Emergence of chaotic scattering in ultracold lanthanides.

Emergence of chaotic scattering in ultracold lanthanides. Phys. Rev. X 5, 041029 arxiv preprint 1506.05221 A. Frisch, S. Baier, K. Aikawa, L. Chomaz, M. J. Mark, F. Ferlaino in collaboration with : Dy group in Stuttgart: T. Maier, H. Kadau, M. Schmitt, M. Wenzel, I. Ferrier-Barbut, T. Pfau Theory group in Temple/Maryland: C. Makrides, A. Petrov, and S. Kotochigova, E. Tiesinga. START Prize 2009

Outline of the talk Toward non-alkali atoms, the quest for atomic richness and the case of magnetic atoms. Specificity of scattering of ultracold lanthanides Random Matrix Theory: Complex systems, dense spectra and chaos Chaos in Feshbach resonances of Lanthanides: First Erbium study and beyond, describing lanthanides chaoticity. Towards a theoretical understanding of the origin of chaos in lanthanides scattering 2 Svetlana s talk

Experimental achievement of BEC 3

3 Experimental achievement of BEC 1998 2001 1995 1995 2001 1995 2009 2004 2009 2002 2011 2012 2003

4 The different condensed atoms Bose-Einstein condensates 1998 1995 cryogenic cooling metastable state 2001 1995 2001 1995 2009 2004 2009 2002 2011 2012 2003

4 The different condensed atoms Bose-Einstein condensates 1998 1995 cryogenic cooling One-electron atom simple electronic structure (L=0), tunable contact interaction! metastable state 2001 1995 2001 1995 2009 2004 2009 2002 2011 2012 2003

The different condensed atoms Bose-Einstein condensates cryogenic cooling One-electron atom simple electronic structure (L=0), tunable contact interaction! metastable state 1998 2001 1995 1995 2001 1995 2002 2009 2004 2009 strong dipolar character V dd = µ 0µ 2 4 1 3 cos 2 r 3 Multi-electron atom complex electronic structure (L very large), tunable contact interaction and DDI! Two-electron atoms simple electronic and nuclear structure (L=0, S=0 and I=0), clock transitions,... 2011 2012 2003 4 More laser cooled species: Ho and Tm

Strongly dipolar systems V dd = µ 0µ 2 4 1 3 cos 2 r 3 5 Review articles: M.A. Baranov, Phys. Rep. 464, 71 (2008); T. Lahaye et al., Rep. Prog. Phys. 72, 126401 (2009);

Strongly dipolar systems V dd = µ 0µ 2 Long Range 4 1 3 cos 2 r 3 B Anisotropy r 5 Review articles: M.A. Baranov, Phys. Rep. 464, 71 (2008); T. Lahaye et al., Rep. Prog. Phys. 72, 126401 (2009);

Strongly dipolar systems V dd = µ 0µ 2 Long Range 4 1 3 cos 2 r 3 B Anisotropy r Few-body Physics anisotropic scattering amplitude, different scattering behavior at low T, partial wave decomposition... 5 Review articles: M.A. Baranov, Phys. Rep. 464, 71 (2008); T. Lahaye et al., Rep. Prog. Phys. 72, 126401 (2009);

Strongly dipolar systems V dd = µ 0µ 2 Long Range 4 1 3 cos 2 r 3 B Anisotropy r Few-body Physics anisotropic scattering amplitude, different scattering behavior at low T, partial wave decomposition... Many-body Physics e.g. ground state properties, spectrum of excitations, novel quantum phases 5 Review articles: M.A. Baranov, Phys. Rep. 464, 71 (2008); T. Lahaye et al., Rep. Prog. Phys. 72, 126401 (2009);

Outline of the talk Toward non-alkali atoms, the quest for atomic richness and the case of magnetic atoms. Specificity of scattering of ultracold lanthanides (Ln) Random Matrix Theory: Complex systems, dense spectra and chaos Chaos in Feshbach resonances of Lanthanides: First Erbium study and beyond, describing lanthanides chaoticity. Towards a theoretical understanding of the origin of chaos in lanthanides scattering 6 Svetlana s talk

