1. Introduction

The recent trapping and laser cooling of magnetic dipolar atomic elements such as chromium [1], erbium [2], dysprosium [3, 4], thulium [5], and holmium [6], with the first three having been cooled to quantum degeneracy [7–13], has opened new avenues of ultracold atomic physics exploration. Specifically, the long-range and anisotropic character of the magnetic dipole-dipole interaction provides a platform to investigate the role dipolar physics can play in quantum simulation. Examples of the latter include proposals to realize topologically non-trivial systems [14–17], and recent progress includes the study of the extended Bose-Hubbard model using erbium [18] and the observation of the arrested implosion of a dipolar dysprosium BEC due to the balance between the mean-field potential and quantum fluctuations [19–24].

Neutral atoms experience a force in an inhomogeneous light field. The resulting trapping force arises from the interaction between the light field and the induced atomic dipole moment and leads to the so-called Stark shift in the atomic energy level [25]. The total Stark shift can vanish at certain wavelengths due to cancellation between multiple atomic transitions. Such “tune-out" wavelengths for various atomic species have been predicted theoretically [26–29] and measured experimentally [30–34]. This knowledge is particularly useful for engineering species-specific trapping potentials. For example, in mixed-species experiments, one can create an optical lattice potential for one species but not the other [35–38], and the interaction between the trapped species and the background species allows for the implementation of novel cooling schemes to realize new quantum phases [39]. This species-specific lattice can also be used to tune the interspecies effective mass ratio of the trapped atoms [26], allowing for the flexible exploration of collective dynamics [40,41]. Lastly, the coexistence of trapped fermions and a background bosonic gas can potentially simulate phonon-like excitations in an optical lattice [26, 42, 43].

Aside from engineering trapping potentials for neutral atoms, knowledge of tune-out wavelengths is also useful in the context of the optical control of Feshbach resonances (OFR), a promising technique to achieve time-varying and/or spatially modulated interatomic interactions [32, 44–46]. One can operate the OFR laser at a far-off-resonance regime to reduce heating and loss rate and, if possible, at the tune-out wavelength of the targeted atomic species so that the parasitic dipole force of the OFR beam is eliminated, as recently demonstrated by [32]. The longer lifetime permits studies of the non-equilibrium dynamics of quantum gases with long observation times when the tune-out wavelength is far-detuned from electronic transitions [32].

The precise determination of atomic transition strengths cannot rest on ab initio quantum-mechanical calculations alone due to the electronic complexity of lanthanides like dysprosium, the atom considered here. The complicated electronic structure of these open- f -shell lanthanide elements presents a significant challenge for such analyses (see [47] and the recent study [48]) and experimental investigations are crucial for understanding their electronic structure. For example, in [49], a semi-empirical approach that utilizes both theoretical calculations and experimental data leads to the prediction of nine unobserved odd-parity energy levels in the erbium atomic spectrum. More generally, improved knowledge of atomic polarizabilities, which are informed by measurements of tune-out wavelengths [34], can guide choices of, e.g., optical dipole trapping wavelengths and laser wavelengths for implementing Raman transitions for realizing synthetic gauge fields [50–52]. It is in this spirit that we present the measurement of the tune-out wavelength for the bosonic ¹⁶²Dy near the narrow 741-nm transition.

We proceed by describing the experimental system in Sec. 2 before introducing the calculation of Stark shifts in Sec. 3 and results of our measurements in Sec. 4.

2. Experimental methods

Using linearly polarized light, we probe the total (scalar plus tensor) light shift in the vicinity of the 741-nm transition by Kapitza-Dirac (KD) lattice diffraction [53], which is a standard tool for optical lattice characterization [54–56]. See Fig. 1. The vector light shift vanishes for linearly polarized light. See Sec. 3. The lattice depth U₀ is measured in units of recoil energy E_r = (ħk_r)²/2m, where m is the mass of the atom, k_r = 2π/λ is the grating wavevector, and λ is the wavelength of the laser.

 

Fig. 1 Schematic of lattice beam (red arrow; with polarization $\widehat{\mathit{ϵ}}$) geometry used to measure lattice depth by KD diffraction of the BEC (blue sphere). Green arrows indicate direction of the applied magnetic field.

Download Full Size | PDF

We perform KD diffraction such that ω_rt ≳ 1, where the recoil angular frequency is ω_r = E_r/ħ and the grating pulse length t is sufficiently long to induce a coherent oscillation between the momentum states [57]. After turning off the light grating diabatically, the atoms project into momentum states |2nħk_r〉, where n = 0, ±1, ±2, … denotes the n^th order of the diffracted matter wave. In the weak-lattice limit, U₀ ⪡ 4E_r, only orders with |n| ≤ 1 effectively participate in the coherent evolution, and the tune-out wavelength can be identified by decreasing population P₁ in the first-order diffraction peaks as U₀ approaches zero.

