Engineering Quantum States of Matter for Atomic Clocks in Shallow Optical Lattices

TL;DR

This paper analyzes how photon scattering in optical lattice clocks limits coherence times and proposes using shallow, state-independent lattices with larger lattice constants to approach the natural lifetime of the clock transition.

Contribution

It introduces a novel approach of using shallow, state-independent optical lattices with larger lattice constants to reduce scattering and dephasing, enhancing atomic clock coherence times.

Findings

Photon scattering limits coherence to less than 12 seconds.

Shallow lattices can potentially extend coherence times to the natural lifetime.

Proposed scheme is compatible with high-density Fermi gases.

Abstract

We investigate the effects of stimulated scattering of optical lattice photons on atomic coherence times in a state-of-the art ${}^{87}\mathrm{Sr}$ optical lattice clock. Such scattering processes are found to limit the achievable coherence times to less than 12 s (corresponding to a quality factor of $1 \times 10^{16}$), significantly shorter than the predicted 145(40) s lifetime of ${}^{87}\mathrm{Sr}$'s excited clock state. We suggest that shallow, state-independent optical lattices with increased lattice constants can give rise to sufficiently small lattice photon scattering and motional dephasing rates as to enable coherence times on the order of the clock transition's natural lifetime. Not only should this scheme be compatible with the relatively high atomic density associated with Fermi-degenerate gases in three-dimensional optical lattices, but we anticipate that certain properties of various quantum states of matter can be used to suppress dephasing due to tunneling.

Equations

$${\sigma }_{\text{QPN}}=\frac{1}{{\omega }_{0}T}\sqrt{\frac{T_{cyc.}}{N}}.$$

$${\gamma }_t=4t|\sin (\varphi /2)|,$$

$$\frac{t(a)}{t(a_0)}=\sqrt{\frac{a_0}{a}}\exp [\sqrt{\frac{V}{E_r}}\left(1-\frac{a}{a_0}\right)].$$

$$\frac{U(a)}{U(a_0)}={\left(\frac{a_0}{a}\right)}^{3/2},$$

$$H/\hslash =$$

$$\displaystyle+\frac{\Omega}{2}e^{i\phi_{x}}c_{x,e}^{\dagger}c_{x,g}\Big{]}-t\sum_{\sigma}c_{\mathrm{L},\sigma}^{\dagger}c_{\mathrm{R},\sigma}+\mathrm{H.c.}$$

$$\gamma =\sqrt{⟨H^2⟩-⟨H{⟩}^2}.$$

Full text

Owing to dramatic improvements in both the precision and accuracy of atomic spectroscopy over the last decade Ludlow et al. (2008); Bloom et al. (2014); McGrew et al. (2018), there is growing interest in the use of atomic clocks as quantum sensors in tests of fundamental physics Blatt et al. (2008); Kolkowitz et al. (2016); Van Tilburg et al. (2015); Pruttivarasin et al. (2015); Sanner et al. (2018). Recent demonstrations of spectroscopic techniques, which are immune to local oscillator noise, promise to dramatically improve the precision of such tests Chou et al. (2011); Nemitz et al. (2016); Schioppo et al. (2016); Marti et al. (2018). In the absence of local oscillator noise, frequency measurements of a single atom follow a binomial distribution and, for Ramsey spectroscopy Ramsey (1956), are spread about its true transition frequency $\omega_{0}$ by an amount $(\omega_{0}T)^{-1}$, given in fractional frequency units with $T$ being the coherent evolution time. In the absence of entanglement, interrogation of a sample of $N$ atoms with an experimental cycle time $T_{\mathrm{cyc.}}$ results in a quantum projection noise (QPN) limit Itano et al. (1993),

[TABLE]

That is, one wants to increase the interrogation time and use a larger number of atoms in order to reduce the measurement noise.

