Inducing spin-order with an impurity: phase diagram of the magnetic Bose polaron

The underlying many-body Hamiltonian

$\begin{equation}\hat{H}=\sum\limits _{\alpha ={\uparrow},{\downarrow}}{\hat{H}}_{\alpha }+\sum\limits _{\alpha ,{\alpha }^{\mathrm{\prime }}={\uparrow},{\downarrow}}\frac{{\hat{H}}_{\alpha {\alpha }^{\mathrm{\prime }}}}{1+{\delta }_{\alpha {\alpha }^{\mathrm{\prime }}}}+{\hat{H}}_{I}+\sum\limits _{\alpha ={\uparrow},{\downarrow}}{\hat{H}}_{\alpha I}+{\hat{H}}_{S},\end{equation} \tag{ 1 }$

with the non-interacting parts of the spinor gas ${\hat{H}}_{\alpha }=\int \mathrm{d}x\;{\hat{{\Psi}}}_{\alpha }^{{\dagger}}(x)\left(-\frac{{\hslash }^{2}}{2{m}_{B}}\frac{{\mathrm{\partial }}^{2}}{\mathrm{\partial }{x}^{2}}+\frac{1}{2}{m}_{B}{\omega }^{2}{x}^{2}\right){\hat{{\Psi}}}_{\alpha }(x)$ where α = ↑, ↓ and the impurity ${\hat{H}}_{I}=\int \mathrm{d}x\,{\hat{{\Psi}}}_{I}^{{\dagger}}(x)\left(-\frac{{\hslash }^{2}}{2{m}_{I}}\frac{{\partial }^{2}}{\partial {x}^{2}}+\frac{1}{2}{m}_{I}{\omega }^{2}{x}^{2}\right){\hat{{\Psi}}}_{I}(x)$. The field operators ${\hat{{\Psi}}}_{\alpha }(x)$ act on the spin-a component of the Bose medium and ${\hat{{\Psi}}}_{I}(x)$ on the impurity atom. Here, we consider ultracold temperatures ensuring the relevance of solely the s-wave interactions that are essentially described by a contact potential [51]. These are characterized by the spin–spin medium effective couplings g_αα' with α, α' = ↑, ↓ and the impurity-spin interaction strengths g_Ia . In this sense, the contact spin–spin interaction term is ${\hat{H}}_{a{a}^{\mathrm{\prime }}}={g}_{\alpha {\alpha }^{\mathrm{\prime }}}\int \mathrm{d}x\;{\hat{{\Psi}}}_{\alpha }^{{\dagger}}(x){\hat{{\Psi}}}_{{\alpha }^{\mathrm{\prime }}}^{{\dagger}}(x){\hat{{\Psi}}}_{{\alpha }^{\mathrm{\prime }}}(x){\hat{{\Psi}}}_{\alpha }(x)$, while the impurity-spin-a interactions are given by ${\hat{H}}_{I\alpha }={g}_{I\alpha }\int \mathrm{d}x\;{\hat{{\Psi}}}_{I}^{{\dagger}}(x){\hat{{\Psi}}}_{\alpha }^{{\dagger}}(x){\hat{{\Psi}}}_{\alpha }(x){\hat{{\Psi}}}_{I}(x)$.

These effective coupling constants are related to the three-dimensional s-wave scattering length, ${a}_{\sigma {\sigma }^{\prime }}^{s}$ (with σ, σ' = ↑, ↓, I being the species index) and the transversal confinement ω_⊥ [51]. Consequently, the interaction coefficients can be experimentally adjusted either via Feshbach resonances [52, 53] through ${a}_{\sigma {\sigma }^{\prime }}^{s}$ or utilizing confinement-induced resonances [51] by tuning ω_⊥.

The spinor contribution facilitating magnetic processes reads

$\begin{equation}{\hat{H}}_{S}=\frac{\hslash {{\Omega}}_{\mathrm{R}}}{2}{\hat{S}}_{x}+\frac{\hslash \delta }{2}{\hat{S}}_{z}.\end{equation} \tag{ 2 }$

The Rabi-coupling Ω_R favors the spin-superposition $(1/\sqrt{2})(\vert {\uparrow}\rangle -\vert {\downarrow}\rangle )$ for the medium atoms and therefore admixes different numbers of spin-↑ and spin-↓ atoms in the ground state. The detuning δ = ν − ν₀ provides the frequency difference of the Rabi-coupling laser from the corresponding non-interacting (g_↑↑ = g_↓↓ = g_↑↓ = 0) resonance transition frequency (ν₀) among the |↑⟩ and |↓⟩ states. Here, we consider δ = 0, corresponding to a resonant driving in the non-interacting case.

The total spin operator is $\hat{\boldsymbol{S}}=\int \mathrm{d}x\;\hat{\boldsymbol{S}}(x)=\int \mathrm{d}x{\;\sum }_{\alpha {\alpha }^{\mathrm{\prime }}}{\hat{{\Psi}}}_{\alpha }^{{\dagger}}(x){\boldsymbol{\sigma }}_{\alpha {\alpha }^{\mathrm{\prime }}}{\hat{{\Psi}}}_{{\alpha }^{\mathrm{\prime }}}(x)$, where σ is the Pauli vector. Accordingly, the total spin magnitude operator is expressed as

$\begin{equation}{\hat{S}}^{2}={\hat{S}}_{+}{\hat{S}}_{-}+{\hat{S}}_{z}({\hat{S}}_{z}-2),\end{equation} \tag{ 3 }$

where the so-called spin ladder operators ${\hat{S}}_{\pm }={\hat{S}}_{x}\pm i{\hat{S}}_{y}$. Importantly, the Hamiltonian of equation (1) preserves the SU(2) symmetry, i.e. $[{\hat{S}}^{2},\hat{H}]=0$ only in the case of g_I↑ = g_I↓ and g_↑↑ = g_↓↓ = g_↑↓. Otherwise, the eigenvalues of ${\hat{S}}^{2}$ are not good quantum numbers. Also, in the presence of Rabi-coupling, Ω_R ≠ 0, S_z is not conserved, allowing the system to become spin-imbalance, i.e. attain a superposition of |↑⟩ and |↓⟩. ⁸ Here, we focus on g_↑↑ = g_↓↓ ≡ g and therefore in the absence of the impurity, the system respects the ${\mathbb{Z}}_{2}$ symmetry implying invariance under spin-inversion along the z-spin axis. However, since the impurity can exhibit different interactions with each of the spin-components, namely g_I↑ ≠ g_I↓, breaking ${\mathbb{Z}}_{2}$ it can thus modify the magnetization of the system because S_z is broken.

In the following, the many-body Hamiltonian described by equation (1) is expressed in terms of ℏω. Then, the length, time, Rabi-coupling and interaction strengths are provided with respect to $\sqrt{\hslash /(m\omega )}$, ω, and $\sqrt{({\hslash }^{3}\omega )/m}$ respectively. Experimentally our 1D multicomponent setup is realizable, for instance, by using two hyperfine states of ⁸⁷Rb (see above) with ${g}_{{\uparrow}{\uparrow}}={g}_{{\downarrow}{\downarrow}}=0.5\sqrt{{\hslash }^{3}\omega /m}\approx 3.55\times 1{0}^{-38}$ J m and a longitudinal (transversal) trap frequency ω = 2π × 100 Hz (ω_⊥ ≈ 2π × 5.1 kHz). The respective temperature effects are suppressed as long as the condition ${k}_{\mathrm{B}}T\ll \frac{{3}^{4/3}}{16}{(\frac{{\alpha }_{\perp }^{2}{N}_{{\uparrow}}^{2}}{{a}_{{\uparrow}{\uparrow}}^{s}\alpha })}^{2/3}\hslash \omega =316\hslash \omega \approx 1.5$ μK is fulfilled [54]. In the latter expression k_B is the Boltzmann constant, ${\alpha }_{\perp }=\sqrt{\hslash /(m{\omega }_{\perp })}$ denotes the transversal confinement length scale and T refers to the temperature of the spinor Bose gas.