We employ a multipulse diffraction scheme to enhance the signal by constructive interference, as demonstrated in [30]. We pulse an optical lattice along one spatial dimension near the 741-nm transition with a detuning Δ = (ω_L − ω₀)/(2π), where ω_L is the laser frequency and ω₀ is the resonant frequency of the 741-nm transition. In the weak lattice limit, the oscillation period between |0〉 and |±2ħk_r〉 approaches τ = h/(4E_r) = 111 µs for ¹⁶²Dy. As in [30], we use a square-wave sequence with n_p pulses since there is an $n_p^2$ enhancement in P₁ in the weakly diffracting limit (n_pU₀ ⪡ 4E_r) given by

The quadratic scaling of P₁ with respect to n_p in Eq. (1) no longer holds after the diffracted peaks of atoms have sufficiently moved in space to lose coherence—and thus the ability to constructively interfere—with the main condensate. This coherence time limit from overlap loss sets the maximum pulse number where the quadratic scaling is valid. As shown in Fig. 2, the P₁ enhancement efficiency starts to drop below quadratic at n_p = 10, after which the atoms in |±2ħk_r〉 have traveled 62% of the Thomas-Fermi radius. Therefore, for accurate light shift measurements, we employ a ten-pulse sequence to amplify P₁. The choice of n_p is appropriate for all Δ used in our measurements, because we operate in the weak-lattice limit, thus the coherence time only depends on k_r.

 

Fig. 2 Fractional first-order diffracted population at a constant laser power and wavelength. Each point is an average of three measurements shown with 1σ standard error. For optimal SNR, we use the maximum pulse number up to which the quadratic enhancement holds (i.e., n_p = 10). The first six data points are fit to Eq. (1). The dashed line is the resulting fit, which yields a measured lattice depth of U₀ = 0.20E_R.

Download Full Size | PDF

The tune-out wavelength is measured in terms of the detuning, denoted Δ₀, from the 741-nm transition frequency. We use a wavelength meter (HighFinesse WS/6 200) to monitor the frequency of the lattice beam derived from a Ti:Sapphire laser. We calibrate the wavelength meter against a frequency-stabilized 741-nm diode cooling laser with a <10 kHz/h drift rate [8] via a beat note setup. While simultaneously monitoring the frequency of the lattice beam and the cooling laser on the wavelength meter, we combine beams of both lasers onto a high-speed photodetector (Electro-Optics Technology ET-4000). We then compare the beat signal frequency, measured up to ±8 GHz, to the frequency difference given by the wavelength meter. Applying this calibration beyond the range of ±8 GHz yields an uncertainty of 10 MHz in the frequency measurement.

We prepare Bose-Einstein condensates (BECs) of ¹⁶²Dy using methods described in [12]. The resulting trap frequencies are [f_x, f_y, f_z] = [62(2), 32(4), 113(2)] Hz, where gravity is along $\widehat{z}$. We initiate KD diffraction with a nearly pure BEC of 5 × 10⁴ atoms in the maximally stretched ground state |J = 8, m_J = −8〉. As illustrated in Fig. 1, the 1D lattice is formed along $\widehat{x}+\widehat{y}$ by retroreflecting a 0.20(2)-W collimated beam with a diameter of 950 µm. The light field polarization is kept linear along $\widehat{z}$, purified by a polarizing beam splitter. Therefore, any anisotropy in Δ₀ should be attributed to the tensor light shift since the vector light shift is identially zero for linearly polarized light (see second term in Eq. (2) below). To probe the anisotropy in Δ₀, we perform the measurement at two different field orientations $\widehat{z}$ and $\widehat{x}-\widehat{y}$, both at a field magnitude of 1.580(5) G, to realize θ = 0 and θ = π/2 in Eq. (2). We note that this field is away from any Feshbach resonances [58].

3. Stark shift calculation

For a single transition from the ground state |F, m_F〉 to an excited state |F′, m_F′〉, the light shift of the ground state with an applied optical field of angular frequency ω is given by

The light shift of alkali atoms at large detuning is due solely to the scalar shift and is spherically symmetric. For these atoms, when the laser detuning is large compared to the hyperfine splitting of the excited state, the hyperfine levels become effectively degenerate and the optical field interacts directly with the fine-structure transition. That is, the hyperfine transition F → F′ is replaced by the fine-structure transition J → J′, ω_FF′ is replaced by ω_JJ′, and the polarizabilities can be directly expressed by the fine-structure dipole matrix elements. For alkali atoms in the ground state |J = 1/2, L = 0〉, the tensor polarizability α⁽²⁾ vanishes at large detuning, and the vector polarizability α⁽¹⁾ is canceled by opposite contributions from the D₁ and D₂ lines [34,60,61].

