Bose polarons in ultracold atoms in one dimension: beyond the Fröhlich paradigm

To arrive at a polaron description of the impurity problem described above, we express the boson field operator $\hat{\phi }(x)$ in terms of Bogoliubov phonons ${\hat{a}}_{k}$ [5, 10, 11, 46]. To this end we write the Fourier components ${\hat{\phi }}_{k}={(2\pi )}^{-1/2}\int {\rm{d}}{x}\,{{\rm{e}}}^{{\rm{i}}{k}{x}}\hat{\phi }(x)$ as

$$\begin{eqnarray}&&{\hat{\phi }}_{k}=\cosh {\theta }_{k}{\hat{a}}_{k}-\sinh {\theta }_{k}{\hat{a}}_{-k}^{\dagger },\end{eqnarray} \tag{ 2 }$$

where ${\theta }_{k}$ is chosen as in the usual Bogoliubov theory for a weakly interacting Bose gas [47],

$$\begin{eqnarray}&&\begin{array}{c}\cosh {\theta }_{k}\\ \sinh {\theta }_{k}\end{array}=\displaystyle \frac{1}{\sqrt{2}}{\left(\displaystyle \frac{{k}^{2}/2{m}_{{\rm{B}}}+{g}_{\mathrm{BB}}{n}_{0}}{{\omega }_{k}}\pm 1\right)}^{1/2}.\end{eqnarray} \tag{ 3 }$$

Here the Bogoliubov dispersion is given by

$$\begin{eqnarray}&&{\omega }_{k}={ck}{\left(1+\displaystyle \frac{1}{2}{k}^{2}{\xi }^{2}\right)}^{-1/2},\end{eqnarray} \tag{ 4 }$$

where $c=\sqrt{{g}_{\mathrm{BB}}{n}_{0}/{m}_{{\rm{B}}}}$ and $\xi =1/\sqrt{2{m}_{{\rm{B}}}{g}_{\mathrm{BB}}{n}_{0}}$ are the speed of sound and the healing length in the limit of weak interactions.

So far we have only applied a basis transformation, allowing us to express the Bose field $\hat{\phi }(x)$ in terms of Bogoliubov phonons ${\hat{a}}_{k}$. Thereby the Hamiltonian in equation (1) can be written as

$$\begin{eqnarray}&&\hat{{ \mathcal H }}={\hat{{ \mathcal H }}}_{{\rm{F}}}+{\hat{{ \mathcal H }}}_{2\mathrm{ph}}+{\hat{{ \mathcal H }}}_{\mathrm{ph}-\mathrm{ph}}\end{eqnarray} \tag{ 5 }$$

without any approximation. The first term corresponds to the effective Fröhlich Hamiltonian,

$$\begin{eqnarray}&&{\hat{{ \mathcal H }}}_{{\rm{F}}}=\int {\rm{d}}{k}\,{\omega }_{k}{\hat{a}}_{k}^{\dagger }{\hat{a}}_{k}+\displaystyle \frac{{\hat{p}}^{2}}{2M}+{g}_{\mathrm{IB}}{n}_{0}+\int {\rm{d}}{k}\,{V}_{k}{{\rm{e}}}^{{\rm{i}}{k}\hat{x}}({\hat{a}}_{k}^{\dagger }+{\hat{a}}_{-k}).\end{eqnarray} \tag{ 6 }$$

Here $\hat{p}$ and $\hat{x}$ denote the momentum and position operators of the impurity in first quantization. The scattering amplitude is given by [5, 11]

$$\begin{eqnarray}&&{V}_{k}=\sqrt{{n}_{0}}{(2\pi )}^{-1/2}{g}_{\mathrm{IB}}{W}_{k},\quad {W}_{k}={\left(\displaystyle \frac{{(\xi k)}^{2}}{2+{(\xi k)}^{2}}\right)}^{1/4}.\end{eqnarray} \tag{ 7 }$$

The second term, ${\hat{{ \mathcal H }}}_{2\mathrm{ph}}$, describes two-phonon scattering processes [23, 24] and reads

$$\begin{eqnarray}&&{\hat{{ \mathcal H }}}_{2\mathrm{ph}}=\displaystyle \frac{{g}_{\mathrm{IB}}}{2\pi }\int {\rm{d}}{k}{\rm{d}}{k}^{\prime} (\cosh {\theta }_{k}{\hat{a}}_{k}^{\dagger }-\sinh {\theta }_{k}{\hat{a}}_{-k})\times (\cosh {\theta }_{k^{\prime} }{\hat{a}}_{k^{\prime} }-\sinh {\theta }_{k^{\prime} }{\hat{a}}_{-k^{\prime} }^{\dagger })\,{{\rm{e}}}^{{\rm{i}}(k-k^{\prime} )\hat{x}}.\end{eqnarray} \tag{ 8 }$$

The two terms above, ${\hat{{ \mathcal H }}}_{{\rm{F}}}$ and ${\hat{{ \mathcal H }}}_{2\mathrm{ph}}$, provide an accurate model for the polaron problem when the Bose gas can be treated within Bogoliubov theory. This MF description of the interacting bosons assumes a macroscopic occupation of the condensate which is absent in one dimension [30, 31]. As pointed out by Lieb and Liniger [48, 49], some quantities including the total energy can nevertheless be calculated accurately using Bogoliubov theory in the regime of weak interactions. This is the case when the dimensionless coupling strength $\gamma \lesssim 2$ is sufficiently small [48], where

$$\begin{eqnarray}&&\gamma =\displaystyle \frac{2}{{n}_{0}| {a}_{\mathrm{BB}}| }=\displaystyle \frac{{m}_{{\rm{B}}}{g}_{\mathrm{BB}}}{{n}_{0}}\end{eqnarray} \tag{ 9 }$$

and the 1D boson–boson s-wave scattering length is given by the relation

$$\begin{eqnarray}&&{a}_{\mathrm{BB}}=-2/({m}_{{\rm{B}}}{g}_{\mathrm{BB}}).\end{eqnarray} \tag{ 10 }$$

In contrast, for strong interactions ${g}_{\mathrm{BB}}$ or small densities n₀, when $\gamma \gtrsim 2$ is large, the Bogoliubov description of the Bose gas breaks down. In this case additional interactions between the Bogoliubov phonons ${\hat{a}}_{k}$ have to be included, which we summarize in ${\hat{{ \mathcal H }}}_{\mathrm{ph}-\mathrm{ph}}$. These terms are of order

$$\begin{eqnarray}&&{\hat{{ \mathcal H }}}_{\mathrm{ph}-\mathrm{ph}}={ \mathcal O }({\hat{a}}_{k}^{3})+{ \mathcal O }({\hat{a}}_{k}^{4}),\end{eqnarray} \tag{ 11 }$$

but do not involve impurity operators. Since we do not need the expression for ${\hat{{ \mathcal H }}}_{\mathrm{ph}-\mathrm{ph}}$ in terms of Bogoliubov phonons in the following, we will not write them out explicitly.

The terms in Hamiltonian (5) give rise to different physical regimes of the Bose polaron. For small ${g}_{\mathrm{IB}}$, the ground state corresponds to a free impurity and phonon contributions are negligible. In this regime the polaron energy is determined by ${g}_{\mathrm{IB}}{n}_{0}$, sometimes referred to as the MF shift. For stronger couplings states with one phonon need to be included, which has been done systematically by using perturbation theory in [25, 26]. This approach is valid when $\gamma \ll 1$ and

$$\begin{eqnarray}&&\eta \ll 2{\gamma }^{3/4}\approx 3{({n}_{0}| {a}_{\mathrm{BB}}| )}^{-3/4}.\end{eqnarray} \tag{ 12 }$$

Beyond this coupling strength multi-phonon states need to be added in the ground state wavefunction to describe correctly the polaron energy its effective mass [16, 44]. To go beyond the Fröhlich model, two-phonon terms need to be included in the Hamiltonian [23, 24], see figure 2. When the polaron cloud contains many phonons and boson–boson interactions become large, their interactions need to be taken into account as well. On a MF level this can be done by using the Gross–Pitaevskii equation, see section 7.2 for a discussion.

Quasi-1D Bose gases have been realized in several experiments (see [50] for a review). Mobile impurities can be realized by using a second atomic level, as in the recent experimental observation of Bose polarons in a 3D Bose–Einstein condensate [9]. Alternatively a second atomic species can be added, as has been demonstrated in [7, 8, 51, 52].

In this paper we analyze the experimental results from [7]. We use the dimensionless parameter

$$\begin{eqnarray}&&\eta ={g}_{\mathrm{IB}}/{g}_{\mathrm{BB}}\end{eqnarray} \tag{ 13 }$$

introduced therein to quantify the interaction strength between impurity and boson with respect to the fixed strength of interactions in the bath. The mass ratio in the experiment was $M/{m}_{{\rm{B}}}=41/87=0.47$ and for n₀ we use the peak density of the Bose gas, which was estimated to be ${n}_{0}=7/\mu {\rm{m}}$ in [7]. The coupling constant in the bath is taken equal to ${g}_{\mathrm{BB}}=2.36\times {10}^{-37}\mathrm{Jm}$ as in [7].

Using Bogoliubov theory this yields the following estimates for the healing length, $\xi =0.15\,\mu {\rm{m}}$ and the speed of sound, $c=3.38\,\mathrm{mm}\,{{\rm{s}}}^{-1}$. Using ξ as a unit of length yields ${n}_{0}\xi =1.05$. A characteristic energy can be defined by $c/\xi =3.6\times 2\pi \,\mathrm{kHz}$. Another important length scale is the 1D boson–boson scattering length ${a}_{\mathrm{BB}}=-3.6\,\mu {\rm{m}}$, see equation (10). The speed of sound calculated from the Bethe ansatz [48, 49] at $\gamma =2/({n}_{0}{a}_{\mathrm{BB}})=0.438$, see equation (9), equals to $c=3.20\,\mathrm{mm}\,{{\rm{s}}}^{-1}$ so that the contribution of quantum fluctuations is about 5%. In order to avoid a possible issue which value (Bogoliubov theory or Bethe ansatz) is used, we take the healing length ξ as a unit of length for comparison with the MF theory and ${a}_{\mathrm{BB}}$ for the comparison with Monte Carlo results.

The temperature of the Bose gas $T=350(50)\mathrm{nK}$ measured in the experiment corresponds to ${k}_{{\rm{B}}}T\approx 2c/\xi$. The transverse confinement frequency of the bosons (the impurity) stated in [7] corresponds to ${\omega }_{\perp }\approx 9.4(12.5)c/\xi$ and justifies a description as a 1D system.

The effective Fröhlich Hamiltonian derived from Bogoliubov theory, see equations (4), (6) and (7), does not require any UV regularization in one dimension and we can simply set the UV cut-off ${{\rm{\Lambda }}}_{0}=\infty$. At momenta $k\gtrsim 1/\xi$ the Bogoliubov dispersion becomes quadratic, ${\omega }_{k}\simeq {k}^{2}/2{m}_{{\rm{B}}}$, making all quantities (the effective mass, the polaron energy, etc) well behaved.

On the other hand, bosonization can be used to derive an effective Fröhlich Hamiltonian [7] for the impurity. It describes the interaction of the impurity with the Luttinger liquid formed by the 1D Bose gas [50, 53]. At long wavelengths, or low energies, the resulting Fröhlich Hamiltonian ${\hat{{ \mathcal H }}}_{{\rm{F}}}^{{\prime} }$ has the same asymptotic form as ${\hat{{ \mathcal H }}}_{{\rm{F}}}$ in equation (6) which was derived from Bogoliubov theory. This provides additional evidence that the use of the Bogoliubov theory for treating the 1D Bose gas is justified.

The Fröhlich Hamiltonian ${\hat{{ \mathcal H }}}_{{\rm{F}}}^{{\prime} }$ is obtained from equation (6) by replacing ${\omega }_{k}$ and V_k with different expressions ${\omega }_{k}^{{\prime} }$ and ${V}_{k}^{{\prime} }$, respectively. The bosonization approach relies on a linear phonon dispersion,

$$\begin{eqnarray}&&{\omega }_{k}^{{\prime} }={ck},\end{eqnarray} \tag{ 14 }$$

and determines the scattering amplitude ${V}_{k}^{{\prime} }$ at small k (long wavelengths). Its ultraviolet (UV) cut-off, required for regularization of the model, is commonly represented by an exponential decay at a characteristic scale k_c [7] which is usually taken to be ${k}_{c}\sim 1/\xi$,

$$\begin{eqnarray}&&{V}_{k}^{{\prime} }=\displaystyle \frac{{g}_{\mathrm{IB}}}{2\pi }\sqrt{K| k| }{{\rm{e}}}^{-k/2{k}_{c}}.\end{eqnarray} \tag{ 15 }$$

The dimensionless Luttinger parameter K can be determined from Bethe-ansatz calculations. It is given by the ratio of the Fermi velocity ${v}_{{\rm{F}}}=\pi {n}_{0}/{m}_{{\rm{B}}}$ and the speed of sound c,

$$\begin{eqnarray}&&K=\displaystyle \frac{{v}_{{\rm{F}}}}{c}=\displaystyle \frac{\pi {n}_{0}}{{{cm}}_{{\rm{B}}}},\end{eqnarray} \tag{ 16 }$$

By using equation (16) one confirms that the asymptotic behaviors of V_k (${\omega }_{k}$) and ${V}_{k}^{{\prime} }$ (${\omega }_{k}^{{\prime} }$) in the infrared (IR) limit $k\ll 1/\xi$ are identical. For weakly interacting bosons, where the gas parameter $\gamma \ll 1$, see equation (9), is small, Bogoliubov theory provides an accurate value for the speed of sound as $c=\sqrt{{g}_{\mathrm{BB}}{n}_{0}/{m}_{{\rm{B}}}}$. For a strongly interacting bath (Tonks–Girardeau regime), the speed of sound equals to the Fermi velocity, $c={v}_{{\rm{F}}}$. For intermediate interactions, Bethe ansatz can be used to obtain the speed of sound c and Luttinger parameter K.

