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arXiv:1502.00933 · Journal of Experimental and Theoretical Physics Letters

Origin of a nonlinear contribution to the shift of the critical temperature in atomic Bose-Einstein condensates

S. Sergeenkov, Fabio Briscese, M. Grether, M. de Llano

Published Mar 1, 2015• 2 citations
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Abstract

We discuss a possible origin of the experimentally observed nonlinear contribution to the shift ΔT c = T c − T 0 of the critical temperature T c in an atomic Bose-Einstein condensate (BEC) with respect to the critical temperature T 0 of an ideal gas. We found that accounting for a nonlinear (quadratic) Zeeman effect (with applied magnetic field closely matching a Feshbach resonance field B 0) in the mean-field approximation results in a rather significant renormalization of the field-free nonlinear contribution b 2, namely, T c /T 0 ≃ b 2* (a/λ T )2 (where a is the s-wave scattering length, λ T is the thermal wavelength at T 0 ) with b 2* = γ2 b 2 and γ = γ(B 0). In particular, we predict b 2* ≃ 42.3 for the B 0 ≃ 403 G resonance observed in the 39K BEC.

DOI
10.1134/s0021364015060119

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We discuss a possible origin of the experimentally observed nonlinear contribution to the shift ΔT c = T c − T 0 of the critical temperature T c in an atomic Bose-Einstein condensate (BEC) with respect to the critical temperature T 0 of an ideal gas. We found that accounting for a nonlinear (quadratic) Zeeman effect (with applied magnetic field closely matching a Feshbach resonance field B 0) in the mean-field approximation results in a rather significant renormalization of the field-free nonlinear contribution b 2, namely, T c /T 0 ≃ b 2* (a/λ T )2 (where a is the s-wave scattering length, λ T is the thermal wavelength at T 0 ) with b 2* = γ2 b 2 and γ = γ(B 0). In particular, we predict b 2* ≃ 42.3 for the B 0 ≃ 403 G resonance observed in the 39K BEC.
We discuss the effect of interatomic interactions on the condensation temperature TcT_cTc​ of a laboratory atomic Bose-Einstein condensate under the influence of an external trapping magnetic field. We predict that accounting for hyperfine interactions mediated Zeeman term in the mean-field approximation produces, in the case of the 403 G403 \, G403G Feshbach resonance in the ∣F,mF>=∣1,1>|F,m_F> = |1,1>∣F,mF​>=∣1,1> hyperfine state of a 39K^{39}K39K condensate, with FFF the total spin of the atom, an experimentally observed (and not yet explained) shift in the condensation temperature ΔTc/Tc0=b0∗+b1∗(a/λT)+b2∗(a/λT)2ΔT_{c}/T_{c}^{0}=b^{*}_0+b^{*}_1 (a/λ_{T}) + b^{*}_2 (a/λ_{T})^2ΔTc​/Tc0​=b0∗​+b1∗​(a/λT​)+b2∗​(a/λT​)2 with b0∗≃0.0002b^{*}_0 \simeq 0.0002b0∗​≃0.0002, b1∗≃−3.4b^{*}_1 \simeq -3.4b1∗​≃−3.4 and b2∗≃47b^{*}_2 \simeq 47b2∗​≃47, where aaa is the s-wave scattering length, and λTλ_TλT​ is the thermal wavelength at Tc0T_{c}^{0}Tc0​. Generic expressions for the coefficients b0∗b^*_0b0∗​, b1∗b^*_1b1∗​ and b2∗b^*_2b2∗​ are also obtained, which can be used to predict the temperature shift for other Feshbach resonances of 39K^{39}K39K or other atomic condensates.
We study the condensed fraction of a harmonically trapped atomic Bose gas at the critical point predicted by mean-field theory. The nonzero condensed fraction f(0) is induced by critical correlations which increase the transition temperature T(c) above T(c) (MF). Unlike the T(c) shift in a trapped gas, f(0) is sensitive only to the critical behavior in the quasiuniform part of the cloud near the trap center. To leading order in the interaction parameter a/λ(0), where a is the s-wave scattering length and λ(0) the thermal wavelength, we expect a universal scaling f(0) proportionally (a/λ(0))(4). We experimentally verify this scaling using a Feshbach resonance to tune a/λ(0). Further, using the local density approximation, we compare our measurements with the universal result obtained from Monte Carlo simulations for a uniform system, and find excellent quantitative agreement.
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In a recent experiment, a Bose-Einstein condensate of 7{}^{7}7Li has been excited by a harmonic modulation of the atomic sss-wave scattering length via Feshbach resonance. By combining an analytical perturbative approach with extensive numerical simulations, we analyze the emerging nonlinear dynamics of the system on the mean-field Gross-Pitaevskii level at zero temperature. Resulting excitation spectra are presented and prominent nonlinear features are found: mode coupling, higher harmonics generation, and significant shifts in the frequencies of collective modes. We indicate how nonlinear dynamical properties could be made clearly observable in future experiments and compared to our results.
