Clock spectroscopy of interacting bosons in deep optical lattices

Our experiment starts with a nearly pure Bose–Einstein condensate (BEC) of about 10^{5 174}Yb atoms in a crossed optical dipole trap (CDT) with initial trap frequencies $\{{\omega }_{x},{\omega }_{y},{\omega }_{z}\}=2\pi \times \{60\,{\rm{Hz}},230\,{\rm{Hz}},260\,{\rm{Hz}}\}$ (see [22, 23] for more details). The BEC is transferred into a three-dimensional cubic optical lattice depicted in figure 1(a). This lattice results from the incoherent superposition of three orthogonal standing waves with orthogonal polarizations [19]. All lattice light beams are derived from the same laser operating at the magic wavelength ${\lambda }_{{\rm{m}}}\approx 759.4$ nm [1, 24], for which the polarizabilities of g and e are equal [25]. The periodic optical lattice potential experienced by the atoms irrespective of their internal state is then

$$\begin{eqnarray}&&{V}_{\mathrm{OL}}=\displaystyle \sum _{\alpha =x,y,z}{V}_{0,\alpha }{\sin }^{2}({k}_{{\rm{m}}}\alpha ),\end{eqnarray} \tag{ 1 }$$

where ${V}_{0,\alpha }$ are the lattice depths and where ${k}_{{\rm{m}}}=2\pi /{\lambda }_{{\rm{m}}}$ is the lattice wavenumber. We calibrate the lattice depths ${V}_{0,\alpha }$ for given laser powers by Kapitza–Dirac diffraction. The vertical lattice (VL) depth ${V}_{0,z}\approx 27\,{E}_{{\rm{R}}}$ is essentially fixed (see section 2.1.2), while the horizontal lattice (HL) depths can be varied from zero to $\{{V}_{0,x},{V}_{0,y}\}\approx \{24,25.4\}\,{E}_{{\rm{R}}}$. Here ${E}_{{\rm{R}}}={{\hslash }}^{2}{k}_{{\rm{m}}}^{2}/2M$ is the recoil energy, with M the atomic mass. In this work, unless stated explicitly, we use the largest available lattice depths. The Gaussian envelope of the lattice lasers also entails an additional, slowly varying harmonic potential

$$\begin{eqnarray}&&{V}_{{\rm{T}}}=\displaystyle \sum _{\alpha =x,y,z}\displaystyle \frac{1}{2}M{{\rm{\Omega }}}_{\alpha }^{2}{\alpha }^{2}.\end{eqnarray} \tag{ 2 }$$

By recording collective mode frequencies of a BEC, we infer the Gaussian beam waists for each standing wave, $\{{w}_{x},{w}_{y},{w}_{z}\}\,\approx$ $\{115\mu {\rm{m}},125\mu {\rm{m}},150\mu {\rm{m}}\}$. The corresponding trap frequencies are $\{{{\rm{\Omega }}}_{x},{{\rm{\Omega }}}_{y},{{\rm{\Omega }}}_{z}\}=2\pi \,\times \{42\,{\rm{Hz}},38\,{\rm{Hz}},33\,{\rm{Hz}}\}$ for lattice depths $\{{V}_{0,x},{V}_{0,y},{V}_{0,z}\}=\{24,25.4,27\}\,{E}_{{\rm{R}}}$.

The BEC is transferred into the optical lattice in three consecutive steps, aiming at preparing a stack of independent 2D Bose gases in the lowest Bloch band of the HL: (i) a fast ramp up of the VL, (ii) a slow extinction of the CDT and (iii) an adiabatic increase of the HL.