The main difference between the bosonization approach and Bogoliubov theory is in the treatment of phonons at high energies. In many transport and dynamic phenomena in 1D systems the main contributions come from the longest wavelength excitations [53] and the results are insensitive to the UV cut-off. In this spirit, [7] focused on low-energy phonons and introduced the exponential cut-off at k_c in equation (15) by hand. In the Bogoliubov theory, on the other hand, the phonon dispersion ${\omega }_{k}$ becomes nonlinear for $k\gtrsim 1/\xi$, providing a natural UV cut-off scale in the model.

To obtain reasonable results from the effective Fröhlich Hamiltonian ${\hat{{ \mathcal H }}}_{{\rm{F}}}^{{\prime} }$, the momentum cut-off k_c has to be of the order $1/\xi$. Even for this choice the properties of UV phonons at momenta around $k\sim 1/\xi$ differ in the two models ${\hat{{ \mathcal H }}}_{{\rm{F}}}$ and ${\hat{{ \mathcal H }}}_{{\rm{F}}}^{{\prime} }$. As has been shown by a dimensional analysis in [16, 40], the high-energy modes with $k\sim 1/\xi$ play a crucial role in determining properties of the Bose polaron. Therefore we expect that there can be sizable quantitative differences between predictions by the two polaron Hamiltonians.

In figure 3 we calculate the polaron mass ${M}_{{\rm{p}}}$ starting from the Fröhlich model ${\hat{{ \mathcal H }}}_{{\rm{F}}}$ and using Feynman's variational path integral formalism [5, 14, 54] (purple dashed-dotted line in the figure). The same calculation was performed in [7], but using the model ${\hat{{ \mathcal H }}}_{{\rm{F}}}^{{\prime} }$ with an exponential UV cut-off (purple solid line in the figure). Surprisingly, the Bogoliubov theory predicts a polaron mass which is about a factor of two larger than expected from assuming an exponential UV cut-off. This also explains the large deviations between the experimental data and theory based on ${\hat{{ \mathcal H }}}_{{\rm{F}}}^{{\prime} }$.

The Fröhlich model is only valid when two-phonon processes can be ignored. This is the case when the depletion of the quasi-1D condensate is small, justifying also the assumption that phonon–phonon interactions cannot modify the polaron cloud substantially in this regime.

To derive an estimate when the Fröhlich model is accurate, we apply standard MF theory [10, 13, 55] (see also section 5.1) to the combined Hamiltonian ${\hat{{ \mathcal H }}}_{{\rm{F}}}+{\hat{{ \mathcal H }}}_{2\mathrm{ph}}$. In appendix A we derive for the phonon number in the polaron cloud,

$$\begin{eqnarray}&&{N}_{\mathrm{ph}}^{\mathrm{MF}}={\beta }_{\mathrm{MF}}^{2}\int {\rm{d}}{k}\,{\left(\displaystyle \frac{{V}_{k}}{{\omega }_{k}+{k}^{2}/2M}\right)}^{2}.\end{eqnarray} \tag{ 17 }$$

Here we assumed that the total momentum carried by the polaron vanishes, p = 0. The result in equation (17) is similar to the expression from the Fröhlich model, which is obtained by setting ${\beta }_{\mathrm{MF}}=1$. When two-phonon terms are included, we obtain

$$\begin{eqnarray}&&{\beta }_{\mathrm{MF}}={\left[1+\displaystyle \frac{{g}_{\mathrm{IB}}}{2\pi }\int {\rm{d}}{k}\displaystyle \frac{{W}_{k}^{2}}{{\omega }_{k}+{k}^{2}/2M}\right]}^{-1}.\end{eqnarray} \tag{ 18 }$$

Therefore the depletion of the quasi-1D condensate can be described accurately by the Fröhlich Hamiltonian provided that ${\beta }_{\mathrm{MF}}\ll 1$.

This yields the condition

$$\begin{eqnarray}&&| {g}_{\mathrm{IB}}| \ll {g}_{\mathrm{IB}}^{c}\,=\,2\pi {\left[\int {\rm{d}}{k}\displaystyle \frac{{W}_{k}^{2}}{{\omega }_{k}+{k}^{2}/2M}\right]}^{-1}\end{eqnarray} \tag{ 19 }$$

for the validity of ${\hat{{ \mathcal H }}}_{{\rm{F}}}$. The integral can be estimated as

$$\begin{eqnarray}&&{\eta }_{c}={g}_{\mathrm{IB}}^{c}/{g}_{\mathrm{BB}}\approx \sqrt{2}\pi c/{g}_{\mathrm{BB}}=\pi \sqrt{{n}_{0}| {a}_{\mathrm{BB}}| }.\end{eqnarray} \tag{ 20 }$$

As indicated in figure 2, the border of the weak coupling region scales as ${\eta }_{c}\sim \sqrt{{n}_{0}| {a}_{\mathrm{BB}}| }$.

For the experimental parameters of [7], the critical coupling strength where the Fröhlich model breaks down is given by ${\eta }_{c}\approx 6$. Indeed, from figure 3 we see that theoretical calculations based on the Fröhlich Hamiltonian only describe the experimental data for $\eta \lesssim 3$, and the deviations become large around ${\eta }_{c}$.

Condition (20) provides an estimate when the Fröhlich Hamiltonian is valid. Note that this is different from condition (12) which describes in which regime the lowest-order perturbation theory in boson–boson and impurity–boson interactions is valid. Comparison of the two expressions shows that there exists a parameter regime at large ${n}_{0}| {a}_{\mathrm{BB}}|$ and $\eta \lt {\eta }_{c}$ where the Fröhlich model is valid but has to be solved non-perturbatively.

In [16–18] the validity of Feynman's variational path integral description of Fröhlich polarons has been questioned. When the impurity mass is small and interactions are moderate, a new regime has been identified where the correlations between phonons in the polaron cloud become important. Because Feynman's method merely interpolates between the two extremes of weak and strong coupling [56], it cannot capture the physics accurately in this situation.

In the Florence experiment [7] the impurity mass is a factor of two smaller than the boson mass; it is then natural to ask how accurate Feynman's approach works in this case. To answer this question we compare predictions by the RG introduced in [16] to Feynman's variational path integral method, see figure 4. Note that we only consider the Fröhlich Hamiltonians ${\hat{{ \mathcal H }}}_{{\rm{F}}}$ and ${\hat{{ \mathcal H }}}_{{\rm{F}}}^{{\prime} }$ in this figure. For the sake of comparison we also present results at strong couplings, beyond the critical value ${\eta }_{c}$ where the Fröhlich model is not sufficient anymore.

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To extract the effective polaron mass ${M}_{{\rm{p}}}$, Feynman suggested a simple estimate where one of his variational parameters serves as a direct approximation for the mass renormalization $\delta M={M}_{{\rm{p}}}-M$. In addition he derived a more accurate expression for the polaronic mass enhancement from the imaginary-time Green's function, see [5, 11, 14, 54] for details. As can be seen from figure 4, the estimated polaron mass has a large relative error, in particular in the regime of weak interactions.

The accurate expression for the polaron mass determined from the imaginary-time Green's function agrees remarkably well with our RG calculation. Only for intermediate couplings, small differences between the two methods can be observed. For weak and strong couplings both predictions coincide exactly. We conclude that Feynman's ansatz provides an accurate description of Fröhlich polarons in the experiment by Catani et al [7] when the Fröhlich Hamiltonian ${\hat{{ \mathcal H }}}_{{\rm{F}}}$ is used.

We start by highlighting some of the polaron properties specific to 1D systems, using MF formalism discussed in [24]. The formalism for 1D Bose polarons is presented in appendix A. In short, firstly one utilizes conservation of the total momentum p by applying the unitary transformation ${\hat{U}}_{\mathrm{LLP}}$ introduced by Lee et al [55]. This transforms the original Hamiltonian ${\hat{{ \mathcal H }}}_{\mathrm{BF}}$ into the new one, ${\tilde{{ \mathcal H }}}_{\mathrm{BF}}={\hat{U}}_{\mathrm{LLP}}{\hat{{ \mathcal H }}}_{\mathrm{BF}}{\hat{U}}_{\mathrm{LLP}}^{\dagger }$. Next, a product wave function of coherent phonon states is assumed to describe the ground state of ${\tilde{{ \mathcal H }}}_{\mathrm{BF}}$. This means that the polaron state can be written as

$$\begin{eqnarray}&&| {\psi }_{\mathrm{MF}}\rangle ={\hat{U}}_{\mathrm{LLP}}^{\dagger }\displaystyle \prod _{k}| {\alpha }_{k}^{\mathrm{MF}}\rangle .\end{eqnarray} \tag{ 22 }$$

To find the minimum variational energy ${E}^{\mathrm{MF}}[{\alpha }_{k}]=\langle {\psi }_{\mathrm{MF}}| {\hat{{ \mathcal H }}}_{\mathrm{BF}}| {\psi }_{\mathrm{MF}}\rangle$ one solves the saddle point equations $\delta {E}_{\mathrm{MF}}[{\alpha }_{k}]/\delta {\alpha }_{k}=0$, which yields

$$\begin{eqnarray}&&{\alpha }_{k}^{\mathrm{MF}}=-\displaystyle \frac{{\beta }_{\mathrm{MF}}{V}_{k}}{{\omega }_{k}+{k}^{2}/2M-k(p-{P}_{\mathrm{ph}}^{\mathrm{MF}})/M}.\end{eqnarray} \tag{ 23 }$$

The total phonon momentum ${P}_{\mathrm{ph}}^{\mathrm{MF}}=\int {\rm{d}}{k}\,k| {\alpha }_{k}^{\mathrm{MF}}{| }^{2}$ and ${\beta }_{\mathrm{MF}}$ have to be determined self-consistently. The MF variational polaron energy is given by

$$\begin{eqnarray}&&{E}_{0}^{\mathrm{MF}}={n}_{0}{g}_{\mathrm{IB}}{\beta }_{\mathrm{MF}}+\displaystyle \frac{{g}_{\mathrm{IB}}}{2\pi }\int {\rm{d}}{k}\,{\sinh }^{2}{\theta }_{k}+\displaystyle \frac{{p}^{2}-{({P}_{\mathrm{ph}}^{\mathrm{MF}})}^{2}}{2M}.\end{eqnarray} \tag{ 24 }$$

Note that MF calculations go beyond a straightforward perturbative treatment of impurity–boson interactions as presented e.g. in [26].

Effective polaron mass. The effective mass of the polaron, ${M}_{{\rm{p}}}$, can be obtained from the momentum dependence of the MF polaron energy. As discussed e.g. in [11], one finds

$$\begin{eqnarray}&&{E}_{0}^{\mathrm{MF}}(p)=\displaystyle \frac{{p}^{2}}{2{M}_{{\rm{p}}}}+{ \mathcal O }({p}^{4}).\end{eqnarray} \tag{ 25 }$$

In figure 3 we compare the MF effective polaron mass ${M}_{{\rm{p}}}$ (dashed line) to the experimental results. At large repulsive couplings around $\eta \approx {\eta }_{c}$ we find a saturation of ${M}_{{\rm{p}}}$ at some large but finite value. Qualitatively, a similar saturation of the polaron mass has been observed in the measurements by Catani et al [7], although it remains unclear how reliable the utilized measurement procedure is at strong couplings, see section 6.4 for a discussion.

Our most reliable DMC simulations in finite-size systems also show a saturation of the effective polaron mass at strong interactions ${g}_{\mathrm{IB}}\gg {g}_{\mathrm{BB}}$, see figure 3. The saturated value ${M}_{{\rm{p}}}(\eta \to \infty )$ is substantially smaller than the result expected from MF theory. In addition, from finite-size scaling we find that the DMC results are consistent with an infinite polaron mass in the thermodynamic limit. Indeed, based on the microscopic model Hamiltonian in equation (1) we expect that ${M}_{{\rm{p}}}\propto N$ is proportional to the number of bosons N when impurity–boson interactions diverge, ${g}_{\mathrm{IB}}\to \infty$. To understand this, note that the average impurity and boson velocities ${v}_{\mathrm{imp}}={v}_{{\rm{B}}}$ are equal because impurities and bosons cannot penetrate each other. By relating them to the average impurity and boson momenta ${p}_{{\rm{I}}}={{Mv}}_{{\rm{I}}}$ and ${p}_{{\rm{B}}}={m}_{{\rm{B}}}{v}_{{\rm{B}}}$ using Ehrenfest's theorem we see that a finite total momentum $p={p}_{{\rm{I}}}+{p}_{{\rm{B}}}={ \mathcal O }({N}^{0})$ is distributed over all particles and it follows that ${v}_{{\rm{I}}}={ \mathcal O }(1/N)$. Because the average impurity velocity ${v}_{{\rm{I}}}={v}_{{\rm{p}}}$ coincides with the average polaron velocity ${v}_{{\rm{p}}}=p/{M}_{{\rm{p}}}$, it follows that ${M}_{{\rm{p}}}={ \mathcal O }(N)$. We also note that an explicit proof that ${M}_{{\rm{p}}}\propto N$ has been given by means of exact Bethe-ansatz calculations in the limit when both impurity–boson and boson–boson interactions diverge, ${g}_{\mathrm{IB}}={g}_{\mathrm{BB}}\to \infty$, see [45].

