# Polaron Problems in Ultracold Atoms: Role of a Fermi Sea across Different Spatial Dimensions and Quantum Fluctuations of a Bose Medium

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## Abstract

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## 1. Introduction

## 2. Fermi Polarons

#### 2.1. T-Matrix Approach to Fermi Polaron Problems

#### 2.2. Spectral Response of Fermi Polarons

#### 2.2.1. Three-Dimensional Case

#### 2.2.2. Spectral Response of Fermi Polarons in Two-Dimensions

#### 2.2.3. Fermi Polarons in One-Dimension

## 3. Bose Polarons

#### 3.1. Bogoliubov Theory for Bose Polaron Problems

#### 3.2. Quantum Depletion around a Bose Polaron

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Pekar, S.I. Local quantum states of electrons in an ideal ion crystal. Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki
**1946**, 16, 341. [Google Scholar] - Landau, L.D.; Pekar, S.I. Effective mass of a polaron. Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki
**1948**, 18, 418. [Google Scholar] - Nascimbène, S.; Navon, N.; Jiang, K.J.; Tarruell, L.; Teichmann, M.; McKeever, J.; Chevy, F.; Salomon, C. Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass. Phys. Rev. Lett.
**2009**, 103, 170402. [Google Scholar] [CrossRef] [Green Version] - Schirotzek, A.; Wu, C.-H.; Sommer, A.; Zwierlein, M.W. Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms. Phys. Rev. Lett.
**2009**, 102, 230402. [Google Scholar] [CrossRef] [PubMed] - Kohstall, C.; Zaccanti, M.; Jag, M.; Trenkwalder, A.; Massignan, P.; Bruun, G.M.; Schreck, F.; Grimm, R. Metastability and Coherence of Repulsive Polarons in a Strongly Interacting Fermi Mixture. Nature
**2011**, 485, 615. [Google Scholar] [CrossRef] - Koschorreck, M.; Pertot, D.; Vogt, E.; Fröhlich, B.; Feld, M.; Köhl, M. Attractive and repulsive Fermi polarons in two dimensions. Nature
**2012**, 485, 619. [Google Scholar] [CrossRef] [Green Version] - Hohmann, M.; Kindermann, F.; Gänger, B.; Lausch, T.; Mayer, D.; Schmidt, F.; Widera, A. Neutral Impurities in a Bose-Einstein Condensate for Simulation of the Fröhlich-Polaron. EPJ Quantum Technol.
**2015**, 2, 23. [Google Scholar] [CrossRef] [Green Version] - Jorgensen, N.B.; Wacker, L.; Skalmstang, K.T.; Parish, M.M.; Levinsen, J.; Christensen, R.S.; Bruun, G.M.; Arlt, J.J. Observation of Attractive and Repulsive Polarons in a Bose-Einstein Condensate. Phys. Rev. Lett.
**2016**, 117, 055302. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hu, M.-G.; Van de Graaff, M.J.; Kedar, D.; Corson, J.P.; Cornell, E.A.; Jin, D.S. Bose Polarons in the Strongly Interacting Regime. Phys. Rev. Lett.
**2016**, 117, 055301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Scazza, F.; Valtolina, G.; Massignan, P.; Recati, A.; Amico, A.; Burchianti, A.; Fort, C.; Inguscio, M.; Zaccanti, M.; Roati, G. Repulsive Fermi Polarons in a Resonant Mixture of Ultracold
^{6}Li Atoms. Phys. Rev. Lett.**2017**, 118, 083602. [Google Scholar] [CrossRef] [Green Version] - Oppong, N.D.; Riegger, L.; Bettermann, O.; Höfer, M.; Levinsen, J.; Parish, M.M.; Bloch, I.; Fölling, S. Observation of Coherent Multiorbital Polarons in a Two-Dimensional Fermi Gas. Phys. Rev. Lett.
**2019**, 122, 193604. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Adlong, H.S.; Liu, W.E.; Scazza, F.; Zaccanti, M.; Oppong, N.D.; Fölling, S.; Parish, M.M.; Levinsen, J. Quasiparticle Lifetime of the Repulsive Fermi Polaron. Phys. Rev. Lett.
**2020**, 125, 133401. [Google Scholar] [CrossRef] [PubMed] - Catani, J.; Lamporesi, G.; Naik, D.; Gring, M.; Inguscio, M.; Minardi, F.; Kantian, A.; Giamarchi, T. Quantum Dynamics of Impurities in a One-Dimensional Bose Gas. Phys. Rev. A
**2012**, 85, 023623. [Google Scholar] [CrossRef] [Green Version] - Scelle, R.; Rentrop, T.; Trautmann, A.; Schuster, T.; Oberthaler, M.K. Motional Coherence of Fermions Immersed in a Bose Gas. Phys. Rev. Lett.
**2013**, 111, 070401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rentrop, T.; Trautmann, A.; Olivares, F.A.; Jendrzejewski, F.; Komnik, A.; Oberthaler, M.K. Observation of the Phononic Lamb Shift with a Synthetic Vacuum. Phys. Rev. X
**2016**, 6, 041041. [Google Scholar] [CrossRef] [Green Version] - Yan, Z.; Patel, P.B.; Mukherjee, B.; Fletcher, R.J.; Struck, J.; Zwierlein, M.W. Boiling a Unitary Fermi Liquid. Phys. Rev. Lett.