In an experiment, our results can be probed by coupling the spin components of the medium via a radiofrequency field [55]. The corresponding implementation involves an adiabatic ramp (within a time-interval $\tau \gg {\omega }_{B}^{-1}$) of Ω_R. The rf field creates a superposition spin state |↑⟩ and |↓⟩ that crucially depends on g_αα' and g_Iα .

In the absence of impurity, the considered setting reduces to a binary Rabi-coupled Bose–Einstein condensate (BEC) [49, 55–57]. The fundamental building block of such a Rabi-coupled mixture is the case of Ω_R = 0, where it boils down to a two-component mixture with particle number conservation in each component. Then, it is well-known [58] that the system enters the phase-separated (miscible) state when ${g}_{{\uparrow}{\downarrow}}^{2} > {g}_{{\uparrow}{\uparrow}}{g}_{{\downarrow}{\downarrow}}$ (in the opposite case). Switching on the Rabi-coupling breaks the S_z spin symmetry of the gas meaning that spin-transfer between the components is allowed. As a general rule the effect of the Rabi-coupling reduces the degree of immiscibility (or spin segregation) even when the spin interactions lie deeply inside the immiscible region. Specifically, for a larger Ω_R the gas features a second-order phase transition entering the miscible state and spin-demixing is prohibited [49, 56, 57], see more details in appendix B and also figure 11.

The spinor bosonic medium is initially in the ground state configuration, characterized by specific spin–spin interactions g_↑↑ = g_↓↓ ≡ g and g_↑↓ and Rabi-coupling Ω_R. In the absence of the impurity, i.e. g_I↑ = g_I↓ = 0 or equivalently $\langle {\hat{H}}_{I\alpha }\rangle =0$, the spins are aligned and the associated spin configuration is almost ⁹ identical to a fully polarized state along the x-axis, see figure 1. It reads $\vert {P}_{x}\rangle ={2}^{-N/2}{\bigotimes}_{i=1}^{N}\left(\vert {{\uparrow}\rangle }_{i}-\vert {{\downarrow}\rangle }_{i}\right)$ with i = 1, 2, ..., N. Then, for g_I↑ ≠ g_I↓ i.e. when the impurity-medium interaction is spin-dependent, the initial polarization vector $\langle \hat{\mathbf{S}}\rangle =-N{\mathbf{e}}_{x}$, with e_x the unit vector, rotates away from the x-spin axis.

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Importantly, the localization of the impurity determined by its spatial width acts as a local perturber distorting the polarization of the medium. The local spin vector $\langle \hat{\mathbf{S}}(x)\rangle \ne \langle \hat{\mathbf{S}}({x}^{\prime })\rangle$, for x ≠ x', is strongly influenced by the degree of localization of the impurity, see figure 1, in turn dressing the impurity with the spin fluctuations. This process gives rise to the magnetic polaron. An adequate measure to identify the degree of local perturbation in the bath is the spin–spin correlation function (see equation (13) below). For cases that the spin-order of the host remains intact by the presence of the impurity, the emergent quasiparticle will be referred to as a non-magnetic polaron.

To address the magnetic polaron properties, we employ the variational ML-MCTDHX approach which has been extensively used to study impurity dynamics and spectroscopy [23, 24, 26, 27, 45]. ML-MCTDHX is based on the expansion of the many-body wave function in terms of a time-dependent and variationally optimized basis set, see appendix D for details. It is tailored to capture interparticle spatial and spin–spin correlations [27], while efficiently truncating the Hilbert space, even for mesoscopic particle numbers. To expose the role of correlations, we compare our results with the predictions of the three-component Gross–Pitaevskii equation (GPE) (equation (16)). An effective potential model is constructed, see in particular appendix C and equation (4), for the interpretation of the mechanisms accompanying the generation and properties of the magnetic polaron.

To elucidate the impact of the impurity on the spatial distribution of the medium's spin components we invoke the one-body density of each component

$\begin{equation}{\rho }_{\sigma }(x)=\langle {\Psi}\vert {\hat{{\Psi}}}_{\sigma }^{{\dagger}}(x){\hat{{\Psi}}}_{\sigma }(x)\vert {\Psi}\rangle .\end{equation} \tag{ 6 }$

Here, ${\hat{{\Psi}}}_{\sigma }(x)$ is the σ = ↑, ↓, I-component bosonic field operator acting at position x and |Ψ⟩ denotes the many-body ground state of the three-component system. This particle density can be observed by single-shot averaging [63]. Notice that ${\rho }_{{\uparrow}}(x)+{\rho }_{{\downarrow}}(x)={\rho }_{\mathrm{T}\mathrm{F}}(x)=(2/(g+{g}_{{\uparrow}{\downarrow}}))({\mu }_{B}-m{\omega }^{2}{x}^{2}+\left\vert {{\Omega}}_{\mathrm{R}}\right\vert /2)$, see also appendix B, and thus below we only present the ρ_↑(x) since the structures emerging in ρ_↓(x) are complementary to it. The dependence of ρ_σ (x) with the bath-impurity coupling is shown in figure 3 for g_I↑ ranging from attractive to strong repulsive with g_I↓ = 0 and Ω_R = 5ω. This value of ${{\Omega}}_{\mathrm{R}} > {{\Omega}}_{\mathrm{R}}^{\text{crit}}\approx 4.8\omega$ enforces the miscibility among the spin components in the absence of the impurity, i.e. when g_I↑ = 0 or equivalently in the case of no polaron, see also section 3 and appendix B.

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Finite repulsive impurity-spin-↑ interactions lead to a depletion of ρ_↑(x) around the trap center, see figure 3(a). Simultaneously, the impurity features a progressive localization tendency for larger g_I↑ (figure 3(b)) thus forming a quasiparticle. This is the first key difference to the non-magnetic repulsive Bose polaron [24, 31] which is known to delocalize for increasingly repulsive impurity-host interactions causing its decay. Turning to g_I↑ < 0, a behavior similar to the attractive non-magnetic Bose polaron [24] is detected. Here, the spin-↑ component shows a sizable density peak at the location of the impurity (figure 3(a)), with the concomitant localization of the impurity (figure 3(b)). However, the most striking distinction of the quasiparticle realized herein from the non-magnetic polaron stems from the response of the spin-↓ state. Indeed, ρ_↓(x) is complementary to ρ_↑(x), with the former being accumulated (depleted) in the spatial extent of the impurity for repulsive (attractive) g_I↑ (not shown). This behavior of ρ_↓(x) is mediated by the immiscible g_↑↓ interactions which lead to an effective attraction, ${g}_{I{\downarrow}}^{\text{eff}}< 0$ (repulsion, ${g}_{I{\downarrow}}^{\text{eff}} > 0$), among the spin-↓ and the impurity for g_I↑ > 0 (g_I↑ < 0), see also figures 4(a₅) and (c₅) and the related discussion in section 4.2. In general, these effective interactions are mediated by the magnonic excitations due to the finite interaction difference g_I↑ − g_I↓. Notably, our observations necessitate the construction of effective Hamiltonians similar to [64] in order to appreciate the strength of induced interactions in three-component settings.

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The above-described behavior of ρ_↑(x) and ρ_↓(x) hints towards the presence of magnetic processes in the system. These processes can be analyzed by studying the local magnetization response, $\langle {\hat{S}}_{z}(x)\rangle ={\rho }_{{\uparrow}}(x)-{\rho }_{{\downarrow}}(x)$ provided in figure 3(c). The emergence of local magnetization in the host is related to a spatially varying effective magnetic field, Δ(x) ∼ (g_I↑ − g_I↓)ρ_I (x), due to the impurity-spin-↑ interactions (see also figure 1) which in turn gives rise to standing spin-waves. Without the impurity (g_I↑ = 0), it holds that $\langle {\hat{S}}_{z}(x)\rangle =0$, due to the polarization of the host in the spin-x axis, while in the case of g_I↑ > 0 (g_I↓ < 0), we find $\langle {\hat{S}}_{z}(x)\rangle < 0$ $(\langle {\hat{S}}_{z}(x)\rangle > 0)$. It is therefore the presence of the impurity that triggers a g_I↑-dependent correlated standing spin-wave, which we analyze further in sections 4.3 and 4.4.