The MF theory is based on the Bogoliubov approximation and ignores phonon–phonon interactions. Thus it cannot fully capture the impenetrable nature of impurities and bosons in the limit when ${g}_{\mathrm{IB}}=\infty$ in equation (1). As discussed in appendix B this leads to an RG flow of the interaction strength which shifts the position of the resonance defined by the divergence of impurity–boson interactions at long wavelengths. MF theory predicts an infinite effective polaron mass at an attractive microscopic interaction strength ${\eta }_{{\rm{c}},\mathrm{MF}}\lt 0$, as indicated in figure 3. When boson–boson interactions are fully taken into account, as in our DMC simulations, such shifts of the impurity–boson resonance cannot occur.

Experimentally [7], the impurity–boson interaction ${g}_{\mathrm{IB}}$ can be tuned by utilizing a confinement-induced (Feshbach) resonance [4]. When formulating the effective Hamiltonian in equation (1), we implicitly assumed that the scattering of an impurity with a boson at low energies is not modified by many-body effects caused by the interactions with the surrounding Bose gas. Similar to the shift of the resonance position predicted by the MF and RG theories, we expect that many-body effects can lead to a shift of the confinement-induced resonance.

In figure 5 we also show results from a quasi-1D DMC calculation. It takes into account the finite extent of the trap in radial direction. The prediction for the effective polaron mass is in excellent agreement with our strictly 1D DMC calculations. This suggests that the large discrepancy between theory and experiment in the strong coupling regime cannot be explained by the influence of radially excited states in the trap. Hence we conclude that the relevant physics remains strictly 1D. Details of this calculation can be found in section 7.1.

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Multi-particle bound states and in-medium Feshbach resonance. For a situation where the polaron momentum p = 0 vanishes, ${\beta }_{\mathrm{MF}}$ is given by equation (18). This MF expression suggests the existence of a transition where ${\beta }_{\mathrm{MF}}\to \infty$ diverges. It takes places at a critical attractive coupling strength

$$\begin{eqnarray}&&{g}_{\mathrm{IB},c}^{\mathrm{MF}}=-2\pi {\left[\int {\rm{d}}{k}\displaystyle \frac{{W}_{k}^{2}}{{\omega }_{k}+{k}^{2}/2M}\right]}^{-1}\lt 0.\end{eqnarray} \tag{ 26 }$$

At this interaction strength the number of phonons in the polaron diverges, see equation (17). This suggests an instability of the system towards a state with a large number of bosons accumulating around the impurity. Such behavior has been associated with multi-particle bound states in higher dimensions [24, 25, 40].

At ${g}_{\mathrm{IB},c}^{\mathrm{MF}}$ the MF polaron energy also diverges and changes sign, see equation (24). This indicates a transition from an attractive polaron with negative energy, to a repulsive polaron with positive energy. In figure 6 the MF polaron energy is shown and the MF critical value ${\eta }_{c,\mathrm{MF}}={g}_{\mathrm{IB},c}^{\mathrm{MF}}/{g}_{\mathrm{BB}}$ is indicated in the plot. In addition, we calculate the density profile of the bosons in figure 7. When ${\eta }_{c,\mathrm{MF}}$ is approached from $\eta =0$, as expected, a large number of bosons accumulates around the impurity. This effect is also observed by our full numerical Monte Carlo calculations.

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To understand the physics of this transition, we first analyze the limits ${g}_{\mathrm{IB}}\to \pm \infty$. Because the MF wavefunction coincides for ${g}_{\mathrm{IB}}=+\infty$ and ${g}_{\mathrm{IB}}=-\infty$, we conclude that the two repulsive polaron branches at strong attraction and repulsion are adiabatically connected. This behavior is reminiscent of the meta-stable super-Tonks–Girardeau state, which can be realized by a strongly interacting Bose gas (without impurities) when the interactions are quickly changed from strongly repulsive to strongly attractive [58, 59].

For an impurity inside a non-interacting Bose gas, ${g}_{\mathrm{BB}}=0$, the critical value becomes zero, ${g}_{\mathrm{IB},c}^{\mathrm{MF}}=0$. In this case, ${g}_{\mathrm{IB},c}^{\mathrm{MF}}$ corresponds to the point where we expect the appearance of an infinite series of multi-particle impurity–boson bound states in 1D, if the impurity mass is infinite. This last condition guarantees that no correlations can be induced between the bosons by scattering off the impurity. Note that such processes are not included in the MF wavefunction, even for finite impurity mass $M\lt \infty$. This explains why ${g}_{\mathrm{IB},c}^{\mathrm{MF}}=0$ vanishes in the non-interacting Bose gas, independent of M. In contrast, the RG approach presented in appendix B includes impurity-induced interactions between the phonons. There we show that ${g}_{\mathrm{IB},c}^{\mathrm{RG}}\ne 0$ becomes non-zero in the non-interacting Bose gas when the impurity mass is finite.

In the limit of a non-interacting Bose gas, ${g}_{\mathrm{BB}}=0$, and for a localized impurity, $M=\infty$, we note that the 1D scattering length ${a}_{\mathrm{IB}}=-1/{m}_{{\rm{B}}}{g}_{\mathrm{IB}}$ [6] diverges when ${g}_{\mathrm{IB}}$ approaches the critical value ${g}_{\mathrm{IB},c}^{\mathrm{MF}}=0$. This effect can be associated with a 1D Feshbach resonance. On the other hand, when the Bose gas is interacting or the impurity becomes mobile, the position of the Feshbach resonance is shifted. It is now located at the bare two-particle interaction ${g}_{\mathrm{IB},c}$. This behavior is reminiscent of the in-medium shift of the Feshbach resonance predicted for Bose polarons in 3D. In this case, too, the Feshbach resonance is associated with the appearance of multi-particle bound states in the spectrum [24, 40].

In general one should be cautious that the inclusion of quantum fluctuations might change the position of the transition compared to the MF prediction. In some cases, transitions predicted by MF theory even disappear completely. We address this problem in appendix B, where quantum fluctuations are included using an RG approach. See also [40] for a detailed discussion.

Logarithmic IR divergence of the polaron energy. In contrast to the 3D case [24], expression (24) for the MF polaron energy is fully convergent when the large-momentum cut-off ${{\rm{\Lambda }}}_{0}$ is sent to infinity. It has a logarithmic divergence when the small-momentum cut-off λ is sent to zero however,

$$\begin{eqnarray}&&\displaystyle \frac{{g}_{\mathrm{IB}}}{2\pi }{\int }_{\lambda }^{{{\rm{\Lambda }}}_{0}}{\rm{d}}{k}\,{\sinh }^{2}{\theta }_{k}\sim -{g}_{\mathrm{IB}}\mathrm{log}\lambda \mathop{\longrightarrow }\limits^{\lambda \to 0}\mathrm{sgn}({g}_{\mathrm{IB}})\times \infty .\end{eqnarray} \tag{ 27 }$$

This IR divergence can be regularized by including quantum fluctuations using the RG approach presented in section 5.2.

In our MF calculations of the polaron energy, for example in figure 6, we ignore the log-divergent term ${g}_{\mathrm{IB}}/2\pi \int {\rm{d}}{k}\,{\sinh }^{2}{\theta }_{k}$ in equation (24). This step can only be justified a posteriori, by showing that the general behavior of the MF polaron energy without the log-divergent term closely resembles the fully regularized RG prediction. Alternatively one can argue that the divergence $\sim \mathrm{log}\lambda$ is only logarithmic in system size $L\sim 1/\lambda$. Hence the correction to the polaron energy from the log-divergent term in equation (27) is not expected to be large for a finite system of an experimentally relevant size. As a consequence, our MF prediction for the polaron energy is not expected to be quantitatively accurate.

Orthogonality catastrophe. In one dimension, the MF theory of Bose polarons has even more noteworthy peculiarities associated with the IR cut-off. First of all, the phonon number in the polaron cloud diverges logarithmically,

$$\begin{eqnarray}&&{N}_{\mathrm{ph}}^{\mathrm{MF}}={\int }_{\lambda }{\rm{d}}{k}\,{\left(\displaystyle \frac{{\beta }_{\mathrm{MF}}{V}_{k}}{{{\rm{\Omega }}}_{k}^{\mathrm{MF}}}\right)}^{2}\sim -\mathrm{log}\lambda \mathop{\longrightarrow }\limits^{\lambda \to 0}\infty .\end{eqnarray} \tag{ 28 }$$

This is directly related to the log-divergence of the MF energy. Depleting the quasi-1D condensate and creating infinitely many phonons costs an interaction energy which scales like $\sim {g}_{\mathrm{IB}}{N}_{\mathrm{ph}}$.

The diverging phonon number can also be understood as a manifestation of Anderson's orthogonality catastrophe [60] for a mobile impurity in a 1D Bose gas. Within MF theory, the quasiparticle residue is determined by the simple relation (see e.g. [11])

$$\begin{eqnarray}&&Z\,=\,{{\rm{e}}}^{-{N}_{\mathrm{ph}}^{\mathrm{MF}}}\simeq \lambda \to 0.\end{eqnarray} \tag{ 29 }$$

Therefore already an infinitesimally small interaction leads to a vanishing quasiparticle weight Z = 0 in an infinite system.

Supersonic polarons. The large number of phonons in the 1D polaron cloud also affects the dependence of polaron properties on the total momentum [61]. While there exists a phase transition from the subsonic to the supersonic regime at large momenta in higher dimensions [44], this transition is absent in 1D. To calculate the critical momentum ${p}_{c}^{\mathrm{MF}}$ where the subsonic to supersonic transition for the impurity takes place, we employ Landau's criterion for superfluidity. It states that the transition takes place when the polaron velocity ${v}_{{\rm{p}}}$ exceeds the speed of sound c in the Bose gas. From the expression ${v}_{{\rm{p}}}=(p-{P}_{\mathrm{ph}}^{\mathrm{MF}})/M$ [44] we derive

$$\begin{eqnarray}&&{p}_{c}^{\mathrm{MF}}={Mc}+\int {\rm{d}}{k}\displaystyle \frac{k{\beta }_{\mathrm{MF}}^{2}{V}_{k}^{2}}{{({\omega }_{k}+{k}^{2}/2M-{kc})}^{2}}\simeq \displaystyle \frac{1}{\lambda }\to \infty .\end{eqnarray} \tag{ 30 }$$

Here λ is the IR momentum cut-off. We find that polarons are subsonic in 1D, due to the dressing of the impurity with an infinite number of phonons.

In appendix B we extend our analysis and include quantum fluctuations on top of the MF solution by applying the non-perturbative RG approach introduced in [16, 39, 40]. The RG procedure allows us to regularize the IR divergence of the polaron energy identified in equation (27). We use this method to make predictions for the energy shift between an interacting and a free impurity, which can be measured directly using radio-frequency spectroscopy [8, 9]. On the other hand, the RG method provides new insights into the polaron phase diagram. We present a detailed discussion of our RG analysis of 1D Bose polarons in appendix B.

From the solution of the RG flow equations (see appendix B) we obtain the fully regularized polaron energy ${E}_{0}^{\mathrm{RG}}$. Explicit calculation in the appendix demonstrates IR divergence of the polaron energy is regularized with this approach. We further benchmark the RG approach by comparing to the DMC calculations and working in the regime of large boson density n₀. More concretely, we assume that the dimensionless interaction strength $\gamma \ll 1$, see equation (9), is small. In this regime the Bogoliubov approximation is justified and the effect of phonon–phonon interactions, included only by DMC, is expected to be weak.

We estimate the importance of nonlinear interaction terms between Bogoliubov phonons by the relative size of corrections ${\rm{\Delta }}{\varepsilon }_{\mathrm{LHY}}$ to the ground state energy ${\varepsilon }_{0}$ of the homogeneous Bose gas caused by quantum fluctuations [48, 49]. Similar to Lee–Huang–Yang corrections in 3D [62, 63], one obtains

$$\begin{eqnarray}&&{\rm{\Delta }}{\varepsilon }_{\mathrm{LHY}}/{\varepsilon }_{0}=-\displaystyle \frac{{2}^{5/2}}{3\pi }\displaystyle \frac{1}{\sqrt{{n}_{0}| {a}_{\mathrm{BB}}| }}=-\displaystyle \frac{4}{3\pi }\sqrt{\gamma },\end{eqnarray} \tag{ 31 }$$

see e.g. [6]. In the second expression, γ is the dimensionless interaction strength in the Bose gas, see equation (9).