**2019**, 122, 093401. [Google Scholar] [CrossRef] [Green Version] - Ness, G.; Shkedrov, C.; Florshaim, Y.; Diessel, O.K.; von Milczewski, J.; Schmidt, R.; Sagi, Y. Observation of a smooth polaron-molecule transition in a degenerate Fermi gas. Phys. Rev. X
**2020**, 10, 041019. [Google Scholar] [CrossRef] - DeSalvo, B.J.; Patel, K.; Cai, G.; Chin, C. Observation of fermion-mediated interactions between bosonic atoms. Nature
**2019**, 568, 61. [Google Scholar] [CrossRef] [Green Version] - Edri, H.; Raz, B.; Matzliah, N.; Davidson, N.; Ozeri, R. Observation of Spin-Spin Fermion-Mediated Interactions between Ultracold Bosons. Phys. Rev. Lett.
**2020**, 124, 163401. [Google Scholar] [CrossRef] - Peyronel, T.; Firstenberg, O.; Liang, Q.-Y.; Hofferberth, S.; Gorshkov, A.V.; Pohl, T.; Lukin, M.D.; Vuletić, V. Quantum nonlinear optics with single photons enabled by strongly interacting atoms. Nature
**2012**, 488, 57. [Google Scholar] [CrossRef] [Green Version] - Ningyuan, J.; Georgakopoulos, A.; Ryou, A.; Schine, N.; Sommer, A.; Simon, J. Observation and characterization of cavity Rydberg polaritons. Phys. Rev. A
**2016**, 93, 041802. [Google Scholar] [CrossRef] [Green Version] - Thompson, J.D.; Nicholson, T.L.; Liang, Q.-Y.; Cantu, S.H.; Venkatramani, A.V.; Choi, S.; Fedorov, I.A.; Viscor, D.; Pohl, T.; Lukin, M.D.; et al. Symmetry-protected collisions between strongly interacting photons. Nature
**2017**, 542, 206. [Google Scholar] [CrossRef] [PubMed] - Chin, C.; Grimm, R.; Julienne, P.; Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys.
**2010**, 82, 1225. [Google Scholar] [CrossRef] - Mistakidis, S.I.; Katsimiga, G.C.; Koutentakis, G.M.; Busch, T.H.; Schmelcher, P. Quench Dynamics and Orthogonality Catastrophe of Bose Polarons. Phys. Rev. Lett.
**2019**, 122, 183001. [Google Scholar] [CrossRef] [Green Version] - Massignan, P.; Zaccanti, M.; Bruun, G.M. Polarons, dressed molecules and itinerant ferromagnetism in ultracold Fermi gases. Rep. Prog. Phys.
**2014**, 77, 034401. [Google Scholar] [CrossRef] - Schmidt, R.; Knap, M.; Ivanov, D.A.; You, J.-S.; Cetina, M.; Demler, E. Universal many-body response of heavy impurities coupled to a Fermi sea: A review of recent progress. Rep. Prog. Phys.
**2018**, 81, 024401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kutschera, M.; Wójcik, W. Proton impurity in the neutron matter: A nuclear polaron problem. Phys. Rev. C
**1993**, 47, 1077. [Google Scholar] [CrossRef] - Forbes, M.M.; Gezerlis, A.; Hebeler, K.; Lesinski, T.; Schwenk, A. Neutron polaron as a constraint on nuclear density functionals. Phys. Rev. C
**2014**, 89, 041301(R). [Google Scholar] [CrossRef] [Green Version] - Tajima, H.; Hatsuda, T.; van Wyk, P.; Ohashi, Y. Superfluid Phase Transitions and Effects Thermal Pairing Fluctuations in Asymmetric Nuclear Matter. Sci. Rep.
**2019**, 9, 18477. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nakano, E.; Iida, K.; Horiuchi, W. Quasiparticle properties of a single α particle in cold neutron matter. Phys. Rev. C
**2020**, 102, 055802. [Google Scholar] [CrossRef] - Vidana, I. Fermi polaron in low-density spin-polarized neutron matter. arXiv
**2021**, arXiv:2101.02941. [Google Scholar] - Sous, J.; Sadeghpour, H.R.; Killian, T.C.; Demler, E.; Schmidt, R. Rydberg impurity in a Fermi gas: Quantum statistics and rotational blockade. Phys. Rev. Res.
**2020**, 2, 023021. [Google Scholar] [CrossRef] [Green Version] - Chevy, F. Universal phase diagram of a strongly interacting Fermi gas with unbalanced spin populations. Phys. Rev. A
**2006**, 74, 063628. [Google Scholar] [CrossRef] [Green Version] - Combescot, R.; Recati, A.; Lobo, C.; Chevy, F. Normal State of Highly Polarized Fermi Gases: Simple Many-Body Approaches. Phys. Rev. Lett.
**2007**, 98, 180402. [Google Scholar] [CrossRef] [Green Version] - Combescot, R.; Giraud, S. Normal State of Highly Polarized Fermi Gases: Full Many-Body Treatment. Phys. Rev. Lett.
