Experimental study of tune-out wavelengths for spin-dependent optical lattice in ⁸⁷Rb Bose-Einstein condensation

Optical lattice for ultracold atoms has become an increasingly important technology in many-body physics Bloch et al. (2008), quantum simulation, quantum computation, quantum information storage and high precision measurements Aidelsburger et al. (2013); Miyake et al. (2013); Chiu et al. (2018); Schnorrberger et al. (2009); Martin et al. (2013). When neutral atoms are trapped in periodic potentials produced by standing wave of light fields, the trapping potentials of various atomic internal states are manipulated by lattice polarizations, which is called the spin-dependent optical lattice Mandel et al. (2003); Duan et al. (2003), which bring more complicated geometry for ultracold atoms such as spin-dependent hexagonal lattice Soltan-Panahi et al. (2011), spin-dependent optical superlattice Yang et al. (2017), and have been used to study many interesting phenomena such as controlled coherent transport Ke et al. (2018); Mandel et al. (2003), spinor BEC Ostrovskaya and Kivshar (2004), spin-orbit coupling and artificial gauge fields Ye et al. (2018); Grusdt et al. (2017), spontaneous emission of matter waves Krinner et al. (2018), twisted-bilayer optical potentials González-Tudela and Cirac (2019).

Tune-out wavelengths that ac Stark shift cancels was initially introduced in species-specific optical manipulation LeBlanc and Thywissen (2007) and can be useful for optical Feshbach resonances Clark et al. (2015) and atomic interferometer Trubko et al. (2015). Since tune-out wavelengths are independent of the light intensity Arora et al. (2011); Jiang et al. (2013); Cheng et al. (2013), it can be precisely measured by various methods Holmgren et al. (2012); Henson et al. (2015); Leonard et al. (2015); Schmidt et al. (2016); Adam and Charles (2016); Trubko et al. (2017); Copenhaver et al. (2019); Décamps et al. (2020). In general, tune-out wavelength is utilized accurately only for the scalar shift by cancelling and neglecting the vector and tensor contribution as much as possible Holmgren et al. (2012); Henson et al. (2015); Leonard et al. (2015); Trubko et al. (2017). In this paper, we investigate the tune-out wavelengths in more general cases of considering the contributions from both the scalar and vector shift. The ac Stark shift of optical lattice is measured by Kapitza-Dirac scattering which diffracts BEC into a number of high momentum states and the characteristics of spin-dependent optical lattice are investigated by scanning the lattice wavelength between the D1 and D2 lines. Kapitza-Dirac scattering becomes a standard tool and shows many applications in calibrating the lattice depth Ovchinnikov et al. (1999); Cahn et al. (1997); Denschlag et al. (2002); Beswick et al. (2019); Chen et al. (2010), detecting the lattice structure Viebahn et al. (2019); Wen et al. (2020), performing high-resolution spectroscopy Stenger et al. (1999) and metrology Gupta et al. (2002); Campbell et al. (2005). The periodic potential originated from scalar shift and vector shift is manipulated by controlling the lattice polarizations, which is used to generate spin-dependent optical lattice. We can design the special spin-dependent optical lattice with the help of tune-out wavelengths.