In more detail, we first ramp up in 20 ms a single standing wave propagating vertically (VL), superimposed on the CDT. This fast increase is used to freeze the vertical motion in the combined potential formed by the optical lattice and gravity. The potential V_T alone barely traps the atoms along gravity with a sag of the cloud much larger than in the CDT (${\rm{\Delta }}{z}_{{\rm{T}}}={g}_{0}/{{\rm{\Omega }}}_{z}^{2}\,\approx$ 230 μm compared to ${\rm{\Delta }}{z}_{\mathrm{CDT}}={g}_{0}/{\omega }_{z}^{2}\,\approx$ 4 μm, respectively, with g₀ the acceleration of gravity). The fast ramp up of the VL (as opposed to a slow, quasi-adiabatic transfer) prevents from inducing any motion along z [26], potentially leading to heating of the cloud. Note, however, that the duration of this first step is still long enough to avoid inter-band transitions. In a second step, the CDT is smoothly extinguished in 200 ms to create a stack of independent two-dimensional condensates in the VL potential alone. This step reduces the atomic density and hence mitigates the rate of three-body losses (see section 2.3). It also avoids differential light shifts between the clock states created by the CDT lasers. In a final step, the two arms of the HL are increased to their desired values in 100 ms, which roughly fulfills the criterion of [27] for adiabatic loading of the lowest Bloch band.

Each planar gas can be described by a 2D Bose–Hubbard Hamiltonian [19]

$$\begin{eqnarray}&&\hat{H}=-\displaystyle \sum _{\langle i,j{\rangle }_{\perp }}{J}_{\perp }{\hat{a}}_{g,j}^{\dagger }{\hat{a}}_{g,i}+\displaystyle \sum _{{i}_{\perp }}\displaystyle \frac{{U}_{{gg}}}{2}{\hat{n}}_{i}({\hat{n}}_{i}-1)+\displaystyle \sum _{{i}_{\perp }}\displaystyle \frac{M}{2}({{\rm{\Omega }}}_{x}^{2}{x}_{i}^{2}+{{\rm{\Omega }}}_{y}^{2}{y}_{i}^{2}){\hat{n}}_{i}.\end{eqnarray} \tag{ 3 }$$

Here the notation ${i}_{\perp }$ indicates summation over all lattice sites at positions ${{\boldsymbol{\rho }}}_{i}=({x}_{i},{y}_{i})$ in the x–y plane, and $\langle i,j{\rangle }_{\perp }$ in-plane tunneling to nearest-neighbors with matrix element ${J}_{\perp }$. Tunneling along the gravity direction has been neglected. The on-site energy U_gg is given by

$$\begin{eqnarray}&&{U}_{{gg}}=\displaystyle \frac{4\pi {{\hslash }}^{2}{a}_{{gg}}}{M}\int {\rm{d}}z\,{\rm{d}}{\boldsymbol{\rho }}\,{| {W}_{z}(z){W}_{\perp }({\boldsymbol{\rho }}-{{\boldsymbol{\rho }}}_{i})| }^{4},\end{eqnarray} \tag{ 4 }$$

where ${\boldsymbol{\rho }}$ denotes a two-dimensional vector in the x–y plane and where W_z (respectively ${W}_{\perp }$) are the Wannier functions for the vertical (resp. horizontal) lattice potentials. The scattering length a_gg describes low-energy s-wave scattering between two atoms in the internal state g, and has been measured in [28], ${a}_{{gg}}=105(2){a}_{0}$ with a₀ the Bohr radius. Using collapse and revival dynamics (see [29] and appendix), we measure ${U}_{{gg}}/h=1475(25)$ Hz, which compares well with the value 1420 Hz calculated from (4) using the calibrated lattice depths. Unless otherwise mentioned, the quoted error bars represent a 68% confidence interval on the optimum fit value.

