# The Variational Reduction for Low-Dimensional Fermi Gases and Bose–Fermi Mixtures: A Brief Review

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## Abstract

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## 1. Introduction

## 2. The Fermi Gas

#### 2.1. The Two-Dimensional Reduction

#### 2.2. The One-dimensional Reduction

## 3. The Bose–Fermi Mixture

#### 3.1. The Two-Dimensional Reduction

#### 3.2. The One-Dimensional Reduction

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A Nonlinear Schrödinger Equation for the Fermionic Superfluid

## References

- Bongs, K.; Sengstock, K. Physics with coherent matter waves. Rep. Prog. Phys.
**2004**, 67, 907. [Google Scholar] [CrossRef] - Jaksch, D.; Zoller, P. The cold atom Hubbard toolbox. Ann. Phys.
**2005**, 315, 52. [Google Scholar] [CrossRef] - Giorgini, S.; Pitaevskii, L.P.; Stringari, S. Theory of ultracold atomic Fermi gases. Rev. Mod. Phys.
**2008**, 80, 1215. [Google Scholar] [CrossRef] - Bloch, I.; Dalibard, J.; Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys.
**2008**, 80, 885. [Google Scholar] [CrossRef] - Spielman, I.B. Light induced gauge fields for ultracold neutral atoms. Rev. Cold At. Mol.
**2013**, 1, 145. [Google Scholar] - Goldman, N.; Juzeliūnas, G.; Öhberg, P.; Spielman, I.B. Light-induced gauge fields for ultracold atoms. Rep. Prog. Phys.
**2014**, 77, 126401. [Google Scholar] [CrossRef] [PubMed] - Zhai, H. Degenerate quantum gases with spin–orbit coupling: A review. Rep. Prog. Phys.
**2015**, 78, 026001. [Google Scholar] [CrossRef] [PubMed] - Malomed, B.A. Creating solitons by means of spin-orbit coupling. EPL
**2018**, 122, 36001. [Google Scholar] [CrossRef] - Snoek, M.; Titvinidze, I.; Bloch, I.; Hofstetter, W. Effect of interactions on harmonically confined Bose-Fermi mixtures in optical lattices. Phys. Rev. Lett.
**2011**, 106, 155301. [Google Scholar] [CrossRef] [PubMed] - Fröhlich, B.; Feld, M.; Vogt, E.; Koschorreck, M.; Zwerger, W.; Köhl, M. Radio-frequency spectroscopy of a strongly interacting two-dimensional Fermi gas. Phys. Rev. Lett.
**2011**, 106, 105301. [Google Scholar] [CrossRef] [PubMed] - Lin, Y.-J.; Jimenez-Garcia, K.; Spielman, I.B. Spin–orbit-coupled Bose–Einstein condensates. Nature
**2011**, 471, 83. [Google Scholar] [CrossRef] [PubMed] - Sakaguchi, H.; Li, B.; Malomed, B.A. Creation of two-dimensional composite solitons in spin-orbit-coupled self-attractive Bose-Einstein condensates in free space. Phys. Rev. E
**2014**, 89, 032920. [Google Scholar] [CrossRef] [PubMed] - Díaz, P.; Laroze, D.; Ávila, A.; Malomed, B.A. Two-dimensional composite solitons in a spin-orbit-coupled Fermi gas in free space. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 70, 372–383. [Google Scholar] [CrossRef] - Salasnich, L.; Parola, A.; Reatto, L. Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates. Phys. Rev. A
**2002**, 65, 043614. [Google Scholar] [CrossRef] - Salasnich, L.; Parola, A.; Reatto, L. Condensate bright solitons under transverse confinement. Phys. Rev. A
**2002**, 66, 043603. [Google Scholar] [CrossRef] - Salasnich, L.; Malomed, B.A. Solitons and solitary vortices in pancake-shaped Bose-Einstein condensates. Phys. Rev. A
**2009**, 79, 053620. [Google Scholar] [CrossRef] - Young-S, L.E.; Salasnich, L.; Adhikari, S.K. Dimensional reduction of a binary Bose-Einstein condensate in mixed dimensions. Phys. Rev. A
**2010**, 82, 053601. [Google Scholar] [CrossRef] - Adhikari, S.K.; Malomed, B.A. Miscibility in a degenerate fermionic mixture induced by linear coupling. Phys. Rev. A
**2006**, 74, 053620. [Google Scholar] [CrossRef] - Adhikari, S.K. Superfluid fermi-fermi mixture: Phase diagram, stability, and soliton formation. Phys. Rev. A
**2007**, 76, 053609. [Google Scholar] [CrossRef] - Adhikari, S.K.; Malomed, B.A. Gap solitons in a model of a superfluid fermion gas in optical lattices. Physica D
**2009**, 238, 1402–1412. [Google Scholar] [CrossRef] - Díaz, P.; Laroze, D.; Schmidt, I.; Malomed, B.A. One-and two-dimensional reductions of the mean-field description of degenerate Fermi gases. J. Phys. B: At. Mol. Opt. Phys.
**2012**, 45, 145304. [Google Scholar] [CrossRef] - Burger, S.; Bongs, K.; Dettmer, S.; Ertmer, W.; Sengstock, K.; Sanpera, A.; Shlyapnikov, G.V.; Lewenstein, M. Dark solitons in Bose-Einstein condensates. Phys. Rev. Lett.
**1999**, 83, 5198. [Google Scholar] [CrossRef] - Becker, C.; Stellmer, S.; Soltan-Panahi, P.; Dörscher, S.; Baumert, M.; Richter, E.-M.; Kronjäger, J.; Bongs, K.; Sengstock, K. Oscillations and interactions of dark and dark–bright solitons in Bose-Einstein condensates. Nat. Phys.
**2008**, 4, 496. [Google Scholar] [CrossRef] - Weller, A.; Ronzheimer, J.; Gross, C.; Esteve, J.; Oberthaler, M.; Frantzeskakis, D.; Theocharis, G.; Kevrekidis, P. Experimental observation of oscillating and interacting matter wave dark solitons. Phys. Rev. Lett.
**2008**, 101, 130401. [Google Scholar] [CrossRef] [PubMed] - Stellmer, S.; Becker, C.; Soltan-Panahi, P.; Richter, E.-M.; Dörscher, S.; Baumert, M.; Kronjäger, J.; Bongs, K.; Sengstock, K. Collisions of dark solitons in elongated Bose-Einstein condensates. Phys. Rev. Lett.
**2008**, 101, 120406. [Google Scholar] [CrossRef] [PubMed] - Antezza, M.; Dalfovo, F.; Pitaevskii, L.P.; Stringari, S. Dark solitons in a superfluid Fermi gas. Phys. Rev. A
**2007**, 76, 043610. [Google Scholar] [CrossRef] - Scott, R.; Dalfovo, F.; Pitaevskii, L.; Stringari, S. Dynamics of dark solitons in a trapped superfluid Fermi gas. Phys. Rev. Lett.
**2011**, 106, 185301. [Google Scholar] [CrossRef] [PubMed] - Liao, R.; Brand, J. Traveling dark solitons in superfluid Fermi gases. Phys. Rev. A
**2011**, 83, 041604. [Google Scholar] [CrossRef] - Yefsah, T.; Sommer, A.T.; Ku, M.J.; Cheuk, L.W.; Ji, W.; Bakr, W.S.; Zwierlein, M.W. Heavy solitons in a fermionic superfluid. Nature
**2013**, 499, 426. [Google Scholar] [CrossRef] [PubMed] - Ku, M.J.; Mukherjee, B.; Yefsah, T.; Zwierlein, M.W. Cascade of solitonic excitations in a superfluid fermi gas: From planar solitons to vortex rings and lines. Phys. Rev. Lett.