A more concrete demonstration of the spatial distributions of the magnetic polaron is obtained by inspecting the underlying two-body configurations. For this reason, we determine the diagonal elements of the two-body reduced density matrix

$\begin{equation}{\rho }_{\sigma {\sigma }^{\prime }}^{(2)}({x}_{1},{x}_{2})=\langle {\Psi}\vert {\hat{{\Psi}}}_{\sigma }^{{\dagger}}({x}_{2}){\hat{{\Psi}}}_{{\sigma }^{\prime }}^{{\dagger}}({x}_{1}){\hat{{\Psi}}}_{{\sigma }^{\prime }}({x}_{1}){\hat{{\Psi}}}_{\sigma }({x}_{2})\vert {\Psi}\rangle .\end{equation} \tag{ 7 }$

This provides the probability to simultaneously detect a σ-component boson at position x₁ and a σ' atom at x₂. The spatially resolved two-body configurations for the immiscible interacting medium with Ω_R/ω = 5 exhibit involved structures for both attractive (figures 4(a₁)–(a₅)) and repulsive (figures 4(b₁)–(b₅)) impurity-spin-↑ interaction strengths. Otherwise, for g_I↑ = 0 they feature a symmetry under the exchange of x₁ ↔ x₂ and a localization around x₁ = x₂ = 0. This implies that the detection of the two atoms of the same or different components are largely independent, while the atoms are likely to reside around the trap center.

As explained above, an attractive impurity-medium coupling enforces the spin-↑ bosons to lie in the vicinity of the impurity. Consequently, ${\rho }_{{\uparrow}{\uparrow}}^{(2)}({x}_{1},{x}_{2})$ is highly localized (figure 4(a₁)) around the trap center demonstrating the portion of spin-↑ atoms bound to the impurity, see in particular ${\rho }_{{\uparrow}{\uparrow}}^{(2)}({x}_{1},{x}_{2})$ (figure 4(a₄)). The low density tails of ${\rho }_{{\uparrow}{\uparrow}}^{(2)}({x}_{1},{x}_{2})$ infer the existence of spin-↑ bosons that remain unbound. On the other hand, ${\rho }_{{\downarrow}{\downarrow}}^{(2)}({x}_{1},{x}_{2})$ exhibits localization in four disjoint spatial domains (figure 4(a₂)) stemming from the phase-separation between the spin components which is evident by the elongated two-hump shape of ${\rho }_{{\uparrow}{\downarrow}}^{(2)}({x}_{1},{x}_{2})$ (figure 4(a₃)). As a consequence, the impurity and the spin-↓ are also phase-separated, despite being non-interacting g_I↓ = 0 (see figure 3(b)). Therefore, ${\rho }_{I{\downarrow}}^{(2)}({x}_{1},{x}_{2})$ shows a finite probability at two distinct domains characterized by |x_↓| > |x_I | (figure 4(a₅)) which certifies the emergence of repulsive induced impurity-spin-↓ interactions caused by the attractive g_I↑.

Turning to strong repulsive g_I↑, an inversion in the roles of the spin-↑ and spin-↓ bosons takes place, compare e.g. figures 4(a₂) and (b₁). Indeed, in this case the spin-↑ atoms and the impurity tend to phase-separate (figure 4(b₄)) due to their strong repulsion g_I↑ ≫ g, while the spin-↓ particles are effectively attracted towards the impurity (figure 4(b₅)), i.e. ${g}_{I{\downarrow}}^{\text{eff}}< 0$ besides the fact that g_I↓ = 0. Similar to the non-magnetic polaron case [24, 31], we observe that also here the quasiparticle character in the repulsive branch is less pronounced than in its attractive counterpart. Hence, the emergent patterns in figure 4(b_i ) are less prominent than those depicted in figure 4(a_i ) with i = 1, ..., 5.

To reveal the imprint of the impurity into the spatial configuration of the spinor gas, we operate in the co-moving impurity frame. This is the natural frame of reference to measure the magnonic or phononic dressing cloud since it avoids effects stemming from the dispersion of the impurity within the confining potential. Such a procedure has been already successfully implemented in the experiment for monitoring the internal structure of magnetic Fermi polarons [65]. Focusing on the bosonic case described above and in order to extract the spatial distribution of the polaron dressing cloud, we consider the following two-body density ansatz

$\begin{equation}{\rho }_{I\alpha }^{(2)}({x}_{\alpha },{x}_{I})\approx [{\rho }_{\alpha }^{0}({x}_{\alpha })+{u}_{\alpha }({x}_{\alpha }-{x}_{I})]{\rho }_{I}^{0}({x}_{I}),\end{equation} \tag{ 8 }$

where ${\rho }_{\alpha }^{0}(x)$, ${\rho }_{I}^{0}(x)$ are the background ¹² densities of the spin-α component and the impurity respectively. These do not directly contribute to the binding of the impurity to the host excitations, but account for the density inhomogeneity originating from the harmonic confinement. The function u_α (x_α − x_I ) captures the impurity dressing by the atoms of its host. It is assumed to be solely a function of x_α − x_I , due to the short range character of the impurity-host interactions. As such, it results in significant variations of the BEC density only for |x_α − x_I | ∼ ξ_α , where ξ_α is the healing length of the spin-a component. For a homogeneous system ${\rho }_{\alpha }^{0}(x)\to {N}_{\alpha }/L$ and ${\rho }_{I}^{0}(x)\to 1/L$, where L is the length of the system, and thus ${\rho }_{I\alpha }^{(2)}({x}_{\alpha },{x}_{I})$ reduces to the form expected within the Lee–Low–Pines mean-field description [16, 44, 66]. Therefore, we anticipate that equation (8) describes adequately the structure of ${\rho }_{I\alpha }^{(2)}({x}_{a},{x}_{I})$ only in the case that the local density approximation is justified, i.e. ξ_α ≪ R_out.

Provided that equation (8) is valid we can extract information regarding the dressing cloud of the impurity by employing the impurity-spin-α correlation function in the comoving frame x_r = x_α − x_I , namely

$\begin{align}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})\equiv \int \mathrm{d}{x}_{I}\;{\rho }_{I\alpha }^{(2)}({x}_{I}+{x}_{r},{x}_{I})& \approx {u}_{\alpha }({x}_{r})+\int \mathrm{d}{x}_{I}\;{\rho }_{\alpha }^{0}({x}_{I}+{x}_{r},{x}_{I}){\rho }_{I}^{0}({x}_{I}).\end{align} \tag{ 9 }$

Then ${\Delta}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})={\bar{\rho }}_{I{\uparrow}}^{(2)}({x}_{r})-{\bar{\rho }}_{I{\downarrow}}^{(2)}({x}_{r})$ is related to the existence of spin-wave excitation (magnon) dressing, captured by u_↑(x_r ) − u_↓(x_r ). Moreover, ${\bar{n}}_{IB}^{(2)}({x}_{r})={\sum }_{\alpha ={\uparrow},{\downarrow}}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})$ is associated to the presence of phonons. Note that for the case of a miscible spinor host (here achieved for Ω_R = 5ω) it holds that $\langle {\hat{S}}_{z}({x}_{\alpha })\rangle =0$ for g_I↑ = 0. Therefore, we expect that any deviation in the magnetization of the system for g_I↑ ≠ 0 stems from the magnetic dressing cloud of the impurity. This implies that ${\rho }_{{\uparrow}}^{0}(x)={\rho }_{{\downarrow}}^{0}(x)$ and consequently ${\Delta}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})={u}_{{\uparrow}}({x}_{r})-{u}_{{\downarrow}}({x}_{r})$. In this way, ${\Delta}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})$ captures the waveform of the standing spin-wave dressing cloud. We present ${\Delta}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})$ for Ω_R = 5ω with respect to g_I↑ in figure 5(a). It features ${\Delta}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})< 0$ $({\Delta}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r}) > 0)$ for g_I↑ > 0 (g_I↑ > 0) indicating the respective magnetization tendency. The shape of ${\Delta}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})$ is found upon fitting to be well-described by

$\begin{equation}{\Delta}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})=-\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({g}_{I{\uparrow}}+{g}_{I{\downarrow}}){A}_{\text{mag}}\;{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^{2}\left(\frac{{x}_{r}}{w}\right),\end{equation} \tag{ 10 }$

with A_mag (w) denoting its amplitude (width). We expect that by operating in the Lee–Low–Pines framework one should be able to extract equation (10) as the solution of the magnetic polaron dressing cloud, an analysis that will be addressed in a future work.

