Observation of coherent multiorbital polarons in a two-dimensional Fermi gas

TL;DR

This paper reports the first experimental observation of multiorbital polarons in a two-dimensional Fermi gas of ytterbium atoms, revealing their energies, coherence, and long-lived nature, supported by a comprehensive many-body theory.

Contribution

It provides the first experimental evidence of multiorbital polarons in 2D Fermi gases and develops a theoretical model that matches the observations.

Findings

Long-lived repulsive polarons with decay rates much lower than their energies

Spectroscopic measurement of attractive and repulsive polaron energies

Theoretical model accurately describing interorbital interactions in 2D

Abstract

We report on the experimental observation of multiorbital polarons in a two-dimensional Fermi gas of $^{173}\mathrm{Yb}$ atoms formed by mobile impurities in the metastable $^3\mathrm{P}_0$ orbital and a Fermi sea in the ground-state $^1\mathrm{S}_0$ orbital. We spectroscopically probe the energies of attractive and repulsive polarons close to an orbital Feshbach resonance and characterize their coherence by measuring the quasiparticle residue. For all probed interaction parameters, the repulsive polaron is a long-lived quasiparticle with a decay rate more than 2 orders of magnitude below its energy. We formulate a many-body theory, which accurately treats the interorbital interactions in two dimensions and agrees well with the experimental results. Our work paves the way for the investigation of many-body physics in multiorbital ultracold Fermi gases.

Tables

Table 1. Table S1: Experimental parameters (Fermi energy, temperature, and minority fraction) for each figure presented in the main text. The values discussed in the main text are the mean of the corresponding row.

	Fig. 2	Fig. 3	Fig. 4
Fermi energy ${ϵ}_F/h$	$3.65\left(22\right)\mathrm{kHz}$	$3.55\left(39\right)\mathrm{kHz}$	$3.39\left(21\right)\mathrm{kHz}$
Reduced temperature $k_{B}T/{ϵ}_F$	$0.17\left(3\right)$	$0.16\left(4\right)$	$0.14\left(3\right)$
Minority fraction $C$	$0.26\left(2\right)$	$0.28\left(3\right)$	$0.31\left(1\right)$

Equations

$$n(x,z)$$

$${ϵ}_F$$

$$E_F(x,z)=\frac{{\hslash }^2}{2m}\left[4\pi n_{\uparrow }(x,z)\right]$$

$${\left(\Delta {ϵ}_F\right)}^2=\frac{1}{N_0(\Delta x)}{\int }_{-\Delta x/2}^{\Delta x/2}dx{\int }_{-\infty }^{\infty }dz{n}_0(x,z){\left[{ϵ}_F-E_F(x,z)\right]}^2.$$

$${\kappa }_F=\frac{\sqrt{2m{ϵ}_F}}{\hslash }.$$

$$C$$

$$n(x)$$

$$T=\frac{E_F/k_B}{\sqrt{-2{\mathrm{Li}}_2(-z)}}.$$

$$\mathrm{OD}(i,j)=\log \left[\frac{I_{\mathrm{in}}(i,j)}{I_{\mathrm{out}}(i,j)}\right]+\frac{I_{\mathrm{in}}(i,j)-I_{\mathrm{out}}(i,j)}{\alpha I_{\mathrm{sat}}}.$$

$$a_{+}$$

$$r_{+}$$

$$\Delta \mu =h\times 554(3)Hz/G.$$

$$\frac{E_{+}}{{ϵ}_F}\approx \frac{2}{\ln ({\epsilon }_b/{ϵ}_F)}=-\frac{1}{\ln ({\kappa }_F{a}_{2D,gg})}.$$

$$a_{2D,gg}=l_y\sqrt{\frac{\pi }{D}}\exp \left(-\sqrt{\frac{\pi }{2}}\frac{l_y}{a_{gg}}\right)\simeq 12.5a_0,$$

$$N(t)=a{e}^{-{\Gamma }_{bg}t}-b{e}^{-{\Gamma }_{R}t}\cos (\Omega t).$$

$$N(t)=a{e}^{-{\Gamma }_{\mathrm{rep}}t}+b,$$

$${ϵ}_{\mathrm{kin}}$$

$$E_{\mathrm{kin},\uparrow }(x,z)=\frac{2\pi }{n_{\uparrow }(x,z)}{\int }_0^{\infty }dpp\frac{p^2}{2m}{f}_{\uparrow }(x,z,p).$$

$${\mu }_0=\mu \left[n(x,z)\right]+\frac{m}{2}\left({\omega }_x^2{x}^2+{\omega }_z^2{z}^2\right),$$

$$\mu (g)=E_F\left[1+2g+4g^2(1-\ln 2)+\mathcal{O}\left(g^3\right)\right].$$

$$\mu (T)=k_{B}T\ln \left(e^{T_F/T}-1\right),$$

$${\hat{H}}_0=\sum _{\mathbf{k},n}2{ϵ}_{\mathbf{k}n}|o,\mathbf{k}n⟩⟨o,\mathbf{k}n|+\sum _{\mathbf{k},n}(2{ϵ}_{\mathbf{k}n}+\delta )|c,\mathbf{k}n⟩⟨c,\mathbf{k}n|,$$

$$\delta \equiv \Delta \mu B$$

$$\hat{V}=\sum _{\mathbf{k},{\mathbf{k}}^{\prime },n,n^{\prime }}{\varphi }_n{\varphi }_{n^{\prime }}\left\{U_{+}|+,\mathbf{k}n⟩⟨+,{\mathbf{k}}^{\prime }{n}^{\prime }|+U_{-}|-,\mathbf{k}n⟩⟨-,{\mathbf{k}}^{\prime }{n}^{\prime }|\right\},$$

