# Pattern Formation in One-Dimensional Polaron Systems and Temporal Orthogonality Catastrophe

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Polaron Hamiltonian and Lee-Low-Pines Transformation

## 3. Gross Ansatz Treatment of the Lee-Low-Pines Hamiltonian

#### 3.1. The Polaron Solution

#### 3.2. The Case of a Static Polaron

#### 3.3. Moving Polaron and the Soliton Solution

_{1})–(d

_{1})), ${m}_{I}={m}_{B}$ (Figure 2(a

_{2})–(d

_{2})) and ${m}_{I}=2{m}_{B}$ (Figure 2(a

_{3})–(d

_{3})). Independently of the impurity mass the velocity of the polaron satisfies ${\beta}_{\mathrm{p}}\le {\beta}_{\mathrm{crit}}$ (Figure 2(a

_{i}), $i=1,2,3$), hinting towards the conclusion that the state described by Equation (7) is stable3 for every ${p}_{I}$ and ${g}_{BI}$. Moreover, the polaron velocity $\beta $ exhibits a non-monotonic behavior since for small momenta ${p}_{I}<{m}_{I}c$, $\beta $ is increasing with ${p}_{I}$ until it reaches a maximum at a ${g}_{BI}$-dependent momentum value ${p}_{I,0}\ge {m}_{I}c$. Beyond that point $\beta $ decreases with increasing ${p}_{I}$ until it reaches the value of $\beta =0$ for ${p}_{I}=\pi \hslash {n}_{0}$. In addition, it can be seen that the solution for ${r}_{0}$ (Figure 2(b

_{i}), $i=1,2,3$) is appreciably larger than 0 only for ${p}_{I}<{m}_{I}c$ (see dashed line) and for ${g}_{BI}<0.5{\hslash}^{2}{n}_{0}/{m}_{B}$.

_{i}), $i=1,2,3$. Here there are two notable effects. For a fixed ${g}_{BI}$, ${E}_{\mathrm{p}}$ depends more weakly on ${p}_{I}$ as the value of ${g}_{BI}$ increases. This is a manifestation of the increase of the effective mass, ${m}^{*}={\left(\right)}^{\frac{{\partial}^{2}{E}_{p}}{\partial {p}_{I}^{2}}}-1$, of the polaron with ${g}_{BI}$ reported in Ref. [57]. Moreover, for momenta ${p}_{I}\to \hslash {n}_{0}\pi $ or large interactions the energy of the polaron tends to saturate to the corresponding energy of the dark-bright soliton solution, Equation (15), ${E}_{\mathrm{b}}={lim}_{{g}_{BI}\to \infty}{E}_{\mathrm{p}}=\frac{4}{3}\hslash {n}_{0}c$. It is evident from Figure 2(c

_{i}), that this asymptotic value of energy decreases with increasing ${m}_{I}$ a fact that can be understood by inspecting Equation (13).

_{i}) with $i=1,2,3$. This quantity compares the momentum contribution of the motion of the impurity, $\langle {\widehat{p}}_{I}^{\mathrm{lab}}\rangle $, to the induced BEC flow, $\langle {\widehat{p}}_{B}\rangle ={p}_{I}-\langle {\widehat{p}}_{I}^{\mathrm{lab}}\rangle $. Therefore, values proximal to 1 indicate that the impurity motion is the dominant contribution and as a consequence, the system is in the polaron regime. On the other hand, values close to 0 signify that the dominant contribution is the BEC flow and accordingly the system behaves as a dark-bright soliton. As Figure 2(d

_{i}) testifies, the polaron regime occurs only for ${p}_{I}<{m}_{I}c$ and ${g}_{BI}<0.5{\hslash}^{2}{n}_{0}/{m}_{B}$, where also ${r}_{0}\gg 0$, see also Figure 2(a

_{i}). Otherwise, the state of the system lies within the dark-bright soliton regime.

