Periodic driving control of Raman-induced spin-orbit coupling in Bose-Einstein condensates: The heating mechanisms

We focus on a technique recently implemented for controlling the magnitude of synthetic spin-orbit coupling (SOC) in ultracold atoms in the Raman-coupling scenario. This technique uses a periodic modulation of the Raman-coupling amplitude to tune the SOC. Specifically, it has been shown that the effect of a high-frequency sinusoidal modulation of the Raman-laser intensity can be incorporated into the undriven Hamiltonian via effective parameters, whose adiabatic variation can therefore be used to tune the SOC. Here, we characterize the heating mechanisms that can be relevant to this method. We identify the main mechanism responsible for the heating observed in the experiments as basically rooted in driving-induced transfer of population to excited states. Characteristics of that process determined by the harmonic trapping, the decay of the excited states, and the technique used for preparing the system are discussed. Additional heating, rooted in departures from adiabaticity in the variation of the effective parameters, is also described. Our analytical study provides some clues that may be useful in the design of strategies for curbing the effects of heating on the efficiency of the control methods.

Figure 1

Heating rate as a function of the driving frequency for ${\mathrm{\Omega }}_R/\omega =1$. The ground state corresponds to the Thomas-Fermi approximation (continuous line) and (in a single-particle description) to the ground state of a harmonic oscillator (dashed line). In the inset the vertical scale is enlarged.

Figure 2

Diagram of energy bands versus quasimomentum (${\delta }_0=0$ and ${\mathrm{\Omega }}_0=0$).

Figure 3

Heating rate as a function of the driving frequency for ${\mathrm{\Omega }}_R/\omega =1$. The ground state corresponds to the Thomas-Fermi approximation (continuous line), to the ground state of a harmonic oscillator (dashed line), and to a plane wave (dotted line). Thick lines incorporate damping, thin lines correspond to the nondissipative system (data from Fig. 1). In the inset the vertical axis is enlarged.

Figure 4

Heating rate as a function of the reduced driving amplitude ${\mathrm{\Omega }}_R/\omega$ for $\omega =10\mathrm{kHz}$. The ground state has been modeled as a plane wave (continuous line) and as the ground state of a harmonic oscillator (dashed line). Damping has been included. In the inset the vertical axis is enlarged.

Figure 5

Heating rate as a function of the reduced driving amplitude ${\mathrm{\Omega }}_R/\omega$ for $\omega =10\mathrm{kHz}$, obtained including only the dominant $H_1$ correction in the perturbative scheme (dashed line; data from Fig. 4) and with all corrections $H_n$ (continuous line). The ground state corresponds to the Thomas-Fermi approximation. Damping has been included. In the inset the vertical axis is enlarged.

Figure 6

Heating rate as a function of the reduced driving amplitude ${\mathrm{\Omega }}_R/\omega$ at three different frequencies: ${\omega }_1=5\mathrm{kHz}$ (dotted line), ${\omega }_2=10\mathrm{kHz}$ (continuous line), and ${\omega }_3=20\mathrm{kHz}$ (dashed line). The ground state corresponds to the Thomas-Fermi approximation. Damping and all $H_n$ corrections have been included. In the inset the vertical axis is enlarged.

Figure 7

Heating rate as a function of the driving frequency $\omega$ and effective amplitude ${\mathrm{\Omega }}_R/\omega$. The ground state is modeled with the Thomas-Fermi approximation (a) and as the ground state of a harmonic oscillator (b). Damping has been included.

Figure 8

Energy gained in the heating process as a function of the driving frequency at ${\mathrm{\Omega }}_R/\omega =2.9$. The ground state corresponds to the Thomas-Fermi approximation (continuous line) and to the ground state of a harmonic oscillator (dashed line). Damping has been included.