This spherical symmetry does not hold for Dy from the ground state |J = 8, L = 6〉 due to its large orbital angular momentum (bosonic Dy has zero nuclear spin). For example, the tensor polarizability ${\alpha }_{741}^{(2)}$ for the 741-nm J = 8 → J′ = 9 transition does not vanish like the alkali J = 1/2 → J′ transitions. On the contrary, it is on the same order of magnitude as the scalar polarizability ${\alpha }_{741}^{(0)}$ for all detunings $({\alpha }_{741}^{(2)}\approx -0.7{\alpha }_{741}^{(0)})$ (α⁽²⁾741 ≈ −0.7α⁽⁰⁾741).

4. Results

We measure P₁ and determine U₀ using Eq. (1). Fig. 3 shows the measured U₀ with respect to the detuning Δ from the 741-nm resonance for both θ = 0 and θ = π/2. We find that P₁ decreases as the laser is detuned further on the blue side of the 741-nm transition. Eventually, P₁ drops below the noise floor in our absorption image. As we further increase Δ, we observe a revival in P₁, indicating a sign change in the total light shift. We observe a clear polarization-dependence in Δ₀, which differs by more than 20 GHz for the two polarizations.

 

Fig. 3 Experimentally measured lattice depth versus detuning from the 741-nm transition for θ = 0 (triangles) and θ = π/2 (squares). Each point is an average of three measurements, and the lines are fits to the expected functional form of the Stark shift from Eq. (3). Error bars are 1σ standard error.

Download Full Size | PDF

We perform a least-squares fit to the data to quantitatively determine Δ₀, using

We also compare the measured Δ₀ values to the calculated Stark shift in Fig. 4. We compute the total Stark shift using the formalism presented in Section 3. To account for all the known transitions of dysprosium, we sum over Eq. (2) for the 26 lines documented in [47], replacing the listed theoretical matrix elements with experimentally measured values when possible (e.g., the 741-nm transition [59]). The calculated values— ${\mathrm{\Delta }}_0^{\text{th}}(\theta =0)=10.4$ GHz and ${\mathrm{\Delta }}_0^{\text{th}}(\theta =\pi /2)=32.2$ GHz—differ from the measured values by 26% and 7%, respectively.

 

Fig. 4 (a) Calculated total light shift (arbitrary unit) versus detuning from the 741-nm line for both θ = 0 and θ = π/2. The arrows indicate the positions of the measured Δ₀ points. The 160-GHz window corresponds to 5 cm⁻¹. (b) Calculated total light shift (arbitrary unit) for both polarizations between laser frequencies (wavelengths) DC and 24,000 cm⁻¹ (417 nm). The cycling transitions (J → J′ = J + 1) are indicated in turquoise [4, 59]. The large anisotropy at far-off resonant frequencies is likely a result of incomplete knowledge of atomic transition data; see main text for details.

Download Full Size | PDF

Furthermore, while [59] reports an experimentally measured linewidth of 1.78(2) kHz for the 741-nm excited state, we extract a linewidth by setting the reduced matrix element for the 741-nm transition µ₇₄₁ as the free parameter and minimize the error function

Figure 4 suggests that the total light shift at θ = 0 and θ = π/2 should differ by around 50% near 1064 nm (9,398 cm⁻¹), a common wavelength for optical dipole traps, and the one employed here. To explore this, we measure the trap frequency along $\widehat{z}$ for fields along $\widehat{z}$ and $\widehat{x}$, which correspond to θ = 0 and θ = π/2, respectively. Nevertheless, we find that the trap shape is isotropic to within 3%, in contradiction to the prediction. This isotropy has also been predicted in the optical trapping of erbium, another lanthanide atom, at 1064 nm using 1284 lines, 33 of which have been observed experimentally [49]. A similar calculation has been recently conducted for dysprosium [48]. In summary, a more complete knowledge of the dysprosium atomic spectrum is required to improve the agreement between theoretically predicted and experimentally measured tune-out wavelengths, and to properly account for the far-off-resonance regime.

5. Conclusions

The measurement of the tune-out wavelengths for ¹⁶²Dy near the narrow 741-nm transition has been presented, along with a calculation that reproduces these wavelengths within a few GHz. As also predicted, we observe an anisotropy in the tune-out wavelength as a function of the relative angle between the optical field polarization and the quantization axis.

While this work focuses on the tune-out wavelengths of ¹⁶²Dy, those of the other bosonic isotopes of Dy are related to ¹⁶²Dy’s by the isotope shifts, 1214(3) MHz and -1338(6) MHz for ¹⁶⁴Dy and ¹⁶⁰Dy, respectively [59], since nuclear effects play little role within the resolution of our measurement and bosonic Dy is I = 0. However, the wavelengths of the fermionic isotopes cannot be accurately determined with this data since those isotopes do possess hyperfine structure on the GHz scale. Analogous measurements using Kapitza-Dirac diffraction for (near) degenerate fermions is much more challenging due to the intrinsically larger momentum spread.