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On the other hand, it is more involved to extract the phononic dressing cloud as the non-vanishing contribution from the last term of equation (9) implies ${u}_{{\uparrow}}({x}_{r})+{u}_{{\downarrow}}({x}_{r})\ne {\bar{n}}_{IB}^{(2)}({x}_{r})$. To account for this correction we employ the following spin-independent pair-correlation ansatz

$\begin{equation}{\bar{n}}_{IB}^{(2)}({x}_{r})={n}_{\text{TF}}({x}_{r})-\underset{={u}_{{\uparrow}}({x}_{r})+{u}_{{\downarrow}}({x}_{r})}{\underbrace{{A}_{\text{ph}}{n}_{\text{TF}}({x}_{r}){sech}^{2}\left(\frac{{x}_{r}}{{w}^{\prime }}\right)}}.\end{equation} \tag{ 11 }$

Here, w' refers to the width of the polaron and A_ph is the amplitude of the phonon dressing cloud. In the above expression we have assumed a TF (Gaussian) background density for the BEC (impurity). With this choice the background two-body density, n_TF(x_r ), reads

$\begin{align}\hfill {n}_{\text{TF}}({x}_{r})& =\int \mathrm{d}{x}_{I}\left[\sum\limits _{\alpha \in \left\{{\uparrow},{\downarrow}\right\}}{\rho }_{\alpha }^{\text{TF}}({x}_{I}+{x}_{r})\right]{\rho }_{I}^{l}({x}_{I})\hfill \\ \hfill & =\frac{m{\omega }^{2}}{4{g}_{\text{eff}}}\left[\left(\text{erf}\frac{{x}_{r}+{R}_{\text{out}}}{l}+\text{erf}\frac{{x}_{r}-{R}_{\text{out}}}{l}\right)\left({R}_{\text{out}}^{2}-\frac{{l}^{2}}{2}-{x}_{r}^{2}\right)+\frac{l({R}_{\text{out}}-{x}_{r})}{\sqrt{\pi }}\,{\mathrm{e}}^{-\frac{{({R}_{\text{out}}+{x}_{r})}^{2}}{{l}^{2}}}\right.\hfill \\ \hfill & \quad \left.+\frac{l({R}_{\text{out}}+{x}_{r})}{\sqrt{\pi }}\,{\mathrm{e}}^{-\frac{{({R}_{\text{out}}-{x}_{r})}^{2}}{{l}^{2}}}\right],\hfill \end{align} \tag{ 12 }$

where ${\rho }_{\alpha }^{\text{TF}}(x)$ corresponds to the TF profile of the spin-α BEC component (see also appendix B) and ${\rho }_{I}^{l}({x}_{I})=\frac{1}{l\sqrt{\pi }}\,{\mathrm{e}}^{-\frac{{x}^{2}}{2{l}^{2}}}$. Also, the fitting parameters R_out and g_eff account for the width and height of the BEC density profile, while l corresponds to the width of the impurity density. Recall that in the un-trapped case using the Lee–Low–Pines transformation leads to a similar form to equation (11) for the dressing cloud of the Bose polaron [44, 66, 67].

Upon fitting the ansatz of equation (10) to ${\Delta}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})$ and the one of equation (11) into ${\sum }_{\alpha ={\uparrow},{\downarrow}}{\bar{\rho }}_{Ia}^{(2)}({x}_{r})$ we determine the width and amplitude of the magnetic polaron and the non-magnetic polaron respectively. Next, in order to discern the dressing cloud stemming from the magnons and the phonons we find the number of medium atoms lying in the waveforms u_↑(x_r ) ± u_↓(x_r ), see equations (10) and (11). These populations, ${N}_{\text{mag}}=\int \mathrm{d}{x}_{r}{\Delta}{\bar{\rho }}_{I\alpha }^{(2)}({x}_{r})$ and ${N}_{\text{ph}}=\int \mathrm{d}{x}_{r}({\bar{n}}_{IB}^{(2)}({x}_{r})-{n}_{\text{TF}}({x}_{r}))$ depicted in figures 5(b) and (c) feature an increasing tendency for a larger magnitude of impurity-spin-↑ interactions testifying quasiparticle formation. It can also be inferred that independently of g_I↑, the magnonic excitations prevail over the phononic ones. This means that the emergent quasiparticle, being genuinely dressed by an admixture of magnons and phonons, possesses a dominant magnetic component. Particularly, the magnetic dressing cloud acquires a maximal value of about 30% of the medium atoms around ${{\Omega}}_{\mathrm{R}}^{\text{crit}}\approx 4.8$, while the phonon branch is at most 3.5%.

For ${{\Omega}}_{\mathrm{R}}=10\omega > {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, the magnonic excitation branch is again pronounced as compared to the phononic one especially for g_I↑ < 0, while both branches are suppressed for g_I↑/g > 3.8 where phase-separation occurs. Naturally, if Ω_R = 0 magnetic dressing is diminished for every g_I↑, while the phononic one is finite for g_I↑ < 0 since only in this case the impurity lies within the spin-↑ host. Regarding ${{\Omega}}_{\mathrm{R}}=40\omega \gg {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, we observe that N_mag is suppressed. In this scenario, the spin degrees-of-freedom are nearly frozen because of the large energy gap for exciting spin-waves [56]. Concluding, from the above we can deduce that the magnonic cloud should be easier experimentally detectable when compared to its phononic counterpart.

To inspect the rise of spin-fluctuations in the medium and thus attest the emergence of the magnetic polaron we track the spin–spin correlation [68]

$\begin{equation}{C}_{{\uparrow}{\downarrow}}^{(2)}=\frac{\langle {\Psi}\vert {\hat{S}}^{2}\vert {\Psi}\rangle -3{\hslash }^{2}N}{{\hslash }^{2}N(N-1)}.\end{equation} \tag{ 13 }$

Here, ${\hat{S}}^{2}$ is the total-spin operator of the system defined in equation (3). This correlation function probes the alignment among two spins and distinguishes ferromagnetic ${C}_{{\uparrow}{\downarrow}}^{(2)}\approx 1$, antiferromagnetic ${C}_{{\uparrow}{\downarrow}}^{(2)}\approx -1$ and paramagnetic ${C}_{{\uparrow}{\downarrow}}^{(2)}=0$ spin configurations ¹³ . It is showcased in figure 6(a) for immiscible spin–spin interactions and several Rabi-couplings as a function of the impurity-spin-↑ interaction strength. As it can be readily seen, for suppressed Rabi-coupling (Ω_R = 0) the medium remains ferromagnetic (i.e. ${C}_{{\uparrow}{\downarrow}}^{(2)}=1$) irrespectively of g_I↑. Recall that the interacting eigenstate of the multicomponent setting corresponds to the one where all atoms are in the spin-↑ (↓) state for g_I↑ > 0 (g_I↑ < 0). In this scenario, a polaron dressed by the phononic excitations of the bosonic bath occurs as long as g_I↑ < g + g_↑↓. Otherwise, an impurity-medium phase-separation takes place evincing the polaron decay [24].

