Atomic Mixtures Alter Temperatures Needed for Bose-Einstein Condensation

Researchers have long known that atomic interactions significantly influence the behaviour of ultracold atomic gases. Pedro M. Gaspar, Vanderlei S. Bagnato, and Patricia C.M. Castilho, all from the Instituto de Física de São Carlos, Universidade de São Paulo, detail in their new work how the critical temperature for Bose-Einstein condensation shifts when a second atomic species is introduced. This study, conducted in collaboration with further researchers at the same institution, provides analytical expressions predicting this shift, considering scenarios where the secondary species exists either above or below its own condensation temperature. The findings are particularly relevant to current experimental setups utilising sodium-potassium bosonic mixtures, and offer broadly applicable insights into the properties of multi-component ultracold atomic gases trapped within various conservative potentials.

Scientists have achieved an advance in understanding how atomic interactions influence the formation of Bose-Einstein condensates, a state of matter where atoms behave as a single quantum entity. Their work details how the critical temperature at which this condensation occurs is altered when two different species of atoms are combined, opening new avenues for manipulating and controlling ultracold atomic gases.

The study builds upon decades of research into ultracold atoms, which have become a powerful tool for exploring fundamental physics, ranging from quantum mechanics to cosmology and condensed matter physics. This research provides analytical expressions detailing the temperature shift, accounting for both the interactions within each species and the crucial interactions between them.

The team focused on a sodium-potassium mixture, a system readily achievable in contemporary experiments, but the principles established are broadly applicable to a wide range of atomic combinations. A key finding is that the presence of a second atomic species demonstrably shifts the critical temperature required for Bose-Einstein condensation in the primary species, arising from the complex interplay between attractive and repulsive forces influencing the density and behaviour of the atoms as they approach the condensation point.

These analytical expressions offer a substantial improvement over previous estimations, particularly for bosonic mixtures, systems where the constituent atoms possess identical quantum properties. The calculations account for both finite-size effects, arising from the confinement of the atoms within a trap, and the direct impact of interspecies interactions on the density distribution.

Applying their model to a mixture of 23Na and 39K atoms, the team demonstrates that the predicted temperature shifts are substantial enough to be measurable with existing experimental technology. To determine the critical temperature shift (δTc,1)12 for species-1, the equations governing this relationship are solved numerically, employing the bisection method to ascertain the chemical potential (μ2) of species-2.

The work begins by establishing a thermal chemical potential, normalised by the number of atoms of species-2 (N2), Planck’s constant (ħ), and the trap frequency (ω2), alongside Boltzmann’s constant (kB) and temperature (T0). This normalization yields a value negative and significantly larger than the thermal energy when the second species is far from its critical temperature, diminishing as it approaches T0,c,2.

Numerical analysis, normalised by the intraspecies shift, tracks the behaviour of this shift as N2 varies from zero to the critical number (Nc,2 = ζ kBT0,c,1 ħω2 3), representing the saturation point beyond which atoms populate the condensed state. Further investigation extends to scenarios where the second species is condensed (T0,c,2 T0,c,1), where the density of species-2 comprises both thermal and condensed fractions, described by a thermal contribution and a condensed fraction approximated using the Thomas-Fermi limit.

The total density is then the sum of these components, allowing for the calculation of the shift in the critical temperature of species-1, incorporating both thermal and condensed contributions, and utilising spherical coordinates to simplify the integral calculations. Analytical expressions reveal a critical temperature shift for Bose-Einstein condensate formation in atomic mixtures, with calculations demonstrating an interspecies interaction-induced shift of 2.6% for a 23Na-39K mixture under specific parameters.

This shift, calculated using a mean-field approximation and expansion around the critical temperature, is comparable in magnitude to the shift caused by intraspecies interactions. The research establishes a general analytical expression for this critical temperature shift, applicable to any bosonic mixture and solvable numerically. Calculations for a 23Na-39K mixture, utilising 5 × 106 atoms and scattering lengths of 54a0, indicate a finite size effect on the critical temperature of approximately 0.4%.

The derived equation for the critical temperature shift, δTc,1 / T0c,1 12, incorporates the thermal wavelengths and trapping potentials of both species, quantifying the shift as being proportional to the interspecies coupling constant, g12, and the densities of both atomic species. Specifically, the research details how manipulating the atom number ratio between the 23Na and 39K species allows for control over the critical temperature shift.

The derived expression, evaluated with a transformation to spherical coordinates, relates the shift to an infinite sum dependent on the chemical potential of the secondary species, solved numerically using the bisection method. The subtle choreography of atomic interactions is yielding increasingly precise control over matter at its coldest. While the principle that interactions shift the critical point is not new, the analytical expressions developed here offer a clearer, more predictable pathway for experimental manipulation.

The significance lies in the potential to engineer bespoke quantum systems, as mixtures of atoms, unlike single-species condensates, offer a richer parameter space for exploring phenomena like miscibility and the formation of self-bound droplets, effectively creating miniature quantum objects. However, extending these calculations beyond idealised scenarios remains a challenge, as real-world experiments are inevitably complicated by factors like trap geometry and finite atom number. Furthermore, the analytical approach relies on certain approximations; the expansion methods used may not hold true under all conditions, particularly at higher temperatures or stronger interactions.