To date, the lowest reported QPN limit ($\sigma_{\mathrm{QPN}}=1.5\times 10^{-17}/\sqrt{\mathrm{Hz}}$) was achieved using a fermi-degenerate gas of $N\approx 10^{4}$ ${}^{87}\mathrm{Sr}$ ($\omega_{0}\approx 2\pi\times 429~{}\mathrm{THz}$) atoms loaded into the Mott-insulating regime of a three-dimensional (3D) optical lattice Campbell et al. (2017); Marti et al. (2018). In these experiments, coherence times were found to be less than $12~{}\mathrm{s}$ and presumed to be limited by Raman scattering of photons from the deep optical lattice Martin (2013); Dörscher et al. (2018). While these scattering processes may be reduced by operating in a shallower optical potential, one then introduces site-to-site tunneling as an additional dephasing mechanism Lemonde and Wolf (2005); Kolkowitz et al. (2017); Bromley et al. (2018).

In this Letter, we discuss a solution that simultaneously addresses both the lattice photon scattering and tunneling induced dephasing problems in 3D optical lattice clocks: shallow optical lattices with increased lattice constants, $a$. We find that not only should the decreased kinetic energies in the ground band of such a lattice be sufficient to suppress motional dephasing in a single atom picture, but additionally, for a nuclear-spin polarized Fermi gas at half-filling, inter-electronic-orbital interactions should provide an additional mechanism for reducing motional dephasing rates. In such a system, atom numbers on the order of $N=10^{5}$ and coherent interrogation times up to $T=140~{}\mathrm{s}$ seem readily achievable and correspond to a QPN limit of $\sigma_{\mathrm{QPN}}<10^{-19}/\sqrt{\mathrm{Hz}}$.

Before describing the details of our proposal, we build upon previous work which investigated trap depth dependent depopulation of the $5s5p~{}{}^{3}P_{0}$ excited clock state ($\ket{e}$) in ${}^{87}\mathrm{Sr}$ one-dimensional optical lattices as a signature of the Raman scattering problem Martin (2013); Dörscher et al. (2018). By leveraging the improved control over motional degrees of freedom Campbell et al. (2017) and imaging techniques Marti et al. (2018) available in a Fermi-degenerate 3D optical lattice clock, we additionally investigate the corresponding loss of Ramsey fringe contrast.

A spin-polarized degenerate fermi gas is created by evaporatively cooling atoms in an equal mixture of the $m_{F}=-5/2,\ldots,9/2$ magnetic sublevels of the $5s^{2}~{}{}^{1}S_{0}$ electronic ground state ($\ket{g}$) before a focused laser beam, detuned from the $5s5p~{}{}^{3}P_{1}$ intercombination line, provides a state-dependent potential, removing nearly all but the $m_{F}=9/2$ atoms from the trap. Approximately $2\times 10^{3}$ atoms with a temperature of $20\%$ of the Fermi temperature are then loaded from the running wave optical dipole trap into a cubic optical lattice. Each arm ($i=x,y,z$) of the lattice is formed by a retroreflected laser at the magic wavelength ($\lambda_{\mathrm{magic}}$) Ye et al. (2008), and is characterized by a variable depth $V_{i}$ and a lattice constant $a_{0}=\lambda_{\mathrm{magic}}/2=407~{}\mathrm{nm}$. The $i=z$ lattice arm is oriented along both gravity and an applied $0.5~{}\mathrm{mT}$ magnetic bias field. We perform an additional step of spin purification by coherently driving $\ket{g,m_{F}=9/2}\rightarrow\ket{e,m_{F}=9/2}$ with $\lambda_{\mathrm{clk}}=698~{}\mathrm{nm}$ clock light, propagating along the $i=x$ lattice axis, then removing all remaining $\ket{g}$ atoms by cycling on the $5s^{2}~{}{}^{1}S_{0}\leftrightarrow 5s6p~{}{}^{1}P_{1}$ transition with resonant $461~{}\mathrm{nm}$ light.