**2016**, 116, 045304. [Google Scholar] [CrossRef] [PubMed] - Syrwid, A.; Delande, D.; Sacha, K. Emergence of dark soliton signatures in a one-dimensional unpolarized attractive Fermi gas on a ring. Phys. Rev. A
**2018**, 98, 023616. [Google Scholar] [CrossRef] - Van Alphen, W.; Lombardi, G.; Klimin, S.N.; Tempere, J. Dark soliton collisions in superfluid Fermi gases. New J. Phys.
**2018**, 20, 053052. [Google Scholar] [CrossRef] - Truscott, A.G.; Strecker, K.E.; McAlexander, W.I.; Partridge, G.B.; Hulet, R.G. Observation of Fermi pressure in a gas of trapped atoms. Science
**2001**, 291, 2570–2572. [Google Scholar] [CrossRef] [PubMed] - Schreck, F.; Khaykovich, L.; Corwin, K.L.; Ferrari, G.; Bourdel, T.; Cubizolles, J.; Salomon, C. Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea. Phys. Rev. Lett.
**2001**, 87, 080403. [Google Scholar] [CrossRef] [PubMed] - Hansen, A.H.; Khramov, A.; Dowd, W.H.; Jamison, A.O.; Ivanov, V.V.; Gupta, S. Quantum degenerate mixture of Ytterbium and Lithium atoms. Phys. Rev. A
**2011**, 84, 011606. [Google Scholar] [CrossRef] - Heinze, J.; Götze, S.; Krauser, J.; Hundt, B.; Fläschner, N.; Lühmann, D.-S.; Becker, C.; Sengstock, K. Multiband spectroscopy of ultracold fermions: Observation of reduced tunneling in attractive Bose-Fermi mixtures. Phys. Rev. Lett.
**2011**, 107, 135303. [Google Scholar] [CrossRef] [PubMed] - Tey, M.K.; Stellmer, S.; Grimm, R.; Schreck, F. Double-degenerate Bose-Fermi mixture of Strontium. Phys. Rev. A
**2010**, 82, 011608. [Google Scholar] [CrossRef] - Best, T.; Will, S.; Schneider, U.; Hackermüller, L.; Van Oosten, D.; Bloch, I.; Lühmann, D.-S. Role of interactions in
^{87}Rb–^{40}K Bose-Fermi mixtures in a 3D optical lattice. Phys. Rev. Lett.**2009**, 102, 030408. [Google Scholar] [CrossRef] [PubMed] - Cumby, T.D.; Shewmon, R.A.; Hu, M.-G.; Perreault, J.D.; Jin, D.S. Feshbach-molecule formation in a Bose-Fermi mixture. Phys. Rev. A
**2013**, 87, 012703. [Google Scholar] [CrossRef] - Deh, B.; Gunton, W.; Klappauf, B.; Li, Z.; Semczuk, M.; Van Dongen, J.; Madison, K. Giant Feshbach resonances in Li 6 - Rb 85 mixtures. Phys. Rev. A
**2010**, 82, 020701. [Google Scholar] [CrossRef] - Tung, S.-K.; Parker, C.; Johansen, J.; Chin, C.; Wang, Y.; Julienne, P.S. Ultracold mixtures of atomic 6 Li and 133 Cs with tunable interactions. Phys. Rev. A
**2013**, 87, 010702. [Google Scholar] [CrossRef] - Park, J.W.; Wu, C.-H.; Santiago, I.; Tiecke, T.G.; Will, S.; Ahmadi, P.; Zwierlein, M.W. Quantum degenerate Bose-Fermi mixture of chemically different atomic species with widely tunable interactions. Phys. Rev. A
**2012**, 85, 051602. [Google Scholar] [CrossRef] - Wu, C.