At zero temperature, the 2D Bose–Hubbard model supports phase transitions to incompressible MI phases [19, 30–32]. For a filling factor $\overline{n}=1$, i.e. 1 atom per lattice site, Monte-Carlo simulations predict a transition for a critical value ${({U}_{{gg}}/{J}_{\perp })}_{{\rm{c}}}\approx 16.7$ [31], corresponding to HL depths around ${V}_{0,x}\approx {V}_{0,y}\approx 9{E}_{{\rm{R}}}$. The smooth harmonic trap V_T leads to a characteristic 'wedding cake' density profile, i.e. density plateaus with integer filling, the denser plateaus occurring near the trap center [19]. In the present work, the HL depths become large enough at the end of the loading procedure that we can safely neglect tunneling altogether (${J}_{\perp }\approx 0$). Introducing a chemical potential μ, the spatial structure is then given, in the local density approximation, by

$$\begin{eqnarray}&&n({{\boldsymbol{\rho }}}_{i})=\mathrm{Int}\left[\displaystyle \frac{\mu -\tfrac{M}{2}({{\rm{\Omega }}}_{x}^{2}{x}_{i}^{2}+{{\rm{\Omega }}}_{y}^{2}{y}_{i}^{2})}{{U}_{{gg}}}\right]+1,\end{eqnarray} \tag{ 5 }$$

where $\mathrm{Int}(x)$ denotes the integer part of x. For finite temperatures and/or tunneling (still small compared to U_gg), the overall density profile remains similar but with smoother edges than predicted by (5). The relative weight of the plateau with $\overline{n}$ atoms can be characterized by its population normalized to the total atom number, noted ${{ \mathcal F }}_{\overline{n}}$ in the remainder of the article.

The optical setup has been presented in detail in [22]. Briefly, a laser resonant with the g–e clock transition near ${\lambda }_{0}\approx 578.4$ nm is locked on a high-finesse cavity well-isolated from its surroundings which serves as frequency reference. We split the optical path into one going to the cavity using an optical fiber with active Doppler noise cancellation [33], and the other going towards the atomic cloud. The wavevector ${{\boldsymbol{k}}}_{\mathrm{cl}}$ is oriented along the ${{\boldsymbol{e}}}_{x}+{{\boldsymbol{e}}}_{y}$ direction (see figure 1(a)).

The g–e electric dipole transition is forbidden at zero field ($J=0\to J^{\prime} =0$ transition, with J the total electronic angular momentum). Following the method pioneered in [34, 35], we use a static magnetic field ${\boldsymbol{B}}={B}_{0}{{\boldsymbol{e}}}_{z}$ (with ${B}_{0}\approx 182$ G) to enable an effective electric dipole coupling between g and e. Neglecting motional degrees of freedom, the effective coupling strength is ${{\rm{\Omega }}}_{{\rm{cl}}}\propto {B}_{0}\sqrt{{I}_{{\rm{cl}}}}$ [34], where ${I}_{\mathrm{cl}}=2{P}_{\mathrm{cl}}/(\pi {w}_{\mathrm{cl}}^{2})$, P_cl and w_cl are respectively the intensity, power and waist ($1/{{e}}^{2}$ radius) of the clock laser beam. For atoms in a deep optical lattice, the complete transition amplitude (hereafter denoted as 'Rabi frequency' for simplicity) is ${\rm{\Omega }}=\kappa {{\rm{\Omega }}}_{\mathrm{cl}}$, where the additional factor $\kappa =| \langle {W}_{\perp }| {{\rm{e}}}^{{\rm{i}}{{\boldsymbol{k}}}_{\mathrm{cl}}\cdot \hat{{\boldsymbol{r}}}}| {W}_{\perp }\rangle | \approx 0.9$ takes into account the overlap between motional states [1]. We find ${\rm{\Omega }}\approx 2\pi \times 1500$ Hz using ${I}_{\mathrm{cl}}\approx 2.4\times {10}^{5}\,$ mW cm⁻² (${P}_{\mathrm{cl}}\approx 18.5\,$mW and ${w}_{\mathrm{cl}}\,\approx$ 70 μm) and a laser linearly polarized along ${{\boldsymbol{e}}}_{z}$, in good agreement with the measured value ${\rm{\Omega }}\approx 2\pi \times 1500\,$ Hz for the experiment of figure 1(b). Hopping transitions to different sites and to higher bands can be safely neglected (the Lamb–Dicke parameter is small, ${({k}_{\mathrm{cl}}{a}_{\mathrm{lat}})}^{2}\approx 0.08$ with ${a}_{\mathrm{lat}}\approx 26\,$ nm the typical extent of the atomic wavefunction ${W}_{\perp }$).