$${\varphi }_n=$$

$$⟨{\mathbf{k}}^{\prime }{n}^{\prime }|\hat{T}(E)|\mathbf{k}n⟩=⟨{\mathbf{k}}^{\prime }{n}^{\prime }|\hat{V}|\mathbf{k}n⟩+\sum _{{\mathbf{k}}^{\prime \prime }{n}^{\prime \prime }}⟨{\mathbf{k}}^{\prime }{n}^{\prime }|\hat{V}|{\mathbf{k}}^{\prime \prime }{n}^{\prime \prime }⟩⟨{\mathbf{k}}^{\prime \prime }{n}^{\prime \prime }|\frac{1}{E-{\hat{H}}_0+i0}|{\mathbf{k}}^{\prime \prime }{n}^{\prime \prime }⟩⟨{\mathbf{k}}^{\prime \prime }{n}^{\prime \prime }|\hat{T}(E)|\mathbf{k}n⟩,$$

$$⟨{\mathbf{k}}^{\prime }{n}^{\prime }|\hat{V}|\mathbf{k}n⟩={\varphi }_n{\varphi }_{n^{\prime }}\left(\begin{array}{cc}{U}_{+}& 0\\ 0& U_{-}\end{array}\right)\equiv {\varphi }_n{\varphi }_{n^{\prime }}\mathbf{V}.$$

$$\sum _{\mathbf{k},n}⟨\mathbf{k}n|\frac{1}{E-{\hat{H}}_0+i0}|\mathbf{k}n⟩=\sum _{\mathbf{k},n}\left(\begin{array}{cc}\frac{{\varphi }_n^2}{E-2{ϵ}_{\mathbf{k}n}+i0}& 0\\ 0& \frac{{\varphi }_n^2}{E-2{ϵ}_{\mathbf{k}n}-\delta +i0}\end{array}\right)\equiv {\mathbf{\Pi }}_{q2D}(E).$$

$${\mathbf{T}}^{(\mathrm{vac})}(n_1,n_2;E)\equiv ⟨{\mathbf{k}}_1{n}_1|\hat{T}(E)|{\mathbf{k}}_2{n}_2⟩={\varphi }_{n_1}{\varphi }_{n_2}{\left[\mathbf{R}{\mathbf{V}}^{-1}\mathbf{R}-{\mathbf{\Pi }}_{q2D}(E)\right]}^{-1}$$

$$\mathbf{R}$$

$${\mathbf{T}}^{(\mathrm{vac})}(0,0;E)=\frac{2\sqrt{2\pi }}{m}{\left[\mathbf{R}\left(\begin{array}{cc}\frac{l_y}{a_{+}}& 0\\ 0& \frac{l_y}{a_{-}}\end{array}\right)\mathbf{R}-\left(\begin{array}{cc}\mathcal{F}(-E/{\omega }_y)& 0\\ 0& \mathcal{F}((-E+\delta )/{\omega }_y)\end{array}\right)\right]}^{-1}$$

Full text

Supplemental Material

Observation of coherent multiorbital polarons in a two-dimensional Fermi gas

N. Darkwah Oppong

  

L. Riegger

  

O. Bettermann

  

M. Höfer

Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 München, Germany

Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

  

J. Levinsen

  

M. M. Parish

School of Physics and Astronomy, Monash University, Victoria 3800, Australia

  

I. Bloch

  

S. Fölling

Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 München, Germany

Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

S.I Experimental techniques

S.I.1 State preparation

After forced evaporation in a magic-wavelength ODT (mODT, $\lambda=759.35\,\mathrm{nm}$), we load the atoms adiabatically into a magic-wavelength optical lattice with a depth of $86E_{\mathrm{rec}}$, which is ramped up within $500\,\mathrm{ms}$. The mODT is kept at a constant value during the optical lattice ramp and holds the atoms against gravity. Typical trap frequencies in the final trap configuration are $\omega_{z}\approx 2\pi\times 250\,\mathrm{Hz}$ along gravity and $\omega_{x}\approx 2\pi\times 65\,\mathrm{Hz}$ along the other transverse axis. Approximating the deep lattice with a harmonic oscillator potential yields an axial trapping frequency $\omega_{y}=2\pi\times 37.1\,\mathrm{kHz}$. Spin mixtures are prepared by a sequence of intensity-stabilized optical pumping pulses on the intercombination line ${{}^{1}\mathrm{S}_{0}^{F=5/2}}\rightarrow{{}^{3}\mathrm{P}_{1}^{F=7/2}}$ at the beginning of evaporation. Usually, we prepare an imbalanced $m_{F}=-3/2,+5/2$ spin mixture in the ${{}^{1}\mathrm{S}_{0}}$ ground state (denoted $\left|{g,0}\right\rangle$ and $\left|{g,\uparrow}\right\rangle$) with a total atom count $N\approx 50\times 10^{3}$ and an overall minority fraction $N_{-3/2}/N\approx 0.24$. Typical temperatures in the mODT before loading into the optical lattice are $T\approx 0.2E_{F}/k_{B}$ determined by fitting density profiles to time-of-flight absorption images. Here, $k_{B}$ is the Boltzmann constant. Ramping up and down the optical lattice leads to an increase of $T$ below $0.05E_{F}/k_{B}$. The specific experimental parameters for all measurements presented in the main text are shown in Table S1.