-H.; Santiago, I.; Park, J.W.; Ahmadi, P.; Zwierlein, M.W. Strongly interacting isotopic Bose-Fermi mixture immersed in a Fermi sea. Phys. Rev. A
**2011**, 84, 011601. [Google Scholar] [CrossRef] - Lelas, K.; Jukić, D.; Buljan, H. Ground-state properties of a one-dimensional strongly interacting Bose-Fermi mixture in a double-well potential. Phys. Rev. A
**2009**, 80, 053617. [Google Scholar] [CrossRef] - Watanabe, T.; Suzuki, T.; Schuck, P. Bose-fermi pair correlations in attractively interacting Bose-Fermi atomic mixtures. Phys. Rev. A
**2008**, 78, 033601. [Google Scholar] [CrossRef] - Kain, B.; Ling, H.Y. Singlet and triplet superfluid competition in a mixture of two-component Fermi and one-component dipolar Bose gases. Phys. Rev. A
**2011**, 83, 061603. [Google Scholar] [CrossRef] - Mering, A.; Fleischhauer, M. Multiband and nonlinear hopping corrections to the three-dimensional Bose-Fermi-Hubbard model. Phys. Rev. A
**2011**, 83, 063630. [Google Scholar] [CrossRef] - Song, J.-L.; Zhou, F. Anomalous dimers in quantum mixtures near broad resonances: Pauli blocking, Fermi surface dynamics, and implications. Phys. Rev. A
**2011**, 84, 013601. [Google Scholar] [CrossRef] - Ludwig, D.; Floerchinger, S.; Moroz, S.; Wetterich, C. Quantum phase transition in Bose-Fermi mixtures. Phys. Rev. A
**2011**, 84, 033629. [Google Scholar] [CrossRef] - Bertaina, G.; Fratini, E.; Giorgini, S.; Pieri, P. Quantum Monte Carlo study of a resonant Bose-Fermi mixture. Phys. Rev. Lett.
**2013**, 110, 115303. [Google Scholar] [CrossRef] [PubMed] - Adhikari, S.K.; Salasnich, L. Superfluid Bose-Fermi mixture from weak coupling to unitarity. Phys. Rev. A
**2008**, 78, 043616. [Google Scholar] [CrossRef] - Maruyama, T.; Yabu, H. Quadrupole oscillations in Bose-Fermi mixtures of ultracold atomic gases made of Yb atoms in the time-dependent Gross-Pitaevskii and Vlasov equations. Phys. Rev. A
**2009**, 80, 043615. [Google Scholar] [CrossRef] - Iskin, M.; Freericks, J. Dynamical mean-field theory for light-fermion–heavy-boson mixtures on optical lattices. Phys. Rev. A
**2009**, 80, 053623. [Google Scholar] [CrossRef] - Nishida, Y.; Son, D.T. Effective field theory of boson-fermion mixtures and bound fermion states on a vortex of boson superfluid. Phys. Rev. A
**2006**, 74, 013615. [Google Scholar] [CrossRef] - Salasnich, L.; Toigo, F. Fermi-bose mixture across a Feshbach resonance. Phys. Rev. A
**2007**, 75, 013623. [Google Scholar] [CrossRef] - Gautam, S.; Muruganandam, P.; Angom, D. Position swapping and pinching in Bose-Fermi mixtures with two-color optical Feshbach resonances. Phys. Rev. A
**2011**, 83, 023605. [Google Scholar] [CrossRef] - Díaz, P.; Laroze, D.; Malomed, B.A. Correlations and synchronization in a Bose–Fermi mixture. J. Phys. B: At. Mol. Opt. Phys.
**2015**, 48, 075301. [Google Scholar] [CrossRef] - Tylutki, M.; Recati, A.; Dalfovo, F.; Stringari, S. Dark–bright solitons in a superfluid Bose–Fermi mixture. New J. Phys.