## 4. Impact of Correlations and Validity of the GA Approximation

## 5. Dynamical Response of the System: The Temporal Orthogonality Catastrophe

#### 5.1. Dynamics of a Subsonic Impurity

#### 5.1.1. Dynamics of Two-Body Correlations

#### 5.1.2. Time-Dependent Overlap: Temporal Orthogonality Catastrophe

#### 5.1.3. Drag Force and Momentum Transfer Mechanism

_{2}), ${\beta}_{f}\le {\beta}_{\mathrm{p}}\left({p}_{I}\right)$.

#### 5.2. Dynamics of a Supersonic Impurity

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Bosonic Momentum Renormalization

**Figure A1.**Convergence of the GA Bose polaron solution to the $N,L\to \infty $ limit. (

**a**) Bath-impurity correlation function, ${\rho}_{IB}^{\left(2\right)}(0;{x}_{B})$ for ${g}_{BI}=1{\hslash}^{2}{n}_{0}^{-1}/{m}_{B}$, ${g}_{BB}=0.1{\hslash}^{2}{n}_{0}/{m}_{B}$, ${p}_{I}=0.1\hslash {n}_{0}$ and ${m}_{I}={m}_{B}$ but different ring lengths L (see legend). In order to keep ${n}_{0}=1$ in our calculations, we demand ${N}_{B}={n}_{0}L$, while the spatial region $|{x}_{B}|>60{n}_{0}^{-1}$ is not depicted, since ${\rho}_{IB}^{\left(2\right)}(0;{x}_{B})\approx {\rho}_{IB}^{\left(2\right)}(0;60{n}_{0}^{-1})$. (

**b**) The phase profile, $\varphi \left({x}_{B}\right)=arg\left(\psi \left({x}_{B}\right)\right)$, of the solutions for the above mentioned parameters and for the same varying values of L.

## Appendix B. The Impact of the Interspecies Interaction Potential

**Figure A2.**Emergence of dispersive shock waves for short-range interaction potentials. (

**a**) Spatiotemporal evolution of ${\rho}_{IB}^{\left(2\right)}(0;{x}_{B})$ for ${g}_{BI}^{f}=0.1{\hslash}^{2}{n}_{0}/{m}_{B}={g}_{BB}$, ${p}_{I}=0$, $L=1600{n}_{0}^{-1}$ and ${N}_{B}=1600$ in the case that a Gaussian bath-impurity interaction potential with width $w=4{n}_{0}^{-1}$ is employed. (

**b**) Same as (

**a**) but for the particular time instant $t=300{m}_{B}/(\hslash {n}_{0}^{2})$. (

**c**) the phase profile corresponding to (

**b**). The insets of (

**b**,

**c**) correspond to $t=0.1{m}_{B}/(\hslash {n}_{0}^{2})$. (

**d**–

**f**) correspond to the same quantities as in (

**a**–

**c**) respectively, calculated for the same set of parameters. However, here the bath-impurity potential corresponds to $\delta $-function. (

**g**–

**i**) the same quantities as in (

**a**–

**c**) but for a wider Gaussian potential $w=20{n}_{0}^{-1}$. The insets of (

**c**,

**f**,

**i**) in addition to the GA numerical results also indicate the approximate profile expected from the phase imprinting of the impurity potential (see text).

## Appendix C. Details on the Computational Techniques

- initialize the system with an ansatz set of single-particle functions ${\varphi}_{k}^{\left(0\right)}\left(r\right)$, where $k=1,\cdots ,M$,
- diagonalize the Hamiltonian within a basis spanned by the single-particle functions,
- set the eigenvector with the lowest energy as the ${A}^{\left(0\right)}$-vector,
- propagate the single-particle functions in imaginary time within a finite time interval $d\tau $,
- update the single-particle functions to ${\varphi}_{k}^{\left(1\right)}\left(r\right)$ and
- repeat steps 2–5 until the state coefficients converge within the prescribed accuracy.