The optical lattice produces an array of quasi-2D traps for each lattice plane where the confinement $\hbar\omega_{y}$ is much larger than the 2D Fermi energy $E_{F}=\hbar\sqrt{2N\omega_{x}\omega_{z}}$. The harmonic confinement of the optical traps causes each of the lattice planes to be occupied by a different number of atoms. Fig. S1 shows such a typical distribution extracted from absorption images of the whole atomic cloud. By imaging the atoms along the transverse direction of the optical lattice, we can select a few lattice planes and reduce the effects of trap inhomogeneity. The minimum number of selected planes is only limited by the imaging resolution $\approx 3\,\text{\textmu}\mathrm{m}$, which is large compared to the lattice spacing ($0.38\,\text{\textmu}\mathrm{m}$). We select a small region with size $\Delta x\times\Delta y=5.6\,\text{\textmu}\mathrm{m}\times 5.6\,\text{\textmu}\mathrm{m}$ near the center of the cloud where we integrate the atom number for all our measurements. This region contains approximately $15$ lattice planes and the variation of atom number per plane is negligible. Finite temperature as well as the non-zero extent of our integration region lead to an effective Fermi energy $\epsilon_{F}<E_{F}$. We calculate $\epsilon_{F}$ from the number density $n(x,z)$ in the local density approximation,

[TABLE]

where $\mathrm{Li}_{s}(x)$ is the polylogarithm function of order $s$, $m$ is the mass of ${{}^{\text{173}}\textrm{Yb}}$, $\mu$ denotes the chemical potential, and $\beta=1/k_{B}T$. The effective background Fermi energy $\epsilon_{F}$ sampled by the minority atoms is then given by

[TABLE]

Here, $N_{0}(\Delta x)=\int_{-\Delta x/2}^{\Delta x/2}dx\int_{-\infty}^{\infty}dz\;n_{0}(x,z)$ is the total number of the $\left|{g,0}\right\rangle$ atoms with $n_{0}(x,z)$ the corresponding number density, and

[TABLE]

is the local Fermi energy of the background atoms with $n_{\uparrow}(x,z)$ the number density of the $\left|{g,\uparrow}\right\rangle$ atoms. We calculate the standard deviation $\Delta\epsilon_{F}$ to quantify the variation of effective Fermi energy in our integration region,

[TABLE]

Typically, we find $\epsilon_{F}\simeq 0.8E_{F}$ and $\Delta\epsilon_{F}\simeq 0.2\epsilon_{F}$. For the effective Fermi wavevector $\kappa_{F}$, we use

[TABLE]

For the minority fraction $C$, we calculate

[TABLE]

S.I.2 In-situ thermometry

For extracting temperatures in the 2D planes of the optical lattice, we use a fit to the column density $n(x)$, which we find by integrating Eq. (S.1) along $z$,

[TABLE]

Here, the fugacity $z=e^{\beta\mu}$ is related to the temperature by

[TABLE]

In the experiment, we cannot reliably measure the in-situ densities of $\left|{g,0}\right\rangle$ and $\left|{g,\uparrow}\right\rangle$ atoms separately. Instead, we fit the distribution of all atoms. Simulating density profiles for imbalanced, non-interacting samples shows that fitting the combined density leads to a small overestimation of the temperature. However, we cannot confirm this result when we compare fitted temperatures at the edge of the cloud (small minority fraction) with the center of the cloud (large minority fraction). This can possibly be explained with the non-zero ground state interaction which leads to a small broadening of the density profile across the trap. We perform the temperature fit across multiple planes of the optical lattice and average the results to retrieve the absolute temperature. The standard deviation is used as the error in the temperature fit. Fig. S2 shows such a fit at the edge of the cloud where the minority fraction is small. We relate the temperature to the effective Fermi energy $\epsilon_{F}$ which yields the reduced temperature $k_{B}T/\epsilon_{F}$ (see Table S1).

S.I.3 Atom number calibration

Measurement of the absolute atom number is crucial for determining the Fermi energies in our experiment. All atom number measurements are done with in-situ imaging at a small magnetic bias field of $1\,\mathrm{G}$. We use the broad ${{}^{1}\mathrm{S}_{0}^{F=5/2}}\rightarrow{{}^{1}\mathrm{P}_{1}^{F=7/2}}$ transition and address the $\sigma^{+}$ as well as $\sigma^{-}$ transition. Short pulses with a duration of $15\,\text{\textmu}\mathrm{s}$ minimize atomic motion during absorption imaging. However, the short imaging pulses require intensities larger than the saturation intensity $I_{\mathrm{sat}}$ to fully penetrate the dense atomic cloud. We account for the saturated atomic transition with the modified Lambert-Beer law Reinaudi et al. (2007)

[TABLE]

Here, $\mathrm{OD}(i,j)$ is the optical column density on camera pixel $(i,j)$ and $I_{\mathrm{in}}$ is the incident intensity whereas $I_{\mathrm{out}}$ is the intensity after absorption. The parameter $\alpha>1$ accounts for imperfect polarization of the imaging light and the multi-level nature of the atom. This particularly applies to ${{}^{\text{173}}\textrm{Yb}}$, since imaging on the ${}^{1}\mathrm{S}_{0}^{F=5/2}\rightarrow\;{{}^{1}\mathrm{P}_{1}^{F=7/2}}$ transition involves up to six $m_{F}$ states. In the experiment, we determine $\alpha$ by imaging almost identical atomic samples with varying intensity. Minimization of the atomic density variance across all pixels then yields $\alpha$. This parameter is independently determined for spin-polarized samples with $m_{F}\in\{-5/2,-3/2,+5/2\}$ and a weighted average is used for imaging spin mixtures. Additionally, we measure $\alpha$ for the spin mixture which results from repumping $m_{F}=-5/2$ in the clock state back to the ground state on the ${{}^{3}\mathrm{P}_{0}^{F=5/2}}\rightarrow{{}^{3}\mathrm{D}_{1}^{F=7/2}}$ transition. We use the methods of Ref. Hung and Chin (2014) to determine the effective intensity $I_{\mathrm{in}}(i,j)$ at the location of the atoms and find agreement within $11\%$ of the directly measured intensity.