The experimental discovery of dilute ultracold quantum gases in 1995 has instigated a broad interest in ultracold atoms and molecules, and paved the way for extensive studies of a wide range of both experimental and theoretical topics. Various theories were developed to describe Bose-Einstein condensates (BECs) for different ranges of temperature and interaction strengths. This thesis focuses on studying the mean-field theory of the Gross-Pitaevskii (GP) equation which provides a good description for the macroscopic wave function when both quantum and thermal fluctuations are negligible. This means that the range of validity is restricted to small interaction strengths at zero temperature. Within the GP theory we study analytically and numerically the nonlinear dynamics of BECs which are induced by a harmonic modulation of either the interaction strength or the harmonic trapping potential. At first a detailed numerical investigation reveals that solving the partial differential equation of Gross and Pitaevskii can be reduced to solving ordinary differential equations for the condensate widths within a Gaussian variational approach. Despite this tremendous simplification the prominent nonlinearity of the underlying GP equation is inherited by the variational equations, which thus allow to describe a rich variety of nonlinear phenomena in different experimental setups. In order to describe them analytically we transfer the Poincar\'e-Lindstedt method, which represents a well-established tool in the field of nonlinear dynamics, to the Gaussian variational approach. Modulation of some of the parameters of a BEC can give rise to prominent nonlinear features, such as shifts in the frequencies of collective oscillation modes, higher harmonics generation, and resonant mode coupling. As a first application we consider a recent Bose-Einstein condensate of Li-7, which has been excited experimentally by a harmonic modulation of the atomic s-wave scattering length with the help of a Feshbach resonance. Combining an analytical perturbative approach with extensive numerical simulations, we analyze the emerging nonlinear dynamics. In particular, we present the resulting shift of collective excitation spectra close to parametric resonances where the driving frequency coincides with a collective mode frequency. Another application of experimental importance is to study geometric resonances in BECs with two-body and three-body contact interactions, where changing the geometry of the trapping potential leads to commensurate collective excitation frequencies with emerging nonlinear effects such as frequency shifts and resonant mode coupling. In this context we also show that a small repulsive three-body interaction is able to extend the stability region of the condensate for an attractive two-body interaction as it increases the critical number of atoms in the trap. Apart from a periodic modulation of a system parameter the dipole mode frequency can be excited by introducing an abrupt change in the potential, where the center of mass oscillates according to the Kohn theorem back and forth in the trapping potential with the natural trap frequency irrespective of both the strength and the type of the two-particle interaction. To this end, we study the collective excitation modes of a Bose-Einstein condensate in a harmonic Ioffe- Pritchard in the vicinity of a Feshbach resonance for experimentally realistic parameters of a Rb-85 BEC, where the dipole mode is excited in z-direction. A linearization of the underlying equations of motions shows that the dipole mode frequency decreases when the bias magnetic field approaches the Feshbach resonance, so the Kohn theorem is violated. Finally within a variational approach we discuss the physical properties of a BEC in an axially-symmetric harmonic trap with a single vortex in the center. At first we analyze the equilibrium configuration and determine the vortex size as well as the Thomas- Fermi radii of the condensate in the Thomas-Fermi limit. Then we calculate the critical rotation frequency for the emergence of the vortex and compare our findings with the literature. Finally, we investigate how the presence of the vortex changes the collective excitation frequencies and we discuss the free expansion of the BEC. All these results indicate how nonlinear dynamics properties could be made clearly observable in future experiments on the basis of our results.
We investigate the possibilities of distinguishing the mean-field and fluctuation effects on the critical temperature of a trapped Bose gas with repulsive interatomic interactions. Since in a direct measurement of the critical temperature as a function of the number of trapped atoms these effects are small compared to the ideal gas results, we propose to observe Bose-Einstein condensation by adiabatically ramping down the trapping frequency. Moreover, analyzing this adiabatic cooling scheme, we show that fluctuation effects can lead to the formation of a Bose-Einstein condensate at frequencies which are much larger than those predicted by the mean-field theory.