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In sharp contrast, switching on the Rabi-coupling results generally to the suppression of the ferromagnetic order as evidenced by the reduction of ${C}_{{\uparrow}{\downarrow}}^{(2)}$ for increasing $\left\vert {g}_{I{\uparrow}}\right\vert$ (figure 6(a)). Particularly, spin correlations become enhanced within the critical magnetization region ${{\Omega}}_{\mathrm{R}}^{\text{crit}}\approx {n}_{B}(0)({g}_{{\uparrow}{\downarrow}}-{g}_{{\uparrow}{\uparrow}})\approx 4.8$ where spin-mixing is dominant (appendix B). This behavior of ${C}_{{\uparrow}{\downarrow}}^{(2)}$ testifies the existence of the magnetic polaron being dressed by the spin-wave excitations of the spinorial medium [37], see also the discussion in section 4.5. Notably, ${C}_{{\uparrow}{\downarrow}}^{(2)}\approx 1$ for ${{\Omega}}_{\mathrm{R}}\gg {{\Omega}}_{\mathrm{R}}^{\mathrm{c}\mathrm{r}}$, see e.g. Ω_R/ω = 10, where impurity spin-↑ spatial separation occurs. On the other hand, ${C}_{{\uparrow}{\downarrow}}^{(2)}\ne 1$ for g_I↑ < 0 (g_I↑ > 0) supporting the occurrence of a self-bound attractive (stable repulsive) magnetic Bose polaron.

A similar to the above-described response of the spin-fluctuations takes place for a heavy impurity (m_I ≫ m_B ) or finite impurity spin-↓ couplings, see figure 6(b). Notice that ${C}_{{\uparrow}{\downarrow}}^{(2)}\to 1$ for g_I↓/g > 1.6, since then a phase-separation between the impurity and the spinor medium is favored as long as g_I↑ > g + g_↑↓ − g_I↓.

Having exemplified the role of spin-fluctuations caused exclusively by the impurity-medium interactions, we next aim to unravel the accompanied spin-demixing mechanisms. These refer to the migration of spin-↑ to spin-↓ particles and vice versa leading ultimately to interaction dependent spin configurations. The latter naturally provide further evidence of the presence of spin-wave excitations identified in figure 5(a). These are quantified herein by the portion of the spin-flipped atoms with respect to the g_I↑ = 0 configuration. To capture the spin-demixing of the bosonic medium, due to g_I↑ ≠ 0, for various Rabi-couplings Ω_R, we monitor the fraction of bosons in each spin-α component

$\begin{equation}{\mathcal{P}}_{\alpha }({{\Omega}}_{\mathrm{R}};{g}_{\alpha {\alpha }^{\mathrm{\prime }}},{g}_{I\alpha })=\frac{\langle {N}_{\alpha }({{\Omega}}_{\mathrm{R}};{g}_{\alpha {\alpha }^{\mathrm{\prime }}},{g}_{I\alpha })\rangle }{N},\end{equation} \tag{ 14 }$

where N = N_↑ + N_↓. The respective fraction of spin-α atoms for several Ω_R is depicted in figure 7(a). In the miscible regime, i.e. ${{\Omega}}_{\mathrm{R}} > {{\Omega}}_{\mathrm{R}}^{\text{crit}}\approx 4.8\omega$, it holds that ${\mathcal{P}}_{\alpha }({{\Omega}}_{\mathrm{R}};g,{g}_{I\alpha }=0)=1/2$, i.e. ${\Delta}\mathcal{P}={\mathcal{P}}_{{\uparrow}}-{\mathcal{P}}_{{\downarrow}}=0$. For g_I↑ > 0 (g_I↑ < 0), ΔP decreases (increases). In the case of ${{\Omega}}_{\mathrm{R}}\gg {{\Omega}}_{\mathrm{R}}^{\text{crit}}$ and g_I↑ > g + g_↑↓, ${\Delta}\mathcal{P}=0$ due to phase-separation, see also the phase diagram of figure 2(a) and the effective potential depicted in figure 2(b₃). For instance, at ${{\Omega}}_{\mathrm{R}}/\omega =10\gg {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, the initial decreasing trend of ${\Delta}\mathcal{P}$ for larger g_I↑/g > 0 is followed by an abrupt saturation towards ${\Delta}\mathcal{P}\to 0$ around g_I↑/g = 1 + g_↑↓/g = 2.6, due to the phase-separation between the impurity and its spinor environment in this region; see light and dark blue lines in figure 7(a). A similar abrupt behavior with respect to g_I↑/g is also visible in other observables such as the residue (figure 8(a)) and the energy (figure 9). In the immiscible scenario, ${{\Omega}}_{\mathrm{R}}< {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, ${\Delta}\mathcal{P}\ne 0$ even for g_I↑ = 0 and it follows the same behavior as in the miscible regime with g_I↑. For Ω_R = 0, ${\Delta}\mathcal{P}=\pm 1$, because the spinor two-component host reduces to a single-component since ${{\Omega}}_{\mathrm{R}}< {{\Omega}}_{\mathrm{R}}^{\text{single}}$ (see also figure 2).

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This behavior of ${\Delta}\mathcal{P}$ in the miscible regime (here Ω_R = 5ω) is not altered also for a finite value of g_I↓ as long as g_I↓ < g, see figure 7(b). Of course, due to g_I↓ ≠ 0, the population balance scenario is achieved for g_I↑ = g_I↓. This becomes more pronounced for g_I↓/g = 1, where we also observe that ${\Delta}\mathcal{P}\to 0$ when g_I↑ > g_↑↓, see appendix C. The fact that ${\Delta}\mathcal{P}\to 0$, is once more a manifestation of the impurity-medium phase-separation but now for finite g_I↓. Consequently, the impurity lies outside of the BEC and thus spin-demixing is diminished. Such a suppression of ${\Delta}\mathcal{P}$ for g_I↑ + g_I↓ > g + g_↑↓ takes equally place upon considering ${{\Omega}}_{\mathrm{R}}\gg {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, which in general produces a reduction of ${\Delta}\mathcal{P}$ for each g_I↑ (not shown).

The quasiparticle residue [11] refers to the overlap between the non-interacting (g_Iα = 0) and the interacting (polaronic) state

$\begin{equation}\mathcal{Z}=\vert \langle {\Psi}({g}_{I\alpha })\vert {\Psi}({g}_{I\alpha }=0)\rangle \vert ,\end{equation} \tag{ 15 }$

with α = ↑, ↓. It can be experimentally tracked e.g. via radiofrequency spectroscopy [3, 5]. The behavior of $\mathcal{Z}$ with respect to g_I↑ and distinct Ω_R is presented in figure 8(a). Apparently, for Ω_R = 0 the residue $\mathcal{Z}=1$ for g_I↑ > 0 implying that the impurity is not dressed by its environment, thus corresponding to a free particle. This is in contrast to g_I↑ < 0, where $\mathcal{Z}$ reduces for stronger attractions being a consequence of the attractive polaron dressed by the phononic excitations of the bosonic medium. Turning to larger Ω_R, in the vicinity of ${{\Omega}}_{\mathrm{R}}^{\text{crit}}$, we find that $\mathcal{Z}$ reduces for increasing $\left\vert {g}_{B{\uparrow}}\right\vert$ signifying a tendency towards a completely deformed interacting state. For repulsive g_I↑ it is caused by the phase-separation, while for attractive interactions it stems from the impurity-medium bound state. This overall behavior of $\mathcal{Z}$ appears to be similar for a heavy impurity. For ${{\Omega}}_{\mathrm{R}} > {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, the polaronic residue is dramatically different. Focusing on g_I↑ < 0, it decreases for larger attractions but it is always larger than for Ω_R/ω = 5. However, repulsive impurity-spin-↑ interactions result in a smooth decrease of $\mathcal{Z}$ for g_I↑ < g + g_↑↓. Otherwise a sharp reduction of $\mathcal{Z}\to 0$ takes place which is associated with the phase-separation among the impurity and the spin-↑ state. A similar response is observed for a heavy medium and Ω_R/ω = 5.

