Quantum lock-in detection of a vector light shift

We demonstrate detection of a vector light shift (VLS) using the quantum lock-in method. The method offers precise VLS measurement with little affect from real magnetic field fluctuation. We detect a VLS on a Bose-Einstein condensate (BEC) of ${}^{87}\mathrm{Rb}$ atoms caused by an optical trap beam with a resolution less than 1 Hz. We demonstrate the elimination of a VLS by controlling the beam polarization to realize a long coherence time of an $F=2$ BEC in the stretched state. Quantum lock-in VLS detection should find wide application, including the study of spinor BECs, electric dipole moment searches, and precise magnetometry.

Figure 1

(a) Experimental configuration. A BEC is trapped in the axial trap beam along the $z$ axis and the radial trap beam along the $x$ axis (not shown). (b) Typical TOF image of a BEC measured after rf pulses for the detection. The spin components ($m_F=-2,-1,0,1,2$) are spatially resolved by the Stern-Gerlach method. (c) Time sequence for the quantum lock-in detection of a VLS. The beam power $P\left(t\right)$ is modulated with a frequency ${\omega }_m$. The phase of the spin vector evolves with an angular frequency of $\omega \left(t\right)$. The accumulated phase $\mathrm{\Phi }={\int }_0^T\omega \left(t\right)dt$ is finally measured.

Figure 2

Detected magnetization after the rf pulses. The error bars represent the sample standard deviation. The red solid line is the fitting curve by $\mathcal{V}Fsinap$. The right axis represents $B_1$, corresponding to the magnetization. It should be noted that the right axis scale is not linear since $B_1$ is proportional to $arcsin\frac{m}{\mathcal{V}F}$.

Figure 3

Polarization dependence of the magnetization signal. (a) Measurement result with $T=7$ ms. (b) Measurement result with $T=28.2$ ms. The blue circles and red squares represent $m_{+}$ and $m_{-}$, respectively. (c) Plot of $\mathrm{\Delta }m$ as a function of $\theta$. Here $\mathrm{\Delta }{B}_1$ is the difference between the fictitious magnetic fields for positive and negative $P_1$. The solid lines in (a) and (c) are the fitting curves of $asinb(\theta -{\theta }^{*})$ (see the text for details). The error bars represent the standard deviation.

Figure 4

Effects of the fictitious magnetic field on a transversally polarized BEC. The vertical displacement of the center of mass in the TOF image is plotted for (a) $\mathrm{\Delta }\theta =-1^{\circ }$, (b) $\mathrm{\Delta }\theta =0.{02}^{\circ }$, (c) $\mathrm{\Delta }\theta =1^{\circ }$, and (d) $\mathrm{\Delta }\theta =4^{\circ }$. The solid and dashed lines are guides for the eyes. (e)–(h) Population evolution corresponding to (a)–(d). The values of $p_0$ (blue circles), $p_1$ (red diamonds), and $p_2$ (yellow squares) are shown ($p_i$ is defined in the text). The error bars represent the standard deviations.

Figure 5

Atom number losses (semilogarithmic plot). The solid line is an exponential fit to the data with the optimized field gradient. The inset shows the population evolution for the optimized case. The dotted lines are the reference curves of the mean-field-driven evolution within the SMA and assuming no inelastic collisional losses [26].