S.I.4 Orbital Feshbach resonance

We use the description in Ref. Zhang et al. (2015); Höfer et al. (2015) for the orbital Feshbach resonance. In this formalism, the Feshbach resonance is fully characterized by the scattering lengths $a_{\pm}$, the effective ranges $r_{\pm}$, and the differential Zeeman shift $\Delta\mu$ between $\left|{g,\uparrow}\right\rangle$ and $\left|{e,\downarrow}\right\rangle$. We use the scattering lengths and effective ranges experimentally determined in Ref. Höfer et al. (2015)

[TABLE]

where $a_{0}$ is the Bohr radius. We measure $\Delta\mu$ with the methods from Ref. Boyd et al. (2007). This differential measurement is independent of the magnetic field calibration and only depends on the nuclear magnetic moment of ${{}^{\text{173}}\textrm{Yb}}$, $\mu_{I}=-0.6776\mu_{N}$ Sansonetti and Martin (2005). We find $\Delta\mu\left/\Delta m_{F}\right.=h\times 110.72(51)\,\mathrm{Hz}/\mathrm{G}$, where $\Delta m_{F}$ is the difference between the ground and excited state $m_{F}$ numbers. We also use this value for the calibration of the magnetic fields in our experiment. For the $m_{F}=\pm 5/2$ states used in all measurements, the differential Zeeman shift is given by

[TABLE]

S.I.5 Clock spectroscopy

We use circularly polarized light to address the $m_{F}=-3/2\rightarrow m_{F}=-5/2$ transition. After the clock excitation pulse, we rapidly lower the magnetic field to $1\,\mathrm{G}$ and turn on a high-intensity ($I\sim I_{\mathrm{sat}}$) “push” beam on the broad ${{}^{1}\mathrm{S}_{0}^{F=5/2}}\rightarrow{{}^{1}\mathrm{P}_{1}^{F=7/2}}$ transition. This removes all ground state atoms from the trap. Subsequently, the impurity atoms in the clock state are pumped back to the ground state with a $0.3\,\mathrm{ms}$ pulse on the ${{}^{3}\mathrm{P}_{0}^{F=5/2}}\rightarrow{{}^{3}\mathrm{D}_{1}^{F=7/2}}$ transition and are imaged. During the repump pulse, atoms can move due to finite photon recoil transferred during the cascade decay ${{}^{3}\mathrm{D}_{1}}\rightarrow{{}^{3}\mathrm{P}_{1}}\rightarrow{{}^{1}\mathrm{S}_{0}}$. We estimate the worst case transverse motion of a single atom during the $0.3\,\mathrm{ms}$ pulse to be $1.7\,\text{\textmu}\mathrm{m}$, which is below our imaging resolution. Our repumping efficiency is $0.86(4)$ which we account for when counting atoms in the clock state. We select a small central region for counting atoms as described in Section S.I.1. When varying the position of the integration region, we can probe planes of the optical lattice with smaller effective Fermi energy $\epsilon_{F}$ and minority fraction which is shown in Fig. S4. However, we do not find any systematic energy shifts within our experimental resolution for regions with lower minority fractions. We also do not find any significant modification of the observed lineshapes for regions with different effective Fermi energies.

The initial ground-state preparation in our measurement has weak repulsive interactions which results in a positive energy shift. We account for this effect by estimating the ground state interaction energy and shifting the spectrum accordingly. For the calculation of this energy shift, we consider the ground state minority atoms as weakly interacting repulsive polarons with energy $E_{+}$. In the limit of a deeply bound dimer ($\varepsilon_{b}\gg\epsilon_{F}$), the energy is given by Ngampruetikorn et al. (2012)

[TABLE]

Here, $\varepsilon_{b}$ is the dimer binding energy and the 2D scattering length of the ground state is given by Petrov and Shlyapnikov (2001); Bloch et al. (2008)

[TABLE]

where $a_{gg}=199.4a_{0}$ is the ground state s-wave scattering length Kitagawa et al. (2008), $a_{0}$ is the Bohr radius, $l_{y}=\hbar/\sqrt{m\omega_{y}}$ is the oscillator length, and $D\simeq 0.905$. We find $\ln(\kappa_{F}a_{\mathrm{2D},\,gg})=-4.9(1)$ and $E_{+}=0.2\epsilon_{F}$ for the repulsive polaron in the ground state. In Fig. 2 of the main text, we use this energy $\Delta E=E_{+}$ to shift the experimental data in the spectrum. However, when comparing theory and experiment at large magnetic fields where the attractive polaron is only weakly interacting, we recover shifts which could be explained by a larger ground state interaction, i.e. $0.4(2)\epsilon_{F}$ at $\ln\left(\kappa_{F}a_{\mathrm{2D}}\right)=8.2(1)$. This disagreement could originate from systematic errors due to the spectral lineshape fit as well as effects of finite momentum and temperature for the polaron energy.

Our clock laser is stabilized to an optical cavity and the residual, non-linear drift of this laser over the course of recording a spectrum such as that shown in Fig. 2 of the main text is typically $\sim 100\,\mathrm{Hz}$ which is below the relevant energy scales of the experiment. We can probe single-particle Fourier-limited line shapes down to $200\,\mathrm{Hz}$ with the configuration discussed in Section S.I.1. The deep magic-wavelength lattice ensures recoil-free clock spectroscopy. Residual misalignment between clock laser and optical lattice is below $26\,\mathrm{mrad}$ and leads to a negligible transverse recoil energy of less than $h\times 2\,\mathrm{Hz}$.