**2008**, 101, 050404. [Google Scholar] [CrossRef] - Cui, X.; Zhai, H. Stability of a fully magnetized ferromagnetic state in repulsively interacting ultracold Fermi gases. Phys. Rev. A
**2010**, 81, 041602(R). [Google Scholar] [CrossRef] [Green Version] - Bruun, G.M.; Massignan, P. Decay of Polarons and Molecules in a Strongly Polarized Fermi Gas. Phys. Rev. Lett.
**2010**, 105, 020403. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mathy, C.J.M.; Parish, M.M.; Huse, D.A. Trimers, Molecules, and Polarons in Mass-Imbalanced Atomic Fermi Gases. Phys. Rev. Lett.
**2011**, 106, 166404. [Google Scholar] [CrossRef] [Green Version] - Schmidt, R.; Enss, T. Excitation spectra and rf response near the polaron-to-molecule transition from the functional renormalization group. Phys. Rev. A
**2011**, 83, 063620. [Google Scholar] [CrossRef] [Green Version] - Trefzger, C.; Castin, Y. Impurity in a Fermi sea on a narrow Feshbach resonance: A variational study of the polaronic and dimeronic branches. Phys. Rev. A
**2012**, 85, 053612. [Google Scholar] [CrossRef] [Green Version] - Baarsma, J.E.; Armaitis, J.; Duine, R.A.; Stoof, H.T.C. Polarons in extremely polarized Fermi gases: The strongly interacting
^{6}Li-^{40}K mixture. Phys. Rev. A**2012**, 85, 033631. [Google Scholar] [CrossRef] [Green Version] - Prokof’ev, N.; Svistunov, B. Fermi-polaron problem: Diagrammatic Monte Carlo method for divergent sign-alternating series. Phys. Rev. B
**2008**, 77, 020408(R). [Google Scholar] [CrossRef] [Green Version] - Prokof’ev, N.V.; Svistunov, B.V. Bold diagrammatic Monte Carlo: A generic sign-problem tolerant technique for polaron models and possibly interacting many-body problems. Phys. Rev. B
**2008**, 77, 125101. [Google Scholar] [CrossRef] [Green Version] - Vlietinck, J.; Ryckebusch, J.; Van Houcke, K. Quasiparticle properties of an impurity in a Fermi gas. Phys. Rev. B
**2013**, 87, 115133. [Google Scholar] [CrossRef] [Green Version] - Kroiss, P.; Pollet, L. Diagrammatic Monte Carlo study of a mass-imbalanced Fermi-polaron system. Phys. Rev. B
**2015**, 91, 144507. [Google Scholar] [CrossRef] [Green Version] - Goulko, O.; Mishchenko, A.S.; Prokof’ev, N.; Svistunov, B. Dark continuum in the spectral function of the resonant Fermi polaron. Phys. Rev. A
**2016**, 94, 051605(R). [Google Scholar] [CrossRef] [Green Version] - Van Houcke, K.; Werner, F.; Rossi, R. High-precision numerical solution of the Fermi polaron problem and large-order behavior of its diagrammatic series. Phys. Rev. B
**2020**, 101, 045134. [Google Scholar] [CrossRef] [Green Version] - Kamikado, K.; Kanazawa, T.; Uchino, S. Mobile impurity in a Fermi sea from the functional renormalization group analytically continued to real time. Phys. Rev. A
**2017**, 95, 013612. [Google Scholar] [CrossRef] [Green Version] - Liu, W.E.; Shi, Z.-Y.; Parish, M.M.; Levinsen, J. Theory of radio-frequency spectroscopy of impurities in quantum gases. Phys. Rev. A
**2020**, 102, 023304. [Google Scholar] [CrossRef] - Liu, W.E.; Shi, Z.-Y.; Levinsen, J.; Parish, M.M. Radio-Frequency Response and Contact of Impurities in a Quantum Gas. Phys. Rev. Lett.
**2020**, 125, 065301. [Google Scholar] [CrossRef] - Pilati, S.; Giorgini, S. Phase Separation in a Polarized Fermi Gas at Zero Temperature. Phys. Rev. Lett.
**2008**, 100, 030401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mora, C.; Chevy, F. Normal State of an Imbalanced Fermi Gas. Phys. Rev. Lett.
**2010**, 104, 230402. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Giraud, S.; Combescot, R. Interaction between polarons and analogous effects in polarized Fermi gases. Phys. Rev. A
**2012**, 85, 013605. [Google Scholar] [CrossRef] [Green Version] - Hu, H.; Mulkerin, B.C.; Wang, J.; Liu, X.-J. Attractive Fermi polarons at nonzero temperatures with a finite impurity concentration. Phys. Rev. A
**2018**, 98, 013626. [Google Scholar] [CrossRef] [Green Version] - Tajima, H.; Uchino, S. Many Fermi polarons at nonzero temperature. New J. Phys.
**2018**, 20, 073048. [Google Scholar] [CrossRef] - Tajima, H.; Uchino, S. Thermal crossover, transition, and coexistence in Fermi polaronic spectroscopies. Phys. Rev. A
**2019**, 99, 063606. [Google Scholar] [CrossRef] [Green Version] - Tajima, H.; Takahashi, J.; Nakano, E.; Iida, K. Collisional dynamics of polaronic clouds immersed in a Fermi sea. Phys. Rev. A
**2020**, 102, 051302(R). [Google Scholar] [CrossRef] - Takahashi, J.; Tajima, H.; Nakano, E.; Iida, K. Extracting non-local inter-polaron interactions from collisional dynamics. arXiv
**2020**, arXiv:2011.07911. [Google Scholar] - Mistakidis, S.I.; Volosniev, A.G.; Schmelcher, P. Induced correlations between impurities in a one-dimensional quenched Bose gas. Phys. Rev. Res.