**2016**, 18, 053014. [Google Scholar] [CrossRef] - Manini, N.; Salasnich, L. Bulk and collective properties of a dilute fermi gas in the BCS-BEC crossover. Phys. Rev. A
**2005**, 71, 033625. [Google Scholar] [CrossRef] - Salasnich, L.; Toigo, F. Extended Thomas-Fermi density functional for the unitary Fermi gas. Phys. Rev. A
**2008**, 78, 053626; ibod Erratum: Extended thomas-fermi density functional for the unitary fermi gas. Phys. Rev. A**2010**, 82, 059902. [Google Scholar] [CrossRef] - Ancilotto, F.; Salasnich, L.; Toigo, F. Dc Josephson effect with fermi gases in the Bose-Einstein regime. Phys. Rev. A
**2009**, 79, 033627. [Google Scholar] [CrossRef] - Ancilotto, F.; Salasnich, L.; Toigo, F. Shock waves in strongly interacting Fermi gas from time-dependent density functional calculations. Phys. Rev. A
**2012**, 85, 063612. [Google Scholar] [CrossRef] - Andreev, P.A. Spin current contribution in the spectrum of collective excitations of degenerate partially polarized spin-1/2 fermions at separate dynamics of spin-up and spin-down fermions. Laser Phys. Lett.
**2018**, 15, 105501. [Google Scholar] [CrossRef] - Kim, Y.E.; Zubarev, A.L. Time-dependent density-functional theory for trapped strongly interacting fermionic atoms. Phys. Rev. A
**2004**, 70, 033612. [Google Scholar] [CrossRef] - Adhikari, S.K. Mixing-demixing in a trapped degenerate fermion-fermion mixture. Phys. Rev. A
**2006**, 73, 043619. [Google Scholar] [CrossRef] - Bragard, J.; Boccaletti, S.; Mendoza, C.; Hentschel, H.; Mancini, H. Synchronization of spatially extended chaotic systems in the presence of asymmetric coupling. Phys. Rev. E
**2004**, 70, 036219. [Google Scholar] [CrossRef] [PubMed] - Shomroni, I.; Lahoud, E.; Levy, S.; Steinhauer, J. Evidence for an oscillating soliton/vortex ring by density engineering of a Bose–Einstein condensate. Nat. Phys.
**2009**, 5, 193. [Google Scholar] [CrossRef] - Cardoso, W.B.; Zeng, J.; Avelar, A.T.; Bazeia, D.; Malomed, B.A. Bright solitons from the nonpolynomial Schrödinger equation with inhomogeneous defocusing nonlinearities. Phys. Rev. E
**2013**, 88, 025201. [Google Scholar] [CrossRef] [PubMed] - Sacha, K.; Delande, D. Proper phase imprinting method for a dark soliton excitation in a superfluid Fermi mixture. Phys. Rev. A
**2014**, 90, 021604. [Google Scholar] [CrossRef] - Donadello, S.; Serafini, S.; Tylutki, M.; Pitaevskii, L.P.; Dalfovo, F.; Lamporesi, G.; Ferrari, G. Observation of solitonic vortices in Bose-Einstein condensates. Phys. Rev. Lett.
**2014**, 113, 065302. [Google Scholar] [CrossRef] [PubMed] - Lee, T.D.; Huang, K.; Yang, C.N. Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Phys. Rev.
**1957**, 106, 1135. [Google Scholar] [CrossRef] - Kim, Y.E.; Zubarev, A.L. Three-body losses in trapped Bose-Einstein-Condensed gases. Phys. Rev. A
**2004**, 69, 023602. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) The 2D radial density, ${n}_{2\mathrm{D}}\left(r\right)$, as obtained from the full 3D equation, and the 2D reduction derived with the help of the variational approximation (VA). (