In figure 8 we compare our result for the polaron energy to DMC calculations. We used the same mass ratio $M/{m}_{{\rm{B}}}=41/87$ as in the experiment by Catani et al [7], but the density was chosen to be ${n}_{0}| {a}_{\mathrm{BB}}| =144$ (corresponding to ${n}_{0}=6/\xi =144/| {a}_{\mathrm{BB}}|$ or $\gamma =0.014$) so that the corrections of quantum fluctuation to the energy is ${\rm{\Delta }}{\varepsilon }_{\mathrm{LHY}}/{\varepsilon }_{0}=0.05\ll 1$. For weak-to-intermediate couplings we obtain excellent agreement between RG and DMC predictions, validating the use of the effective beyond-Fröhlich Hamiltonian ${\hat{{ \mathcal H }}}_{\mathrm{BF}}$ in this regime. For the case of repulsive microscopic interactions, ${g}_{\mathrm{IB}}\gt 0$, we obtain sizable deviations in the regime of very strong couplings. Yet the correction by the RG accounts for about half of the deviation between MF and DMC results in this case.

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In the case of attractive microscopic interactions ${g}_{\mathrm{IB}}\lt 0$, see figure 8(a), an additional regularization of the polaron energy ${E}_{0}({\rm{\Lambda }})$ is required in the RG to deal with divergencies of one of the coupling constants (${G}_{-}$) during the RG. This procedure is described in detail in [40]. The agreement for the energy of attractive polarons is reasonable. It should also be noted that the attractive polaron is not expected to be the ground state of the Hamiltonian ${\hat{{ \mathcal H }}}_{\mathrm{BF}}$ based on the Bogoliubov approximation for the Bose gas in this case, as discussed in detail in appendix B. Instead it has been argued in [40] that the true ground state contains a large number of bosons accumulating around the impurity. This prediction is consistent with our DMC calculations in this regime.

In figure 1(b) we calculate how the density profile of the bosons is modified by impurity–boson interactions. Again we consider the high-density regime where the Bogoliubov approximation for the Bose gas is justified. In contrast to the energies shown in figure 8, the depletion of the Bose gas is accurately described by the RG approach in this case, even for very large interactions ($\eta ={10}^{3}$ in the figure). This serves as another important benchmark for the RG and the use of the effective beyond-Fröhlich Hamiltonian ${\hat{{ \mathcal H }}}_{\mathrm{BF}}$ in the regime where quantum fluctuations in the bath are weak.

In contrast, for the experiment by Catani et al [7] the correction to the energy due to quantum fluctuations is sizable, ${\rm{\Delta }}{\varepsilon }_{\mathrm{LHY}}/{\varepsilon }_{0}\approx -0.30$, indicating that Bogoliubov theory is no longer quantitatively accurate. Indeed the predictions for the effective polaron mass in the strong coupling regime differ by a factor of 4, see figure 5. Below, in figure 15, we also compare predictions for the polaron energy in this case and find large quantitative deviations between the two approaches on the repulsive side.

To describe the dynamics of strongly coupled Bose polarons inside a shallow trapping potential, we employ a time-dependent variant of MF theory as in [24, 44]. We supplement this approach with the local-density approximation (LDA). This allows us to treat the problem in the Lee–Low–Pines (LLP) frame despite the external potential.

Our variational ansatz corresponds to a product wave function of coherent states,

$$\begin{eqnarray}&&| {\psi }_{\mathrm{MF}}(t)\rangle ={{\rm{e}}}^{-{\rm{i}}\chi (t)}{\hat{U}}_{\mathrm{LLP}}(P(t))\displaystyle \prod _{k}| {\alpha }_{k}(t)\rangle ,\end{eqnarray} \tag{ 32 }$$

in the frame which is co-moving with the impurity, see appendix A. This is achieved by the unitary transformation ${\hat{U}}_{\mathrm{LLP}}(P(t))$ [55], where P(t) denotes the time-dependent total system momentum. $\chi (t)$ denotes an overall phase which guarantees conservation of the total energy. Similar to the MF ground state, this wavefunction ignores quantum fluctuations of phonons induced by the mobile impurity.

The position of the impurity $X(t)=\langle \hat{x}\rangle$ can be calculated in the LLP frame by using the Ehrenfest theorem,

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}X(t)=(P(t)-{P}_{\mathrm{ph}}(t))/M,\end{eqnarray} \tag{ 33 }$$

where ${P}_{\mathrm{ph}}(t)=\int {\rm{d}}{k}\,k| {\alpha }_{k}(t){| }^{2}$ is the expectation value of the phonon momentum at time t. The equations of motion for ${\alpha }_{k}(t)$ can be derived from the Hamiltonian ${\hat{{ \mathcal H }}}_{\mathrm{BF}}$, see equation (21), using Dirac's time-dependent variational principle [64]. As discussed in [24] this yields

$$\begin{eqnarray}{\rm{i}}{\partial }_{t}{\alpha }_{k} & = & {V}_{k}+\left({\omega }_{k}+\displaystyle \frac{{k}^{2}}{2M}-\displaystyle \frac{k}{M}(P(t)-{P}_{\mathrm{ph}}(t))\right){\alpha }_{k}\\ & & +\displaystyle \frac{{g}_{\mathrm{IB}}}{2\pi }\displaystyle \int {{\rm{d}}{k}}^{{\prime} }\left[\mathrm{Re}({\alpha }_{k^{\prime} }){W}_{k}{W}_{k^{\prime} }+{\rm{i}}\,\displaystyle \frac{\mathrm{Im}({\alpha }_{k^{\prime} })}{{W}_{k}{W}_{k^{\prime} }}\right].\end{eqnarray} \tag{ 34 }$$

We note that the log-divergent term in the MF polaron energy, see equation (27), does not enter in the equations of motion (34). Therefore the time-dependent MF theory is well-behaved in the long-wavelength limit. We note that starting from a free impurity we observe numerically that the phonon number grows logarithmically in time,

$$\begin{eqnarray}&&{N}_{\mathrm{ph}}(t)\simeq \mathrm{log}t.\end{eqnarray} \tag{ 35 }$$

This can be understood as a dynamical manifestation of the orthogonality catastrophe in a 1D Bose gas. In the polaron ground state the number of phonons diverges logarithmically even after inclusion of quantum fluctuations by the RG, see appendix B.2. By analogy we expect that the dynamical divergence of the phonon number in time is not an artifact of the MF theory but a physical effect observable also in the presence of quantum fluctuations.

In the absence of an external force, the total system momentum is conserved, ${\rm{d}}{P}(t)/{\rm{d}}{t}=0$. Now we consider the effect of an additional (confinement) potential V(x) on the impurity. In the limit when its changes are small over a length scale set by the size of the polaron wavepacket, we can still treat the system as a homogeneous one. In this LDA, the equation of motion for the total momentum is given by

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}P(t)=-{\partial }_{x}V(X(t)).\end{eqnarray} \tag{ 36 }$$

Potentials from the Florence experiment. In the following we will consider different potentials, which have all played a role in the experiment by Catani et al [7]. Firstly, both the impurity and the bosons are trapped inside a shallow parabolic potential,

$$\begin{eqnarray}&&{V}_{{\rm{I}},{\rm{B}}}(x)=\displaystyle \frac{1}{2}{m}_{{\rm{I}},{\rm{B}}}{{\rm{\Omega }}}_{{\rm{I}},{\rm{B}}}^{2}{x}^{2},\end{eqnarray} \tag{ 37 }$$

where ${m}_{{\rm{I}}}=M$ is the impurity mass and ${{\rm{\Omega }}}_{{\rm{I}},{\rm{B}}}$ are the trapping frequencies. The oscillator lengths in [7] are ${{\ell }}_{{\rm{I}}}\approx 11\xi$ and ${{\ell }}_{{\rm{B}}}\approx 9\xi$ respectively, justifying the LDA in this case because the size of the polaron is of the order $\xi \ll {{\ell }}_{{\rm{I}},{\rm{B}}}$.

In addition, Catani et al [7] used a tight species-selective dipole trap (SSDP) to localize the impurities in the beginning. This gives rise to a steeper harmonic trap, ${V}_{\mathrm{SSDP}}(x)=M{\omega }_{{\rm{I}}}^{2}{x}^{2}/2$, where ${\omega }_{{\rm{I}}}=0.28c/\xi$ in our units. In this case the oscillator length is ${\ell }\approx 3.3\xi$ and the LDA is less justified.

Finally, the impurity energy depends on the density n(x) of the Bose gas. This gives rise to an effective potential which is determined by the interaction strength ${g}_{\mathrm{IB}}$. Assuming that the impurity can adiabatically follow its ground state at P = 0, we obtain the following polaronic potential,

$$\begin{eqnarray}&&{V}_{\mathrm{pol}}(x)\equiv {E}_{0}^{\mathrm{MF}}(x)={g}_{\mathrm{IB}}{\beta }_{\mathrm{MF}}(x)n(x).\end{eqnarray} \tag{ 38 }$$

To calculate ${\beta }_{\mathrm{MF}}(x)$ we note that the healing length ξ and the speed of sound c both depend on the density n(x), and thus on x. For a weakly interacting quasi-1D condensate in a shallow trap we can apply the Thomas–Fermi approximation to estimate the boson density,

$$\begin{eqnarray}&&n(x)\approx {n}_{0}-{V}_{{\rm{B}}}(x)/{g}_{\mathrm{BB}}.\end{eqnarray} \tag{ 39 }$$

In figure 9 the potentials are shown for parameters as in the experiment by Catani et al [7] and assuming repulsive impurity–boson interactions. We note that in the strong coupling regime, for $\eta \gtrsim 1$, the effective polaronic potential ${V}_{\mathrm{pol}}(x)$ has its minimum at the edge of the Bose gas. In this case we expect that impurities become trapped in this region after they have had enough time to equilibrate with the quasi-1D condensate. Such behavior has indeed been observed in [7] for $\eta \gtrsim 1$.

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We begin by studying the dynamics of an impurity interacting with a homogeneous Bose gas, where the impurity is subject to a shallow trapping potential. Experimentally this situation corresponds to the assumption that ${{\rm{\Omega }}}_{{\rm{B}}}\ll {{\rm{\Omega }}}_{{\rm{I}}}$ is small. In this case we can ignore the polaronic potential in equation (38), and we will consider only ${V}_{{\rm{I}}}(x)$ now.

Adiabatic limit. In the limit where the polaron follows its local ground state adiabatically, it can be described by an effective Hamiltonian

$$\begin{eqnarray}&&{\hat{{ \mathcal H }}}_{\mathrm{eff}}=\displaystyle \frac{{\hat{p}}^{2}}{2{M}_{{\rm{p}}}}+\displaystyle \frac{1}{2}M{{\rm{\Omega }}}_{{\rm{I}}}^{2}{\hat{x}}^{2}.\end{eqnarray} \tag{ 40 }$$

If we start from a wavepacket in the origin, with momentum p₀, it will undergo harmonic oscillations. Their frequency is renormalized,

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{p}}}={{\rm{\Omega }}}_{{\rm{I}}}\sqrt{M/{M}_{{\rm{p}}}},\end{eqnarray} \tag{ 41 }$$

due to the enhanced polaron mass.

The amplitude of the harmonic oscillations, σ, is easily obtained from energy conservation, ${p}_{0}^{2}/2{M}_{{\rm{p}}}=M{{\rm{\Omega }}}_{{\rm{I}}}^{2}{\sigma }^{2}/2$. By the same polaronic mass enhancement, this amplitude is also renormalized compared to the case of a free impurity,

$$\begin{eqnarray}&&\sigma =\sqrt{\displaystyle \frac{M}{{M}_{{\rm{p}}}}}\displaystyle \frac{{p}_{0}}{M{{\rm{\Omega }}}_{{\rm{I}}}}.\end{eqnarray} \tag{ 42 }$$

Now we will use full dynamical simulations to show that both the frequency and the amplitude renormalization of polaron oscillations can serve as indicators of the effective polaron mass. In the case of a homogeneous Bose gas, equations (41), (42) provide an accurate description of polaron trajectories at weak couplings.

Initial conditions. We use initial conditions as close as possible to the experimental situation described in [7]. There the impurities were tightly confined in the SSDP, before they were released and their breathing oscillations were recorded. To model this situation, we consider polaron wavepackets localized in the origin x = 0. We assume that the distribution $f({p}_{0})$ of their momenta ${p}_{0}=P(0)$ can be derived from the thermal state of the impurity in the initial, tight trapping potential. Then we perform numerical calculations for different initial system momenta $P(0)={p}_{0}$ and add up the resulting trajectories with their respective thermal weights.