S.I.6 Rabi oscillations

After the clock excitation pulse, we detect the remaining $\left|{g,0}\right\rangle$ atoms by first removing the $\left|{g,\uparrow}\right\rangle$ atoms with a resonant pulse on the intercombination line ($250\,\text{\textmu}\mathrm{s}$ duration). The detection of remaining $\left|{g,0}\right\rangle$ atoms works more robustly for us than repumping the impurities in $\left|{e,\downarrow}\right\rangle$. We fit the detected atom number $N$ as a function of the pulse duration $t$ to

[TABLE]

Here, $\Gamma_{bg}$ is the background decay rate accounting for losses in the excited state, $\Gamma_{R}$ is the coherence decay rate, $\Omega$ is the Rabi frequency, and $a$, $b$ are two dimensionless parameters. After every experimental cycle, we prepare a non-interacting, spin-polarized sample ($m_{F}=-3/2$), and measure the single-particle Rabi frequency $\Omega_{0}$. For this, we use a fit function without a background decay rate and for the decoherence rate we typically find $\Gamma_{R}\approx 0.3\,\mathrm{kHz}$. The non-zero $\Gamma_{R}$ can be attributed to finite laser linewidth and other experimental imperfections. We find $\Omega\propto\Omega_{0}$ for various single-particle Rabi frequencies $\Omega_{0}$ both on the repulsive and attractive side as shown in Fig. S5. On the repulsive side at small magnetic fields, the large Rabi frequencies are problematic since the clock transitions of other spin states can be addressed as well. The detuning of the $m_{F}=+5/2\rightarrow{m_{F}}^{\prime}=+3/2$ transition is $\simeq 0.55\,\mathrm{kHz/G}$. We verify that addressing of this additional transition is negligible for magnetic fields $\geq$ $20\,\mathrm{G}$ by detecting atoms driven to the $m_{F}=+3/2$ clock state in an independent measurement.

S.I.7 Repulsive polaron lifetime

We measure the repulsive polaron lifetime with two sequential clock pulses. Right after the first pulse, we remove any remaining $\left|{g,0}\right\rangle$ atoms with a sequence of “push” pulses on the intercombination line ($70\,\text{\textmu}\mathrm{s}$ duration). The total number of impurities transferred back to $\left|{g,0}\right\rangle$ is detected after the second pulse and after removing all majority atoms ($\left|{g,\uparrow}\right\rangle$) using another “push” pulse ($100\,\text{\textmu}\mathrm{s}$ duration). We fit the detected atom number $N$ with

[TABLE]

where $\Gamma_{\mathrm{rep}}$ is the decay rate and $a$, $b$ are dimensionless fit parameters. Fig. S6(a) shows such a fit for longer times $t$ compared to the inset in Fig. 4 of the main text. Here, the fast initial decay and the slow decay of the plateau is clearly visible. In Fig. S6(b), we plot the data of Fig. 4 of the main text along with theoretically expected contributions to the repulsive polaron decay rate. We note that the width of the repulsive polaron branch predicted from our polaron theory model in Eq. (S.49) (see below) approximately matches the slope, but its mean amplitude disagrees with the experimental data. We also compare with the 2D three-body recombination rate $K_{3}$ from Ref. Ngampruetikorn et al. (2013), where we assume that the background density $n_{\uparrow}$ is constant such that we obtain the decay rate through $dn_{\downarrow}/dt=n_{\downarrow}[n_{\uparrow}^{2}K_{3}(E)]=n_{\downarrow}\Gamma_{\rm rep}$ for the impurity density $n_{\downarrow}$. To estimate the collision energy $E$, we assume a zero momentum impurity and approximate $E$ as the mean kinetic energy of the background sampled by the minority atoms,

[TABLE]

where the local kinetic energy $E_{\mathrm{kin,\uparrow}}(x,z)$ is given by

[TABLE]

Here, $f_{\uparrow}(x,z,p)=\left\{1+e^{\beta\left[m/2\,\left(\omega_{x}^{2}x^{2}+\omega_{z}^{2}z^{2}\right)+p^{2}/(2m)-\mu\right]}\right\}^{-1}$ is the $\left|{g,\uparrow}\right\rangle{}$ number density in phase space.

S.I.8 Polarons with dual Fermi seas

For probing the effect of an additional Fermi sea in $\left|{g,\downarrow}\right\rangle$, we follow the same state preparation as described in Section S.I.1. The only difference is the optical pumping sequence which we adjust to yield a nearly balanced spin mixture of $m_{F}=\pm 5/2$. Minority atoms are still prepared in $m_{F}=-3/2$. The two Fermi seas interact repulsively which modifies the local density compared to the case without an additional Fermi sea. We use the local density approximation and an expansion of the chemical potential for weak repulsive interactions to find the local Fermi energy $E_{F}(x,z)$. In our system, the Thomas-Fermi equation is given by

[TABLE]

where $n(x,z)$ is the local atom density, $\mu(n)$ is the local chemical potential, and $\mu_{0}$ is fixed by $\int n(x,z)=N$, where $N$ is the total number of atoms. We use the expansion of the chemical potential for weak repulsive interactions in 2D up to second order from Ref. Engelbrecht et al. (1992),

[TABLE]

Here, $E_{F}$ is the local Fermi energy and $g$ is a dimensionless interaction parameter which is related to $\ln(\kappa_{F}a_{\mathrm{2D}})$. In 2D, we use the following form to calculate the finite temperature chemical potential Kittel (2004)

[TABLE]

where $T_{F}=E_{F}/k_{B}$ is the local Fermi temperature. Numerically solving Eqs. (S.19), (S.20), and (S.21) yields the local density $n(x,z)$ and effective Fermi energy $\epsilon_{F}$ in the presence of interactions and finite temperature. For the experimental parameters, we find an effective Fermi energy $\epsilon_{F}=h\times 2.99(19)\,\mathrm{kHz}$ compared to $h\times 3.26(20)\,\mathrm{kHz}$ in the absence of an additional Fermi sea. The effective Fermi energy of the additional Fermi sea is given by $\epsilon_{F}^{\downarrow}=h\times 2.73(19)\,\mathrm{kHz}$. For the temperature of the $\left|{g,\downarrow}\right\rangle$ Fermi sea, we find $k_{B}T/\epsilon_{F}=0.23(4)$.

