Focusing of Bragg-outcoupled paraxial atom lasers

We theoretically study the focusing of a quasicontinuous atom laser beam of rubidium-85 $\left({}^{85}\mathrm{Rb}\right)$. A two-state model analysis based on the Gross-Pitaevskii equation is used which comprises the effects of two-body atom-atom interactions and three-body recombination losses. Utilizing optical focusing potentials such as harmonic potentials, the essential factors such as the width, peak density, and atom loss rate of the focused atom laser beam profile are investigated. Our analysis predicts that using an atom laser offers a dramatic improvement in resolutions of up to 8 nm.

Figure 1

Schematic of the arrangement of the laser fields in Bragg-outcoupling event.

Figure 2

Schematic illustration of $n$ two-photon Bragg scattering process. The upper and lower solid curves ($5{}^{2}P_{3/2}$ and $5{}^{2}S_{1/2}$ states) show the internal states in terms of energy and momentum for ${}^{87}\mathrm{Rb}$. The single black dot on the energy-momentum parabola indicates the final momentum state arising from the total energy conservation corresponding to the Bragg resonant transition. The magnitudes of energy and momentum transferred to atoms in each two-photon event are dependent on the value of the resultant wave number $q$ and, consequently, the angle between lasers, $\alpha$. For every two-photon process, the associated detuning from resonance is depicted by ${\mathrm{\Delta }}_i$, where $i=1,...,n$.

Figure 3

Schematic of the atom laser operation in three different regions: (I), (II), and (III) indicate the WKB inside the BEC, Kirchhoff, and paraxial regimes, respectively. The solid blue sphere shows the BEC, while the solid dark-blue arrow along the $z$ axis depicts the atom laser beam. The harmonic focusing potential has been illustrated from the top view ($x-z$ plane) by the two shaded oval areas at the bottom of the figure. The intensity of the potential reaches its maximum value at the black spots, whereas the minimum intensity occurs between the two peaks.

Figure 4

(a),(c) BEC and (b),(d) atom laser density profiles, indicated by the color map, in position space in the $x-z$ plane (cross-section view), where (a),(b) ${\left(a_s\right)}_{\text{Laser}}=100a_0$ and (c),(d) ${\left(a_s\right)}_{\text{Laser}}=-100a_0$. The profile has been integrated over the $y$ axis in position space. The atom laser is outcoupled from the BEC using two Bragg photons, kicked by $p=2\hslash k$, and travels for a total distance of $d_z=300\mu \mathrm{m}$. (e) beam width, (f) beam momentum (transverse velocity) width, and (g) beam quality factor as a function of longitudinal path $z$ for three different $s$-wave interaction strengths, ${\left(a_s\right)}_{\text{Laser}}=100a_0$, 0, and $-100a_0$, within the atom laser beam. In (e)–(g), the Fresnel-Kirchhoff ($0\le z\le 140\mu \mathrm{m}$) and paraxial ($-140\le z<0\mu \mathrm{m}$) zones are distinguished by the blue and red colors, respectively. Each black solid line in (e) is a linear fit to the corresponding curve estimating the associated beam divergence, $\theta$, in both zones. Parameters used in the simulations are $N_0={10}^5$, ${\left(a_s\right)}_{\text{BEC}}=100a_0$, $p=2\hslash k$, ${\mathrm{\Delta }}_z=50$ nm, $\mathrm{\Omega }=0.669$ kHz, $a_0=5.29\times {10}^{-11}$ m, and $K=4\times {10}^{-41}{\mathrm{m}}^6{\mathrm{s}}^{-1}$.

Figure 5

Simulation of an atom laser beam outcoupled from a BEC optimally focused by a harmonic optical focusing potential. (a),(b) The trapped BEC and atom laser density profiles highlighted by the corresponding color maps. (c) The focusing potential intensity profile determined by the color map. The BEC and atom laser profiles have been integrated over the $y$ axis. The atom laser is outcoupled from the BEC at $z=150\mu \mathrm{m}$ by a Bragg kick of $p=2\hslash k$ corresponding to an initial velocity of $v_i=1.18$ cm/s. It is then accelerated under gravity and arrives by $v_f=7.76$ cm/s at $z=-150\mu \mathrm{m}$ to be focused by the potential whose maximum intensity is $I_0=9.91\times {10}^6\mathrm{W}/{\mathrm{m}}^2$ at $x=\pm \lambda /4=\pm 78\mu \mathrm{m}$. The parameters used in the simulation are $N_0={10}^5$, ${\mathrm{\Delta }}_z=40$ nm, $\mathrm{\Omega }=0.535$ kHz, ${\sigma }_z=25\mu \mathrm{m}$, $\lambda =312\mu \mathrm{m}$, $I_s=16.7\mathrm{W}/{\mathrm{m}}^2$, $\gamma =38$ MHz, $\mathrm{\Delta }=200$ GHz, ${\left(a_s\right)}_{\text{BEC}}=100a_0$, ${\left(a_s\right)}_{\text{Laser}}=-300a_0$, $a_0=5.29\times {10}^{-11}$ m, and $K=4\times {10}^{-41}{\mathrm{m}}^6{\mathrm{s}}^{-1}$.