For the excited state lifetime measurement, we insert a variable hold time before a series of $5~{}\mu\mathrm{s}$ pulses of $461~{}\mathrm{nm}$ light form an absorption image of the $\ket{g}$ atoms on a CCD camera, providing a count of the $\ket{g}$ atoms, $N_{g}$, while also removing the imaged atoms from the trap. We obtain a count of the remaining atoms, $N_{\tilde{e}}$, by optically pumping $5s5p~{}{}^{3}P_{0},{}^{3}P_{2}\rightarrow 5s5p~{}{}^{3}P_{1}$ with light resonant on the $5s5p~{}{}^{3}P_{0},{}^{3}P_{2}\leftrightarrow 5s6s~{}{}^{3}S_{1}$ transitions at $679~{}\mathrm{nm}$ and $707~{}\mathrm{nm}$. Atoms then rapidly decay to the ground state, via the $21~{}\mu\mathrm{s}$ lived $5s5p~{}{}^{3}P_{1}$ Nicholson et al. (2015), where they are subsequently imaged with $461~{}\mathrm{nm}$ light. We note that this readout method counts not only atoms in $\ket{e,m_{F}=9/2}$, but all atoms in the metastable $5s5p~{}{}^{3}P_{0},{}^{3}P_{2}$ manifold in the quantity $N_{\tilde{e}}$. The decay of the excited population $\rho_{\tilde{e}\tilde{e}}=N_{\tilde{e}}/(N_{g}+N_{\tilde{e}})$ is then fit to extract a $1/e$ lifetime.

These lifetimes are measured for various lattice depths, $V_{z}$, ranging from $5E_{r}$ to $310E_{r}$, while fixing $V_{x}=59(2)E_{r}$ and $V_{y}=70(2)E_{r}$, where $E_{r}=h^{2}/8ma_{0}^{2}\approx 3.5~{}h\cdot\mathrm{kHz}$ is the lattice photon recoil energy, $h$ the Planck constant, and $m$ the atomic mass. Fig. 1(a) shows the trap depth dependence of the extracted lifetimes. We find the measured lifetimes to be significantly shorter than the predicted $\tau_{0}=145(40)~{}\mathrm{s}$ natural lifetime Boyd et al. (2007), yet largely consistent with numerical simulations with no free parameters (shaded red region) in which two-photon Raman transitions, stimulated by the lattice light, distribute atoms amongst the $5s5p~{}{}^{3}P_{J}$ manifold where the atoms can then spontaneously decay to the ground state from $5s5p~{}{}^{3}P_{1}$. The vacuum limited lifetime of atoms prepared in $\ket{g}$ is independently measured to be $>100~{}\mathrm{s}$. An energy level diagram depicting the Raman scattering processes, and the master equation used in the simulation can be found in Ref. Sup, .

Such scattering events are detrimental to clock operation as they destroy the coherence $\rho_{eg}$ between the two clock states Cohen-Tannoudji et al. (1998). Using imaging spectroscopy Campbell et al. (2017); Marti et al. (2018), we observe this loss in coherence as a reduction in the Ramsey fringe contrast for increasing dark time, $T$. The contrast decay at a given lattice depth is then fit to extract a $1/e$ coherence time. The results of such measurements are shown in Fig. 1(b) for the same lattice conditions as in Fig. 1(a). The observed coherence times are found to scale proportionally to $V_{\mathrm{tot.}}^{-1}$ ($V_{\mathrm{tot.}}=\sum_{i}V_{i}$) for $V_{z}>20E_{r}$, yet they fall significantly below the predicted decoherence rate due to Raman scattering (shaded red region) Sup .

This suggests that other, lattice depth dependent, decoherence mechanisms are present in the system. Rayleigh scattering is not expected to directly contribute as a dephasing mechanism since the scattering amplitudes are identical for both clock states in a magic wavelength trap Uys et al. (2010); Martin (2013). However, both Raman and Rayleigh scattering processes can heat atoms out of the ground band of the lattice Gerbier and Castin (2010) at which point, we suspect, they are able to tunnel around and dephase through contact interactions.