We then compare our variational findings with the results from the mean-field approximation. Such a direct comparison is especially motivated by recent studies on the 1D Bose polarons whose ground state characteristics have been argued to be described (at least to some extent [69–71]) within a mean-field framework [67]. This issue has been carefully benchmarked, for instance, against quantum Monte Carlo techniques [21], the Lee–Low–Pines transformation [17, 44] as well as the ML-MCTDHX and the flow equation (IM-SRG) methods [69, 70].

The respective mean-field approximation for our system is described by the following system of three coupled GPEs

$\begin{equation}\begin{aligned}\hfill & \left[-\frac{{\hslash }^{2}}{2{m}_{B}}\frac{{\mathrm{d}}^{2}}{\mathrm{d}{x}^{2}}+\frac{1}{2}{m}_{B}{\omega }_{B}^{2}{x}^{2}-{\mu }_{B}+{\tilde{g}}_{{\uparrow}{\uparrow}}{\left\vert {\psi }_{{\uparrow}}(x)\right\vert }^{2}+{\tilde{g}}_{{\uparrow}{\downarrow}}{\left\vert {\psi }_{{\downarrow}}(x)\right\vert }^{2}+{g}_{I{\uparrow}}{\left\vert {\psi }_{I}(x)\right\vert }^{2}\right]{\psi }_{{\uparrow}}(x)+\frac{{{\Omega}}_{\mathrm{R}}}{2}{\psi }_{{\downarrow}}(x)=0\hfill \\ \hfill & \left[-\frac{{\hslash }^{2}}{2{m}_{B}}\frac{{\mathrm{d}}^{2}}{\mathrm{d}{x}^{2}}+\frac{1}{2}{m}_{B}{\omega }_{B}^{2}{x}^{2}-{\mu }_{B}+{\tilde{g}}_{{\downarrow}{\downarrow}}{\left\vert {\psi }_{{\downarrow}}(x)\right\vert }^{2}+{\tilde{g}}_{{\uparrow}{\downarrow}}{\left\vert {\psi }_{{\uparrow}}(x)\right\vert }^{2}+{g}_{I{\downarrow}}{\left\vert {\psi }_{I}(x)\right\vert }^{2}\right]{\psi }_{{\downarrow}}(x)+\frac{{{\Omega}}_{\mathrm{R}}}{2}{\psi }_{{\uparrow}}(x)=0\hfill \\ \hfill & \left[-\frac{{\hslash }^{2}}{2{m}_{I}}\frac{{\mathrm{d}}^{2}}{\mathrm{d}{x}^{2}}+\frac{1}{2}{m}_{I}{\omega }_{I}^{2}{x}^{2}-{\mu }_{I}+{g}_{I{\uparrow}}{\left\vert {\psi }_{{\uparrow}}(x)\right\vert }^{2}+{g}_{I{\downarrow}}{\left\vert {\psi }_{{\downarrow}}(x)\right\vert }^{2}\right]{\psi }_{I}(x)=0,\hfill \end{aligned}\end{equation} \tag{ 16 }$

where ${\tilde{g}}_{\alpha {\alpha }^{\mathrm{\prime }}}=(1-1/N){g}_{\alpha {\alpha }^{\mathrm{\prime }}}$. The chemical potential of the medium μ_B is inherently related with the particle number ¹⁴ N = N_↑ + N_↓, see also appendix C. Importantly, the mean-field framework accounts for the hybridization of the spin and spatial degrees-of-freedom, ignoring the excited state contributions or effects originating from quantum fluctuations [71]. Comparing the quasiparticle weight between the many-body approach (appendix D) and the mean-field theory (equation (16)), reveals only small deviations for weak boson–boson interactions, see figure 8(a). From this we can conclude that spin-spatial correlations play the dominant role in the behavior of $\mathcal{Z}$ and thus on the generation of the magnetic polaron. Spatial correlations giving rise to deviations from the mean-field picture become relevant for stronger boson–boson interactions.

Indeed, deviations from the mean-field become in particular noticeable for repulsive impurity-medium interactions and around ${{\Omega}}_{\mathrm{R}}^{\text{crit}}$, implying that in this region spatial correlations become non-negligible. The imprint of spatial correlations on $\mathcal{Z}$ is especially pronounced by considering stronger interparticle interactions, compare the many-body and the mean-field results in figure 8(b). In particular, the presence of correlations lead to smaller residue for repulsive g_I↑ due to the superposition of the many-body wave function as compared to the mean-field value. For attractive g_I↓, the situation is reversed because the impurity is less spatially localized in the mean-field case.

Since the mean-field solution is adequate for weak medium interactions, we have also studied the behavior of $\mathcal{Z}$ for substantially larger atom numbers. For this investigation, we systematically approach the thermodynamic limit ¹⁵ observing that $\mathcal{Z}\sim \vert \langle {\psi }_{{\uparrow}}\vert {\psi }_{{\uparrow}0}\rangle +\langle {\psi }_{{\downarrow}}\vert {\psi }_{{\downarrow}0}\rangle {\vert }^{{N}_{B}}\vert \langle {\phi }_{I}\vert {\phi }_{I0}\rangle {\vert }^{{N}_{I}}$, where |ϕ_σ ⟩ and |ϕ_σ0⟩, with σ ∈ {↑, ↓, I} are the mean-field order parameters for g_I↑ ≠ 0 and g_I↑ = 0 respectively. Thus, $\mathcal{Z}({N}_{B},{N}_{I})\approx {\tilde{\mathcal{Z}}}^{\frac{{N}_{B}}{100}}\to 0$, for large N_B and N_I , where $\tilde{\mathcal{Z}}$ corresponds to the residue for N = 100. This exponentially decaying behavior of $\mathcal{Z}$ manifests the Anderson catastrophe of the magnetic polaron in the thermodynamic limit in 1D [11, 47]. It further supports the generalization of our results and their experimental detection in the large atom number limit as long as the impurities density is low to ensure that their interactions are negligible.

The energy of the quasiparticle is naturally determined by the energy difference among the interacting impurity-medium setting and the non-interacting one,

$\begin{equation}{E}_{\mathrm{p}\mathrm{o}\mathrm{l}}=\langle \hat{H}({g}_{I\alpha })\rangle -\langle \hat{H}({g}_{I\alpha }=0)\rangle .\end{equation} \tag{ 17 }$

As it can be verified by inspecting figure 9, the polaron energy shows a continuously increasing (decreasing) trend for stronger repulsive (attractive) g_I↑ as long as Ω_R ≠ 0. Moreover, for ${{\Omega}}_{\mathrm{R}} > {{\Omega}}_{\mathrm{R}}^{\text{crit}}$ and in the case of g_I↑ > g + g_↑↓, the energy saturates due to the impurity-spin-↑ phase-separation process (see also figure 3). The saturation value of the polaron energy caused by the impurity-medium phase-separation can also be predicted within V_eff(x) (see equation (C4) in appendix C) and it corresponds roughly to ${E}_{\text{ph.}\;\text{s.}}\sim (1/2){m}_{I}{\omega }_{I}^{2}{R}_{\text{out}}^{2}$ with R_out being the TF radius of the interacting component. However, for ${{\Omega}}_{\mathrm{R}}\approx {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, the magnetic polaron energy does not exhibit an upper bound due to the absence of phase-separation. Notice that in the case of Ω_R = 0 the energy E_pol = 0 in the repulsive g_I↑ > 0 regime since there is no dressing, while E_pol < 0 for g_I↑ < 0 due to the self-bound attractive Bose polaron.