## Notes

1 | |

2 | The lower bound corresponds to a ${}^{6}$Li impurity immersed in a ${}^{176}$Yb bath and the upper bound to a ${}^{176}$Yb impurity immersed in a Bose medium of ${}^{7}$Li. |

3 | Importantly, by explicitly evaluating the Hessian matrix for the numerical solutions presented in Figure 2 we can prove that ${\left(\right)}_{{H}_{{E}_{\mathrm{p}}}}ij$ is positive definite, where $i,j=1,2$, with ${a}_{1}=\beta $, ${a}_{2}={r}_{0}$. This supports the stability of the solution within the subspace spanned by Equation (7). |

4 | The bath density is expelled from the vicinity of the impurity and accumulates in the spatial region away from it ${x}_{B}>10{n}_{0}^{-1}$, see also Figure 3c. This density increase leads to greater kinetic energy scaling quadratically with the bath density. However, since the number of expelled atoms is roughly constant as the perimeter of the ring L increases this correction becomes negligible for $L\to \infty $. |

5 | |

6 | This means that due to the presence of the potential, the phase of the BEC shifts, leading to a flow of the bosonic density away from the repulsive potential. Note here that the amplitude of this phase disturbance increases with the decrease of the width of the perturbing potential. This effect is maximized for a $\delta $-shape as the one corresponding to the bath-impurity interactions. |

7 | This term denotes the excitation of the BEC due to the locally supersonic motion of the impurity. This effect is analogous to the emission of electromagnetic radiation when electrons move through a dielectric medium with a velocity greater than the phase velocity of light. |

8 | Note here that the phase of $\psi (r;t)$ is shifted so that $arg\left[\psi \right(r=0;t\left)\right]=0$ and the rest of the parameters in Equation (7) are fixed to their corresponding values in the thermodynamic limit, namely $\xi ={\left(0.1\right)}^{-1/2}{n}_{0}^{-1}$. |

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**Figure 1.**Analytical predictions for the energy and critical velocity of the Bose polaron within GA. (

**a**) Polaron energy, ${E}_{\mathrm{p}}$ [Equation (12)] and (

**b**) critical velocity of the polaronic solution, ${\beta}_{\mathrm{crit}}$, [Equation (14)] for varying bath-impurity interaction strength, ${g}_{BI}$. In both cases the solid lines indicate the exact results while the dashed lines correspond to weak (leftmost line), $\mathcal{O}{\left(\right)}^{\frac{{g}_{BI}}{2\hslash c}}2$, and strong (rightmost line), $\mathcal{O}{\left(\right)}^{\frac{2\hslash c}{{g}_{BI}}}2$, asymptotic Taylor expansions.

**Figure 2.**Characteristic properties of the moving Bose polaron within GA. (

**a**${}_{\mathbf{1}}$–

**a**${}_{\mathbf{3}}$) Velocity of the polaron over its critical one, ${\beta}_{\mathrm{p}}/{\beta}_{\mathrm{crit}}$, (

**b**${}_{\mathbf{1}}$–

**b**${}_{\mathbf{3}}$) offset parameter, ${r}_{0}$, of the polaron solution, (

**c**${}_{\mathbf{1}}$–

**c**${}_{\mathbf{3}}$) polaron energy, ${E}_{\mathrm{p}}$ and (

**d**${}_{\mathbf{1}}$–

**d**${}_{\mathbf{3}}$) expectation value ratio of the impurity momentum between the laboratory and the impurity frames, $\langle {\widehat{p}}_{I}^{\mathrm{lab}}\rangle /{p}_{I}$, for different values of ${g}_{BI}$ and ${p}_{I}$. The distinct columns correspond to different impurity masses, ${m}_{I}={m}_{B}/2$ (left panels), ${m}_{I}={m}_{B}$ (middle panels) and ${m}_{I}=2{m}_{B}$ (right panels). In all cases, the data correspond to thermodynamic limit calculations, $N,L\to \infty $, with ${g}_{BB}=0.1{\hslash}^{2}{n}_{0}/{m}_{B}$ and dashed lines represent ${p}_{I}={m}_{I}c$.