We also need the initial phonon configuration in the polaron frame. Because the SSDP provides a rather deep trapping potential, ${\omega }_{{\rm{I}}}=0.28c/\xi$, we think it is reasonable to assume that the impurity is initially localized, corresponding to $M=\infty$. Therefore the phonon cloud at time t = 0 is chosen as the MF solution at $M=\infty$,

$$\begin{eqnarray}&&{\alpha }_{k}(0)=-{\beta }_{\mathrm{MF}}^{\infty }\displaystyle \frac{{V}_{k}}{{\omega }_{k}},\quad {\beta }_{\mathrm{MF}}^{\infty }={\left(1+\displaystyle \frac{{g}_{\mathrm{IB}}}{2\pi }\int {\rm{d}}{k}\displaystyle \frac{{W}_{k}^{2}}{{\omega }_{k}}\right)}^{-1}.\end{eqnarray} \tag{ 43 }$$

We study the influence of the initial phonon distribution on the long-time behavior in figure 10. We compared cases where the MF solution at finite mass M, infinite mass $M=\infty$, or the phonon vacuum ${\alpha }_{k}=0$, was chosen initially. In figure 10 we show typical polaron trajectories calculated within LDA for moderate interactions. Unless the MF solution at finite $M\gt 0$ is chosen as initial state, the polaron loses a substantial part of its momentum at short times (see inset), until its velocity drops below the speed of sound c. This dissipation leads to a reduced oscillation amplitude, which manifests itself in the trajectories even at long times.

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Numerical results. To study polaron oscillations systematically, we simulated the experiment by Catani et al [7] but assumed a homogeneous Bose gas first. In figure 10 a typical polaron trajectory is shown, starting from an initial momentum P(0) which is characteristic for the temperatures in the experiment. Note that it corresponds to a supersonic impurity with velocity $\dot{X}(0)\gt c$. At long times we observe long-lived but decaying oscillations with polaron velocities which are always below the speed of sound c.

We repeated these calculations for different interactions η and averaged over the thermal distribution of initial momenta P(0) corresponding to [7]. Then the resulting trajectories

$$\begin{eqnarray}&&\sigma (t)=\sqrt{\langle {X}^{2}(t){\rangle }_{T}},\end{eqnarray} \tag{ 44 }$$

where $\langle \cdot {\rangle }_{T}$ corresponds to thermal averaging, were fitted to a function,

$$\begin{eqnarray}&&\sigma (t)=\sigma | \sin (\omega t)| {{\rm{e}}}^{-\gamma t}.\end{eqnarray} \tag{ 45 }$$

In figure 11 the fit parameters σ, ω and γ are shown as a function of the interaction strength η. Some of the trajectories from which these plots were derived are plotted in figure 12.

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In figure 11(a) we compare results for the frequency renormalization with predictions by the adiabatic approximation, see equation (41), and obtain excellent agreement. Our findings demonstrate that a frequency measurement of polaron oscillations provides an accurate method to detect the effective polaron mass. In particular, it also works in the strongly interacting regime and shows the saturation of the polaron mass in this case.

In figure 11(b) we compare results for the amplitude renormalization to predictions by the adiabatic approximation, see equation (42). The amplitude ${\sigma }_{0}$ is determined by the temperature of the impurity in this case; It is on the order of ${\sigma }_{0}\approx {{\rm{\Omega }}}_{{\rm{I}}}^{-1}\sqrt{2{k}_{{\rm{B}}}T/M}$. Both for weak and strong impurity–boson interactions we find that the amplitude of impurity oscillations is directly related to the effective polaron mass according to the adiabatic expression in equation (42). In the intermediate regime large deviations from this prediction can be observed.

To understand this behavior we take a closer look at the impurity trajectories. In figure 12(b) we observe that, for small interactions, the impurity oscillates back and forth several times at supersonic speeds. It is slowed down continuously until eventually it becomes subsonic. In this regime the amplitude of oscillations is well approximated by the adiabatic expression (42). At intermediate interactions the impurity quickly loses kinetic energy and becomes subsonic, see figures 10 and 12(d). This dissipative effect goes beyond the adiabatic approximation and explains the strongly enhanced amplitude renormalization found in figure 11(b) for this regime. For strong interactions, the impurity velocity drops below the speed of sound almost instantly for all initial velocities, see figure 12(f). This suggests that the initial impurity momentum is almost reversibly transformed into polaron momentum in this regime. Afterwards the propagation of the heavy polaron can be described by the adiabatic approximation again.

In figure 11(c) we present results for the damping rate of polaron oscillations in a harmonic trap. As before, three regimes of weak, intermediate and strong interactions can be identified. In agreement with our previous analysis, the decay is largest for moderate interactions. In this regime the non-adiabatic corrections are maximal: for stronger coupling the impurity can follow the phonon cloud more easily, suppressing non-adiabatic processes. For weaker couplings the probability for a non-adiabatic process where a phonon is emitted is strongly reduced because it scales as ${\eta }^{2}$.

Relation to the Florence experiment. In this section we neglect the additional complication arising from an inhomogeneous Bose gas and compare experimental results to the theoretical analysis that assumes uniform boson density. We discuss the role of inhomogeneity in the following section 6.4.

Our analysis in figure 11 suggests that the oscillation frequency is the most useful observable for measurement of the effective polaron mass. Surprisingly no frequency renormalization has been found by Catani et al [7]. On the other hand, the amplitude renormalization has been observed and as shown in figure 3 its comparison with our different theoretical calculations is excellent at weak couplings. At larger interactions the agreement is not as good, and the deviations are opposite from what we would have expected according to figure 11(b).

From figure 11(c) we read off a typical damping rate $\gamma \approx {10}^{-3}c/\xi$ around $\eta =1$. The breathing frequency ${\omega }_{b}$, which is twice the oscillation frequency ω, is on the order of ${\omega }_{b}=0.04c/\xi$ in this regime. Therefore we expect a friction coefficient $\tilde{\gamma }:= \gamma /2{\omega }_{b}$, as introduced in [7], of $\tilde{\gamma }\approx 0.025$. This value agrees well with the measured value of $\tilde{\gamma }=0.03(2)$ for this interaction strength.

As we discussed above, the experimental observations in [7] are not consistent with our predictions for polaron oscillations inside a homogeneous Bose gas. Now we discuss possibilities how the inhomogeneity of the Bose gas can affect the oscillations of the impurity.

First we start by including the effective polaronic potential from equation (38) in the simulations. Some comments are in order. The use of this potential can only be justified in the limit where the polaron cloud follows the impurity adiabatically. Moreover, the treatment becomes meaningless close to the boundary of the Bose gas. In this regime the boson density is small and the Bogoliubov approximation breaks down. Moreover the Thomas–Fermi approximation is not accurate in this regime. Therefore the following results should be considered as a qualitative guide to understanding experiments rather than accurate quantitative analysis.

The polaron trajectories in the presence of the polaronic trapping potential are shown in figure 13. For weak impurity–boson interactions, the result is similar to the case of a homogeneous Bose gas and we obtain long-lived oscillations, slightly renormalized compared to the free impurity. For larger interactions all trajectories quickly become localized at the boundary of the Bose gas. Similar behavior has been reported in [7] for long times, although pronounced oscillations have also been seen in that case.

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We now discuss that absence of frequency renormalization observed in [7] in the strongly interacting regime may be a result of inhomogeneous density of bosons. Let us consider an impurity trajectory with an amplitude which exceeds the size of the Bose gas by a sizable amount. Moreover, when the particle is interacting with the bosons, let us apply the adiabatic approximation. We may thus assume that all the initial momentum of the impurity is adiabatically transferred into polaron momentum. Indeed, we have shown for the case of a homogeneous gas that this simple picture explains why the amplitude of polaron oscillations is renormalized by a factor of $\sqrt{M/{M}_{{\rm{p}}}}$ for very strong couplings. This argument only relies on the conservation of energy, and therefore it also applies when the impurity exists the region where the Bose gas is confined.

For sufficiently large interactions, figure 9 shows that the effective polaronic potential can be steeper than the harmonic confining potential for the impurity. Therefore it is conceivable that the impurity spends more time outside the Bose gas, than inside. In an extreme case it may even be reflected off the boundary. In such a situation the breathing frequency ${\omega }_{b}$ is given by twice the bare trap frequency, ${\omega }_{b}=2{{\rm{\Omega }}}_{{\rm{I}}}$, as observed in [7]. The oscillation amplitude, on the other hand, is determined by the initial energy of the impurity inside the Bose gas, which can be related to the effective polaron mass.

Monte Carlo methods provide an efficient tool for the evaluation of multidimensional integrals. Expectation values of quantum mechanical operators can be written as integrals over the ground state wave function and can be efficiently evaluated using Monte Carlo techniques. Here we resort to the DMC method in order to obtain the ground state properties of the Bose polaron problem. For a general reference on the DMC method, see for example [42]. An important advantage of the non-perturbative DMC method is that it can be applied to arbitrary interaction strength η between the impurity and the surrounding Bose gas. This is true for arbitrary values of the gas parameter γ of the bath, see equation (9).

Another question which can be addressed by performing DMC simulations concerns the dimensionality of the system. Specifically, we study the effect of the transverse confinement potential in the Florence experiment [7] on the energetic and dynamical properties of the system. To this end we consider both the effective 1D Hamiltonian with contact interactions, see equation (1), and a full 3D Hamiltonian. The latter takes into account the geometry of the optical lattice potential as well as the three-dimensional scattering length. Even though the experimental geometry is rather complicated, with different optical lattice potentials felt by the Rb and K atoms, the DMC method can still be applied.

We focus on two densities, first corresponding to strong interactions in the bath, ${n}_{0}| {a}_{\mathrm{BB}}| =4.56$ or $\gamma =0.44$, as in the Florence experiment [7]. The second considered value, ${n}_{0}| {a}_{\mathrm{BB}}| =144$, corresponds to $\gamma =0.014$ deep in the Bogoliubov regime. For the first case we consider impurities (Rb) and bath particles (K) of different mass. In the second case we will also consider the limit of an infinitely repulsive pinned impurity, in which Bethe ansatz can be used in order to find the ground state energy of the Bose polaron. This permits us to verify the consistency of the DMC energy with the Bethe ansatz result in this exactly integrable limit. In addition we obtain the correlation functions and the density profile of bosons around the impurity from DMC calculations.

Three-dimensional DMC calculations. In first quantization the 3D model Hamiltonian is

$$\begin{eqnarray}&&\hat{{ \mathcal H }}=-\displaystyle \frac{{{\hslash }}^{2}}{2M}{{\rm{\Delta }}}_{{\rm{I}}}-\displaystyle \frac{{{\hslash }}^{2}}{2{m}_{{\rm{B}}}}\displaystyle \sum _{i=1}^{N}{{\rm{\Delta }}}_{i}+\displaystyle \sum _{i\lt j}^{N}{V}_{\mathrm{BB}}({{\bf{r}}}_{i}-{{\bf{r}}}_{j})+\displaystyle \sum _{i=1}^{N}{V}_{\mathrm{IB}}({{\bf{r}}}_{i}-{{\bf{r}}}_{{\rm{I}}})+{V}_{{\rm{I}}}^{\mathrm{ext}}({{\bf{r}}}_{{\rm{I}}})+\displaystyle \sum _{i=1}^{N}{V}_{{\rm{B}}}^{\mathrm{ext}}({{\bf{r}}}_{i}).\end{eqnarray} \tag{ 46 }$$

Here ${V}_{\mathrm{BB}}(r)$ and ${V}_{\mathrm{IB}}(r)$ are the boson–boson and impurity–boson interaction potentials, and ${{\rm{\Delta }}}_{{\rm{I}}}$, ${{\rm{\Delta }}}_{i}$ denote the Laplace operators with respect to the impurity and the boson labeled by i, respectively. The external potentials, ${V}_{{\rm{B}}}^{\mathrm{ext}}$ and ${V}_{{\rm{I}}}^{\mathrm{ext}}$, are felt by bosons and the impurity respectively. In the experiment [7] they have been created by two-dimensional lattices forming an array of 1D tubes. We consider the case when a single tube in the array is populated.

The depth of the Rb lattice potential ${V}_{\mathrm{latt}}(r)$ is s = 60 recoils with the lattice wavelength $\lambda =1064\,\mathrm{nm}$, ${V}_{{\rm{B}}}(r)=s{{\hslash }}^{2}/(2{m}_{\mathrm{Rb}}{\lambda }^{2})$. We have checked that similar result can be obtained by considering a simple harmonic external potential, ${V}_{\mathrm{osc}}(r)$, with the transverse oscillator frequency ${\omega }_{\perp }/2\pi =34(45)$ kHz for Rb(K) atoms. We ignore the residual shallow trapping along the longitudinal direction in DMC calculations.

The relation of the three-dimensional s-wave scattering length, ${a}_{3{\rm{D}}}$, to the 1D one, ${a}_{1{\rm{D}}}$, for the tight transverse confinement with oscillator length ${a}_{\mathrm{osc}}$, is given by Olshanii's formula ${a}_{1{\rm{D}}}=-{a}_{\mathrm{osc}}^{2}/{a}_{3{\rm{D}}}(1-1.0326{a}_{3{\rm{D}}}/{a}_{\mathrm{osc}})$ [4]. In an optical lattice geometry no simple analytical result is known and the corresponding relation is obtained following [65]. In the Florence experiment [7] the boson–boson s-wave scattering length was fixed to ${a}_{1{\rm{D}},\mathrm{Rb}}=-652.2$ nm and was not changed. The boson–impurity s-wave scattering length, in contrast, is tunable over a wide range by changing the strength of the applied magnetic field.