In the clock spectroscopy measurement shown in Fig. S7, we follow the same methods described in the main text and Section S.I.5 except for the modified state preparation. We can still clearly identify the repulsive and attractive polaron branches but do not find any striking features in the spectrum compared to the configuration without the additional Fermi sea. However, at intermediate magnetic fields we find a small shift of the attractive polaron energy towards smaller energies (see Fig. S7). This shift towards smaller energies is in contrast to the results of our theoretical model (see Section S.II.6). We point out that the energy shift is on the same level as our experimental resolution as well as the correction due to the repulsive interactions of the $\left|{g,\downarrow}\right\rangle{}$ and $\left|{g,\uparrow}\right\rangle$ Fermi sea discussed above.

For the Rabi oscillations, we can only probe the repulsive polaron due to the finite tuning range of our laser used for the “push” pulses on the intercombination line. For the repulsive polaron, the normalized Rabi frequency ${(\Omega/\Omega_{0})}^{2}$ is slightly larger in the presence of an additional Fermi sea which could indicate a larger quasiparticle residue in this case. The decoherence of Rabi oscillations $\Gamma_{R}$ is comparable for both configurations as shown in Fig. S8.

S.II Theoretical description

S.II.1 Two-body problem in a quasi-two-dimensional geometry

In our theoretical modeling, we assume that the 173Yb atoms move in a uniform two-dimensional plane, while in the direction transverse to the plane we assume a harmonic potential $V(y)=\frac{1}{2}m\omega_{y}^{2}y^{2}$. We define the associated oscillator length $l_{y}=1/\sqrt{m\omega_{y}}$ (here and throughout this section we work in units where $\hbar=1$). In this geometry, the center-of-mass and relative motion decouple. Therefore, to describe the two atoms we consider the Hamiltonian of the relative motion, $\hat{\mathcal{H}}=\hat{H_{0}}+\hat{V}$. Here, the non-interacting part is

[TABLE]

where ${\bf k}$ is the in-plane relative momentum, $n$ denotes the transverse harmonic oscillator quantum number for the relative motion, and $2\epsilon_{{\bf k}n}={\bf k}^{2}/m+n\omega_{y}$ is the non-interacting energy (measured from the zero-point energy). The 173Yb system features two electronic orbitals ($\left|{g}\right\rangle$ and $\left|{e}\right\rangle$) as well as two nuclear spin states ($\left|{\downarrow}\right\rangle$ and $\left|{\uparrow}\right\rangle$). In principle, there are four different configurations for two atoms, but for the situation of interest we only have two: The open channel $\left|{o}\right\rangle\equiv\left|{g\uparrow,e\downarrow}\right\rangle$ and the closed channel $\left|{c}\right\rangle\equiv\left|{e\uparrow,g\downarrow}\right\rangle$. Here, $\delta$ is the detuning of the closed channel,

[TABLE]

with $\Delta\mu$ given by Eq. (S.12) and $B$ the magnetic field. The interaction does not preserve the orbital configuration, but rather proceeds via the triplet ($+$) and singlet ($-$) channels, with $\left|{\pm}\right\rangle\equiv\frac{1}{\sqrt{2}}\left(\left|{o}\right\rangle\pm\left|{c}\right\rangle\right)$. Thus, the interaction part of the Hamiltonian takes the form

[TABLE]

where we assume contact interactions of strength $U_{\pm}$. Here,

[TABLE]

is the wave function of the transverse relative motion at zero separation.

The renormalization of the contact interaction in a quasi-two-dimensional geometry was carried out in Refs. Petrov and Shlyapnikov (2001); Bloch et al. (2008). Here, we mainly follow the notation outlined in the review Levinsen and Parish (2015), properly generalized to account for the orbital structure of the interactions. We start by writing down the Lippmann-Schwinger equation for the two-body $T$ matrix

[TABLE]

where $E$ is the energy, and the infinitesimal positive imaginary part $+i0$ ensures that we consider the outgoing scattered part of the wave function. Note that we suppress the indices related to the orbital configuration since we will consider this in a matrix representation. Since the interaction part of the Hamiltonian, Eq. (S.24), is independent of momentum, Eq. (S.28) factorizes as a function of momentum. Thus, in the triplet-singlet basis we have

[TABLE]

On the other hand, the polarization bubble is most straightforward to evaluate in the open-closed channel basis where it takes the form

[TABLE]

With these definitions, it is straightforward to formally obtain the vacuum scattering $T$ matrix, which we write in the open-closed basis

[TABLE]

for arbitrary relative momenta ${\bf k}_{1,2}$. Here,

[TABLE]

is the involutory matrix that transforms between the triplet-singlet basis and the open-closed channel basis.