**2020**, 2, 023154. [Google Scholar] [CrossRef] - Mistakidis, S.I.; Katsimiga, G.C.; Koutentakis, G.M.; Busch, T.; Schmelcher, P. Pump-probe spectroscopy of Bose polarons: Dynamical formation and coherence. Phys. Rev. Res.
**2020**, 2, 033380. [Google Scholar] [CrossRef] - Mukherjee, K.; Mistakidis, S.I.; Majumder, S.; Schmelcher, P. Induced interactions and quench dynamics of bosonic impurities immersed in a Fermi sea. Phys. Rev. A
**2020**, 102, 053317. [Google Scholar] [CrossRef] - Compagno, E.; De Chiara, G.; Angelakis, D.G.; Palma, G.M. Tunable Polarons in Bose-Einstein Condensates. Sci. Rep.
**2017**, 7, 2355. [Google Scholar] [CrossRef] [Green Version] - Sous, J.; Berciu, M.; Krems, R.V. Bipolarons bound by repulsive phonon-mediated interactions. Phys. Rev. A
**2017**, 96, 063619. [Google Scholar] [CrossRef] [Green Version] - Nakano, E.; Yabu, H.; Iida, K. Bose-Einstein-Condensate Polaron in Harmonic Trap Potentials in the Weak-Coupling Regime: Lee-Low-Pines–Type Approach. Phys. Rev. A
**2017**, 95, 023626. [Google Scholar] [CrossRef] [Green Version] - Watanabe, K.; Nakano, E.; Yabu, H. Bose Polaron in Spherically Symmetric Trap Potentials: Ground States with Zero and Lower Angular Momenta. Phys. Rev. A
**2019**, 99, 033624. [Google Scholar] [CrossRef] [Green Version] - Peña Ardila, L.A.; Jorgensen, N.B.; Pohl, T.; Giorgini, S.; Bruun, G.M.; Arlt, J.J. Analyzing a Bose polaron across resonant interactions. Phys. Rev. A
**2019**, 99, 063607. [Google Scholar] [CrossRef] [Green Version] - Peña Ardila, L.A.; Astrakharchik, G.E.; Giorgini, S. Strong coupling Bose polarons in a two-dimensional gas. Phys. Rev. Res.
**2020**, 2, 023405. [Google Scholar] [CrossRef] - Cucchietti, F.M.; Timmermans, E. Strong-Coupling Polarons in Dilute Gas Bose-Einstein Condensates. Phys. Rev. Lett.
**2006**, 96, 210401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sacha, K.; Timmermans, E. Self-Localized Impurities Embedded in a One-Dimensional Bose-Einstein Condensate and Their Quantum Fluctuations. Phys. Rev. A
**2006**, 73, 063604. [Google Scholar] [CrossRef] [Green Version] - Kalas, R.M.; Blume, D. Interaction-Induced Localization of an Impurity in a Trapped Bose-Einstein Condensate. Phys. Rev. A
**2006**, 73, 043608. [Google Scholar] [CrossRef] [Green Version] - Bruderer, M.; Bao, W.; Jaksch, D. Self-trapping of impurities in Bose-Einstein condensates: Strong attractive and repulsive coupling. EuroPhys. Lett.
**2008**, 82, 30004. [Google Scholar] [CrossRef] [Green Version] - Boudjemâa, A. Self-localized state and solitons in a Bose-Einstein-condensate-impurity mixture at finite temperature. Phys. Rev. A
**2014**, 90, 013628. [Google Scholar] [CrossRef] [Green Version] - Sous, J.; Chakraborty, M.; Adolphs, C.P.J.; Krems, R.V.; Berciu, M. Phonon-mediated repulsion, sharp transitions and (quasi)self-trapping in the extended Peierls-Hubbard model. Sci. Rep.
**2017**, 7, 1169. [Google Scholar] [CrossRef] [Green Version] - Cai, Z.; Wang, L.; Xie, X.C.; Wang, Y. Interaction-induced anomalous transport behavior in one-dimensional optical lattices. Phys. Rev. A
**2010**, 81, 043602. [Google Scholar] [CrossRef] [Green Version] - Theel, F.; Keiler, K.; Mistakidis, S.I.; Schmelcher, P. Many-body collisional dynamics of impurities injected into a double-well trapped Bose-Einstein condensate. arXiv
**2020**, arXiv:2009.12147. [Google Scholar] - Keiler, K.; Mistakidis, S.I.; Schmelcher, P. Doping a lattice-trapped bosonic species with impurities: From ground state properties to correlated tunneling dynamics. New J. Phys.
**2020**, 22, 083003. [Google Scholar] [CrossRef] - Siegl, P.; Mistakidis, S.I.; Schmelcher, P. Many-body expansion dynamics of a Bose-Fermi mixture confined in an optical lattice. Phys. Rev. A
**2018**, 97, 053626. [Google Scholar] [CrossRef] [Green Version] - Mistakidis, S.I.; Grusdt, F.; Koutentakis, G.M.; Schmelcher, P. Dissipative correlated dynamics of a moving impurity immersed in a Bose–Einstein condensate. New J. Phys.