In our quasi-1D simulations we model the three-dimensional interaction potential by hard-spheres, ${V}_{\mathrm{HS}}({\bf{r}})=+\infty$ when $| {\bf{r}}| \lt {a}_{\mathrm{HS}}$ and ${V}_{\mathrm{HS}}({\bf{r}})=0$ otherwise. The diameter of the hard-sphere potential coincides with its s-wave scattering length and is set to reproduce Rb–Rb (Rb–K) value for the boson–boson (boson–impurity) scattering amplitude. In our simulations we consider a single impurity and impose periodic boundary conditions along the longitudinal direction of the tube.

The statistical fluctuations in Monte Carlo simulation can be greatly reduced by using importance sampling. This is done on the basis of a distribution function which we derive from a trial guiding wave function. Motivated by our physical insights, it is chosen as a product of one- and two-body terms,

$$\begin{eqnarray}&&{\psi }_{T}({{\bf{r}}}_{1},\,\cdots ,\,{{\bf{r}}}_{N};{{\bf{r}}}_{{\rm{I}}})={h}_{{\rm{I}}}({{\bf{r}}}_{{\rm{I}}})\displaystyle \prod _{i=1}^{N}{h}_{{\rm{B}}}({{\bf{r}}}_{i})\displaystyle \prod _{i\lt j}^{N}{f}_{\mathrm{BB}}(| {{\bf{r}}}_{i}-{{\bf{r}}}_{j}| )\displaystyle \prod _{i=1}^{N}{f}_{\mathrm{IB}}(| {{\bf{r}}}_{i}-{{\bf{r}}}_{{\rm{I}}}| ).\end{eqnarray} \tag{ 47 }$$

The Gaussian one-body terms ${h}_{{\rm{I}}}({\bf{r}})=\exp [-({x}^{2}+{y}^{2})/2{a}_{\mathrm{osc},{\rm{K}}}^{2}]$ and ${h}_{{\rm{B}}}({\bf{r}})=\exp [-({x}^{2}+{y}^{2})/2{a}_{\mathrm{osc},\mathrm{Rb}}^{2}]$ localize the particles inside the central tube. The two-body Jastrow terms are taken in the following form

$$\begin{eqnarray}&&{f}_{\alpha }(r)=\left\{\begin{array}{l}0,r\lt {a}_{\alpha }\\ \tfrac{A}{r}\sin (B(r-{a}_{\alpha }))\\ 1,r\gt L/2\end{array}\right.,\end{eqnarray} \tag{ 48 }$$

where parameters A and B are chosen such that both ${f}_{\alpha }(r)$ and its derivative are continuous at the half size of the box, $r=L/2$. Here $\alpha =\mathrm{BB}$ corresponds to boson–boson and $\alpha =\mathrm{IB}$ to the case of impurity–boson scattering. The Jastrow terms (48) are obtained as the solutions of the two-body scattering problem on the hard sphere potential. Although the guiding wave function (47) and (48) does not contain any variational parameters, it provides a sufficient quality.

1D DMC calculations. We also perform calculations for the 1D Hamiltonian from equation (1). In this case the guiding wave function can be conveniently written in a pair-product form

$$\begin{eqnarray*}&&{\psi }_{T}({x}_{1},\,\cdots ,\,{x}_{N};{x}_{{\rm{I}}})=\displaystyle \prod _{i\lt j}^{N}{f}_{\mathrm{BB}}(| {x}_{i}-{x}_{j}| )\displaystyle \prod _{i=1}^{N}{f}_{\mathrm{IB}}(| {x}_{i}-{x}_{{\rm{I}}}| ).\end{eqnarray*}$$

We chose the two-body terms as

$$\begin{eqnarray}&&{f}_{\alpha }(x)=\left\{\begin{array}{l}{A}_{\alpha }\cos ({k}_{\alpha }(x-{B}_{\alpha })),\quad | x| \leqslant {R}_{\alpha }^{\mathrm{par}}\\ | \sin (\pi x/L){| }^{1/{K}_{\alpha }^{\mathrm{par}}},\qquad {R}_{\alpha }^{\mathrm{par}}\lt | x| \end{array}\right.,\end{eqnarray} \tag{ 49 }$$

where $\alpha \,=$ BB, BI denotes boson–boson (boson–impurity) terms. Parameters A_α and B_α are chosen in such a way that ${f}_{\alpha }(x)$ and its first derivative are continuous for $| x| \gt 0$ and satisfy Bethe–Peierls boundary condition at the contact, ${f}_{\alpha }^{{\prime} }(0)=-{f}_{\alpha }(0)/{a}_{\alpha }$. Here a_α denotes the corresponding 1D s-wave scattering length. The short-range part in equation (49), for $| x| \leqslant {R}_{\alpha }^{\mathrm{par}}$, is the two-body scattering solution for a contact δ-function potential. Instead the 'phononic' long-range part in equation (49), for $| x| \gt {R}_{\alpha }^{\mathrm{par}}$, is obtained from the hydrodynamic approach [66].

The variational parameter ${R}_{\alpha }^{\mathrm{par}}$ corresponds to the crossover distance between the two-body and phononic regimes. It is optimized by minimizing the variational energy, which leads to the VMC results presented earlier in this paper. The variation parameter ${K}_{\mathrm{BB}}^{\mathrm{par}}$ for large system size coincides with the Luttinger parameter of the bath K. Its dependence on the gas parameter, $K(\gamma )$, is known from the Bethe ansatz solution to the Lieb–Liniger model [67]. We use the thermodynamic value of parameter ${K}_{\mathrm{BB}}^{\mathrm{par}}$ for the bath and optimize parameter ${K}_{\mathrm{IB}}^{\mathrm{par}}$ by minimizing the variational energy.

VMC method. The VMC method evaluates averages over the trial wave function ${\psi }_{T}$. The Metropolis algorithm [68] is used to sample its square, $| {\psi }_{T}{| }^{2}$ by generating a Markov chain with corresponding probability distribution. The average of the Hamiltonian, ${E}_{\mathrm{var}}=\langle {\psi }_{T}| \hat{{ \mathcal H }}| {\psi }_{T}\rangle /\langle {\psi }_{T}| {\psi }_{T}\rangle$, provides an upper bound to the ground state energy E₀. It is interesting to compare the value of ${E}_{\mathrm{var}}$ with prediction of MF theory.

DMC method. The DMC method [42] is based on solving the Schrödinger equation in imaginary time. For large times, the contribution to the energy from the excited states is exponentially suppressed, permitting to obtain the exact ground state energy E₀. The density profile of the polaron can be calculated using the technique of pure estimators [69, 70]. The effective mass of the polaron, ${M}_{{\rm{p}}}$, is obtained by calculating the diffusion coefficient D of the impurity in the imaginary time τ, $D={\mathrm{lim}}_{\tau \to \infty }\langle {[{r}_{{\rm{I}}}(\tau )-{r}_{{\rm{I}}}(0)]}^{2}\rangle /\tau$, according to the relation $D={{\hslash }}^{2}/2{M}_{{\rm{p}}}$. The variational value for ${M}_{{\rm{p}}}$ is obtained from DMC algorithm without branching, which is an alternative method to the Metropolis algorithm to generate the probability distribution according to $| {\psi }_{T}{| }^{2}$.

A physically important limit is that of an impurity with an infinite mass, $M=\infty$. It corresponds to two realistic situations: (i) a pinned impurity, (ii) a static potential created by a focused laser beam. Furthermore this limit is interesting as it is allows to obtain physical insights to the effects of phonon–phonon interactions on the polaron cloud.

When the Bogoliubov approximation is justified—which is the case for $\gamma \ll 1$ or ${n}_{0}| {a}_{\mathrm{BB}}| \gg 1$—we can study this situation using the Gross–Pitaevskii mean-field equation (GPE) [47]:

$$\begin{eqnarray}&&{E}_{{\rm{B}}}\phi (x)=\left[-\displaystyle \frac{{\partial }_{x}^{2}}{2{m}_{{\rm{B}}}}+{g}_{\mathrm{BB}}| \phi (x){| }^{2}+{g}_{\mathrm{IB}}\delta (x)\right]\phi (x).\end{eqnarray} \tag{ 50 }$$

Here $\phi (x)=\langle \hat{\phi }(x)\rangle$ describes the boson field in the 1D system, and ${E}_{{\rm{B}}}$ denotes the total energy ${E}_{{\rm{B}}}$ of the combined impurity–boson system.

The GPE is valid even for strong impurity–boson interactions and goes beyond perturbative expansions in orders of ${g}_{\mathrm{IB}}$. This is different from perturbative theories which linearize boson–impurity interactions, which is justified when ${g}_{\mathrm{IB}}$ is small. Because the boson–boson interactions are treated on a MF level, the validity of GPE is limited to the Bogoliubov regime ${n}_{0}| {a}_{\mathrm{BB}}| \gg 1$ ($\gamma \ll 1$) as we discuss below.

Dark soliton solution. The repulsive nonlinear GPE possesses a famous class of solutions known as gray solitons. These correspond to a depletion in the boson density which maintains its shape while propagating with a constant speed. In the case of zero velocity the density completely vanishes in a single point, and the solution is referred to a a dark soliton. Thus we expect that the effect of a massive impenetrable impurity, with $M,{g}_{\mathrm{IB}}\to \infty$, is to localize a dark soliton in the Bose gas.

The wave function of a dark soliton is given by ${\phi }_{0}(x)=\sqrt{{n}_{0}}\tanh (x/2\xi )$ with the corresponding energy ${E}_{{\rm{B}}}={g}_{\mathrm{BB}}{n}_{0}N/2-2{n}_{0}c/3$. If we take into account that ${\rm{\Delta }}{N}_{\mathrm{sol}}=2\sqrt{2}{n}_{0}\xi$ bosons are repelled from the homogeneous Bose gas, we find that the energy E₀ of the impurity in a system with fixed total particle number N and density n₀ is

$$\begin{eqnarray}&&{E}_{0}^{\mathrm{sol}}=\displaystyle \frac{4}{3}{n}_{0}c.\end{eqnarray} \tag{ 51 }$$

This energy has the same physical meaning as other polaron energies E₀ which we calculated before.

On the other hand, we can calculate the polaron energy starting from an impurity and using MF theory as described in section 5.1. In that case we obtain

$$\begin{eqnarray}&&{E}_{0}^{\mathrm{MF}}=2{n}_{0}c=\displaystyle \frac{3}{2}{E}_{0}^{\mathrm{sol}}\end{eqnarray} \tag{ 52 }$$

by sending ${g}_{\mathrm{IB}}\to \infty$ and $M\to \infty$. The obtained expression overestimates the correct energy by the factor of 3/2. The reason is that in our MF theory we ignore the nonlinearity in equation (50). By making the Bogoliubov approximation we effectively linearize $| \phi {| }^{2}\phi \approx {n}_{0}^{3/2}+2{n}_{0}\delta \phi (x)+{n}_{0}\delta {\phi }^{* }(x)$.

Comparison to DMC. To benchmark our theoretical methods we calculated the energy of an impenetrable, localized impurity for different parameters of the Bose gas. Our comparison in figure 14 shows that DMC is in perfect agreement with the exact result in the limit where Bogoliubov theory is valid, ${n}_{0}| {a}_{\mathrm{BB}}| \gg 1$ or $\gamma \ll 1$. As expected, MF theory and RG differ by the factor 2/3 in this regime.

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In section 5.2 we suggested to use the relative size of LHY corrections to the ground state energy of the Bose gas as an indicator where the Bogoliubov theory can be applied to describe polarons. This leads to the condition that ${n}_{0}| {a}_{\mathrm{BB}}| \gg 1$ (or $\gamma \ll 1$). Although in figure 14 we never obtain quantitative agreement of the Bogoliubov approximation with the numerically exact DMC results, we find that the qualitative behavior of the impurity energy E₀ is correctly described in this framework for ${n}_{0}| {a}_{\mathrm{BB}}| \gg 1$ (or small $\gamma \ll 1$). In the opposite limit ${n}_{0}| {a}_{\mathrm{BB}}| \lesssim 1$ (or $\gamma \gtrsim 1$) in contrast, the Bogoliubov description completely fails. This shows that the LHY corrections provide a reliable measure for the accuracy of the truncated description of the polaron cloud using only the beyond-Fröhlich Hamiltonian ${\hat{{ \mathcal H }}}_{\mathrm{BF}}$.

In figure 15 we compare the polaron energy for repulsive interactions ${g}_{\mathrm{IB}}\gt 0$ and parameters as in the experiment of [7]. For weak-to-intermediate impurity–boson interactions, $| \eta | \lesssim 1$, where the Fröhlich model is valid and the density modulation in the Bose gas is small, all theoretical predictions agree with each other. For stronger couplings we observe sizable quantitative differences. Nevertheless, the qualitative behavior of all results going beyond the Fröhlich Hamiltonian is the same. The corrections of the RG to the MF results is pronounced at large couplings, but DMC predicts even smaller polaron energies in this regime.

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In figure 3 we compared predictions for the effective polaron mass to the experimental results from analyzing polaron oscillations. There we have found large deviations for $\eta \gtrsim 3$. Interestingly this is exactly where the strong coupling regime starts and DMC predicts different energies than our RG approach. In this regime the density modulations of the Bose gas around the impurity are expected to become large. As a result, our analysis of polaron oscillations showed that large deviations from the adiabatic result can be expected. We speculate that this may be related to the large difference between theory and experiment at strong couplings in figure 3.