Referring to miscible spinor components (R_in = 0), according to equation (B6), the effective potential reads

$\begin{equation}{V}_{\text{eff}}(x)=\left\{\frac{1}{2}{m}_{B}{D}_{\text{misc}}{\omega }_{B}^{2}{x}^{2}+\frac{{g}_{I{\uparrow}}+{g}_{I{\downarrow}}}{g+{g}_{{\uparrow}{\downarrow}}}\left({\mu }_{B}+\frac{\vert {{\Omega}}_{\mathrm{R}}\vert }{2}\right)\quad \quad \text{for}\enspace \vert x\vert \leqslant {R}_{\text{out}},\\ \frac{1}{2}{m}_{I}{\omega }_{I}^{2}{x}^{2},\quad \quad \text{for}\enspace \vert x\vert > {R}_{\text{out}},\right.\end{equation} \tag{ C2 }$

here the factor D_misc determines the shape of the effective potential and reads

$\begin{equation}{D}_{\text{misc}}=\frac{{m}_{I}{\omega }_{I}^{2}}{{m}_{B}{\omega }_{B}^{2}}-\frac{{g}_{I{\uparrow}}+{g}_{I{\downarrow}}}{g+{g}_{{\uparrow}{\downarrow}}}.\end{equation} \tag{ C3 }$

Therefore, V_eff(x) possesses a harmonic oscillator shape for D_misc > 0, or equivalently ${g}_{I{\uparrow}}+{g}_{I{\downarrow}}< {g}_{BI\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=(g+{g}_{{\uparrow}{\downarrow}})\frac{{m}_{I}{\omega }_{I}^{2}}{{m}_{B}{\omega }_{B}^{2}}$ deforming into a double-well in the opposite case, D_misc < 0, with minima located at x_min = ±R_out. As such, in line with the results of references [24, 27] for a single-component host, we expect that the impurity and the medium bath become immiscible in the case of g_I↑ + g_I↓ > g_BIcrit. This ultimately results in the instability of the repulsive Bose polaron being inherently related to its decaying character in this interspecies interaction regime [24, 31]. This provides the first threshold for the stability of the Bose polaron indicated in figure 12(a), see in particular the edge of the blue region for g_↑↓ = g_I↑ − g. This is equivalent to the edge of the repulsive Bose polaron regime for g_I↑ = g + g_↑↓ appearing in figure 2(a).

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It should be emphasized that within V_eff(x) we completely neglect backaction effects of the impurity to the BEC, which in the case of a pseudo-spinor Rabi-coupled BEC can be important for determining the stability of the Bose polaron. Nevertheless, by comparing V_eff(x) with the corrected ${V}_{\text{eff}}^{\prime }(x)$ potential incorporating the effect of impurity-medium correlations we do not observe significant deviations among the two approaches as long as, the Rabi-coupling is strong enough, see figure 2(b₂) for the repulsive polaron and figure 2(b₃) for the immiscible impurity-medium case. Similar findings are obtained for g_↑↓≪ g (not shown here for brevity). The above can be interpreted in view of reference [56], where it was demonstrated that the spin-excitations possess a sizable gap when the system is deep into the miscible regime of the pseudospinor BEC. Therefore, within the latter regime the generation of spin-wave excitations into the host is prohibited. Recall that also the phonon-impurity coupling is much weaker than the V_eff(x) contribution, thus suppressing possible density excitations of the medium. As such, both excitations pathways are essentially frozen within this miscible BEC regime and consequently the impurity-medium correlations are expected to be insignificant. This explains the fact that ${V}_{\text{eff}}(x)\approx {V}_{\text{eff}}^{\prime }(x)$ as shown in figures 2(b₂) and (b₃).

As the miscibility–immiscibility threshold is approached, i.e. for ${{\Omega}}_{\mathrm{R}}\approx {{\Omega}}_{\mathrm{R}}^{\text{crit}}$ or ${g}_{{\uparrow}{\downarrow}}\approx {g}_{{\uparrow}{\downarrow}}^{\text{crit}}$, the above-mentioned energy gap among the spin and the phononic excitations closes [56] allowing for the coupling of the impurity state with the spin-fluctuations of its host. As a consequence we expect pronounced impurity-medium correlations in this regime. Indeed, by incorporating this correction into a new effective potential, ${V}_{\text{eff}}^{\prime }(x)$, we observe the emergence of an additional well around x = 0 which is responsible for binding the impurity into the Bose gas provided that ${V}_{\text{eff}}^{\prime }(0)< {V}_{\text{eff}}^{\prime }({R}_{\text{out}})$, see figure 2(b₄). This behavior is reminiscent of the so-called Pekkar polaron according to which the density excitations of the host are maximized in the vicinity of the impurity leading to its spatial localization [2, 75]. Importantly though, in our case the excitations dressing the impurity possess a magnetic character (see section 4.4) and therefore we refer to the structure emerging in this regime as the Pekkar-magnetic polaron.

For a medium with immiscible spin components one has to consider a larger number of cases since depending on the sign of g_I↑ − g_I↓ distinct medium configurations are favored. Indeed, if g_I↑ > g_I↓ (g_I↑ < g_I↓) it is preferable that n_↑ > n_↓, Δ(x) < 0 (n_↑ < n_↓, Δ(x) > 0). This leads to a discontinuity in n_α at g_I↑ = g_I↓. Then the effective potential in the TF approximation becomes

$\begin{equation}{V}_{\text{eff}}(x)=\begin{cases}\begin{aligned}\hfill & \left(\frac{{m}_{I}{\omega }_{I}^{2}}{{m}_{B}{\omega }_{B}^{2}}-\frac{{g}_{I{\uparrow}}+{g}_{I{\downarrow}}}{2g}\right)\frac{1}{2}{m}_{B}{\omega }_{B}^{2}{x}^{2}+\frac{{g}_{I{\uparrow}}+{g}_{I{\downarrow}}}{2g}{\mu }_{B}\hfill \\ \hfill & -\frac{\vert {g}_{I{\uparrow}}-{g}_{I{\downarrow}}\vert }{2({g}_{{\uparrow}{\downarrow}}-g)}\sqrt{{\left[\frac{{g}_{{\uparrow}{\downarrow}}-g}{g}\left({\mu }_{B}-\frac{1}{2}{m}_{B}{\omega }_{B}^{2}{x}^{2}\right)\right]}^{2}-{{\Omega}}_{\mathrm{R}}^{2}},\hfill \end{aligned}\quad \hfill & \quad \text{for}\enspace \vert x\vert \leqslant {R}_{\text{in}},\hfill \\ \left(\frac{{m}_{I}{\omega }_{I}^{2}}{{m}_{B}{\omega }_{B}^{2}}-\frac{{g}_{I{\uparrow}}+{g}_{I{\downarrow}}}{g+{g}_{{\uparrow}{\downarrow}}}\right)\frac{1}{2}{m}_{B}{\omega }_{B}^{2}{x}^{2}+\frac{{g}_{I{\uparrow}}+{g}_{I{\downarrow}}}{g+{g}_{{\uparrow}{\downarrow}}}\left({\mu }_{B}+\frac{\vert {{\Omega}}_{\mathrm{R}}\vert }{2}\right),\quad \hfill & \quad \text{for}\enspace {R}_{\text{in}}< \vert x\vert \leqslant {R}_{\text{out}},\hfill \\ \frac{1}{2}{m}_{I}{\omega }_{I}^{2}{x}^{2},\quad \hfill & \quad \text{for}\enspace \vert x\vert > {R}_{\text{out}}.\hfill \end{cases}\end{equation} \tag{ C4 }$

To further elucidate the form of V_eff(x) we perform an expansion with respect to x around |x| ⩽ R_in. Within the harmonic approximation (i.e. dropping terms ∝xⁿ for n > 2) we obtain

