Narrow line cooling and momentum-space crystals

Narrow line laser cooling is advancing the frontier for experiments ranging from studies of fundamental atomic physics to high precision optical frequency standards. In this paper, we present an extensive description of the systems and techniques necessary to realize $689\mathrm{nm}$ ${}^{1}S_0\text{−}{}^{3}P_1$ narrow line cooling of atomic ${}^{88}\mathrm{Sr}$. Narrow line cooling and trapping dynamics are also studied in detail. By controlling the relative size of the power broadened transition linewidth and the single-photon recoil frequency shift, we show that it is possible to smoothly bridge the gap between semiclassical and quantum mechanical cooling. Novel semiclassical cooling processes, some of which explicitly depend on the relative size of gravity and the radiative force, are also explored. Moreover, for laser frequencies tuned above the atomic resonance, we demonstrate momentum-space crystals containing up to 26 well defined lattice points. Gravitationally assisted cooling is also achieved with blue-detuned light. Theoretically, we find the blue detuned dynamics are universal to Doppler limited systems. This paper offers the most comprehensive study of narrow line laser cooling to date.

Figure 1

(Color online) (a) Partial ${}^{88}\mathrm{Sr}$ energy level diagram. Numbers in parentheses give Einstein A coefficients (in ${\mathrm{s}}^{-1}$). Wavelengths are in vacuum. (b) Top ($x\text{−}y$ plane) view of the Sr cooling and trapping apparatus. Blue (red) arrows represent $461\mathrm{nm}$ ($689\mathrm{nm}$, $679\mathrm{nm}$, and $707\mathrm{nm}$) trapping (trapping and repumping) beams. M, mirror; $\lambda ∕4$, dual wavelength quarter-wave plate; $\lambda ∕2$, half-wave plate; DBS, dichroic beamsplitter; PBS, polarization beamsplitter; BS, beamsplitter; SMPM, single-mode polarization maintaining; ECDL, external-cavity diode laser; BA, Bayard-Alpert vacuum gauge; NI, nude vacuum gauge; TSP, titanium sublimation pump; CC, compensation coil.

Figure 2

${}^{3}P_2$ magnetic trap population versus (a) hold time for a fixed $2\mathrm{s}$ loading time and (b) loading time at a hold time of $400\mathrm{ms}$. Solid lines are exponential fits. ${\tau }_D\left({\tau }_L\right)$ is the measured exponential decay (loading) time.

Figure 3

(Color online) $689\mathrm{nm}$ laser system. Solid (dashed) lines are optical beams (electrical connections). $\lambda ∕4$, quarter-wave plate; $\lambda ∕2$, half-wave plate; M, mirror; CM, curved mirror; PBS, polarization beamsplitter; BS, beamsplitter; OI, optical isolator; FR, Faraday rotator; SMPM, single-mode polarization maintaining; ECDL, external-cavity diode laser; DL, diode laser; TSVI, temperature stabilized and vibration isolated; PD, photodiode; AOM, acousto-optic modulator; EOM, electro-optic modulator; DBM, double balanced mixer; INT, integrator; AM, amplitude modulation; CUR (PZT), current (piezoelectric mounted grating) modulation input.

Figure 4

${}^{1}S_0\text{−}{}^{3}P_1$ MOT timing diagram. Images at top show the atomic cloud at each stage of the cooling and compression process. From left to right, the first four frames are in situ images while the last frame shows the cloud after $25\mathrm{ms}$ of free expansion from the $\delta =-520\mathrm{kHz}$, $s=75$ single-frequency MOT. $N$ $\left(T_M\right)$ is the cloud population (temperature).

Figure 5

(Color online). (a) Semiclassical radiative force versus position (bottom axis, $v_x=v_y=0$) and velocity (upper axis, $x=y=0$) for a range of detunings. Corresponding (b) trap potential energy in the $z$ direction and (c) in situ images of the ${}^{1}S_0\text{−}{}^{3}P_1$ MOT. Dashed lines in (c) are calculated maximum force contours. For each, $s=248$.

Figure 6

(Color online) (a) Semiclassical radiative force versus position (bottom axis, $v_x=v_y=0$) and velocity (upper axis, $x=y=0$) for a range of intensities. Corresponding (b) trap potential energy in the $z$ direction and (c) in situ images of the ${}^{1}S_0\text{−}{}^{3}P_1$ MOT. Dashed lines in (c) are calculated maximum force contours. For each, $\delta =-520\mathrm{kHz}$.

Figure 7

Vertical cloud position versus $\delta$ for $s=248$ and $d{B}_z=10\mathrm{G}∕\mathrm{cm}$. The solid line is a linear fit.

