Strongly Interacting Quantum Particles in One Dimension - Research at Aarhus Universitet

Researchers at Aarhus Universitet delved into the dynamics of confined few-body quantum systems in one dimension, focusing on the interactions and properties of strongly interacting particles. Through experimental realizations and theoretical models, they explored the behavior of bosons and fermions under different conditions, shedding light on impenetrability, fermionization, and Tonks-Girardeau gases. The study also investigated the crossover between repulsive and attractive interactions in the strongly-interacting regime. This groundbreaking research, conducted by Nikolaj Thomas Zinner, contributes to the understanding of quantum mechanics and its applications in various fields.

• Quantum particles
• One dimension
• Aarhus Universitet
• Strong interactions
• Nikolaj Thomas Zinner

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1. AARHUS UNIVERSITET OCTOBER 17 2014 STRONGLY INTERACTING QUANTUM PARTICLES IN ONE DIMENSION TAILORED DYNAMICS OF CONFINED FEW-BODY SYSTEMS NIKOLAJ THOMAS ZINNER DEPARTMENT OF PHYSICS AND ASTRONOMY Critical Stability 2014 Santos, Brazil October 17th 2014 VERSITET UNI

2. AARHUS UNIVERSITET A ONE DIMENSIONAL WORLD Identical bosons r Distinguishable fermions r Relative wave function Interaction Source: G. Z rn, thesis Strong interactions -> Impenetrability! STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 2

3. AARHUS UNIVERSITET STRONGLY INTERACTING BOSONS |g1D| limit Tonks (1936)-Girardeau (1960) gas of impenetrable bosons Mapping identical bosons to spin-polarized fermions. Girardeau (1960). Impenetrable bosons Antisymmetrized fermions Lieb-Liniger (1963) used Bethe ansatz to solve N boson problem for any g>0 STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 3

4. AARHUS UNIVERSITET EXPERIMENTAL REALIZATION Optical lattices Confinement-induced resonances Maxim Olshanii Phys. Rev. Lett. 81, 938 (1998) Divergent at specific point depending on lattice and 3D Feshbach resonance I. Bloch, Nature Physics 1, 23 (2005) STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 4

5. AARHUS UNIVERSITET EXPERIMENTAL REALIZATION Nature 429, 277 (2004) Science 305, 1125 (2004) Experimentally produced and probed the Tonks-Girardeau gas on the repulsive side g>0 STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 5

6. AARHUS UNIVERSITET EXPERIMENTAL REALIZATION Study of the crossover from g>0 to g<0 in the strongly-interacting regime. Science 325, 1224 (2009) STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 6

7. AARHUS UNIVERSITET 1D FERMIONS A FRONTIER Two kinds of relative motion for two-body states! Source: G. Z rn, thesis Fermionization of two fermions in a 1D harmonic trap: G. Z rn et al., Phys. Rev. Lett. 108, 075303 (2012). STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 7

8. AARHUS UNIVERSITET EXPERIMENTAL REALIZATION Two-body tunneling experiments Fermionization of two fermions in a 1D harmonic trap: G. Z rn et al., Phys. Rev. Lett. 108, 075303 (2012). STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 8

9. AARHUS UNIVERSITET THREE FERMIONS Relative wave functions. What should we take? or ??? Conjecture: Use the symmetric choice for non- identical pairs for any N-body system STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 M.D. Girardeau, Phys. Rev. A 82, 011607(R) (2010). 9

10. AARHUS UNIVERSITET THREE FERMIONS Two strict conditions: Let s keep an open mind! 1) functions have not vanish at zero for identical and non-identical pairs! In limit g1D-> , relative wave 2) Identical fermions must have odd relative wave functions! a1 a2 IDEA: Keep a1 and a2 as free parameters and do a variation! r 0 Non-identical relative wave function STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 A.G. Volosniev et al., arXiv:1306.4610 (2013) Nature Communications, in press (October 2014). 10

11. AARHUS UNIVERSITET THREE FERMIONS - SOLUTION Split space in patches Spectrum on resonance Important: Antisymmetric state! General solution Optimize derivative! Pauli and parity reduces problem to a1, a2, and a3. STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 11

