STRONGLY INTERACTING QUANTUM PARTICLES IN ONE DIMENSION AN EXACT APPROACH TO LOW-DIMENSIONAL QUANTUM MAGNETISM

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1 OCTOBER STRONGLY INTERACTING QUANTUM PARTICLES IN ONE DIMENSION AN EXACT APPROACH TO LOW-DIMENSIONAL QUANTUM MAGNETISM DEPARTMENT OF PHYSICS AND ASTRONOMY Universidade Estadual Paulista Júlio de Mesquita Filho Campus de São Paulo Instituto de Física Teórica Colloquium October 8 th 2014 UNI V E R S I T E T

2 A ONE DIMENSIONAL WORLD Identica l bosons r Distinguisha ble fermions r Relative wave function Intera ction Source: G. Zürn, thesis Strong intera c tions -> Impenetra bility! 2

3 STRONGLY INTERACTING BOSONS g 1D limit Tonks (1936)-Girardeau (1960) gas of impenetrable bosons Ma pping identica l bosons to spin-pola rized fermions. G ira rdea u (1960). Impenetrable bosons Antisymmetrized fermions Lieb-Liniger (1963) used Bethe ansatz to solve N boson problem for any g>0 3

4 EXPERIMENTAL REALIZATION O ptica l la ttice s Confinement-induced resonances Maxim Olshanii Phys. Rev. Lett. 81, 938 (1998) I. Bloch, Na ture Physics 1, 23 (2005) Divergent at specific point depending on lattice and 3D Feshbach resonance 4

5 EXPERIMENTAL REALIZATION Na ture 429, 277 (2004) Science 305, 1125 (2004) Experimentally produced and probed the Tonks-Girardeau gas on the repulsive side g>0 5

6 EXPERIMENTAL REALIZATION Science 325, 1224 (2009) Study of the crossover from g>0 to g<0 in the strongly-interacting regime. 6

7 1D FERMIO NS A FRONTIER Two kinds of rela tive motion for two-body sta tes! Source: G. Zürn, thesis Fermioniza tion of two fermions in a 1D ha rmonic tra p: G. Zürn et al., Phys. Rev. Lett. 108, (2012). 7

8 EXPERIMENTAL REALIZATION Tw o-body tunneling experiments Fermioniza tion of two fermions in a 1D ha rmonic tra p: G. Zürn et al., Phys. Rev. Lett. 108, (2012). 8

9 THREE FERMIONS Relative wave functions. What should we take? or Conjecture: Use the symmetric choice for nonidentica l pa irs for a ny N-body system??? M.D. G ira rdea u, Phys. Rev. A 82, (R) (2010). 9

10 THREE FERMIONS Let s keep an open mind! Two strict conditions: 1) In limit g 1D ->, relative wave functions ha ve not va nish a t zero for identica l a nd non-identica l pa irs! 2) Identica l fermions must ha ve odd relative wave functions! a 1 a 2 Non-identical relative wave function 0 r IDEA: Keep a 1 a nd a 2 as free parameters and do a variation! A.G. Volosniev et al., arxiv: (2013) Nature Communications, in press (October 2014). 10

11 THREE FERMIONS - SOLUTION Split space in patches Spectrum on resona nce Importa nt: Antisymmetric sta te! G enera l solution O ptimize deriva tive! Pauli and parity reduces problem to a 1, a 2, a nd a 3. 11

12 THREE FERMIONS - SOLUTION Extremizing solutions a re: a 1 =a 2 =a 3 a 1 =a 3 a nd a 2 =0 2a 1 =2a 3 =-a 2 Non-intera cting sta te Excited sta te, even pa rity G round sta te, odd pa rity IMPO RTANT: C oefficients a re generally NOT the same! A.G. Volosniev et al., arxiv: (2013) Nature Communications, in press (October 2014). 12

13 HARMONICALLY TRAPPED SYSTEMS Sta nda rd style Elega nt style E.J. Lindgren et al., New J. Phys. 16, (2014). S.E. Gharashi and D. Blume, Phys. Rev. Lett. 111, (2013) 13

14 G ROUND STATE PROPERTIES Tra p de nsity Occupa tion numbers E.J. Lindgren et al. New Journal of Physics 16, (2014). 14

15 FERMIONIZATIO N OF FERMIONS It is different from identica l bosons a nd spin-pola rized fermions! The democratic solution or trivia l Bose-Fermi ma pping uses: between all nonidentica l pa irs. In the 2+1 ca se it is NOT a relevant eigensta te but ra ther a linear combina tion! ψ BF = (8 1/2 ψ gs + ψ non )/3 BUT can we tell the difference in experiments? S.E. G ha ra shi a nd D. Blume, Phys. Rev. Lett. 111, (2013) A.G. Volosniev et al., arxiv: (2013) 15