**b**) The 2D radial density, ${n}_{2\mathrm{D}}\left(r\right)$, as obtained from the full 3D equation, and the 2D reduction derived with the help of the VA, assuming that the Gaussian width is a constant: ${\xi}_{0}=\sqrt{\hslash /2{m}_{\mathrm{F}}{\omega}_{z}}$. Different curves correspond to the indicated values of ${a}_{s}=(0,-50,-100)\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$. The other parameters are $N=1000$, ${\omega}_{x}={\omega}_{y}=1050\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$, ${\omega}_{z}=21\phantom{\rule{3.33333pt}{0ex}}\mathrm{kHz}$, and $A=0$. The panel (a) is taken from Ref. [21].

**Figure 2.**Density ${n}_{2D}$ as a function of coordinates x and y for four different angles between the triangular OLs $\theta $. The fixed parameters are $N=1000$, ${\omega}_{x}={\omega}_{y}=1050\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$, ${\omega}_{z}=52.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{kHz}$, $A=1.74\times {10}^{-29}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}$, $\lambda =10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$, and ${a}_{s}=200\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$. (

**a**) $\theta =5\xb0$; (

**b**) $\theta =10\xb0$; (

**c**) $\theta =15\xb0$; (

**d**) $\theta =20\xb0$.

**Figure 3.**The initial 1D density for one soliton (

**a**) and eighteen dark solitons (

**b**). In both cases, ${\Delta}_{s}=0.8\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}m$ and ${n}_{b}=10$ are used. The other fixed parameters are ${a}_{\mathrm{F}}=-5$ nm and ${\omega}_{t}=1000$ Hz.

**Figure 4.**(

**a**) The spatiotemporal diagram for the density, ${n}_{1D}$, when the initial core-core separation between the two dark solitons is $d=4$ $\mathsf{\mu}$m. (

**b**) The speed of the solitons at $t=90$ ms as a function of d. The other fixed parameters are the same as in Figure 3.

**Figure 5.**The spatiotemporal diagram for the density, ${n}_{1D}$ for different numbers of dark solitons: (

**a**) ${N}_{s}=6$, (

**b**) ${N}_{s}=10$, (

**c**) ${N}_{s}=14$, and (

**d**) ${N}_{s}=18$. In all the cases the initial distance between the solitons is $d=4$ $\mathsf{\mu}$m. The other fixed parameters are the same as in Figure 3.

**Figure 6.**The distance between the central part and the edge at $z>0$ of the dark-soliton gas, $\delta {z}_{e}$, as a function of time for different numbers of the dark solitons, ${N}_{s}$. The other fixed parameters are the same as in Figure 3.

**Figure 7.**Dependence of the speed of the dark solitons for ${N}_{s}=18$. (

**a**) The distribution of the speed of each soliton at $t=90$ ms for different initial distances d. (

**b**) The speed of the dark soliton near the edge at $t=90$ ms as a function of d. The other fixed parameters are the same as in Figure 3.

**Figure 8.**Spatiotemporal diagrams of the density ${n}_{1D}$ for ${N}_{s}=18$ for two different initial conditions in the presence of random perturbation $\u03f5$. Panels (

**a**) and (

**b**) display the results for $\u03f5$ taking values in the ranges of $[-{\u03f5}_{max},{\u03f5}_{max}]$, with ${\u03f5}_{max}=0.4$ $\mathsf{\mu}$m and $0.8$ $\mathsf{\mu}$m, respectively. (

**c**) ${E}_{\u03f5}$ normalized to ${E}_{0}$ as a function of ${\u03f5}_{max}$, i.e., the amplitude of the randomly varying variable.

**Figure 9.**The radial profile of the 2D particle density, and the respective width for different values of interaction strength ${a}_{\mathrm{BF}}$. (

**a**) ${n}_{2\mathrm{D},\mathrm{B}}$, (

**b**) ${n}_{2\mathrm{D},\mathrm{F}}$, (

**c**) ${\xi}_{\mathrm{B}}$, and (

**d**) ${\xi}_{\mathrm{F}}$. The parameters are ${N}_{\mathrm{B}}=5\times {10}^{4}$, ${N}_{\mathrm{F}}=2.5\times {10}^{3}$, ${a}_{\mathrm{B}/\mathrm{F}}=5$ nm, ${\omega}_{z,\mathsf{B}/\mathsf{F}}=1000$ Hz, and ${\omega}_{x,\mathrm{B}/\mathrm{F}}={\omega}_{y,\mathrm{B}/\mathrm{F}}=30$ Hz. The inset in panel (a) shows the difference between the VA and full 3D simulations, by means of $\Delta {n}_{2\mathrm{D}}\equiv {\overline{n}}_{2\mathrm{D}}-{n}_{2\mathrm{D}}$. This figure is taken from Ref. [57].

