Majorana Decomposition and Spinor Condensates

1. Majorana Decomposition and Spinor Condensates Kalle-Antti Suominen Department of Physics University of Turku Finland With Harri Mäkelä Supported by the Academy of Finland and the Finnish Academy of Science and Letters

2. Atomic condensates • Magnetically or optically trapped neutral atoms • (typically alkali atoms) which are boson-like. • Bose-Einstein condensation has been achieved. • Dilute gases, mean-field approach • (Gross-Pitaevskii) works very well in most cases. • Low magnetic fields: atomic spin = hyperfine quantum number F. • Magnetic substates (Zeeman states) mF: • ”spinor” structure, 2F+1 components.

3. Magnetic fields The magnetic field creates a shift in the energy levels, mostly linear.

4. Experimental background Magnetic traps are usually based on the Ioffe-Pritchard model, where for a specific spin state one obtains a parabolic trapping potential. Obviously, this setup is not the best one for studies of spinors.

5. Optical traps Optical traps are based on light forces, that are equal to all mF states. Trap potential = spatial intensity variation of highly off-resonant laser beams. Suitable for spinor studies. Magnetic field can be added if its effects need to be studied, and for manipulation and detection purposes. Note: In cold atom physics interactions are often tuned using Feshbach resonances. Ths method requires magnetic fields and is thus not very useful for spinor studies.

6. Interactions and spinors • Dilute gases, low temperatures: s-wave interaction only. • Short distance details -> Contact potential & scattering length. • Negative or positive a: stability issues for condensates and vortices created by rotation, trapping potential plays a role • [see e.g. E. Lundh, A. Collin, and K.-A. Suominen: Phys. Rev. Lett. 92, 070401 (2004)] • Generalization of the order parameter to spinor systems

7. Physics of spinors • The multistate structure with interactions leads to • a) Non-trivial ground states, ordered structures: • For example, in the F=1 case we can have either ferromagnetic or antiferromagnetic ordering. • We seek ground states by minimising the spinor energy functional. • b) Possibility for topological defects • vortices and coreless vortices • monopoles • Here we must investigate the stability of such defects • topological stability • energetics • See e.g. the theses of Jani-Petri Martikainen (Helsinki 2001), • Anssi Collin (Helsinki 2006) and Harri Mäkelä (Turku 2007).

8. Spinor energy functional The contact potential changes in the multistate case: For two identical atoms total spin is Ftot=F1+F2: Energy minimization (to seek ground states) concentrates on For details see the thesis of Harri Mäkelä at Doria: https://oa.doria.fi/handle/10024/29116

9. Spinor energy functional: F=1 & F=2 F=1: F=2:

10. Spinor energy functional: Phases

11. Special situations 1 The presence of magnetic field changes the energy functional. We need to keep the magnetic field sufficiently weak so that F remains a good quantum number. Normally one needs to consider only the linear and quadratic Zeeman shifts. Example: F=1 (-) and F=2 (+), b = normalized B-field: General aspect: reduces symmetry and usually reduces also the set of possible ground states (and defect classes). In practice, most of the interesting phenomena relate to the case of B=0.

12. Special situations 2 • Another case is if the atom has a permanent dipole moment. This • applies to Cr (spin-3 system), and a spin direction-dependent long- • range term needs to be added to the energy functional. • Typically leads to favouring the situation where the spin is aligned with • the long axis of the typically cigar-shaped condensate. • H. Mäkelä & K.-A. Suominen, Phys. Rev. A 75, 033610 (2007). • – ground states for fixed magnetization.

13. Experiments? Spinor experiments are few so far, mainly F=1 and F=2: – Ketterle group at MIT – Chapman group at Georgia Inst. of Technology – Sengstock group at Hamburg – Stamper-Kurn at UC Berkeley These involve 23Na and 87Rb, where for 87Rb the F=2 state is relatively stable. Relaxation i.e. spin-mixing is usually slow (orders of a second) so ground states are hard to observe. Spurious magnetic fields cause fragmention of spinor states. Possibility for F≥3 studies: – 85Rb (F=2 & F=3); F=3 is not very stable – Cs (F=3 & F=4); hard to condense – Cr (S=3); permanent electric dipole moment

14. Experiments: F=2 example 87Rb: Polar (antiferromagnetic) state for F=2, ferromagnetic for F=1. – This is very much as expected from theoretical studies on these cases. The cyclic state can be prepared, and its decay into the polar state was very slow. – 87Rb is close to the borderline between polar and cyclic phases so this is also expected. For a discussion on F=2 ground states see e.g. J.-P. Martikainen and K.-A. Suominen, J. Phys. B 34, 4091 (2001).