**2019**, 21, 103026. [Google Scholar] [CrossRef] - Strinati, G.C.; Pieri, P.; Röpke, G.; Schuck, P.; Urban, M. The BCS-BEC crossover: From ultra-cold Fermi gases to nuclear systems. Phys. Rep.
**2018**, 738, 1. [Google Scholar] [CrossRef] [Green Version] - Ohashi, Y.; Tajima, H.; van Wyk, P. BCS-BEC crossover in cold atomic and nuclear systems. Prog. Part. Nucl. Phys.
**2020**, 111, 103739. [Google Scholar] [CrossRef] - Kashimura, T.; Watanabe, R.; Ohashi, Y. Spin susceptibility and fluctuation corrections in the BCS-BEC crossover regime of an ultracold Fermi gas. Phys. Rev. A
**2012**, 86, 043622. [Google Scholar] [CrossRef] [Green Version] - Tajima, H.; van Wyk, P.; Hanai, R.; Kagamihara, D.; Inotani, D.; Horikoshi, M.; Ohashi, Y. Strong-coupling corrections to ground-state properties of a superfluid Fermi gas. Phys. Rev. A
**2017**, 95, 043625. [Google Scholar] [CrossRef] [Green Version] - Horikoshi, M.; Koashi, M.; Tajima, H.; Ohashi, Y.; Kuwata-Gonokami, M. Ground-State Thermodynamic Quantities of Homogeneous Spin-1/2 Fermions from the BCS Region to the Unitarity Limit. Phys. Rev. X
**2017**, 7, 041004. [Google Scholar] [CrossRef] [Green Version] - Haussmann, R. Crossover from BCS superconductivity to Bose-Einstein condensation: A self-consistent theory. Z. Phys. B
**1993**, 91, 291. [Google Scholar] [CrossRef] - Haussmann, R.; Rantner, W.; Cerrito, S.; Zwerger, W. THermodynamics of the BCS-BEC crossover. Phys. Rev. A
**2007**, 75, 023610. [Google Scholar] [CrossRef] [Green Version] - Takahashi, J.; Imai, R.; Nakano, E.; Iida, K. Bose Polaron in Spherical Trap Potentials: Spatial Structure and Quantum Depletion. Phys. Rev. A
**2019**, 100, 023624. [Google Scholar] [CrossRef] [Green Version] - Pethick, C.J.; Smith, H. Bose-Einstein Condensation in Dilute Gases; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar]
- Morgan, S.A.; Lee, M.D.; Burnett, K. Off-shell T matrices in one, two and three dimensions. Phys. Rev. A
**2002**, 65, 022706. [Google Scholar] [CrossRef] - Fetter, A.L.; Walecka, J.D. Quantum Theory of Many-Particle Systems; Dover: New York, NY, USA, 2003. [Google Scholar]
- Törmä, P. Spectroscopies–Theory. In Quantum Gas Experiments: Exploring Many-Body States; Törmä, P., Sengstock, K., Eds.; Imperial College: London, UK, 2015. [Google Scholar]
- Frank, B.; Lang, J.; Zwerger, W. Universal phase diagram and scaling functions of imbalanced Fermi gases. J. Exp. Theor. Phys.
**2018**, 127, 812. [Google Scholar] [CrossRef] [Green Version] - Pini, M.; Pieri, P.; Strinati, G.C. Fermi gas throughout the BCS-BEC crossover: Comparative study of t-matrix approaches with various degrees of self-consistency. Phys. Rev. B
**2019**, 99, 094502. [Google Scholar] [CrossRef] [Green Version] - Punk, M.; Dumitrescu, P.T.; Zwerger, W. Polaron-to-molecule transition in a strongly imbalanced Fermi gas. Phys. Rev. A
**2009**, 80, 053605. [Google Scholar] [CrossRef] [Green Version] - Cui, X. Fermi polaron revisited: Polaron-molecule transition and coexistence. Phys. Rev. A
**2020**, 102, 061301(R). [Google Scholar] [CrossRef] - Thouless, D.J. Perturbation theory in statistical mechanics and the theory of superconductivity. Ann. Phys.
**1960**, 10, 553. [Google Scholar] [CrossRef] [Green Version] - Liu, X.-J.; Hu, H. BCS-BEC crossover in an asymmetric two-component Fermi gas. Europhys. Lett.
**2006**, 75, 364. [Google Scholar] [CrossRef] - Grudst, F.; Seetharam, K.; Shchadilova, Y.; Demler, E. Strong-coupling Bose polarons out of equilibrium: Dynamical renormalization group approach. Phys. Rev. A
**2018**, 97, 033612. [Google Scholar] - Nielsen, K.; Peña Ardila, L.A.; Bruun, G.M.; Pohl, T. Critical slowdown of non-equilibrium polaron dynamics. New J. Phys.
**2019**, 21, 043014. [Google Scholar] [CrossRef] - Sekino, Y.; Tajima, H.; Uchino, S. Mesoscopic spin transport between strongly interacting Fermi gases. Phys. Rev. Res.