Figure 6

Three-dimensional (3D) view of the (a) ${\left[a_s\right]}_{\text{Laser}}=100a_0$ and (b) ${\left[a_s\right]}_{\text{Laser}}=-300a_0$ atom laser profiles in the focusing process as well as their focused profiles along the $x$ axis at $z=-150\mu \mathrm{m}$ for (c) ${\left[a_s\right]}_{\text{Laser}}=100a_0$ and (d) ${\left[a_s\right]}_{\text{Laser}}=-300a_0$. In (c) and (d), the solid blue curves indicate the 1D density profiles and the dashed red curves illustrate the Gaussian fits. The values of peak density and FWHM for ${\left[a_s\right]}_{\text{Laser}}=100a_0$ and ${\left[a_s\right]}_{\text{Laser}}=-300a_0$ are, respectively, achieved as 296.3 atoms/$\mu {\mathrm{m}}^2$, 13.2 nm and 605.1 atoms/$\mu {\mathrm{m}}^2$, 8.04 nm. The parameters used in the simulations are $N_0={10}^5$, ${\mathrm{\Delta }}_z=40$ nm, $\mathrm{\Omega }=0.535$ kHz, ${\sigma }_z=25\mu \mathrm{m}$, $\lambda =312\mu \mathrm{m}$, $I_s=16.7\mathrm{W}/{\mathrm{m}}^2$, $\gamma =38$ MHz, $\mathrm{\Delta }=200$ GHz, ${\left(a_s\right)}_{\text{BEC}}=100a_0$, ${\left(a_s\right)}_{\text{Laser}}=-300a_0$, $a_0=5.29\times {10}^{-11}$ m, and $K=4\times {10}^{-41}{\mathrm{m}}^6{\mathrm{s}}^{-1}$.

Figure 7

Results for (a) FWHM, (b) peak density, and (c) atom number within the focused atom laser beam as a function of the atom laser interaction strength. The blue dots, red squares, and green crosses refer to ${\sigma }_z=100$, 50, and $25\mu \mathrm{m}$, respectively. All data for FWHM and peak density magnitudes are taken at the optimal focus spot, $z=-150\mu \mathrm{m}$, and the values for the beam atom numbers are obtained by integrating the square absolute value of the density profile from $z=150$ to $-150\mu \mathrm{m}$. The simulation parameters are $N_0={10}^5$, ${\mathrm{\Delta }}_z=40$ nm, $\mathrm{\Omega }=0.535$ kHz, $p=2\hslash k$, $\lambda =312\mu \mathrm{m}$, $I_s=16.7\mathrm{W}/{\mathrm{m}}^2$, $\gamma =38$ MHz, $\mathrm{\Delta }=200$ GHz, ${\left(a_s\right)}_{\text{BEC}}=100a_0$, $a_0=5.29\times {10}^{-11}$ m, and $K=4\times {10}^{-41}{\mathrm{m}}^6{\mathrm{s}}^{-1}$.

Figure 8

Results for (a) FWHM, (b) peak density, and (c) atom number within the focused atom laser beam as a function of the atom laser interaction strength. The blue dots, red squares, and green crosses refer to $\lambda =400{\lambda }_{{\text{D}}_2}$, $100{\lambda }_{{\text{D}}_2}$, and $25{\lambda }_{{\text{D}}_2}$ corresponding to $D=156$, 39, and $19.5\mu \mathrm{m}$, respectively. All data for FWHM and peak density magnitudes are taken at the optimal focus spot, $z=-150\mu \mathrm{m}$, and the values for the beam atom numbers are obtained by integrating the square absolute value of the density profile from $z=150$ to $-150\mu \mathrm{m}$. The simulation parameters are $N_0={10}^5$, ${\mathrm{\Delta }}_z=40$ nm, $\mathrm{\Omega }=0.535$ kHz, $p=2\hslash k$, ${\sigma }_z=50\mu \mathrm{m}$, $I_s=16.7\mathrm{W}/{\mathrm{m}}^2$, $\gamma =38$ MHz, $\mathrm{\Delta }=200$ GHz, ${\left(a_s\right)}_{\text{BEC}}=100a_0$, $a_0=5.29\times {10}^{-11}$ m, and $K=4\times {10}^{-41}{\mathrm{m}}^6{\mathrm{s}}^{-1}$.