We detect atoms in the ground state g using standard resonant absorption imaging on the ${}^{1}{S}_{0}\to {}^{1}{P}_{1}$ transition. Atoms in the excited state e are detected using a repumping laser on the ${}^{3}{P}_{0}\to {}^{3}{D}_{1}$ transition near 1389 nm. Here and in the remainder we denote by ${N}_{g/e}$ the atom number in the state g/e. The auxiliary ${}^{3}{D}_{1}$ state can decay to the ³P_J states ($J=0,1,2$), where ${}^{3}{P}_{\mathrm{0,2}}$ are metastable and where ³P₁ decays to the ground state by emitting a photon at 556 nm. The metastable ³P₂ state is a dark state for both the repumping and the imaging lasers. However, the branching ratio ${}^{3}{D}_{1}\to {}^{3}{P}_{2}$ is small [36], so that the repumping efficiency ${\eta }_{e}$ from ³P₀ to ¹S₀ is close to unity for a 500 μs long repumping pulse. Comparing N_g and N_e for Rabi oscillations similar to figure 2(a), we estimate ${\eta }_{e}\approx 80$%, consistent with a calculation based on optical Bloch equations. In addition, atoms in the ground state are very far off-resonant and hardly affected by the repumping laser. Hence, applying the repumping pulse allows us to detect the total population ${N}_{g}+{N}_{e}$. To detect selectively atoms in e and measure only N_e, we apply an additional 5 μs-long removal pulse on the ${}^{1}{S}_{0}\to {}^{1}{P}_{1}$ transition before the repumping pulse. Atoms in g scatter many photons and are pushed away from the imaging region. Atoms in e are mostly unaffected by the removal pulse, although we measure a reduced detection efficiency by approximately 10%. This is possibly due to secondary scattering between trapped e atoms and untrapped g atoms leaving the sample. We have also observed a slight influence of the atomic density on the repumping efficiency. The observed efficiency is reduced by roughly 10% when N_e is higher than about 3 × 10⁴. We do not correct experimental atom numbers in e for the repumping efficiency but we take it into account in the analysis of sections 3.2 and 3.3.

For atoms in singly-occupied lattice sites, the time evolution consists of textbook Rabi oscillations between g and e, described by the Hamiltonian

$$\begin{eqnarray}{\hat{H}}_{\mathrm{eff}}^{(\overline{n}=1)}=\left[\begin{array}{cc}0 & \displaystyle \frac{{\hslash }{\rm{\Omega }}}{2}\\ \displaystyle \frac{{\hslash }{\rm{\Omega }}}{2} & -{\hslash }\delta \end{array}\right],\end{eqnarray} \tag{ 6 }$$

with $\delta ={\omega }_{\mathrm{cl}}-{\omega }_{0}$ the laser detuning, ${\omega }_{0}=2\pi c/{\lambda }_{0}$ the atomic Bohr frequency of the transition, ${\omega }_{\mathrm{cl}}$ the clock laser frequency and ${\rm{\Omega }}$ the Rabi frequency introduced before. Starting from a sample in the ground state and switching on the coupling laser, the atomic populations oscillate between g and e at the frequency $\sqrt{{{\rm{\Omega }}}^{2}+{\delta }^{2}}$. In the remainder we characterize the coupling laser pulse by its duration T, or equivalently by its area ${\rm{\Omega }}T$. We show in figure 2(a) Rabi oscillations for a cloud prepared with only singly-occupied sites $[{N}_{\mathrm{at}}\approx 8\times {10}^{3}$], where almost full contrast is observed up to 10 ms. However, for higher atom numbers as in figure 1(b) $[{N}_{\mathrm{at}}\approx 8\times {10}^{4}$], Rabi oscillations are damped on the same time scale. For both cases the Rabi frequency is ${\rm{\Omega }}/(2\pi )\approx 1500\,$ Hz. In the following we consider possible dephasing mechanisms to explain this difference.

