detailed treatment.

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Notes

  1. 1.

    Note that integrability is only well defined for the classical case. It means that a system with n degrees of freedom is restricted by n independent conserved quantities. The translation of this concept to quantum mechanics is not trivial [6]. Nevertheless, the term is generally used in the literature in reference to 1d Bose gases.

  2. 2.

    This particular value is obtained if both atoms are in the internal state \(|F=2,m_F=2\rangle \), which is the case in our setup (see Sect. 3.1.1).

  3. 3.

    This is the length scale on which a condensate wave function returns to its bulk value next to a local perturbation [14].

  4. 4.

    Assuming both expansion parameters to be of the same size.

  5. 5.

    Whenever convenient we are suppressing the z and t dependence of \(\delta \hat{n}\), \(\hat{\theta }\), \(n_\mathrm {1d}\) and the quantities derived from them to simplify the notation and aid readability.

  6. 6.

    For the forth term we used \(-\int \mathop {}\!\mathrm {d}{z}\, \hat{\theta }\partial ^2_z \hat{\theta }= \int \mathop {}\!\mathrm {d}{z}\, (\partial _z \hat{\theta })^2.\)

  7. 7.

    In the case of periodic boundary conditions only every second of these frequencies would be supported due to the continuity conditions \(\hat{\theta }(0) = \hat{\theta }(L)\) and \(\delta \hat{n}(0) = \delta \hat{n}(L)\).

  8. 8.

    This is equivalent to the two mode approximation of the double well potential [30].

  9. 9.

    The constant first term being absorbed by a shift of the energy minimum that does not affect the dynamics.

  10. 10.

    This corresponds to the ground state energy of the 2d harmonic potential.

  11. 11.

    Note that Eq. (2.36) is not exactly converging to the 3d result \(c_\mathrm {3d}= \sqrt{\hbar \omega _{\!\perp \!}/ m}(a_{\mathrm {s}}n_\mathrm {1d})^{{1}/{4}}\) due to the approximation made by taking the broadened Gaussian ansatz.

  12. 12.

    In order to avoid spurious soliton excitations in the relative degrees of freedom of two coupled gases the seed state needs to have a finite occupation.

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  1. Laboratoire Kastler Brossel, École Normale Supérieure, Paris, France

    Bernhard Rauer

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Rauer, B. (2019). Introduction and Theoretical Basics. In: Non-Equilibrium Dynamics Beyond Dephasing. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-18236-6_2

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