Figure 8

(Color online). Modematching broadband cooled atoms to the single-frequency MOT. (a) Broadband modulation spectra. Dashed vertical lines give the effective detuning corresponding to the measured cloud size. (b) In situ images of the broadband cooled atoms. For each, the temperature is $8.5\left(0.8\right)\mu \mathrm{K}$. Dashed lines give the maximum force contour for the $\delta =-520\mathrm{kHz}$, $s=75$ single-frequency MOT shown in the bottom frame. (c) Broadband to single-frequency MOT transfer efficiency versus broadband spectra number.

Figure 9

Relative trap population versus $\delta$ when $r_z$ and $r_h$ are optimized at each point (open circles) or fixed at $r_z=0.79\mathrm{mm}$ and $r_h=1.56\mathrm{mm}$ (solid circles). For both, $s=248$ and $d{B}_z=10\mathrm{G}∕\mathrm{cm}$.

Figure 10

(Color online) ${}^{1}S_0\text{−}{}^{3}P_1$ MOT temperature versus (a) detuning $\delta$ for various intensities and (b) intensity $s$ at $\delta =-520\mathrm{kHz}$. In (a), DT is standard Doppler theory. In (b), solid circles are experimental data while the solid, dashed, and short dashed lines give standard Doppler theory, the Doppler limit $(\hslash {\Gamma }_E∕2k_B)$, and the single-photon recoil limit $(2\hslash {\omega }_R∕k_B)$, respectively. (c) Global scaling factor applied to Doppler theory in order to match regime (II) data versus intensity.

Figure 11

(a) Schematic depiction of regime (I) cooling. Atoms sag to the bottom of the potential energy (PE) curve where interactions occur primarily with the upward traveling trapping beam. The inset shows the excitation geometry. (b) Numerical factor $N_R$ in Eq. (6) versus intensity.

Figure 12

(Color online) The approach to thermal equilibrium at $\delta =-520\mathrm{kHz}$. (a) Measured temperature versus time for $s=248$. The inset gives a semilog plot for times greater than $5\mathrm{ms}$. The solid line is a linear fit. $T$, instantaneous temperature; $T_0$, final equilibrated temperature. (b) Measured exponential equilibration time versus $s$. The dashed, short dashed, and solid lines are predictions from standard Doppler theory, the single-beam theory outlined in the text, and quantum theory, respectively.

Figure 13

One dimensional velocity bunching due to single-beam $\delta >0$ absorption for (a) $\delta =80\mathrm{kHz}$ and (b) $\delta =200\mathrm{kHz}$. Solid, short dash, and dashed lines correspond to initial velocities of $0.1\mathrm{cm}∕\mathrm{s}$, $5\mathrm{cm}∕\mathrm{s}$, and $10\mathrm{cm}∕\mathrm{s}$, respectively. (c) Three-dimensional $\delta >0$ momentum space structure. Final momenta corresponding to the black, gray, and white circles arise from three-, two-, and single-beam interactions, respectively.

Figure 14

Top view (tilted slightly relative to gravity) in situ images for a range of $\delta$ and $s$ at $\mid \overrightarrow{dB}\mid =0$ and $t_H=t_V=25\mathrm{ms}$. Arrows in the uppermost center frame show the propagation direction for the horizontal molasses beams. Note the horizontal beams are not completely orthogonal to the field of view. The initial cloud temperature is $11.5\left(2\right)\mu \mathrm{K}$.

Figure 15

(Color online) Calculated horizontal $\delta >0$ final versus initial velocity for (a) various $\delta$ at $s=30$ and (b) various $s$ at $\delta =120\mathrm{kHz}$. (c) and (d) give the corresponding spatial distributions. For each, $t_H=25\mathrm{ms}$ and the $t_H=0$ temperature is $11.5\mu \mathrm{K}$.

Figure 16

(Color online) Broad line momentum space crystals. Calculated horizontal (a) final versus initial velocity and (b) corresponding spatial distributions for $461\mathrm{nm}$ $\delta =0$ optical molasses. For each, $t_H=500\mu \mathrm{s}$ and the initial temperature is $2.5\mathrm{mK}$.

Figure 17

(Color online) Side view in situ images for $s=30$, $\mid \overrightarrow{dB}\mid =0$, $\delta =140\mathrm{kHz}$, $t_H=25\mathrm{ms}$, and a range of $t_V$ times. The $t_H=t_V=0$ cloud temperature is $11.5\left(2\right)\mu \mathrm{K}$. Calculated (b) final versus initial velocity for the same set of parameters. (c) Corresponding spatial distributions.

Figure 18

Measured vertical position of the upward moving layer versus $t_V$ at $\delta =140\mathrm{kHz}$ and $s=30$. The solid line is a fit to the model described in the text.

Figure 19

(Color online) $\delta =0$ cooling. (a) Composite gravitational and $\delta >0$ radiative force versus $v_z$. Arrows show the direction for force induced velocity changes. Stable cooling occurs around a velocity $v_0$, labeled by the circle, where gravity cancels the radiative force. (b) Measured $v_0$ versus $\delta$ for $s=10$ and $s=75$. Solid lines are linear fits.