$\begin{equation}{V}_{\text{eff}}^{\text{approx}}(x)=\frac{1}{2}{m}_{B}{D}_{\text{im}}{\omega }_{B}^{2}{x}^{2}+\frac{{\mu }_{B}}{2g}\left[{g}_{I{\uparrow}}+{g}_{I{\downarrow}}-\left\vert {g}_{I{\uparrow}}-{g}_{I{\downarrow}}\right\vert \sqrt{1-{\left(\frac{{{\Omega}}_{\mathrm{R}}}{{{\Omega}}_{\mathrm{R}}^{\text{crit}}}\right)}^{2}}\right]+\mathcal{O}\left[{\left(\frac{x}{\sqrt{{\mu }_{B}/({m}_{B}{\omega }_{B}^{2})}}\right)}^{4}\right].\end{equation} \tag{ C5 }$

Consequently, whether V_eff(x) exhibits a double- or single-well structure depends on the sign of

$\begin{equation}{D}_{\text{im}}=\frac{{m}_{I}{\omega }_{I}^{2}}{{m}_{B}{\omega }_{B}^{2}}-\frac{1}{2g}\left[{g}_{I{\uparrow}}+{g}_{I{\downarrow}}-\frac{\vert {g}_{I{\uparrow}}-{g}_{I{\downarrow}}\vert }{\sqrt{1-{\left(\frac{{{\Omega}}_{\mathrm{R}}}{{{\Omega}}_{\mathrm{R}}^{\text{crit}}}\right)}^{2}}}\right].\end{equation} \tag{ C6 }$

Indeed, D_im > 0 designates a single-well potential around x = 0, while D_im < 0 implies a double-well with minima at x = ±R_out. A first simplification occurs in the case of g_I↓ = 0 (due to the ${\mathbb{Z}}_{2}$ symmetry we can equivalently assume g_I↑ = 0), where D_im > 0 for all g_I↑. This means that in the immiscible case and for g_I↓ = 0, the impurity can not escape from the BEC within the TF approximation and the polaron exists for every g_I↑.

By comparing, V_eff(x) with ${V}_{\text{eff}}^{\prime }(x)$ within the immiscible regime, we observe a qualitative agreement between these two approaches see figure 2(b₅). The notable differences lie in the modification of the effective potential around x = 0 owing to the magnon dressing. This leads to a stronger binding of the impurity in the BEC when impurity-medium correlations are incorporated. As such, the Bose polaron in this parameter range possesses a similar structure as the magnetic one identified in the miscible regime, despite that the former does not show a self-bound character. For this reason we do not differentiate among these two regimes in the phase diagram of figures 2(a) and (b) and refer to the emerging structures as magnetic Bose polarons. An additional alteration of ${V}_{\text{eff}}^{\prime }(x)$ occurs near x = ±R_in, originating from the non-negligible kinetic energy of the bosons in this spatial region which is neglected within the TF approximation. Further, for g_I↑ = 0 (or g_I↓ = 0), we note that deep in the immiscible regime, i.e. for ${{\Omega}}_{\mathrm{R}}\ll {{\Omega}}_{\mathrm{R}}^{\text{crit}}$ (or equivalently for ${g}_{{\uparrow}{\downarrow}}\gg {g}_{{\uparrow}{\downarrow}}^{\text{crit}}$) the BEC tends to be fully polarized. Then, all atoms occupy the spin-state that is non-interacting with the impurity and therefore in this case no polaron exists. Of course, as the system approaches this regime it exhibits a crossover character and consequently no phase-boundary emerges. In order to estimate the parameter region where the behavior of the system changes we define, ${g}_{{\uparrow}{\downarrow}}^{\text{single}}$ possessing the property that for ${g}_{{\uparrow}{\downarrow}} > {g}_{{\uparrow}{\downarrow}}^{\text{single}}$ and fixed Ω_R less than a single BEC atom occupies the component that is interacting with the impurity, see figure 12(a). Similarly, we have defined ${{\Omega}}_{\mathrm{R}}^{\text{single}}$, see figure 2(a), such that for ${{\Omega}}_{\mathrm{R}}< {{\Omega}}_{\mathrm{R}}^{\text{single}}$ and fixed g_↑↓ the component interacting with the impurity is occupied by less than one atom.

Let us now briefly comment on the case of g_I↑ ≠ 0 ≠ g_I↓, by examining the expected regimes where a polaron appears within the TF approximation (i.e. D_im > 0). The behavior of the system is the simplest for ${{\Omega}}_{\mathrm{R}}\ll {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, where the spin components are strongly imbalanced and almost only one of them is occupied (see also figure 11). It naturally follows from equation (C6) that the polaron exists for $\mathrm{min}({g}_{I{\uparrow}},{g}_{I{\downarrow}})< g\frac{{m}_{I}{\omega }_{I}^{2}}{{m}_{B}{\omega }_{B}^{2}}$. These results are in agreement to the ones regarding an impurity immersed in a spinless BEC [24, 27, 31]. On the contrary, in the region ${{\Omega}}_{\mathrm{R}}\approx {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, i.e. close to the miscibility–immiscibility threshold, D_imm < 0 signifying that a potential minimum at x = 0 always exists independently of the impurity-medium interaction strength. Finally, in the intermediate case, $0< {{\Omega}}_{\mathrm{R}}< {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, there is a threshold

$\begin{equation}\mathrm{min}({g}_{I{\uparrow}},{g}_{I{\downarrow}})-g\frac{{m}_{I}{\omega }_{I}^{2}}{{m}_{B}{\omega }_{B}^{2}}< \lambda ({{\Omega}}_{\mathrm{R}})\left(\mathrm{max}({g}_{I{\uparrow}},{g}_{I{\downarrow}})-g\frac{{m}_{I}{\omega }_{I}^{2}}{{m}_{B}{\omega }_{B}^{2}}\right),\end{equation} \tag{ C7 }$

below which the Bose polaron exists. Here, the function λ(Ω_R) features an increasing behavior for $0< {{\Omega}}_{\mathrm{R}}< {{\Omega}}_{\mathrm{R}}^{\text{crit}}$, and exhibits the extrema λ(Ω_R = 0) = 0 and $\lambda ({{\Omega}}_{\mathrm{R}}={{\Omega}}_{\mathrm{R}}^{\text{crit}})=1$. The precise values of λ(Ω_R) in the above-mentioned domain, depend solely on the parameters characterizing the medium, namely g, g_↑↓ and N. For illustration, figure 12(b) provides λ(Ω_R), alongside with an example stability phase diagram in the inset, for the host parameters employed within this work.

Concluding, V_eff(x) provides insight into the existence of polaronic states, but not whether they correspond to the system's ground state. Indeed, if the impurity remains within the BEC its energy increases without bound for a larger g_Iα . In contrast, the phase-separated states possess an energy that saturates for strongly repulsive impurity-medium interaction strengths since the interspecies spatial overlap is small. Therefore, it is expected that the impurity energy in these cases will saturate towards ${E}_{\text{ph.}\;\text{s.}}\sim \frac{1}{2}{m}_{I}{\omega }_{I}^{2}{R}_{\text{out}}^{2}$. This implies that even in the cases that the Bose polaron persists for large interactions, the phase-separated states might be the overall ground states of the system and therefore explicit investigations are required to determine the energy of both branches. Here a hint towards the stability of the polaronic states relies on the fact that the state of the BEC is highly perturbed for the cases of the magnetic polaron and almost completely intact for a phase-separated impurity-medium configuration. As a consequence, the corresponding coupling among the polaronic and phase-separated states that would render the former configuration unstable is small. This can be understood from the fact that the state overlap typically scales as $\propto {[{\sum }_{\alpha }\int \mathrm{d}x\sqrt{{\rho }_{\alpha }(x){\rho }_{\alpha }^{\prime }(x)}]}^{N}$, where ρ_α (x), ${\rho }_{\alpha }^{\prime }(x)$ are the original and perturbed single-particle densities respectively.