**2020**, 2, 023152. [Google Scholar] [CrossRef] - Parish, M.M.; Adlong, H.S.; Liu, W.E.; Levinsen, J. Thermodynamic signatures of the polaron-molecule transition in a Fermi gas. Phys. Rev. A
**2021**, 103, 023312. [Google Scholar] [CrossRef] - Schmidt, R.; Enss, T.; Pietilä, V.; Demler, E. Fermi polarons in two dimensions. Phys. Rev. A
**2012**, 85, 021602(R). [Google Scholar] [CrossRef] [Green Version] - Klawunn, M.; Recati, A. The Fermi-polaron in two dimensions: Importance of the two-body bound state. Phys. Rev. A
**2011**, 84, 033607. [Google Scholar] [CrossRef] [Green Version] - Tajima, H.; Tsutsui, S.; Doi, T.M. Low-dimensional fluctuations and pseudogap in Gaudin-Yang Fermi gas. Phys. Rev. Res.
**2020**, 2, 033441. [Google Scholar] [CrossRef] - Mistakidis, S.I.; Katsimiga, G.C.; Koutentakis, G.M.; Schmelcher, P. Repulsive Fermi polarons and their induced interactions in binary mixtures of ultracold atoms. New J. Phys.
**2018**, 21, 043032. [Google Scholar] [CrossRef] [Green Version] - Kwasniok, J.; Mistakidis, S.I.; Schmelcher, P. Correlated dynamics of fermionic impurities of an emsemble of fermions. Phys. Rev. A
**2020**, 101, 053619. [Google Scholar] [CrossRef] - Guan, X.-W.; Batchelor, M.T.; Lee, C. Fermi gases in one dimension: From Bethe ansatz to experiments. Rev. Mod. Phys.
**2013**, 85, 1633. [Google Scholar] [CrossRef] [Green Version] - Doggen, E.V.H.; Kinnunen, J.J. Energy and Contact of the One-Dimensional Fermi Polaron at Zero and Finite Temperature. Phys. Rev. Lett.
**2013**, 111, 025302. [Google Scholar] [CrossRef] - Lampo, A.; Charalambous, C.; García-March, M.Á.; Lewenstein, M. Non-Markovian Polaron Dynamics in a Trapped Bose-Einstein Condensate. Phys. Rev. A
**2018**, 98, 063630. [Google Scholar] [CrossRef] [Green Version] - Mistakidis, S.I.; Volosniev, A.G.; Zinner, N.T.; Schmelcher, P. Effective Approach to Impurity Dynamics in One-Dimensional Trapped Bose Gases. Phys. Rev. A
**2019**, 100, 013619. [Google Scholar] [CrossRef] [Green Version] - Pitaevskii, L.; Stringari, S. Bose-Einstein Condensation; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Dalfovo, F.; Giorgini, S.; Pitaevskii, L.P.; Stringari, S. Theory of Bose-Einstein Condensation in Trapped Gases. Rev. Mod. Phys.
**1999**, 71, 463. [Google Scholar] [CrossRef] [Green Version] - Bogoliubov, N.N. On the theory of superfluidity. J. Phys.
**1947**, 11, 32. [Google Scholar] - De Gennes, P.G. Superconductivity of Metals and Alloys; CRC Press: Boca Raton, FL, USA, 1966. [Google Scholar]
- Katsimiga, G.C.; Mistakidis, S.I.; Bersano, T.M.; Ome, M.K.H.; Mossman, S.M.; Mukherjee, K.; Schmelcher, P.; Engels, P.; Kevrekidis, P.G. Observation and Analysis of Multiple Dark-Antidark Solitons in Two-Component Bose-Einstein Condensates. Phys. Rev. A
**2020**, 102, 023301. [Google Scholar] [CrossRef] - Katsimiga, G.C.; Mistakidis, S.I.; Schmelcher, P.; Kevrekidis, P.G. Phase Diagram, Stability and Magnetic Properties of Nonlinear Excitations in Spinor Bose-Einstein Condensates. New J. Phys.
**2021**, 23, 013015. [Google Scholar] [CrossRef] - Mueller, E.J.; Ho, T.L.; Ueda, M.; Baym, G. Fragmentation of bose-einstein condensates. Phys. Rev. A
**2006**, 74, 033612. [Google Scholar] [CrossRef] [Green Version] - Shchadilova, Y.E.; Schmidt, R.; Grusdt, F.; Demler, E. Quantum Dynamics of Ultracold Bose Polarons. Phys. Rev. Lett.
**2016**, 117, 113002. [Google Scholar] [CrossRef] [PubMed] - Drescher, M.; Salmhofer, M.; Enss, T. Theory of a resonantly interacting impurity in a Bose-Einstein condensate. Phys. Rev. Res.
**2020**, 2, 032011(R). [Google Scholar] [CrossRef] - Guenther, N.-E.; Schmidt, R.; Bruun, G.M.; Gurarie, V.; Massignan, P. Mobile impurity in a Bose-Einstein condensate and the orthogonality catastrophe. Phys. Rev. A
**2021**, 103, 013317. [Google Scholar] [CrossRef] - Tempere, J.; Klimin, S.N.; Devreese, J.T. Effect of population imbalance on the Berezinskii-Kosterlitz-Thouless phase transition in a superfluid Fermi gas. Phys. Rev. A
**2009**, 79, 053637. [Google Scholar] [CrossRef] [Green Version] - Levinsen, J.; Baur, S.K. High-polarization limit of the quasi-two-dimensional Fermi gas. Phys. Rev. A
**2012**, 86, 041602(R). [Google Scholar] [CrossRef] [Green Version] - Nishimura, K.; Nakano, E.; Iida, K.; Tajima, H.; Miyakawa, T.; Yabu, H. The ground state of polaron in an ultracold dipolar Fermi gas. arXiv
**2020**, arXiv:2010.15558. [Google Scholar]

