Resonant energy transfer in Bose–Einstein condensates
Introduction
Bose–Einstein condensates (BECs) [1] are one of the most appealing systems for nonlinear science due to their unprecedented experimental maneuverability and the supporting theoretical modeling. A BEC is comprised of a dilute gas of magnetically (or optically) trapped bosons that, when cooled to extremely low temperatures, occupy their lowest-energy quantum state. BECs can be manipulated in time and space. On the spatial side, one has good experimental control over the shape and the strength of the trapping potential, while on the temporal side one can modify in time the strength of the two-body scattering length [2], [3] to the extent of probing the region between attractive and repulsive condensates. Theoretically, the dynamics of Bose-condensed gases is given by the so-called Gross–Pitaevskii (GP) equation [4], [5], a cubic Schrödinger equation describing the dynamics of the condensate. In one spatial dimension (with , a convention followed throughout the rest of the paper, being the mass of each boson) the adimensional GP equation is given by where is the trapping potential, here taken as . The coefficient of the cubic term is , where is the two-body scattering length. Notice that , the number of particles, is a dynamical invariant of Eq. (1) for general, time dependent, and/or . The GP Lagrangian is given by
Due to the advent of the Feshbach resonance [2], [3] it is now possible to modulate in time the scattering length. While there are numerous results concerning the parametric resonances that take place when the scattering length is modulated as , where is a fixed frequency (see Refs. [6], [7], [8] and references therein), very little is known about the case when the driving frequency is time dependent. Here we focus on the case when the driving frequency increases linearly in time, i.e., with .
In this paper we show that, apart from a set of intrinsically nonlinear resonances exhibited by the Gross–Pitaevskii equation, the dynamics of the condensate is given to leading order by , where is a rescaled value of the time-dependent part of the width of the condensate, is the rescaled time and is proportional to . Comparing this simplified equation with full GP numerics we see that for small and it captures accurately both the mode-locking and the resonant energy transfer that take place when the effective frequency of the driving field matches the eigenfrequency of the condensate, from now on called . More explicitly, for , the width of the condensate shows oscillations close to the effective frequency of the drive (i.e., ). At resonance (i.e., ) there is a sudden increase of the amplitude of oscillations due to resonant energy transfer, while at later time the oscillations are on the condensate eigenfrequency and not that of the driving field (i.e., ). Drawing from previous studies on dissipative systems we refer to this phenomenon as mode locking.
We consider a dilute, magnetically-trapped condensate and a Gaussian wave-function ansatz (see Refs. [9], [10], [11], [12] for the main results regarding the use of a Gaussian ansatz for the cubic nonlinear Schrödinger equation). After the usual variational recipe we linearize the ensuing ordinary differential equations around the equilibrium/ground-state value of the width of the condensate. The final equation (in the time-dependent part of the width of the condensate) is that of a driven harmonic oscillator. The rest of the paper is structured as follows. Section 2 is dedicated to the variational method that simplifies the condensate dynamics to an ODE. In Section 3 we analyze the equation of the driven harmonic oscillator, while in Section 4 we make a one-to-one comparison between the reduced ODE dynamics and full PDE numerics of the GP equation. Section 5 gathers our conclusions.
Section snippets
Variational recipe
We consider a dilute, magnetically-trapped BEC and a one-dimensional Gaussian-like profile as ansatz where is the number of atoms in the cloud, is an overall phase (the canonical conjugate of ), is the width of the condensate while , the so-called chirp, is the canonical conjugate of [9], [10], [11], [12]. The above trial wave-function yields the Lagrangian The Euler–Lagrange equations
Analytical solution
In this section we analyze the equation of the driven harmonic oscillator. The solution of Eq. (13) is given by where and and are the well-known Fresnel functions defined as and
Before stepping into the analysis of Eq. (15) let us first
Numerical results
In this section we compare the width of the condensate obtained from the full partial-differential GP equation with the approximate formula derived from the harmonic-oscillator picture. Our main result is that for small amplitudes of the driving field the harmonic oscillator picture captures quantitatively both the mode-locking and the resonant energy transfer. In Fig. 1 we depict the dynamics of the width of a typical low-density condensate obtained from the full GP simulation (upper panel)
Conclusions
We have investigated the dynamics of a dilute, magnetically-trapped one-dimensional Bose–Einstein condensate whose scattering length is driven as , where increases linearly in time, i.e., . Solving numerically the GP equation we have shown that the response frequency of the condensate locks to its eigenfrequency at resonance (for small values of and ). The locking is accompanied by a sudden increase in the oscillations amplitude due to resonance energy transfer.
Acknowledgments
The authors thank Henrik Smith and Christopher J. Pethick for seminal discussions. A.I.N. thanks Kim Ø. Rasmussen for discussions and acknowledges the hospitality of Los Alamos National Laboratory where part of this work was carried out. R.C.G. acknowledges the support of NFS-DMS-0505663.
References (18)
- et al.
Bose–Einstein Condensation in Dilute Gases
(2001) Nature
(1998)- et al.
Phys. Rev. Lett.
(2000) Sov. Phys. JETP
(1961)Nuovo Cimento
(1961)- et al.
Phys. Rev. A
(2003) - et al.
Phys. Rev. E
(2005) - et al.
Phys. Rev. E
(2004) - et al.
Phys. Rev. Lett.
(1996)
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