**Figure 3.**Comparison of the Bose polaron characteristics between the GA and the correlated MCTDHB framework. (

**a**) Bath-impurity correlations, ${\rho}_{IB}^{\left(2\right)}(0;{x}_{B})$, for varying interspecies interaction strength, ${g}_{BI}$, within MCTDHB. (

**b**) Bath-impurity correlations at coincidence, ${\rho}_{IB}^{\left(2\right)}(0;0)$, for different ${g}_{BI}$ and for all employed approaches (see legend). (

**c**) Comparison of the correlation profile ${\rho}_{IB}^{\left(2\right)}(0;{x}_{B})$ within the MCTDHB and the GA for ${g}_{BI}=2{\hslash}^{2}{n}_{0}/{m}_{B}$. The inset of (

**c**) provides a magnification of ${\rho}_{IB}^{\left(2\right)}(0;{x}_{B})$, showing the behavior of the system away from the impurity. Comparison of (

**d**) the polaron energy, ${E}_{\mathrm{p}}$, (

**e**) the inverse effective mass, ${m}_{I}/{m}^{*}$ and (

**f**) the polaron residue among the different approaches and for varying ${g}_{BI}$. To elucidate the comparison between GA and MCTDHB, the insets of (

**b**,

**d**,

**e**) provide the difference of the corresponding observables between the distinct approaches (see legend). The inset of (

**f**) indicates the many-body overlap between the MCTDHB and the GA many-body states for varying ${g}_{BI}$. In all cases, ${m}_{I}={m}_{B}$, ${p}_{I}=0$ and ${g}_{BB}=0.1{\hslash}^{2}{n}_{0}/{m}_{B}$. The relevant ring confined setups are characterized by ${N}_{B}=100$ and $L=100{n}_{0}^{-1}$.

**Figure 4.**Quench dynamics of an impurity in a homogeneous Bose gas. (

**a**,

**b**) Spatiotemporal evolution of the two-body interspecies correlations, ${\rho}_{IB}^{\left(2\right)}(0;{x}_{B})$, for different initial impurity momenta, ${p}_{I}$ (see column labels). Here the postquench interaction is ${g}_{BI}^{f}=0.1{\hslash}^{2}{n}_{0}/{m}_{B}={g}_{BB}$. The dashed lines indicate ${x}_{B}=(\pm 1-{\beta}_{f})ct$, with ${\beta}_{f}$ the final velocity of the generated polaron provided as an inset label. (

**c**,

**d**) The time-dependent overlap, $|\langle {\mathsf{\Psi}}_{0}|\mathsf{\Psi}\left(t\right)\rangle |$, of the post-quench many-body wavefunction, Equation (3), with the initial state, for varying ${g}_{BI}$. The insets of (

**c**,

**d**) provide the time-evolution of $|\langle {\mathsf{\Psi}}_{0}|\mathsf{\Psi}\left(t\right)\rangle |$ within a more extensive ${g}_{BI}$ range. (

**e**,

**f**) present the modulus and phase of the GA bath-wavefunction, $\psi ({x}_{B};t)$ respectively for ${g}_{BI}^{f}=0.1{\hslash}^{2}{n}_{0}/{m}_{B}$, ${p}_{I}=0.4\hslash {n}_{0}$ and $t=400\frac{{m}_{B}}{\hslash {n}_{0}^{2}}$. For comparison (

**e**,

**f**) also provide the equilibrium profile of the polaron, Equation (7), with $\beta ={\beta}_{f}=0.64$. In all cases the system is confined in a ring of $L=1600{n}_{0}^{-1}$ and contains ${N}_{B}=1600$ while ${m}_{I}={m}_{B}$.