For $V_{z}<20E_{r}$, coherence times are seen deviate from the $V_{\mathrm{tot.}}^{-1}$ scaling and instead rapidly decay. This decay is accompanied by a loss in atom number which we attribute to significant tunneling rates along the $i=z$ lattice and inelastic collisions Bishof et al. (2011). This demonstrates the difficulty in overcoming the Raman scattering problem in conventional optical lattice clocks. One would like to operate in an optical trap shallow enough to make scattering induced decoherence rates comparable to the natural lifetime — one requires $V_{i}\lesssim 4E_{r}$ for ${}^{87}\mathrm{Sr}$ — but then one finds additional, tunneling enabled dephasing mechanisms due to the increased kinetic energy scale.

We address this issue in the following proposal. Hereafter, we assume a uniform lattice, $V_{i}=V$, and for the time being, a non-interacting gas. For a Ramsey type experiment in an inertial reference frame, tunneling at a rate $t$ along the direction of the probe laser results in a loss in contrast of the spectroscopic signal according to $\rho_{eg}=|\mathcal{J}_{0}(4tT\sin(\phi/2))|$, $\mathcal{J}_{0}$ being the zeroth order Bessel function of the first kind Bromley et al. (2018), and $\phi=2\pi a/\lambda_{\mathrm{clk}}$, the site-to-site phase shift of the clock light where we now allow for a variable lattice constant, $a$, as depicted in Fig. 2(a) and Fig. 2(b). For the purpose of comparing different energy scales of the system, we define the argument of the Bessel function,

[TABLE]

as the “motional dephasing rate”. As a practical example, such a variable spacing lattice can be engineered, while restricting the wavelength of the trapping light to be magic by interfering the lattice beams at an arbitrary angle, $\theta$, giving a spacing $a=\lambda_{\mathrm{magic}}/|2\sin(\theta/2)|$ Morsch et al. (2001); Huckans et al. (2009); Al-Assam et al. (2010), or with an optical tweezer array Norcia et al. (2019).

Under the harmonic approximation, the tunneling rate for fixed $V$ scales exponentially with $a$ as Bloch et al. (2008)

[TABLE]

One can think of this intuitively as a change in the lattice constant rescaling the lattice recoil energy, $E_{r}\rightarrow(a_{0}/a)^{2}E_{r}$, such that the lattice depth in units of the new recoil energy can be made quite large for modest increases in $a/a_{0}$. Numerical values of $\gamma_{t}$ and the total kinetic energy , $t^{\ast}=12t$, for a $V=4E_{r}$ lattice are shown in Fig. 2(c). For a sufficiently large lattice constant, the atomic limit ($t\rightarrow 0$) is achieved where tunneling related effects can be neglected. We find that both $\gamma_{t}$ and $t^{\ast}$ are suppressed below $\tau_{0}^{-1}$ for lattice spacings $a\gtrsim 2~{}\mu\mathrm{m}$.

Additionally, $\gamma_{t}$ is found to sharply drop to zero upon matching the condition $a/\lambda_{\mathrm{clk}}\mod 1=0$. These resonances can be understood in a momentum space picture where the clock photon recoil is matched to a reciprocal lattice vector and thus absorption or emission of a clock photon leaves each atom’s motional state unchanged. In this case, for a nuclear-spin polarized gas at half filling, the system behaves as a band insulator throughout clock spectroscopy as the indistinguishability of all atoms is preserved. This scheme, however, requires an accuracy in $\theta$ beyond the $2\times 10^{-5}$, and $2\times 10^{-2}$ levels for the $a/\lambda_{\mathrm{clk}}=1,2$ configurations, respectively. Throughout this range of parameters ($1<a/a_{0}<5$) the lattice bandgap is greater than $h\cdot 2~{}\mathrm{kHz}$, and the effective Rabi coupling is suppressed by no more than 60% of the bare Rabi coupling such that carrier resolved spectroscopy is easily achievable; line-pulling effects from off-resonant excitation of motional sidebands can be suppressed below the $10^{-19}$ level for Rabi frequencies below 10 Hz, as shown in Fig. S4 Sup .