Following the steps in Ref. Levinsen and Parish (2015) (see also Refs. Petrov and Shlyapnikov (2001); Bloch et al. (2008)) to evaluate the polarization bubble, we obtain

[TABLE]

where we specialize to scattering between the lowest transverse levels, and consider excited states only in the intermediate virtual processes. This is a valid assumption since in our experiment the transverse confinement frequency greatly exceeds all other relevant energy scales. The function $\mathcal{F}$ takes the form Bloch et al. (2008)

[TABLE]

The quantities $a_{+}$ and $a_{-}$ are the triplet and singlet scattering lengths, respectively. While the contact interaction model, Eq. (S.24), does not contain effective range corrections to the 3D scattering phase shifts, we could in principle have used a two-channel model for each of the singlet and triplet interactions to include this. This procedure is straightforward, but cumbersome to write down. The end result is, however, simple. Indeed, we just have to make the replacement

[TABLE]

with $r_{\pm}$ the singlet and triplet effective ranges. In practice, all theory curves are calculated by making this replacement and using the scattering lengths and effective ranges from Ref. Höfer et al. (2015).

Finally, we extract the two-body binding energy and the effective 2D scattering length from the $T$ matrix. The two-body binding energy $\varepsilon_{b}>0$ is found from the pole of the $T$ matrix, i.e., from the condition $\det[\mathbf{T}^{(\rm{vac})}(0,0;\varepsilon_{b})]^{-1}=0$. Thus, $\varepsilon_{b}$ satisfies

[TABLE]

To extract an effective low-energy open-channel 2D scattering length, we compare the open-channel element of Eq. (S.33) with the expression for the low-energy scattering:

[TABLE]

where the subscript on $\mathbf{T}$ indicate the corresponding matrix element. Assuming that $\delta\gg|E|$ (i.e., that we are not close to $B=0$) we find

[TABLE]

Again, we can replace $a^{-1}_{\pm}\to a^{-1}_{\pm}-\frac{1}{4}r_{\pm}m(\omega_{y}-\delta)$ to include the effective ranges in the triplet-singlet channels. Fig. S9 shows the functional dependence of $a_{\mathrm{2D}}$ for our typical experimental parameters. Here, we also see that the universal formula in the 2D limit, $\varepsilon_{b}=1/(ma_{\mathrm{2D}}^{2})$, breaks down at small $\delta$ (low magnetic field) where the dimer binding energy becomes comparable to the transverse confinement.

S.II.2 Impurity Green’s function

To investigate the many-body dressing of an impurity, we evaluate its Green’s function which, in general, also takes the form of a matrix. However, we will work under the assumption that we create at most a single excitation of the fermionic medium. Such an approximation has proven extremely accurate since it was first introduced in Ref. Chevy (2006); Combescot et al. (2007). Within this approximation, the open and closed channel parts of the impurity Green’s function completely decouple, and since our clock laser pulse injects the impurity into the open channel, we restrict ourselves to considering this channel. Furthermore, since the transverse confinement frequency greatly exceeds both the Fermi energies and temperature, we assume that the impurity is only virtually scattered into excited bands of the transverse confinement such that we can consider an effective 2D Green’s function. We note that higher bands of the transverse confinement can in principle be taken into account using the formalism developed in Levinsen and Baur (2012).

Within our approximations, the open-channel retarded impurity Green’s function satisfies the Dyson equation

[TABLE]

where we take the impurity to be in the lowest harmonic oscillator state, and we use the 2D single-particle energy $\epsilon_{{\bf k}}\equiv\epsilon_{{\bf k}0}$. The open-channel self energy is

[TABLE]

with $n_{o}(q)=\left\{1+e^{\beta\left[q^{2}/(2m)-\mu\right]}\right\}^{-1}$ the finite-temperature Fermi-Dirac distribution of the background Fermi sea in $\left|{g,\uparrow}\right\rangle$ and $\mathbf{T}$ the in-medium $T$ matrix. The latter is related to the vacuum $T$ matrix, Eq. (S.33), via

[TABLE]

Note that we have dropped the transverse harmonic oscillator indices on the vacuum $T$ matrix since these have been set to 0. The functions $\Delta\Pi_{o,c}$ are the differences between the medium and vacuum $T$ matrices:

[TABLE]

where $n_{c}(q)$ is the Fermi-Dirac distribution of the additional Fermi sea in state $\left|{g,\downarrow}\right\rangle$ which must be taken into account for the configuration described in Sec. S.I.8. We extract all quasiparticle properties from the impurity Green’s function.

S.II.3 Polaron spectral response

The spectral response measured in experiment corresponds to the impurity spectral function $A(E)$, which we calculate from the Green’s function in Eq. (S.39) as

[TABLE]

where we neglect the initial distribution of impurity momenta. We extract the attractive and repulsive polaron energies $E_{-}$ and $E_{+}$, respectively, from the peak values of the spectral function. The results are shown in Figs. 1(a) and 2(a) of the main text.

Fig. S10 shows sample spectra recorded at fixed magnetic fields of (a) 45G and (b) 210G compared with the results of our theory. We see that while the width of the repulsive polaron is approximately captured within our model, the attractive polaron peak is somewhat broader in experiment than in the theoretical model.

S.II.4 Quasiparticle residue

From the Rabi oscillations observed in experiment, we extract the quasiparticle residue, $Z$, defined as the squared overlap between the non-interacting state of the impurity plus medium with that of the strongly interacting system. Similar to the scenario in alkali-atom experiments Kohstall et al. (2012); Scazza et al. (2017), this overlap in reality compares states involving the impurity atom in two different spin states. However, the Rabi coupling provides the spin-flip operator that is necessary to simply relate the overlap to that of interacting and non-interacting states, without making explicit reference to the spin state. In contrast to the case of alkali atoms, the interacting state of the impurity atoms involves two orbitals, but since the overlap with the non-interacting state only involves the open channel it can be extracted from the open-channel Green’s function.

Specifically, the Rabi frequency $\Omega$ is reduced by a factor $\sqrt{Z}$ Kohstall et al. (2012) from its bare value $\Omega_{0}$ due to the dressing of the impurity by excitations of the medium. We calculate the residue using the standard expression

[TABLE]

The results are shown in Fig. 3(a) of the main text, where we see that the measured repulsive (attractive) polaron residues are systematically above (below) the theoretical prediction.