**Figure 1.**Feynman diagrams for (

**a**) the T-matrix approach (TMA), (

**b**) the extended T-matrix approach (ETMA), and (

**c**) the self-consistent T-matrix approach (SCTMA). $\Gamma $ and ${\Gamma}_{\mathrm{S}}$ are the many-body T-matrices, whose perturbative expansions are shown schematically in (

**d**,

**e**), consisting of bare and dressed propagators ${G}_{\sigma}^{0}$ and ${G}_{\sigma}$, respectively. While in TMA, all the lines in the self-energy (

**a**) consist of ${G}_{\sigma}^{0}$, they are replaced with ${G}_{\sigma}$ partially (upper loop of (

**b**)) in ETMA and fully in SCTMA (

**c**) (see also (

**e**) where ${G}_{\sigma}^{0}$ is replaced by ${G}_{\sigma}$ compared to (

**d**)), respectively.

**Figure 2.**Feynman diagrams for induced (

**a**) two- and (

**b**) three-body interactions ${V}_{\mathrm{eff}}^{(2,3)}$ among polarons. The arrows represent the direction of momentum and energy transfer in each propagator.

**Figure 3.**Zero -momentum spectral functions ${A}_{\sigma}(\mathit{p}=\mathbf{0},\omega )$ of (

**a**) the majority (medium) and (

**b**) the minority (impurities) fermions for varying energy $\omega $ at unitarity, ${\left({p}_{\mathrm{F}}a\right)}^{-1}=0$. We consider a temperature $T=0.3{T}_{\mathrm{F}}$ and an impurity concentration $x=0.1$. The solid, dashed, and dash-dotted lines represent the results of the TMA, ETMA, and SCTMA approaches respectively. While ${A}_{\uparrow}(\mathit{p}=\mathbf{0},\omega )$ is almost the same among the three approaches, ${A}_{\downarrow}(\mathit{p}=\mathbf{0},\omega )$ within the SCTMA experiences a sizable difference compared to the response obtained in the TMA and the ETMA approaches.

**Figure 4.**(

**a**) Polaron spectral function ${A}_{\downarrow}(\mathit{p}=\mathbf{0},\omega )$ for several coupling strengths ${\left({p}_{\mathrm{F}}a\right)}^{-1}$. The spectrum is calculated within the ETMA at temperature $T=0.03{T}_{\mathrm{F}}$ and impurity concentration $x=O\left({10}^{-4}\right)$ [55]. Panel (

**b**) represents the attractive and repulsive polaron energies, namely, ${E}_{\mathrm{P}}^{\left(\mathrm{a}\right)}$ and ${E}_{\mathrm{P}}^{\left(\mathrm{r}\right)}$, respectively, as a function of ${\left({p}_{\mathrm{F}}a\right)}^{-1}$. The polaron energies have been extracted from the peak position of ${A}_{\downarrow}(\mathit{p}=\mathbf{0},\omega )$, that is, the pole of ${G}_{\downarrow}^{\mathrm{R}}(\mathit{p}=\mathbf{0},\omega )$. The experimental data of Ref. [10] are plotted in black circles for direct comparison with the theoretical predictions.

**Figure 5.**Polaron spectral function ${A}_{\downarrow}(\mathit{p},\omega )$ as a function of the momentum $\mathit{p}$ and the energy $\omega $ of the impurities at temperature $T=0.2{T}_{\mathrm{F}}$, impurity concentration $x=0.1$, and interaction ${\left({p}_{\mathrm{F}}a\right)}^{-1}=0$. ${A}_{\downarrow}(\mathit{p},\omega )$ is calculated within (

**a**) the ETMA and (

**b**) the SCTMA approaches. The vertical dashed line marks the Fermi momentum $p={p}_{\mathrm{F}}$ of the medium. While the two approaches predict qualitatively similar spectra with a sharp peak at low momenta and broadening above $p={p}_{\mathrm{F}}$, the SCTMA result (

**b**) shows a relatively broadened peak at low momenta compared to the ETMA one (

**a**).

**Figure 6.**Spectral function of the medium ${A}_{\uparrow}(\mathit{p}=\mathbf{0},\omega )$ within the ETMA approach at zero momentum of the impurity and for different impurity-medium couplings ${\left({p}_{\mathrm{F}}a\right)}^{-1}=-0.4$, 0, $0.4$, $0.7$, and $1.0$. The temperature and the impurity concentration are given by $T=0.4{T}_{\mathrm{F}}$ and $x=0.1$, respectively. The inset shows the corresponding impurity spectral functions ${A}_{\downarrow}(\mathit{p}=\mathbf{0},\omega )$. While the sharp peak at $\omega +{\mu}_{\uparrow}\simeq 0$ in ${A}_{\uparrow}(\mathit{p}=0,\omega )$ is associated with the bare state, the small amplitude side peaks at positive ($\omega +{\mu}_{\uparrow}\simeq {E}_{\mathrm{F}}$) and negative energies ($\omega +{\mu}_{\downarrow}\simeq -3{E}_{\mathrm{F}}$ for the case with ${\left({p}_{\mathrm{F}}a\right)}^{-1}=1$) originate from the backaction due to the impurities.

**Figure 7.**Spectral function ${A}_{\sigma}(\mathit{p},\omega )$ of the Fermi (

**a1**) medium and (

**a2**) impurities in two-dimensions for different momenta and energies of the impurities. We consider a temperature $T=0.3{T}_{\mathrm{F}}$, impurity concentration $x=0.1$, and dimensionless coupling parameter $ln\left({p}_{\mathrm{F}}{a}_{2\mathrm{D}}\right)=0.4$. The vertical dashed line indicates the Fermi momentum $p={p}_{\mathrm{F}}$ of the majority component atoms. While the majority component (a) exhibits a sharp peak with quadratic dispersion $\omega +{\mu}_{\uparrow}={p}^{2}/\left(2m\right)$, the minority atoms (b) form the attractive polaron at negative energies ($\omega +{\mu}_{\downarrow}<0$) and a broadened peak associated with the repulsive impurity branch at positive energies $(\omega +{\mu}_{\downarrow}>0)$. For comparison, we provide the spectral functions of the medium (

**b1**) and the impurities (

**b2**) in the case of $T=0.3{T}_{\mathrm{F}}$, $x=0.3$ and $ln\left({p}_{\mathrm{F}}{a}_{2\mathrm{D}}\right)=0$. Evidently, the feedback on the medium from the impurities is enhanced in the low-momentum region ($p\simeq 0$).