In [7] it was moreover suggested that $\eta \approx 15$ could also be the point where higher transverse modes are important. Population of such modes would imply that the system can no longer be treated as strictly 1D. However, the argument of [7] was based on a comparison of the bare energy ${g}_{\mathrm{IB}}{n}_{0}$ with the transverse trapping frequency ${\omega }_{\perp }\approx 126/{m}_{{\rm{B}}}{a}_{\mathrm{BB}}^{2}$. As can be seen from figure 15, the relevant polaron energies are well below ${\omega }_{\perp }$ for all interaction strengths. Therefore we conclude that a cross-over into a higher-dimensional regime cannot explain the experimental observations. This supplements our analysis of higher transverse modes in figure 6, where we arrived at the same conclusion.

We conclude by noting that phonon–phonon interactions play an important role for understanding polarons at strong couplings in the experiment by Catani et al [7]. Not only do two-phonon terms beyond the Fröhlich Hamiltonian become important, but also the nonlinear interactions between phonons are required to describe correctly the polaron energy for repulsive interactions. On the attractive side, where ${g}_{\mathrm{IB}}\lt 0$, we expect their influence to be even more dramatic, because larger deformations of the Bose gas around the impurity are possible.

Starting from the MF polaron solution ${\alpha }_{k}^{\mathrm{MF}}$ one can rewrite the Hamiltonian ${\hat{{ \mathcal H }}}_{\mathrm{BF}}={\hat{{ \mathcal H }}}_{{\rm{F}}}+{\hat{{ \mathcal H }}}_{2\mathrm{ph}}$ by an effective Hamiltonian $\tilde{{ \mathcal H }}$ describing quantum fluctuations around the MF solution. Then the high-energy phonons, with momenta larger than a running UV cut-off Λ, are eliminated step by step in the RG. This modifies the MF solution ${\alpha }_{k}({\rm{\Lambda }})$ at small energies [39], and gives rise to an RG flow of the coupling constants when Λ changes. Using the same notation and following the derivation of [40] we obtain an effective Hamiltonian

$$\begin{eqnarray}\tilde{{ \mathcal H }}({\rm{\Lambda }}) & = & {E}_{0}({\rm{\Lambda }})+{\displaystyle \int }^{{\rm{\Lambda }}}{\rm{d}}{k}\left\{{{\rm{\Omega }}}_{k}{\hat{a}}_{k}^{\dagger }{\hat{a}}_{k}+{\displaystyle \int }^{{\rm{\Lambda }}}{\rm{d}}{k}^{\prime} \displaystyle \frac{{kk}^{\prime} }{2{ \mathcal M }}:\,{\hat{{\rm{\Gamma }}}}_{k}{\hat{{\rm{\Gamma }}}}_{k^{\prime} }\,:\right.\\ & & +\left.\displaystyle \frac{{G}_{+}}{2{n}_{0}}\,:\delta {\hat{n}}_{k}\delta {\hat{n}}_{k^{\prime} }\,:+2{n}_{0}{G}_{-}\,:{\hat{\vartheta }}_{k}{\hat{\vartheta }}_{k^{\prime} }\,:\right\}.\end{eqnarray} \tag{ B1 }$$

A few explanations are in order. First of all, note that the impurity operators $\hat{x}$ and $\hat{p}$ have been eliminated by applying the LLP transformation [55] and considering a polaron with vanishing total momentum. The polaron energy ${E}_{0}({\rm{\Lambda }})$, starting at ${E}_{0}^{\mathrm{MF}}$ for ${\rm{\Lambda }}={{\rm{\Lambda }}}_{0}$, decreases until it reaches the ground state energy of the polaron for ${\rm{\Lambda }}\to 0$. The effective phonon frequency in the frame co-moving with the impurity is given by

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{k}({\rm{\Lambda }})={\omega }_{k}+{k}^{2}/2{ \mathcal M }({\rm{\Lambda }}),\end{eqnarray} \tag{ B2 }$$

where ${ \mathcal M }({\rm{\Lambda }})$ denotes the renormalized mass of the impurity. Note that ${ \mathcal M }({\rm{\Lambda }}\to 0)\approx {M}_{{\rm{p}}}$ can be used as an approximation for the effective polaron mass [39].

The last term in the first line of equation (B1) describes phonon–phonon interactions induced by the mobile impurity, where the operators ${\hat{{\rm{\Gamma }}}}_{k}$ are defined as ${\hat{{\rm{\Gamma }}}}_{k}={\hat{a}}_{{\boldsymbol{k}}}^{\dagger }{\hat{a}}_{{\boldsymbol{k}}}+{\alpha }_{k}({\hat{a}}_{{\boldsymbol{k}}}^{\dagger }+{\hat{a}}_{{\boldsymbol{k}}})$. Note that the MF amplitude is flowing in the RG,

$$\begin{eqnarray}&&{\alpha }_{k}({\rm{\Lambda }})=-\displaystyle \frac{\beta ({\rm{\Lambda }}){V}_{k}}{{{\rm{\Omega }}}_{k}({\rm{\Lambda }})}.\end{eqnarray} \tag{ B3 }$$

The second line in equation (B1) is an alternative formulation of the two-phonon scattering terms ${\hat{{ \mathcal H }}}_{2\mathrm{ph}}$ in equation (8), with coupling constants ${G}_{\pm }({\rm{\Lambda }})$ running in the RG. We introduced the following pairs of conjugate operators,

$$\begin{eqnarray}&&\delta {\hat{n}}_{k}=\sqrt{{n}_{0}}{W}_{k}({\hat{a}}_{k}+{\hat{a}}_{-k}^{\dagger }),\end{eqnarray} \tag{ B4 }$$

$$\begin{eqnarray}&&{\hat{\vartheta }}_{k}=\displaystyle \frac{1}{2{\rm{i}}\sqrt{{n}_{0}}}{W}_{k}^{-1}({\hat{a}}_{k}-{\hat{a}}_{-k}^{\dagger }),\end{eqnarray} \tag{ B5 }$$

describing particle number and phase fluctuations of the Bose gas.

The initial conditions for the coupling constants flowing in the RG are given by

$$\begin{eqnarray}&&{G}_{\pm }({{\rm{\Lambda }}}_{0})=\displaystyle \frac{{g}_{\mathrm{IB}}}{4\pi },{ \mathcal M }({{\rm{\Lambda }}}_{0})=M\end{eqnarray} \tag{ B6 }$$

$$\begin{eqnarray}&&{E}_{0}({{\rm{\Lambda }}}_{0})={E}_{0}^{\mathrm{MF}},\beta ({{\rm{\Lambda }}}_{0})={\beta }_{\mathrm{MF}}.\end{eqnarray} \tag{ B7 }$$

In one dimension the RG flow equations read [40],

$$\begin{eqnarray}&&{\partial }_{{\rm{\Lambda }}}{ \mathcal M }=-4{{\rm{\Lambda }}}^{2}{\alpha }_{{\rm{\Lambda }}}^{2}/{{\rm{\Omega }}}_{{\rm{\Lambda }}},\end{eqnarray} \tag{ B8 }$$

$$\begin{eqnarray}&&{\partial }_{{\rm{\Lambda }}}{G}_{\pm }^{-1}=-4{W}_{{\rm{\Lambda }}}^{\pm 2}/{{\rm{\Omega }}}_{{\rm{\Lambda }}},\end{eqnarray} \tag{ B9 }$$

and the RG flow of $\beta ({\rm{\Lambda }})$ is described by

$$\begin{eqnarray}&&\beta ({\rm{\Lambda }})={G}_{+}^{-1}({{\rm{\Lambda }}}_{0}){\left[{G}_{+}^{-1}({\rm{\Lambda }})+2{\int }_{-{\rm{\Lambda }}}^{{\rm{\Lambda }}}{\rm{d}}{p}{W}_{{\rm{p}}}^{2}/{{\rm{\Omega }}}_{{\rm{p}}}\right]}^{-1}.\end{eqnarray} \tag{ B10 }$$

The ground state energy can be determined from

$$\begin{eqnarray}{\partial }_{{\rm{\Lambda }}}{E}_{0} & = & \displaystyle \frac{1}{2}{\partial }_{{\rm{\Lambda }}}{{ \mathcal M }}^{-1}{\displaystyle \int }_{-{\rm{\Lambda }}}^{{\rm{\Lambda }}}{\rm{d}}{p}\,{\alpha }_{{\rm{p}}}^{2}{p}^{2}\\ & & +{\displaystyle \int }_{-{\rm{\Lambda }}}^{{\rm{\Lambda }}}{\rm{d}}{p}\,\displaystyle \frac{2}{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}{({W}_{{\rm{\Lambda }}}{W}_{{\rm{p}}}{G}_{+}-{W}_{{\rm{p}}}^{-1}{W}_{{\rm{\Lambda }}}^{-1}{G}_{-})}^{2}.\end{eqnarray} \tag{ B11 }$$

In three dimensions [40] all coupling constants converge when the cut-off ${\rm{\Lambda }}\to 0$. In that case the RG flows stop when the dispersion relation ${\omega }_{k}$ becomes linear for ${\rm{\Lambda }}\approx 1/\xi$. Except for ${G}_{-}$, this is also true in one dimension. Here the coupling constant ${G}_{-}$ always flows to the weak coupling fixed-point in the IR limit,

$$\begin{eqnarray}&&{G}_{-}({\rm{\Lambda }}\to 0)={0}^{+}.\end{eqnarray} \tag{ B12 }$$

To see this, note that we obtain a divergent RG flow ${\partial }_{{\rm{\Lambda }}}{G}_{-}^{-1}\simeq -1/{{\rm{\Lambda }}}^{2}$ when ${\rm{\Lambda }}\to 0$.

The MF polaron energy ${E}_{0}^{\mathrm{MF}}$ from equation (24) diverges when the IR cut-off λ is sent to zero. The reason for this divergence is the unphysical assumption of MF theory that the coupling constant ${g}_{\mathrm{IB}}$ is unmodified by quantum fluctuations. Now we show that the RG flow of ${G}_{-}$ to the universal weak coupling fixed point ${G}_{-}=0$, see equation (B12), leads to a regularized polaron energy.

We assume an IR cut-off λ, where the RG flow is stopped. As discussed around equation (27) the MF energy ${E}_{0}^{\mathrm{MF}}(\lambda )$ has a contribution

$$\begin{eqnarray}&&{E}_{0}^{\mathrm{MF}}(\lambda )\mathop{\simeq }\limits^{\mathrm{IR}}\displaystyle \frac{{g}_{\mathrm{IB}}}{\pi }{\int }_{\lambda }^{{{\rm{\Lambda }}}_{0}}{\rm{d}}{k}\,{\sinh }^{2}{\theta }_{k}\,\mathop{\simeq }\limits^{\mathrm{IR}}-\displaystyle \frac{{g}_{\mathrm{IB}}\mathrm{log}(\lambda )}{2\sqrt{2}\pi \xi },\end{eqnarray} \tag{ B13 }$$

which diverges logarithmically when $\lambda \to 0$. From the RG we can calculate the polaron energy ${E}_{0}^{\mathrm{RG}}(\lambda )$ by solving equation (B11) for Λ flowing from ${{\rm{\Lambda }}}_{0}$ to λ, i.e. ${E}_{0}^{\mathrm{RG}}(\lambda )={E}_{0}({\rm{\Lambda }}=\lambda )$. From the terms in the second line of equation (B11) we obtain a contribution

$$\begin{eqnarray}&&{\partial }_{{\rm{\Lambda }}}{E}_{0}\,\mathop{\simeq }\limits^{\mathrm{IR}}\,\displaystyle \frac{4{G}_{-}^{2}({\rm{\Lambda }})}{{W}_{{\rm{\Lambda }}}^{2}{{\rm{\Omega }}}_{{\rm{\Lambda }}}}{\displaystyle \int }_{\lambda }^{{\rm{\Lambda }}}{\rm{d}}{p}\,{W}_{{\rm{p}}}^{-2}=({\partial }_{{\rm{\Lambda }}}{G}_{-}){\displaystyle \int }_{\lambda }^{{\rm{\Lambda }}}{\rm{d}}{p}\,{W}_{{\rm{p}}}^{-2}\\ &&\quad \mathop{\simeq }\limits^{\mathrm{IR}}-({\partial }_{{\rm{\Lambda }}}{G}_{-})\sqrt{2}{\xi }^{-1}\mathrm{log}(\lambda ).\end{eqnarray} \tag{ B14 }$$

Integrating this equation yields

$$\begin{eqnarray}&&{E}_{0}^{\mathrm{RG}}(\lambda )\mathop{\simeq }\limits^{\mathrm{IR}}{E}_{0}^{\mathrm{MF}}(\lambda )+\displaystyle \frac{\sqrt{2}\mathrm{log}(\lambda )}{\xi }[{G}_{-}({{\rm{\Lambda }}}_{0})-{G}_{-}(\lambda )].\end{eqnarray} \tag{ B15 }$$

From the explicit solution of the RG flow of ${G}_{-}^{-1}({\rm{\Lambda }})$ we obtain the exact expression

$$\begin{eqnarray}&&{G}_{-}^{-1}(\lambda )={G}_{-}^{-1}({{\rm{\Lambda }}}_{0})+4{\int }_{\lambda }^{{{\rm{\Lambda }}}_{0}}{\rm{d}}{\rm{\Lambda }}\,\displaystyle \frac{{W}_{{\rm{\Lambda }}}^{-2}}{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}\mathop{\simeq }\limits^{\mathrm{IR}}{\lambda }^{-1},\end{eqnarray} \tag{ B16 }$$

from which it follows that ${G}_{-}(\lambda )={ \mathcal O }(1/\lambda )$. I.e. ${G}_{-}$ approaches the weak coupling fixed point ${G}_{-}=0$ with a power-law in λ. Because $\mathrm{log}(\lambda )/\lambda \to 0$ for $\lambda \to 0$, the last term in equation (B15) is irrelevant in the IR limit.