16 Selim Jochim experiments in Heidelberg. From Few to Many: Observing the Formation of a Fermi Sea One Atom at a Time A. N. Wenz et al., Science 342, 457 (2013) Green solid line from S.E. Gharashi, K.M. Daily, and D. Blume, Phys. Rev. A 86, (2012). Orange many-body line J.B. McGuire, J. Math. Phys. 6, 432 (1965). G.E. Astrakharchik and I. Brouzos, Phys. Rev. A 88, (2013). 16

17 EXPERIMENTAL SIGNATURE Do tunneling experiments! F. Serwane et al., Science 332, 336 (2011). G. Zürn et al., Phys. Rev. Lett. 108, (2012). Source: G. Zürn 17

18 THEORY VS. EXPERIMENT NOT 33% Theory by Lindgren a nd Volosniev. Data from Jochim group (G. Zürn) 18

19 FOUR-BODY SYSTEMS A.G. Volosniev et al., arxiv: (2013) Nature Communications, in press (October 2014). 19

20 A.G. Volosniev et al., arxiv: (2013) Nature Communications, in press (October 2014). 20

21 THREE TW O-COMPONENT BOSONS Strong AB interactions No AA interactions No BB interactions N.T. Zinner et al., EPL 107, (2014). 21

22 Stochastic variational calculations N.T. Zinner et al., EPL 107, (2014). 22

23 AAB Perfect ferromagnet? ABA Perfect antiferromagnet? G round sta te First excited sta te BAA 2nd exc ited sta te with node! N.T. Zinner et al., EPL 107, (2014). G round sta te for 2+1 fermions 23

24 FOUR-BODY BOSONS G round sta te structure AABB+BBAA A.S. Dehkha rgha ni et al., arxiv: (2014), in review at Phys. Rev. X. 24

25 BALANCED N-BODY SYSTEM G round sta te structure for N=10 AAAAABBBBB+ BBBBBAAAAA A B Ta king linea r combina tions of the doubly degenerate ground sta te, we obta in perfectly separated densities. C onfirms physica l picture! A.S. Dehkha rgha ni et al., arxiv: (2014), in review at Phys. Rev. X. 25

26 BOSE POLARONS G round sta te structure ABB.BB+BB.BBA In the ground sta te, impurities will NEVER penetra te the ma jority component! A.S. Dehkha rgha ni et al., arxiv: (2014), in review at Phys. Rev. X. 26

27 ENERGETICS Ba la nced systems Bi-polaron Polaron Numerical and semi-analytical 27

28 MOMENTUM DISTRIBUTIONS 28

29 SPIN MO DELS We can map strongly interacting two-component 1D systems in a trap to a spin model of XXZ type and do ENGINEERING! Nearest-neighbor interactions are tunable via external trap! Note: It is not a lattice index! It is a particle index. Confinement is taken into account exactly. A.G. Volosnievet al., arxiv: (2014). 29

30 STATE TRANSFER Use trap to manipulate dynamics example of quantum state transfer Fermions or hard-core bosons Bosons kappa=1/ 2 Fidelity of quantum state transfer Bosons kappa=2 A.G. Volosnievet al., arxiv: (2014). 30

31 MAIN MESSAG ES C omplete theory goes beyond Bose-Fermi ma pping Must conne ct sta te s to e ige nsta te s in the spectrum Magnetic correlations are accessible Good agreement with experimental data Fermions a nd bosons ca n be VERY different even in the ha rd-core limit! Engineering of ferro- a nd a ntife rroma gne tic sta te s! W a ve functions a nd not energies a re the most importa nt obje cts! 31

32 ACKNOWLEDGEMENTS Artem Volosniev, postdoctora l resea rcher (Aa rhus) Amin Dehkha rgha ni, gra dua te student (Aa rhus) Dmitri Fedorov a nd Aksel Jensen (Aa rhus) Manuel Valiente (Heriot-W a tt Unive rsity, Edinburgh) Jona tha n Lindgren, gra dua te student (C ha lmers) C hristia n Forssén a nd Jimmy Roturea u (C ha lmers) Jochim group in Heidelberg: Selim Jochim, G erha rd Zürn, Thoma s Lompe, Simon Murma nn, Andre W enz Tha nk you for your a ttention! 33