The main source of discrepancy between theory and experiment for the residue shown in Fig. 3(a) is likely to be the repulsive interactions between the initial $\left|{g,0}\right\rangle$ state and the Fermi sea in state $\left|{g,\uparrow}\right\rangle$. Such repulsive interactions will respectively enhance and reduce the overlap of the initial state with the repulsive and attractive polarons in the final state. We can estimate the maximum amount this overlap can change as follows. Assume that a polaron $\left|{\Psi_{p}}\right\rangle$ with residue $Z_{p}$ has a wave function consisting of only two terms, the state that overlaps with the non-interacting ground state $\left|{0}\right\rangle$ and an incoherent background $\left|{\rm inc}\right\rangle$:

[TABLE]

where we assume that both $\left|{0}\right\rangle$ and $\left|{\rm inc}\right\rangle$ are normalized and $e^{i\varphi_{p}}$ is a phase. Within this approximation, we would thus arrive at the maximum (minimum) overlap between the repulsive (attractive) polaron states $\left|{\Psi_{+}}\right\rangle$ ($\left|{\Psi_{-}}\right\rangle$) and the initial (repulsive polaron) state $\left|{\Psi_{i}}\right\rangle$:

[TABLE]

with the initial-state residue $Z_{i}\simeq 0.98$. We show these estimates in Fig. S11, and we see that this can improve the agreement between theory and experiment substantially. A similar effect was found in Ref. Scazza et al. (2017) for 6Li atoms.

S.II.5 Damping of Rabi oscillations

Another observable is the width $\Gamma_{\pm}$ of each polaron peak in the spectral function, which can be computed from the self energy for the attractive and repulsive branches:

[TABLE]

The result $\Gamma_{+}$ for the repulsive polaron is plotted in Fig. 3(b) of the main text and is seen to match the damping rate of the Rabi oscillations observed in experiment. However, for the attractive polaron, the damping rate is much larger than that predicted from Eq. (S.49). This enhanced damping could be because the Rabi frequency in experiment is comparable to or larger than the Fermi energy, which has been theoretically shown to affect the Rabi oscillations of the attractive polaron Parish and Levinsen (2016).

To understand the damping rate of the attractive polaron, we consider a simple three-state model of the Rabi oscillations that involves the non-interacting initial state $\left|{\Psi_{i}}\right\rangle$, the final attractive polaron state $\left|{\Psi_{-}}\right\rangle$, and a state $\left|{\Psi_{\rm con}}\right\rangle$ that represents the continuum of states in the final spectrum that are orthogonal to $\left|{\Psi_{-}}\right\rangle$. Taking the clock laser to be resonant with the attractive polaron, the Hamiltonian for the three-state system is, in matrix form,

[TABLE]

Here, we have assumed that the continuum is dominated by its spectral width $\Gamma_{\rm con}\sim\epsilon_{F}$ since it does not correspond to a well-defined peak in the spectrum. We have also required that the spectrum obeys the sum rule $|\langle\Psi_{i}|\Psi_{-}\rangle|^{2}+|\langle\Psi_{i}|\Psi_{\rm con}\rangle|^{2}=1$, so that $|\langle\Psi_{i}|\Psi_{\rm con}\rangle|=\sqrt{1-Z}$ with $Z=|\langle\Psi_{i}|\Psi_{-}\rangle|^{2}$.

When the final state is non-interacting, then $Z=1$ and we obtain the standard result for Rabi oscillations of the initial-state fraction $N_{i}$ as a function of time $t$,

[TABLE]

However, the continuum can significantly affect the oscillations when $Z<1$ and $\Omega_{0}$ is comparable or larger than $\epsilon_{F}$, as is the case in experiment. In particular, if we assume that $\Omega_{0}\gg\Gamma_{\rm con}$ and $Z$ is close to 1, a perturbative calculation yields

[TABLE]

where the Rabi damping rate $\Gamma_{R}\simeq(1-Z)\Gamma_{\rm con}$. This is consistent with the results shown in Fig. 3(b) of the main text. While we do not have a precise value for the width of the continuum $\Gamma_{\rm con}$, we expect it to be comparable to the Fermi energy $\epsilon_{F}$ of the medium.

S.II.6 Dual Fermi sea configuration

Finally, we comment on the dual Fermi sea configuration discussed experimentally in Sec. S.I.8. The motivation for this investigation is that this scenario corresponds to an effective frustration of interactions close to the orbital Feshbach resonance. Indeed, while the impurity in state $\left|{g,0}\right\rangle$ does not directly (strongly) interact with the Fermi sea in state $\left|{g,\downarrow}\right\rangle$, the presence of a Fermi sea in the weakly-detuned closed channel provides an effective Pauli blocking of the open-channel interactions. This is fundamentally different from the usual Feshbach resonances Chin et al. (2010), where the closed channel is detuned by an energy that far exceeds all scales relevant to the physics of interest. Thus, our experiment provides the first steps towards realizing a “frustrated” Feshbach resonance.

Fig. S12 shows our predicted energy shifts in the “frustrated” configuration. We see that we expect the energy of both polaron branches to increase. For the experimental Fermi energy in the closed channel, the expected energy shifts (solid lines in Fig. S12) are below $0.15\epsilon_{F}$ within the range of magnetic fields where we have sufficient contrast, which may be the reason why these shifts are not observed in experiment (see Fig. S7). However, as also illustrated in Fig. S12, we expect the energy shift to strongly increase with the $\left|{g,\downarrow}\right\rangle$ Fermi energy. In particular, the increased energy of the repulsive branch could potentially stabilize a ferromagnetic phase Massignan et al. (2014).