12. AARHUS UNIVERSITET THREE FERMIONS - SOLUTION a1=a2=a3 Non-interacting state Extremizing solutions are: a1=a3 and a2=0 Excited state, even parity 2a1=2a3=-a2 Ground state, odd parity IMPORTANT: Coefficients are generally NOT the same! STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 A.G. Volosniev et al., arXiv:1306.4610 (2013) Nature Communications, in press (October 2014). 12

13. AARHUS UNIVERSITET HARMONICALLY TRAPPED SYSTEMS Standard style Elegant style E.J. Lindgren et al., New J. Phys. 16, 063003 (2014). S.E. Gharashi and D. Blume, Phys. Rev. Lett. 111, 045302 (2013) STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 13

14. AARHUS UNIVERSITET GROUND STATE PROPERTIES Trap density Occupation numbers STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 E.J. Lindgren et al. New Journal of Physics 16, 063003 (2014). 14

15. AARHUS UNIVERSITET FERMIONIZATION OF FERMIONS It is different from identical bosons and spin-polarized fermions! The democratic solution or trivial Bose-Fermi mapping uses: between all non- identical pairs. In the 2+1 case it is NOT a relevant eigenstate but rather a linear combination! BF= (81/2 gs+ non)/3 BUT can we tell the difference in experiments? S.E. Gharashi and D. Blume, Phys. Rev. Lett. 111, 045302 (2013) A.G. Volosniev et al., arXiv:1306.4610 (2013) STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 15

16. AARHUS UNIVERSITET MAPPING TO SPIN MODEL Consider the slope of the energy Solution obtained by extreming is an eigenvalue problem Can we interpret the right-hand side matrix as the effective Hamiltonian? STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 16

17. AARHUS UNIVERSITET MAPPING TO SPIN MODEL Diagonal entries map a configuration to itself Indeed we can! Go back to the physical meaning of coefficients in terms of the spins! Mapping a2 to a3 or: Off-diagonal elements correspond to magnetic exchange terms! STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 17

18. AARHUS UNIVERSITET SPIN MODELS 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. STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 A.G. Volosnievet al., arXiv:1408.3414 (2014). 18

19. AARHUS UNIVERSITET Fidelity of quantum state transfer STATE TRANSFER Fermions or hard-core bosons Use trap to manipulate dynamics example of quantum state transfer Bosons kappa=1/2 Bosons kappa=2 STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 A.G. Volosnievet al., arXiv:1408.3414 (2014). 19

20. AARHUS UNIVERSITET FIDELITY FOR LARGER SYSTEMS Calculation by Jonatan Midtgaard (Aarhus University) Constant J Heisenberg model transfer: S. Bose, Phys. Rev. Lett. 91, 207901 (2003) Approximate expression for harmonic oscillator J coefficients from J. Levinsen et al. arXiv:1408.7096 (2014). Neither constant (box potential) or harmonic potential are good for large fidelity transfers beyond about N particles! Need to tailor potential to optimize transfer protocols! STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 A.G. Volosnievet al., arXiv:1408.3414 (2014). 20

21. AARHUS UNIVERSITET MAIN MESSAGES Complete theory goes beyond Bose-Fermi mapping Must connect states to eigenstates in the spectrum Magnetic correlations are accessible Good agreement with experimental data Fermions and bosons can be VERY different even in the hard-core limit! Engineering of ferro- and antiferromagnetic states! Wave functions and not energies are the most important objects! STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 21

22. AARHUS UNIVERSITET ACKNOWLEDGEMENTS Artem Volosniev, postdoctoral researcher (Aarhus) Amin Dehkharghani, graduate student (Aarhus) Dmitri Fedorov and Aksel Jensen (Aarhus) Manuel Valiente (Heriot-Watt University, Edinburgh) Jonathan Lindgren, graduate student (Chalmers) Christian Forss n and Jimmy Rotureau (Chalmers) Jochim group in Heidelberg: Selim Jochim, Gerhard Z rn, Thomas Lompe, Simon Murmann, Andre Wenz Thank you for your attention! STRONG INTERACTIONS IN 1D NIKOLAJ THOMAS ZINNER OCTOBER 17 2014 22