**Figure 5.**Characterization of the drag force exerted on the impurity by the Bose gas. (

**a**) The critical, ${\beta}_{\mathrm{crit}}$ (dashed line) and equilibrium, ${\beta}_{\mathrm{p}}\left({p}_{I}\right)$ (solid lines) velocities of the Bose polaron with ${g}_{BI}={g}_{BI}^{f}$, compared to the final velocity of the polaron formed after the quench, ${\beta}_{f}\left({p}_{I}\right)$ (data points) for varying ${g}_{BI}^{f}$. The parameters of the system are as in Figure 4 and ${p}_{I}$ is given in the legend. (

**b**) Temporal evolution of the drag force exerted to an impurity, initially possessing ${p}_{I}=0.4\hslash {n}_{0}$, for different values of the post-quench interspecies interaction strength, ${g}_{BI}^{f}$. (

**c**) Time-evolution of the impurity momentum for ${p}_{I}=0.4\hslash {n}_{0}$ and varying ${g}_{BI}^{f}$. The solid lines in (

**b**,

**c**) indicate the time that ${p}_{I}^{\mathrm{lab}}$ becomes equal to the corresponding value for the equilibrium polaron with ${g}_{BI}={g}_{BI}^{f}$.

**Figure 6.**Dynamics of an initially supersonically moving impurity. (

**a**–

**c**) Spatiotemporal evolution of the bath-impurity correlation function, ${\rho}_{IB}^{\left(2\right)}(0;{x}_{B})$, for a quench to ${g}_{BI}^{f}=0.07{\hslash}^{2}{n}_{0}/{m}_{B}$ and ${p}_{I}=1\hslash {n}_{0}$. The inset of (

**a**) provides a magnification of the corresponding bath-impurity correlation function in the vicinity of the impurity. The mass of the impurity is provided in the corresponding labels, while ${g}_{BB}=0.1{\hslash}^{2}{n}_{0}/{m}_{B}$, ${N}_{B}=3200$ and $L=3200{n}_{0}^{-1}$. The light dashed lines indicate ${x}_{I}=(\pm c-{p}_{I}/{m}_{I})t$ and the dark dashed lines in (

**a**) correspond to ${x}_{I}=(\pm 1-{\beta}_{f})ct+{x}_{0}$, with ${\beta}_{f}=0.7464$ the final velocity of the polaron and ${x}_{0}=64{n}_{0}^{-1}$ an offset selected for illustration purposes. (

**d**) The time evolution of ${\rho}_{IB}^{\left(2\right)}(0;{x}_{B})$, for the same parameters as in (

**a**) except for ${g}_{BI}^{f}=1{\hslash}^{2}{n}_{0}/{m}_{B}$. The trajectories indicated by the dashed lines correspond to ${x}_{I}=(\pm 1-{\beta}_{f})ct$, with ${\beta}_{f}=0.08$.

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**MDPI and ACS Style**

Koutentakis, G.M.; Mistakidis, S.I.; Schmelcher, P.
Pattern Formation in One-Dimensional Polaron Systems and Temporal Orthogonality Catastrophe. *Atoms* **2022**, *10*, 3.
https://doi.org/10.3390/atoms10010003

**AMA Style**

Koutentakis GM, Mistakidis SI, Schmelcher P.
Pattern Formation in One-Dimensional Polaron Systems and Temporal Orthogonality Catastrophe. *Atoms*. 2022; 10(1):3.
https://doi.org/10.3390/atoms10010003

**Chicago/Turabian Style**

Koutentakis, Georgios M., Simeon I. Mistakidis, and Peter Schmelcher.
2022. "Pattern Formation in One-Dimensional Polaron Systems and Temporal Orthogonality Catastrophe" *Atoms* 10, no. 1: 3.
https://doi.org/10.3390/atoms10010003