Solvable Model of a Generic Driven Mixture of Trapped Bose–Einstein Condensates and Properties of a Many-Boson Floquet State at the Limit of an Infinite Number of Particles

• Main Author: Ofir E. Alon
• Format: Article
• Language: English
• Published: MDPI AG 2020-11-01
• Series: Entropy
• Subjects: solvable models; time-dependent Schrödinger equation; driven Bose–Einstein dondensates; mixtures; Floquet Hamiltonian; time-dependent reduced density matrices
• Online Access: https://www.mdpi.com/1099-4300/22/12/1342

A solvable model of a periodically driven trapped mixture of Bose–Einstein condensates, consisting of <inline-formula><math display="inline"><semantics><msub><mi>N</mi><mn>1</mn></msub></semantics></math></inline-formula> interacting bosons of mass <inline-formula><math display="inline"><semantics><msub><mi>m</mi><mn>1</mn></msub></semantics></math></inline-formula> driven by a force of amplitude <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mrow><mi>L</mi><mo>,</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>N</mi><mn>2</mn></msub></semantics></math></inline-formula> interacting bosons of mass <inline-formula><math display="inline"><semantics><msub><mi>m</mi><mn>2</mn></msub></semantics></math></inline-formula> driven by a force of amplitude <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mrow><mi>L</mi><mo>,</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula>, is presented. The model generalizes the harmonic-interaction model for mixtures to the time-dependent domain. The resulting many-particle ground Floquet wavefunction and quasienergy, as well as the time-dependent densities and reduced density matrices, are prescribed explicitly and analyzed at the many-body and mean-field levels of theory for finite systems and at the limit of an infinite number of particles. We prove that the time-dependent densities per particle are given at the limit of an infinite number of particles by their respective mean-field quantities, and that the time-dependent reduced one-particle and two-particle density matrices per particle of the driven mixture are <inline-formula><math display="inline"><semantics><mrow><mn>100</mn><mo>%</mo></mrow></semantics></math></inline-formula> condensed. Interestingly, the quasienergy per particle does not coincide with the mean-field value at this limit, unless the relative center-of-mass coordinate of the two Bose–Einstein condensates is not activated by the driving forces <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mrow><mi>L</mi><mo>,</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mrow><mi>L</mi><mo>,</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula>. As an application, we investigate the imprinting of angular momentum and its fluctuations when steering a Bose–Einstein condensate by an interacting bosonic impurity and the resulting modes of rotations. Whereas the expectation values per particle of the angular-momentum operator for the many-body and mean-field solutions coincide at the limit of an infinite number of particles, the respective fluctuations can differ substantially. The results are analyzed in terms of the transformation properties of the angular-momentum operator under translations and boosts, and as a function of the interactions between the particles. Implications are briefly discussed.