Many-body effects arise through an on-site interaction energy $U$ parameterized by the anti-symmetric inter-electronic-orbital $s$-wave scattering length, $a_{eg^{-}}=69.1(0.9)~{}a_{B}$ Goban et al. (2018), where $a_{B}$ is the Bohr radius. This energy scale decreases algebraically with an increasing lattice constant,

[TABLE]

such that for sufficiently large lattice spacings, the system enters the Mott-insulating regime ($t^{\ast}/U\ll 1$) Jördens et al. (2008); Schneider et al. (2008). Here, for a sufficiently cold gas at half-filling, the only available excitations below the energy gap $U$ are of the order of the superexchange energy $J=4t^{2}/U$. Thus we expect, for a sufficiently weak probe pulse, the motional dephasing rates to be suppressed by a factor of $t/U$ as compared to the non-interacting case. Numerical values for $U$ in a $V=4E_{r}$ lattice are shown in Fig. 2(c).

We investigate these effects with a “toy model” consisting of a double well potential. We assign the following Hamiltonian in the rotating wave approximation,

[TABLE]

to such a system. Here $c_{x,\sigma}^{\dagger}$ ($c_{x,\sigma}$) creates (destroys) a fermion with internal state $\sigma\in\{g,e\}$ in well $x\in\{\mathrm{L},\mathrm{R}\}$, $n_{x,\sigma}=c^{\dagger}_{x,\sigma}c_{x,\sigma}$, $\Delta=\omega-\omega_{0}$ is the difference between the frequency of the driving field $\omega$ from the atomic resonance, $\omega_{0}$, $\Omega$ is the Rabi coupling strength, and $\phi_{x}=2\pi\frac{a}{\lambda_{\mathrm{clk}}}\delta_{x,\mathrm{R}}$ is the site-dependent phase shift of the clock light, with $\delta_{i,j}$ being the Kronecker delta function. The two atom spectrum of this Hamiltonian with $\Omega=0$, is shown in Fig. 3(a).

We simulate Ramsey spectroscopy of one and two atoms in the double well by numerically integrating the Schrödinger equation. A resonant ($\Delta=0$) $\pi/2$-pulse with $\Omega\gg t$ places each atom in an equal superposition of ground and excited electronic states. For $a/\lambda_{\mathrm{clk}}\mod 1\neq 0$, this pulse also changes the system’s motional state. During field-free evolution ($\Omega=0$), the different motional states beat against each other causing a dephasing of the spectroscopic feature. We quantify this effect with the following relation,

[TABLE]

For a single atom, this quantity approximates the dephasing rate in an infinite lattice, $\gamma_{t}$, falling off with an envelope proportional to $t$ as shown in Fig. 3(b) (red dashed line). For two atoms, we observe that as one begins to resolve the interaction energy ($\Omega\ll U$) the dephasing rate becomes proportional to the superexchange energy, falling off with an envelope proportional to $t^{2}/U$ as shown in Fig. 3(b) (solid blue line). While the exact numerical prefactor differs slightly from what one would get for an infinite lattice Essler et al. (2005), the general conclusion is the same: the minimum lattice spacing such that $\gamma\tau_{0}<1$ is significantly relaxed as compared to the non-interacting case.

We have identified scattering of lattice photons as a dominant decoherence mechanism in a state-of-the-art 3D optical lattice clock and proposed a number of ways in which quantum materials may be engineered to overcome such limits. The improved clock stability associated with longer coherence times will directly enable new searches for time variation of fundamental constants and tests of general relativity on sub-mm length scales. Additionally, the shallow optical potentials described in this Letter will help reduce systematic clock shifts related to the traps themselves, especially terms that are nonlinear in trap depth Porsev et al. (2018); Ushijima et al. (2018). Furthermore, increasing the distance between atoms should aid in reducing systematic clock shifts arising from collective radiative effects Chang et al. (2004); Bromley et al. (2016). Future work can investigate the use of atomic collisions to create metrologically useful entanglement He et al. (prep).