15. Experiments: F=2 spin dynamics Spin-mixing dynamics time scales ~40 ms Two-body hyperfine loss and three-body recombination loss step in at later times. For F=2 Na and Rb collisional stability issues, see K.-A. Suominen, E. Tiesinga, and P.S. Julienne, Phys. Rev. A 58, 3983 (1998). Slow decay seen for but these states can be obtained from each other by rotation (see Mäkelä’s thesis).

16. Majorana decomposition In 1932 Ettore Majorana considered what happens when a beam of atoms with spin-S passes a point in which the magnetic field vanishes [Nuovo Cimento 9, 43 (1932)]. -> Majorana spin flips This work [see also F. Bloch and I.I. Rabi, Rev. Mod. Phys. 17, 237 (1945)] provides a general tool for understanding spin-S systems as a collection of 2S spin-1/2 particles (not limited to integer S). Examples:

17. Spin-S vs 2S spin-1/2 In general: For a spin-1/2 particle labelled k: Now we define So we have a mapping between any superposition state of a spin-S system into the superposition states of the 2S spin-1/2 systems.

18. Uses of the decomposition The mapping can be used to describe the internal dynamics of the spin-F atoms. a) The mapping survives the presence of a linear Zeeman shift. b) The action of an external B-field that couples the different mF states can be seen as a spin rotation c) The field-induced transitions between the |F,mF> states can be mapped to spin-1/2 dynamics. Thus, if we apply time-dependent fields (pulsed or chirped) to a spin-F system, the dynamics is obtained if the corresponding spin-1/2 model has a solution. Application: Condensate output coupling

19. Condensate output coupling N. Vitanov & K.-A. Suominen, Phys. Rev. A 56, R4377 (1997).

20. Majorana flips One starts from the extreme state |F,±mF>, applies the interaction, and obtains the populations Pi of the 2F+1 states, in terms of the population p of the initially unoccupied spin-1/2 state. Example: F=2 system with linearly chirped but otherwise constant B-field.

21. MIT condensate output coupling

22. Majorana & spinor condensates A standard method for describing the state and the dynamics of spin-1/2 particles is the Bloch sphere. E. Demler and co-workers [PRL 97, 180412 (2006)] took the notation (apparently unaware of the Majorana work) and mapped a spin-F system onto 2F points on a unit sphere (spin-1/2 particles). When the points are connected, they form geometric shapes. This allows classification of the phases of the spin-F systems.

23. Inert states As F increases, it becomes very hard to minimize the energy functional in respect to all possible spinor configurations and combinations of scattering lengths. Inert states are stationary states of the energy functional for all parameters. Whether they are global minima or maxima, can change with parameters. But not all stationary states are also inert. Example:

24. Inert states for spinors • In any case finding inert states provides possible candidates for stable • ground states. • Encouraging: for F=1 and F=2 all inert states are also ground states. • F≥3: finding inert states is hard [S.K. Yip, Phys. Rev. A 75, 023625 (2007)]. • Our solution: use the Majorana/Demler approach. • It can be shown that • If any infinitesimal change in the configuration of the 2F points on the unit sphere changes the symmetry group of the configuration, the configuration defines an inert state. • H. Mäkelä and K.-A. Suominen: Phys. Rev. Lett. 99, 190408 (2007).

25. Inert states: Examples

26. Inert states: S = 1 - 4

27. Conclusions • The Majorana decomposition of large spins into a group of spin-1/2 systems • is an useful tool for describing spinor systems and spin dynamics. • Especially when mapped into the Bloch sphere it provides a simple method for visualisation of topological properties. • Further work? • Majorana decomposition and topological defects? • Extension into quantum information (symmetric subspaces, state estimation and universal quantum cloning)?

28. Turku group Wiley 2005