The schematic of experimental setup is shown in Fig. 2(a). Ultracold ⁸⁷Rb atoms in $|F=2,{{m}_{F}}=2\rangle$ hyperfine state are loaded in a crossed optical dipole trap Xiong et al. (2010). Forced evaporation in the optical trap is used to create the BEC with up to $5\times 10^{5}$ atoms. In order to obtain the atoms in the different single spin states, BEC is transferred from $|F=2,{{m}_{F}}=2\rangle$ into $|F=1,{{m}_{F}}=1\rangle$ via a rapid adiabatic passage induced by a microwave-frequency field with duration of 10 ms at 3.9 G of the bias magnetic field, where the frequency of center is 6.842935 GHz and the width is 0.25 MHz. Then the atoms in $|F=1,{{m}_{F}}=1\rangle$ can further be transferred into $|F=1,{{m}_{F}}=0\rangle$ state using a rapid adiabatic passage induced by a radio frequency (rf) field at 28 G of the bias magnetic field. The lattice beam is derived from a single frequency Ti:sapphire laser with broad tuning range of the frequency. An acousto-optical modulator is used to control the intensity of the lattice beam. The lattice beam passes through the polarizing beam splitter to generate the perfect polarization. The polarization extinction ratio of the polarizing beam splitter can reach 500:1. Therefore the linear polarization purity of the lattice beam is about 0.2$\%$. Furthermore the circular polarization purity of the lattice beam can reach about 0.5$\%$. A lattice beam propagates with z axis and converges on BEC with waist of 100 $\mu m$ by a lens (f = 300 mm). Then the beam is reflected by a concave mirror ( curvature radius r=300 mm) and refocused on BEC with the almost same waist size. The advantage of this configuration can reduce phase jitter significantly. Here, we employ Kapitza-Dirac (or Raman-Nath) scattering to measure the ac Stark shift. Kapitza-Dirac scattering is used to diffract the BEC into a number of momentum states by a standing light wave, in which the interaction is sufficiently short and strong Gould et al. (1986). In this process, BEC is kept in a crossed optical dipole trap and the lattice potential imprints a phase modulation on matter wave in position space. Then the phase modulation on matter wave is measured in momentum space via the time-of-flight (TOF) absorption image. It is obvious that higher momentum orders $\pm 2N\hbar k$ appear in the atomic density distribution of the TOF absorption image, which depends on the potential depth and interaction time. Here, we apply a 1D optical lattice short pulse for 4 $\mu s$ with the power of 80 mW on BEC. Then immediately turn off the optical trap, let the atoms ballistically expand in 12 ms and take the absorption images. We obtain the lattice depth from the absorption images by applying the lattice at a fixed laser power for different intervals of time and by observing the interval at which the $n=0$ order atoms in the lattice vanish Ovchinnikov et al. (1999); Cahn et al. (1997); Denschlag et al. (2002); Beswick et al. (2019); Chen et al. (2010). We define the recoil momentum $\hbar k=2\pi\hbar/\lambda$ and recoil energy $E_{r}=(\hbar k)^{2}/2m=h\times 3.67$ kHz as the nature momentum and energy units, where $m$ is the mass of ⁸⁷Rb atom, $\lambda$ is the wavelength of the lattice laser.

For case 1, a 1D optical lattice with the linear parallel polarization produces the spatial intensity modulation, which only comes from the scalar shift. Thus it is a spin-independent optical lattice and the potential depth as the function of the lattice wavelength are plotted in Fig. 3(a). Here, the positive and negative periodic potentials correspond to blue and red detuned lattice laser. The higher momentum orders $\pm 2N\hbar k$ are observed in the atomic density distribution of the TOF absorption images as shown in Fig. 3(b), which depends on the potential depth. We measure the tune-out wavelengths by changing the lattice wavelength and find its location at $\sim$790.020 nm for all spin states, which is in good agreement with the previous works  Leonard et al. (2015); Schmidt et al. (2016).

For case 2, a laser beam with the linear polarization pass through a quarter-wave plate and is reflected by a concave mirror, which produces the linear orthogonal polarization configuration. We plot the potential depth vs different wavelength as shown in Fig. 4(a). This periodic potential is a spin-dependent lattice, which only comes from the contribution of the vector shift. Therefore, there is no lattice potential for the $|F=1,{{m}_{F}}=0\rangle$ state ($\Delta{{m}_{F}}=0$) for all wavelength as shown in Fig. 4(b). The $|{{m}_{F}}=\pm 2\rangle$ and $|{{m}_{F}}=\pm 1\rangle$ states ($\Delta{{m}_{F}}\neq 0$) always experience the lattice potential when adjusting the lattice wavelength.

For case 3, a 1D optical lattice with the circular parallel polarization produces the spatial intensity modulation, which includes the contribution of the scalar and vector shift simultaneously. The potential depth of $|F=1,{{m}_{F}}=1\rangle$, $|F=1,{{m}_{F}}=0\rangle$ and $|F=2,{{m}_{F}}=2\rangle$ are given in Fig. 5(a) and (b). The tune-out wavelengths are generated by balancing the scalar and vector shift, which are 792.462(22)nm, 790.020(25)nm and 785.146(12)nm for $|F=1,{{m}_{F}}=1\rangle$, $|F=1,{{m}_{F}}=0\rangle$ and $|F=2,{{m}_{F}}=2\rangle$ respectively as shown in Fig. 5(c). Note that the tune-out wavelengths in this case are sensitive to the ellipticity of the lattice polarization, the alignment between the direction of the 1D optical lattice and the external bias magnetic field. This case provides us more controlled ways to generate the different kinds of spin-dependent optical lattice. There is an interesting phenomenon that two neighbouring spin states have the opposite lattice potential (blue and red detuning respectively) by choosing the appropriate wavelength, for example, $|F=1,{{m}_{F}}=1\rangle$ and $|F=1,{{m}_{F}}=0\rangle$ states have the opposite lattice potential at the wavelength of 791.24 nm.