**Figure 8.**Spectral function ${A}_{\sigma}(\mathit{p},\omega )$ of the fermionic (

**a1**) background and (

**a2**) impurity atoms of concentration $x=0.326$ with an attractive medium-impurity interaction for varying momenta and energies of the impurities in one-dimension. The system is at temperature $T=0.157{T}_{\mathrm{F}}$ and dimensionless coupling parameter ${\left({p}_{\mathrm{F}}{a}_{1\mathrm{D}}\right)}^{-1}=0.28$. ${P}_{T}=\sqrt{2mT}$ is the momentum scale associated with the temperature T. The vertical dashed line marks the Fermi momentum $p={p}_{\mathrm{F}}$ of the background atoms. The majority component (

**a1**) is largely broadened due to the backaction from the impurities in the low-momentum region ($p\lesssim {p}_{T}$). On the other hand, the minority component (

**a2**) exhibits a sharp peak in the low-momentum region below $p={p}_{\mathrm{F}}$ and it is broadened above $p={p}_{\mathrm{F}}$ For comparison, we show the (

**b1**) medium and (

**b2**) impurity spectral functions in the case of repulsive medium-impurity interaction characterized by ${\left({p}_{\mathrm{F}}{a}_{1\mathrm{D}}\right)}^{-1}=-0.55$, where the temperature and the impurity concentraion are given by $T=0.598{T}_{\mathrm{F}}$ and $x=0.264$ Although the impurity quasiparticle peak in the low-energy region ($\omega +{\mu}_{\downarrow}\simeq 0$) is shifted upward, the tendency of a spectral broadening is similar to the attractive case.

**Figure 9.**Radial profiles of (

**a**) the order parameter $\overline{\varphi}\left(r\right)=\varphi (r;{g}_{\mathrm{IB}}=0)/\sqrt{{N}_{0}/4\pi}$ and (

**c**) the density of depletion ${\overline{n}}_{\mathrm{ex}}\left(r\right)={n}_{\mathrm{ex}}(r;{g}_{\mathrm{IB}}=0)$ in the absence of an impurity. Differences of the radial profiles of (

**b**) the order parameter $\delta \Phi \left(r\right)=(\varphi (r;{g}_{\mathrm{IB}})-\varphi (r;{g}_{\mathrm{IB}}=0))/\sqrt{{N}_{0}/4\pi}$ and (

**d**) the density of depletion $\delta {n}_{\mathrm{ex}}\left(r\right)={n}_{\mathrm{ex}}(r;{g}_{\mathrm{IB}})-{n}_{\mathrm{ex}}(r;{g}_{\mathrm{IB}}=0)$ in the presence of an impurity from the result depicted in (

**a**) and (

**c**), respectively.

**Table 1.**The number of depletion ${N}_{\mathrm{ex}}$ and its deviation $\delta {N}_{\mathrm{ex}}=4\pi \int \phantom{\rule{-0.166667em}{0ex}}dr\phantom{\rule{0.166667em}{0ex}}{r}^{2}\delta {n}_{\mathrm{ex}}\left(r\right)$ from the case of zero impurity-medium interaction. It is evident that degree of depletion increases (decreases) for attractive (repulsive) interactions.

$1/\left({\mathit{a}}_{\mathbf{IB}}{\mathit{n}}_{\mathbf{B}}^{1/3}\right)$ | ∞ | +1 | −1 |
---|---|---|---|

${N}_{\mathrm{ex}}$ | 24.244 | 24.220 | 24.270 |

$\delta {N}_{\mathrm{ex}}$ | 0 | −2.361$\phantom{\rule{0.166667em}{0ex}}\times $${10}^{-2}$ | 2.584$\phantom{\rule{0.166667em}{0ex}}\times $ ${10}^{-2}$ |

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## Share and Cite

**MDPI and ACS Style**

Tajima, H.; Takahashi, J.; Mistakidis, S.I.; Nakano, E.; Iida, K.
Polaron Problems in Ultracold Atoms: Role of a Fermi Sea across Different Spatial Dimensions and Quantum Fluctuations of a Bose Medium. *Atoms* **2021**, *9*, 18.
https://doi.org/10.3390/atoms9010018

**AMA Style**

Tajima H, Takahashi J, Mistakidis SI, Nakano E, Iida K.
Polaron Problems in Ultracold Atoms: Role of a Fermi Sea across Different Spatial Dimensions and Quantum Fluctuations of a Bose Medium. *Atoms*. 2021; 9(1):18.
https://doi.org/10.3390/atoms9010018

**Chicago/Turabian Style**

Tajima, Hiroyuki, Junichi Takahashi, Simeon I. Mistakidis, Eiji Nakano, and Kei Iida.
2021. "Polaron Problems in Ultracold Atoms: Role of a Fermi Sea across Different Spatial Dimensions and Quantum Fluctuations of a Bose Medium" *Atoms* 9, no. 1: 18.
https://doi.org/10.3390/atoms9010018