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Shift of the Bose-Einstein condensation transition in the presence of a second atomic species

Pedro M. Gaspar, Vanderlei S. Bagnato, Patricia C. M. Castilho

TL;DR

The paper addresses how a second bosonic species shifts the BEC transition temperature of a principal species in Bose-Bose mixtures. It develops analytical expressions by extending the single-species mean-field framework to two coupled Gross-Pitaevskii equations and treats both thermal and condensed second-species regimes to derive analytic expressions for the interspecies contribution to $T_c$ shifts. The results show that interspecies shifts can be comparable to intraparticle shifts and can be tuned by the atom-number ratio and interspecies scattering length, demonstrated with a realistic $^{23}$Na-$^{39}$K example. This provides a practical tool for designing and interpreting multi-component ultracold gas experiments and enabling phase diagram exploration across arbitrary trapping geometries.

Abstract

Atomic interactions play an important role in the properties of ultracold atomic gases. In single component bosonic systems, its effect is already present at the critical point for the Bose-Einstein condensate phase transition by shifting it to lower temperatures as a consequence of effective repulsion between the atoms. When considering atomic bosonic mixtures, interesting effects arise from the competition between intra- and interspecies interactions such as the miscible-immiscible phase transition and the particular case of self-bounded quantum droplets. In such a scenario, it is natural to expect that these interactions will also affect the critical point of each species composing the mixture. In this paper, we obtain analytical expressions for the critical temperature shift of the phase transition to a Bose-Einstein condensate in the presence of a second species. We treat differently the cases in with the second species is above or below its own critical temperature and apply the obtained relations to the case of a $^{23}$Na-$^{39}$K bosonic mixture which can be realized in current running experimental setups. Our findings can be easily extended to other atomic mixtures trapped by arbitrary conservative traps.

Shift of the Bose-Einstein condensation transition in the presence of a second atomic species

TL;DR

The paper addresses how a second bosonic species shifts the BEC transition temperature of a principal species in Bose-Bose mixtures. It develops analytical expressions by extending the single-species mean-field framework to two coupled Gross-Pitaevskii equations and treats both thermal and condensed second-species regimes to derive analytic expressions for the interspecies contribution to shifts. The results show that interspecies shifts can be comparable to intraparticle shifts and can be tuned by the atom-number ratio and interspecies scattering length, demonstrated with a realistic Na-K example. This provides a practical tool for designing and interpreting multi-component ultracold gas experiments and enabling phase diagram exploration across arbitrary trapping geometries.

Abstract

Atomic interactions play an important role in the properties of ultracold atomic gases. In single component bosonic systems, its effect is already present at the critical point for the Bose-Einstein condensate phase transition by shifting it to lower temperatures as a consequence of effective repulsion between the atoms. When considering atomic bosonic mixtures, interesting effects arise from the competition between intra- and interspecies interactions such as the miscible-immiscible phase transition and the particular case of self-bounded quantum droplets. In such a scenario, it is natural to expect that these interactions will also affect the critical point of each species composing the mixture. In this paper, we obtain analytical expressions for the critical temperature shift of the phase transition to a Bose-Einstein condensate in the presence of a second species. We treat differently the cases in with the second species is above or below its own critical temperature and apply the obtained relations to the case of a Na-K bosonic mixture which can be realized in current running experimental setups. Our findings can be easily extended to other atomic mixtures trapped by arbitrary conservative traps.
Paper Structure (6 sections, 18 equations, 2 figures)

This paper contains 6 sections, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Single bosonic species
  3. Two bosonic species
  4. Not condensed second species
  5. Condensed second species
  6. Conclusions

Figures (2)

  • Figure 1: Comparing the shift due to single species and due to the second species for species-1 equal to species-2 when varying $N_2$. The curve reaches exactly 0.5 at $N_2/N_1=1$.
  • Figure 2: Shift in the critical temperature of species-1 in therms of number of atoms of species-2 for a $^{23}$Na-$^{39}$K mixture castilho2017new. The blue line represents Eq. \ref{['eq: drift termico']} (before dashed line) and \ref{['eq:drift_cond']} (after dashed line), while the orange line represents Eq. \ref{['eq: drift 11']}. The dashed line represents the BEC transition point for species-2.