For case 4, two counter-propagating laser beams with the orthogonal circular polarization can not generate any spatial modulation on the BEC. The potential depth for $|F=1,{{m}_{F}}=1\rangle$, $|F=1,{{m}_{F}}=0\rangle$ and $|F=2,{{m}_{F}}=2\rangle$ are given in Fig. 6(a) and (b), showing no effective potential (no density modulation) for these states.

Furthermore, we study the dependence of the tune-out wavelengths on the strength of the external bias magnetic field in more detail. The intersection angle $\phi$ between $\hat{e}_{k}$ and $\hat{e}_{B}$ is expressed as

where $B_{Bi}$ is the external bias magnetic field, $B_{Re}$ is the residual magnetic field (such as the earth magnetic field). We measure the tune-out wavelengths as the function of the strength of the external bias magnetic field as shown in Fig. 7. By changing the bias magnetic field to small value, the direction of the total magnetic field and the intersection angle $\phi$ can be changed. Therefore, the tune-out wavelengths change when the strength of the bias magnetic field is near to the residual magnetic field value. The strength of the external bias magnetic field in one direction is gradually decreased to zero and then increased in the opposite direction. We find that the tune-out wavelengths for $\xi=1$ jumps into $\xi=-1$ due to inversion the external bias magnetic field direction for the spin $|F=1,{{m}_{F}}=1\rangle$ and $|F=2,{{m}_{F}}=2\rangle$ states as shown in Fig. 7(a) and (b). The slope is sensitive to the strength of residual magnetic field in the perpendicular direction of z axis. Here, three pairs of Helmholtz coils are employed to compensate the background magnetic field. When the residual magnetic field in the (x, y, z) directions are suppressed from (0.4 G, 0.46 G, 0.38 G) to (0.25 G, 0.28 G, 0.2 G) which are measured by a triaxial fluxgate magnetometer, the slope is changed from -4.054 nm/G to -6.584 nm/G for $|F=1,{{m}_{F}}=1\rangle$ and 8.108 nm/G to 13.168 nm/G for $|F=2,{{m}_{F}}=2\rangle$ as shown in Fig. 7(a) and (b) respectively. Therefore, this method can be utilized to calibrate and measure the residual magnetic field.

In conclusion, we present an experiment to measure the ac Stark shift around the tune-out wavelengths of ⁸⁷Rb BEC in the three different hyperfine ground states $|F=1,{{m}_{F}}=1\rangle$, $|F=1,{{m}_{F}}=0\rangle$ and $|F=2,{{m}_{F}}=2\rangle$ between D1 and D2 lines. Four different polarization configurations of one-dimensional optical lattice, which are originated from scalar shift, vector shift, both scalar and vector shift respectively by manipulating the lattice polarizations, are investigated. Kapitza-Dirac scattering technique is employed to probe the ac Stark shift of atoms in optical lattice and the characteristics of spin-dependent optical lattice are presented by scanning the lattice wavelength. We present the tune-out wavelengths in more general cases of considering the contributions of both the scalar and vector shift. We further study the dependence of the tune-out wavelengths on the strength of the external bias magnetic field in more detail. Our work provides a clear interpretation for spin-dependent optical lattice and can be used for the realization of two species system, or the same atoms (Rb) with different spin states, in which one of them move freely while the others are trapped to different degrees of optical lattice potential. This system can be a test-bed for observing or simulating phenomena such as entropy cooling Catani et al. (2009); Lamporesi et al. (2010); Mckay and Demarco (2010); Arora et al. (2011); Bause et al. (2020), Kondo effect Zhang et al. (2016); Yao et al. (2019) etc, and even have application in Sr optical lattice clock Heinz et al. (2020).

This research was supported by the MOST (2016YFA0301602, 2018YFA0307601), NSFC (Grant No. 11804203, 11974224, 11704234), the Fund for Shanxi ”1331 Project” Key Subjects Construction.