We first consider a possible deviation from the exact magic wavelength, inducing a position-dependent differential light shift caused by the lattice potential. A calculation based on [24] indicates a differential shift across the cloud smaller than 10 Hz for a wavelength mismatch of 0.1 nm. This effect can be neglected in the present work.

Frequency or intensity fluctuations of the clock laser also induce dephasing over time. Careful monitoring shows that intensity fluctuations remain below 1% and cannot explain the observed dephasing. Regarding frequency fluctuations, we recorded spectra at low atom numbers (to ensure almost unity filling across the cloud), π-pulse areas and long pulse times. When the pulse time exceeds ∼10 ms, we observe substantial shot-to-shot fluctuations of the measured transition probability with identical parameters. This could be explained by shot-to-shot fluctuations of the clock laser frequency with a standard deviation of about 100 Hz, possibly due to the high-finesse cavity. This value can certainly be improved in future work. In order to evaluate the impact of such frequency fluctuations on the coherent dynamics in figure 1(b), we modeled them by a random detuning δ with a Gaussian distribution function of width ${\rm{\Delta }}{\omega }_{\mathrm{cl}}$. We verified numerically that a width ${\rm{\Delta }}{\omega }_{\mathrm{cl}}$ greater than $2\pi \times 600\,\mathrm{Hz}$ would be required to account entirely for the observed damping, a value incompatible with the narrowest measured spectra.

Additional dephasing mechanisms come from the Gaussian profile of the coupling beam. This profile entails a non-uniform Ω and also induces a position-dependent differential light shift, leading to a non-uniform δ [35]. Close to resonance, the first effect is dominant. Inhomogeneities cause a dephasing over time between the center and the edges of the cloud, and thus to an apparent damping when averaging over the whole cloud. The dephasing time ${\tau }_{{\rm{d}}}$ thus depends on the cloud size. Using a parabolic approximation for the Gaussian profile, we estimate ${\tau }_{{\rm{d}}}\approx \alpha {w}_{\mathrm{cl}}^{2}/({\rm{\Omega }}{R}^{2})$, where R is the radius of the cloud and α a numerical factor depending on the atomic density distribution⁴ . We used the model for the spatial atomic density of section 2.3 to obtain the cloud size for a given N_at and to estimate $\alpha \approx 6$. For the highest atom number ${N}_{\mathrm{at}}\approx 8\times {10}^{4}$ $[R\,\approx$ 19 μm] we find ${\tau }_{{\rm{d}}}\approx 9$ ms, close to the observed damping time in figure 1(b). For the lowest atom number ${N}_{\mathrm{at}}\approx 8\times {10}^{3}$ $[R\,\approx$ 7.6 μm] we find ${\tau }_{{\rm{d}}}\approx 50\,$ ms, consistent with almost no decay observed in figure 2(a).

We conclude that inhomogeneous dephasing is likely responsible for the observed damping for large N_at but quickly becomes negligible when N_at decreases due to the quadratic dependence ${\tau }_{{\rm{d}}}\sim {R}^{-2}$. In the low N_at limit, the damping is then most likely dominated by frequency fluctuations on a time scale $\gtrsim \,10$ ms. We note that the single-particle damping discussed in this paragraph has little influence on the shape and position of the spectra shown in section 3.3, where the pulse time T is shorter than the damping time. However, it limits the uncertainty on our measurement of collisional parameters presented in section 3.3.