Figure 9

Results for (a) FWHM, (b) peak density, and (c) atom number within the focused atom laser beam as a function of the BEC scattering length. The blue dots, red squares, green crosses, purple stars, and black triangles correspond to ${\left(a_s\right)}_{\text{Laser}}=100a_0$, 0, $-100a_0$, $-200a_0$, and $-300a_0$, respectively. All data for FWHM and peak density magnitudes are evaluated at the optimal focus spot, $z=-150\mu \mathrm{m}$, and the values for the beam atom numbers are obtained by integrating the square absolute value of the density profile from $z=150$ to $-150\mu \mathrm{m}$. The simulation parameters are $N_0={10}^5$, ${\mathrm{\Delta }}_z=40$ nm, $\mathrm{\Omega }=0.535$ kHz, $p=2\hslash k$, ${\sigma }_z=50\mu \mathrm{m}$, $\lambda =400{\lambda }_{{\text{D}}_2}$, $I_s=16.7\mathrm{W}/{\mathrm{m}}^2$, $\gamma =38$ MHz, $\mathrm{\Delta }=200$ GHz, $a_0=5.29\times {10}^{-11}$ m, and $K=4\times {10}^{-41}{\mathrm{m}}^6{\mathrm{s}}^{-1}$.

Figure 10

Results for (a) FWHM, (b) peak density, and (c) atom number within the focused atom laser beam as a function of the outcoupling resonance. The blue dots, red squares, green crosses, purple stars and black triangles correspond to ${\left(a_s\right)}_{\text{Laser}}=100a_0$, 0, $-100a_0$, $-200a_0$, and $-300a_0$, respectively. All data for FWHM and peak density magnitudes are evaluated at the optimal focus spot, $z=-150\mu \mathrm{m}$, and the values for the beam atom numbers are obtained by integrating the square absolute value of the density profile from $z=150$ to $-150\mu \mathrm{m}$. The simulation parameters are $N_0={10}^5$, ${\left(a_s\right)}_{\text{BEC}}=100a_0$, $\mathrm{\Omega }=0.535$ kHz, $p=2\hslash k$, ${\sigma }_z=50\mu \mathrm{m}$, $\lambda =400{\lambda }_{{\text{D}}_2}$, $I_s=16.7\mathrm{W}/{\mathrm{m}}^2$, $\gamma =38$ MHz, $\mathrm{\Delta }=200$ GHz, $a_0=5.29\times {10}^{-11}$ m, and $K=4\times {10}^{-41}{\mathrm{m}}^6{\mathrm{s}}^{-1}$.

Figure 11

Results for (a) FWHM, (b) peak density, and (c) atom number within the focused atom laser beam as a function of the momentum kick. The blue dots, red squares, green crosses, purple stars, and black triangles correspond to ${\left(a_s\right)}_{\text{Laser}}=100a_0$, 0, $-100a_0$, $-200a_0$, and $-300a_0$, respectively. All data for FWHM and peak density magnitudes are taken at the optimal focus spot, $z=-150\mu \mathrm{m}$, and the values for the beam atom numbers are obtained by integrating the square absolute value of the density profile from $z=150$ to $-150\mu \mathrm{m}$. The simulation parameters are $N_0={10}^5$, ${\left(a_s\right)}_{\text{BEC}}=100a_0$, ${\mathrm{\Delta }}_z=40$ nm, $\mathrm{\Omega }=0.535$ kHz, ${\sigma }_z=50\mu \mathrm{m}$, $\lambda =400{\lambda }_{{\text{D}}_2}$, $I_s=16.7\mathrm{W}/{\mathrm{m}}^2$, $\gamma =38$ MHz, $\mathrm{\Delta }=200$ GHz, $a_0=5.29\times {10}^{-11}$ m, and $K=4\times {10}^{-41}{\mathrm{m}}^6{\mathrm{s}}^{-1}$.