**Figure 11.**Spatial correlation ${C}_{s}$ of the GS of the 2D mixture as a a function of ${a}_{\mathrm{BF}}$, for three fermionic regimes: polarized, BCS, and unitarity. The fixed parameters are: ${N}_{\mathrm{B}}=5\times {10}^{4}$, ${N}_{\mathrm{F}}=2.5\times {10}^{3}$, ${a}_{\mathrm{B}/\mathrm{F}}=5$ nm, ${\omega}_{x,\mathrm{B}/\mathrm{F}}={\omega}_{y,\mathrm{B}/\mathrm{F}}=30$ Hz, and ${\omega}_{z,\mathrm{B}/\mathrm{F}}=1000$ Hz. This figure is taken from Ref. [57].

**Figure 12.**Profiles of the particle density and the width in the confined direction as a function the z-coordinate for different values of the interaction strength ${a}_{\mathrm{BF}}$. (

**a**) ${n}_{1\mathrm{D},\mathrm{B}}$, (

**b**) ${n}_{1\mathrm{D},\mathrm{F}}$, (

**c**) ${\sigma}_{\mathrm{B}}$, and (

**d**) ${\sigma}_{\mathrm{F}}$. The parameters are ${N}_{\mathrm{B}}=5\times {10}^{4}$, ${N}_{\mathrm{F}}=2.5\times {10}^{3}$, ${a}_{\mathrm{B}/\mathrm{F}}=5$ nm, ${\omega}_{z,\mathrm{B}/\mathrm{F}}=30$ Hz, and ${\omega}_{t,\mathrm{B}/\mathrm{F}}=1000$ Hz. The inset in panel (a) shows the difference between the VA and full 3D simulations, by means of $\Delta {n}_{1\mathrm{D},\mathrm{B}}={\overline{n}}_{1\mathrm{D},\mathrm{B}}-{n}_{1\mathrm{D},\mathrm{B}}$. This figure is taken from Ref. [57].

**Figure 13.**Space-time diagrams of the densities of bosons (top) and fermions (bottom), for three different values of the interspecies scattering parameter: (

**a**,

**b**) ${a}_{\mathrm{BF}}=-18$ nm, (

**c**,

**d**) ${a}_{\mathrm{BF}}=-26$ nm, and (

**e**,

**f**) ${a}_{\mathrm{BF}}=-34$ nm. The initial conditions are the same in all the cases, see the text. The other parameters are the same as in Figure 12.

**Figure 14.**Comparison of the dynamics, as produced by the 1D VA, and from the 3D simulations. Spatiotemporal diagrams for bosons (

**a**) and fermions (

**b**) are obtained from the 3D simulations. The other panels show spatial profiles for: (

**c**,

**d**) $t=0\phantom{\rule{3.33333pt}{0ex}}$ ms, (

**e**,

**f**) $t=25$ ms, and (

**g**,

**h**) $t=50$ ms. Here ${a}_{\mathrm{BF}}=-10$ nm, the initial conditions and other fixed parameters being the same as in Figure 12. This figure is taken from Ref. [57].

Regime | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | $\mathit{\beta}$ | ${\mathit{s}}_{\mathit{F}}$ |
---|---|---|---|---|

Polarized | 1 | 1 | 1 | 0 |

BCS | 2 | 4 | 1 | 1/2 |

Unitary | 2 | 4 | 0.44 | 1/2 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Díaz, P.; Laroze, D.; Malomed, B.A.
The Variational Reduction for Low-Dimensional Fermi Gases and Bose–Fermi Mixtures: A Brief Review. *Condens. Matter* **2019**, *4*, 22.
https://doi.org/10.3390/condmat4010022

**AMA Style**

Díaz P, Laroze D, Malomed BA.
The Variational Reduction for Low-Dimensional Fermi Gases and Bose–Fermi Mixtures: A Brief Review. *Condensed Matter*. 2019; 4(1):22.
https://doi.org/10.3390/condmat4010022

**Chicago/Turabian Style**

Díaz, Pablo, David Laroze, and Boris A. Malomed.
2019. "The Variational Reduction for Low-Dimensional Fermi Gases and Bose–Fermi Mixtures: A Brief Review" *Condensed Matter* 4, no. 1: 22.
https://doi.org/10.3390/condmat4010022