Finally, combining equations (B13), (B15) and using ${G}_{-}({{\rm{\Lambda }}}_{0})={g}_{\mathrm{IB}}/4\pi$ we obtain

$$\begin{eqnarray}&&{E}_{0}^{\mathrm{RG}}(\lambda )\mathop{\simeq }\limits^{\mathrm{IR}}\displaystyle \frac{\sqrt{2}\mathrm{log}(\lambda )}{\xi }\left[{G}_{-}({{\rm{\Lambda }}}_{0})-\displaystyle \frac{{g}_{\mathrm{IB}}}{4\pi }\right]=0\times \mathrm{log}(\lambda ).\end{eqnarray} \tag{ B17 }$$

I.e. the two log-divergent terms cancel exactly and the polaron energy obtained from the RG is fully convergent when the IR cut-off $\lambda \to 0$ becomes small.

Before moving on, a comment is in order about the number of phonons in the polaron cloud, which according to MF theory diverges logarithmically with the IR cut-off. In section 5.1 we argued that this is directly connected to the log-divergence of the MF polaron energy. Now we have proven that the coupling constant ${g}_{\mathrm{IB}}$ gives rise to two different couplings ${G}_{\pm }$ flowing in the RG, where the renormalization of ${G}_{-}$ to zero at low energies regularizes the polaron energy.

We emphasize that the number of phonons in the polaron cloud is still diverging as $\lambda \to 0$, as can be readily checked from analyzing the IR behavior of the renormalized MF amplitudes in equation (B3). As a consequence we expect that the quasiparticle weight Z = 0 for $\lambda \to 0$ also within the RG formalism. Therefore we conclude that the orthogonality catastrophe [60] also exists for mobile impurities interacting with 1D quantum gases. As a direct way of detecting this effect for ultracold atoms, Ramsey interferometry can be used as suggested in [73].

Now we analyze the RG flows of the coupling constants more closely and derive the polaron phase diagram. We work in a regime where phonon–phonon interactions ${\hat{{ \mathcal H }}}_{\mathrm{ph}-\mathrm{ph}}$ can be neglected. We will show that the phase diagram shares all qualitative features with the 3D case discussed in [40].

Static impurity in a non-interacting Bose gas. Let us start by considering the exactly solvable case of an infinitely heavy impurity, $M=\infty$, localized in the origin. Furthermore we assume that the bosons are non-interacting. For repulsive impurity–boson interactions, ${g}_{\mathrm{IB}}\gt 0$, the ground state corresponds to a wave function where all bosons populate the same single-particle state, forming a repulsive polaron. For arbitrarily weak attraction, ${g}_{\mathrm{IB}}\lt 0$, a bound state ${\psi }_{b}(x)$ of bosons to the impurity always exists in one dimension. In this regime the spectrum is unbounded, because ${\psi }_{b}(x)$ can be occupied by any integer number of bosons. Note that the MF and RG theories provide a description of the polaron state at finite energy, where no bosons are bound to the impurity [40]. The polaron is meta-stable on the attractive side because it can decay and form a molecule.

In the RG theory the existence of a bound state is indicated by a divergence of the effective interaction strength during the RG flow, ${G}_{\pm }({\rm{\Lambda }})\to \infty$. For the case without boson–boson interactions described above it holds ${G}_{+}({\rm{\Lambda }})={G}_{-}({\rm{\Lambda }})$. As shown in equation (B12), ${G}_{-}\to {0}^{+}$ always flows to the repulsive weak coupling fixed point. Because the RG flow starts at ${G}_{\pm }({{\rm{\Lambda }}}_{0})={g}_{\mathrm{IB}}/4\pi$, there always exists a divergence ${G}_{-}({{\rm{\Lambda }}}_{c})\to \infty$ at some intermediate ${{\rm{\Lambda }}}_{c}$ on the attractive side ${g}_{\mathrm{IB}}\lt 0$. As explained in detail in [40] this is a direct manifestation for the bound state existing at low energies in this regime.

When the mass of the impurity is slowly decreased, we expect the bound states to remain stable because their energy spacings are sizable. Note however that the finite mass of the impurity introduces correlations between the bosons and requires us to solve a full many-body problem.

Stable repulsive polarons. Now we extend our discussion to finite mass $M\lt \infty$ and non-vanishing boson–boson interactions ${g}_{\mathrm{BB}}\gt 0$. The RG flows of ${G}_{\pm }({\rm{\Lambda }})$, which differ in this case, are shown in figure 16. On the repulsive side, ${g}_{\mathrm{IB}}\gt 0$, the only qualitative change is that ${G}_{+}(0)\gt 0$ saturates at a finite value in the IR limit. In this regime the ground state is a repulsive polaron. In figure 7 we show the density profile of the Bose gas around the impurity, and indeed the impurity repels bosons in this regime. For very strong repulsive interactions, the quasi-1D Bose gas is completely depleted around the impurity, reminiscent of the bubble polarons predicted in this regime in [74] or, equivalently, a dark soliton as described in section 7.2.

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Attractive polarons. As discussed above, ${G}_{-}$ diverges during the RG flow for arbitrary attractive interactions. On the other hand, the flow of the coupling constant ${G}_{+}({\rm{\Lambda }})$ stops in the IR limit due to finite ${g}_{\mathrm{BB}}\gt 0$. For sufficiently weak attraction,

$$\begin{eqnarray}&&0\gt {g}_{\mathrm{IB}}\gt {g}_{\mathrm{IB},{c}}^{\mathrm{RG}}=-2\pi {\left({\int }_{-{{\rm{\Lambda }}}_{0}}^{{{\rm{\Lambda }}}_{0}}{\rm{d}}{\rm{\Lambda }}{W}_{{\rm{\Lambda }}}^{2}/{{\rm{\Omega }}}_{{\rm{\Lambda }}}\right)}^{-1},\end{eqnarray} \tag{ B18 }$$

this effect is sufficient to prevent ${G}_{+}({\rm{\Lambda }})$ from diverging during the RG and we obtain ${G}_{+}(0)\lt 0$. This corresponds to attractive interactions of the impurity with density fluctuations in the Bose gas and gives rise to an attractive polaron. Compared to the MF result for the critical interaction strength ${g}_{\mathrm{IB},c}^{\mathrm{MF}}$, see equation (26), the RG predicts a transition already somewhat earlier,

$$\begin{eqnarray}&&0\geqslant {g}_{\mathrm{IB},c}^{\mathrm{RG}}\geqslant {g}_{\mathrm{IB},c}^{\mathrm{MF}},\end{eqnarray} \tag{ B19 }$$

as a consequence of the renormalized mass ${ \mathcal M }\geqslant M$.

In figure B1 the attractive polaron regime, where $0\gt \eta \gt {\eta }_{{c},\mathrm{RG}}={g}_{\mathrm{IB},{c}}^{\mathrm{RG}}/{g}_{\mathrm{BB}}$, can be identified. Comparison with figure 6 shows that, indeed, this is the regime where the polaron energy is negative. The density profile of the Bose gas around the impurity also shows a pronounced peak for this set of parameters, see figure 7.

It is worth emphasizing that the existence of attractive polarons in one dimension is due solely to non-vanishing boson–boson interactions. As pointed out before, when ${g}_{\mathrm{BB}}=0$ the interactions are always repulsive, ${G}_{\pm }(0)\gt 0$, in the long-wavelength limit ${\rm{\Lambda }}\to 0$. The effect manifests in the relation ${g}_{\mathrm{IB},c}^{\mathrm{RG}}={g}_{\mathrm{IB},c}^{\mathrm{MF}}=0$ for ${g}_{\mathrm{BB}}=0$. In view of this conclusion, the agreement between attractive polaron energies in figure 8(a) is remarkable, because it suggests that indeed a meta-stable polaronic eigenstate exists which has negative energy ${E}_{0}\lt 0$.

The divergence of ${G}_{-}$ during the RG in the attractive polaron regime suggests that there exists a mode bound to the impurity at energies below the polaron. In [40] this effect is discussed in detail and it is shown that the polaron becomes dynamically unstable in this case. Because only ${G}_{-}$ is negative, while ${G}_{+}$ remains positive, the spectrum of ${\hat{{ \mathcal H }}}_{\mathrm{BF}}$ is continuous and unbounded in the attractive polaron regime [40]. In our DMC calculations we fully included phonon–phonon interactions, which are expected to stabilize the attractive polaron [40]. Indeed we find a nodeless state at energies corresponding to the attractive polaron, see figure 8(a).

Break-down of the RG. We find that the number of phonons increases dramatically when ${g}_{\mathrm{IB}}$ approaches ${g}_{\mathrm{IB},c}^{\mathrm{RG}}$. In the regime

$$\begin{eqnarray}&&{g}_{\mathrm{IB},c}^{\mathrm{RG}}\gt {g}_{\mathrm{IB}}\gt {g}_{\mathrm{IB},c}^{\mathrm{MF}},\end{eqnarray} \tag{ B20 }$$

the RG breaks down because the MF amplitude diverges at a finite value of Λ, $\beta ({\rm{\Lambda }})\to \infty$. This is a result of quantum fluctuations of the mobile impurity, because for $M=\infty$ it holds ${g}_{\mathrm{IB},c}^{\mathrm{RG}}={g}_{\mathrm{IB},c}^{\mathrm{MF}}$. Unlike for ${G}_{\pm }({\rm{\Lambda }})$, the divergence of β cannot be regularized. As discussion further in [40] phonon–phonon interactions are required to stop the divergence of the MF amplitude.

Metastable repulsive polarons. When ${g}_{\mathrm{IB}}\lt {g}_{\mathrm{IB},c}^{\mathrm{MF}}$ both coupling constants ${G}_{\pm }({\rm{\Lambda }})$ diverge during the RG flow, see figure B1. While they both start out attractive at high energies, they become repulsive in the long-wavelength limit. Therefore the polaron energy ${E}_{0}\gt 0$ is positive in this regime, corresponding to a repulsive polaron, see figure 6. The repulsive polaron branch is adiabatically connected to the polaronic states realized for repulsive microscopic interactions, similar to the physics of the super-Tonks–Girardeau metastable state [58, 59].

The accumulation of bosons around the impurity is an indicator for molecule formation at ${g}_{\mathrm{IB}}\lt {g}_{\mathrm{IB},c}^{\mathrm{MF}}$. Already for ${g}_{\mathrm{IB}}\gtrsim {g}_{\mathrm{IB},c}^{\mathrm{RG}}$ we find pronounced oscillations in the impurity–boson correlation function, which decay with the distance from the impurity. Indeed, when both coupling constants ${G}_{\pm }({\rm{\Lambda }})$ diverge during the RG, the appearance of a bound state with a discrete energy is expected [24, 40]. This state is adiabatically connected to the molecular bound state discussed above at $M=\infty$ and ${g}_{\mathrm{BB}}=0$. In the spectral function it is expected to give rise to a series of peaks separated by the bound state energy [24].

The possibility to decay into molecular states leads to a finite life-time of repulsive polarons when ${g}_{\mathrm{IB}}\lt 0$, a well-known phenomenon close to a Feshbach resonance in three dimensions [8, 9, 23, 24]. This makes a direct calculation of the polaron energy using DMC difficult, because the polaron is no longer the ground state.

To observe repulsive polarons at ${g}_{\mathrm{IB}}\lt 0$ experimentally, we suggest to study quenches from the strongly repulsive side ${g}_{\mathrm{IB}}=+\infty$ to $-\infty$ where the microscopic interactions are attractive. This approach has successfully been used to realize the super-Tonks–Girardeau regime of an interacting 1D gas, see [58, 59]. We expect that the finite life-time of the repulsive polaron at ${g}_{\mathrm{IB}}\lt 0$ should be observable, for example by using Ramsey interferometry between two spin states which interact differently with the bosons.

Comparison to the experiment. In figure 3 we plotted the critical values ${\eta }_{{c},\mathrm{RG}}$ and ${\eta }_{{c},\mathrm{MF}}$ corresponding to the parameters in [7]. For weakly attractive interactions, $0\gt \eta \gtrsim {\eta }_{{c},\mathrm{RG}}$, the measured values for the effective mass are in good agreement with predictions for an attractive polaron. Around ${\eta }_{{c},\mathrm{RG}}$ the qualitative behavior of the data changes. The range of parameters ${\eta }_{{c},\mathrm{MF}}\lt \eta \lt {\eta }_{{c},\mathrm{RG}}$ where the RG breaks down and we expect a polaron cloud with many phonons is too narrow to draw any conclusions from the comparison.