Specific structure of Ln atoms laser-cooled members Lanthanide (Ln) family look at Tb? MOT? partially filled 4f electron shell, submerged below a filled 6s shell. ground state with vacancies of large ml of the f-shell (l=3) f orbital basis 7 Large magnetic moment!! θ r Magnetic Anisotropy (long range) (Erbium µ 7µB, Dysprosium µ 10µB) Large total angular momentum j! Large ground state degeneracy + Orbital Anisotropy (short range) (Erbium j=6, Dysprosium j=8)

Interactions of lanthanides atoms θ r Ĥ rel = ~2 2µ r d 2 d 2 ~r + ˆV (~r) + ĤZ(B) µ r : reduced mass Zeeman energy Which specific interaction potential lanthanide atoms? ˆV (~r) for 8 Review: Svetlana Kotochigova, Report on progress in physics, 77, 093901 (2014)

Interactions of lanthanides atoms θ r Ĥ rel = ~2 2µ r d 2 d 2 ~r + ˆV (~r) + ĤZ(B) µ r : reduced mass Zeeman energy Which specific interaction potential lanthanide atoms? ˆV (~r) for Dipole-Dipole interaction (DDI): V dd (~r) = C 3 r 3 3 cos2 1 C 3 = µ 0µ 2 4 8 Review: Svetlana Kotochigova, Report on progress in physics, 77, 093901 (2014)

Interactions of lanthanides atoms θ r Ĥ rel = ~2 2µ r d 2 d 2 ~r + ˆV (~r) + ĤZ(B) µ r : reduced mass Zeeman energy Which specific interaction potential lanthanide atoms? ˆV (~r) for Dipole-Dipole interaction (DDI): V dd (~r) = C 3 r 3 3 cos2 1 C 3 = µ 0µ 2 4 Anisotropic van der Waals dispersion: 8 V vdw (~r) = C 6 r 6 C ani 6 r 6 3 cos 2 1 C 6 (u.a)!c 6 (u.a) Er 1703 174 Dy 2003 188 Review: Svetlana Kotochigova, Report on progress in physics, 77, 093901 (2014)

9 Interactions and magnetic field θ r B Ĥ rel = ~2 2µ r d 2 d 2 ~r + ˆV (~r) +ĤZ(B)

9 Interactions and magnetic field ˆ~J = ˆ~J 1 + ˆ~J 2 + ˆ~` total angular momentum of the pair angular momentum of relative motion tot. angular momentum of atom i=1, 2 ˆ~j = ˆ~J 1 + ˆ~J 2 atomic angular momentum of the pair (without relative motion) θ r B Ĥ rel = ~2 2µ r d 2 d 2 ~r + ˆV (~r) +ĤZ(B) Ĥ Z (B) = g j µ B ˆ~j. ~B ~ Commutes with j 2 and j z ˆV (~r) = ˆV iso (r) + ˆV ani (~r) Commutes with J 2 and J z but not with j 2 and j z due to the anisotropic part Vani

Feschbach resonances in Ln closed channel coupling entrance channel Ĥ rel = ~2 2µ r d 2 d 2 ~r + ˆV (~r) +ĤZ(B) Alkali atoms 10 coupling btw Zeeman states induced by hyperfine coupling btw electron and nuclear spins. perturbative, weak coupling isotropic few channels are coupled

Feschbach resonances in Ln closed channel coupling entrance channel Ĥ rel = ~2 2µ r d 2 d 2 ~r + ˆV (~r) +ĤZ(B) Scattering scheme! Lanthanides atoms Alkali atoms ˆV (~r) = ˆV iso (r) + ˆV ani (~r) 10 coupling btw Zeeman states induced by hyperfine coupling btw electron and nuclear spins. perturbative, weak coupling isotropic few channels are coupled coupling btw Zeeman states induced by interaction potential itself (Vani ). typically stronger coupling. anisotropic: bound-states can have l 0. a large number of channels are coupled (large j). [Partial wave are not a natural basis]

Feshbach Spectroscopy in Ln 11

Feshbach Spectroscopy in Ln 168 Er Er: K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm, and F. Ferlaino, Phys. Rev. Lett. 108, 210401 (2012). + Theory works: A. Petrov, E. Tiesinga, S. Kotochigova, PRL 109 103002 (2012) for a review, see Svetlana Kotochigova, Report on progress in physics, 77, 093901 (2014) Dy: K. Baumann, N. Q. Burdick, M. Lu, and B. L. Lev, Phys. Rev. A 89, 020701 (2014) 11

Feshbach Spectroscopy in Ln Er: A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn, C. Makrides, A. Petrov, and S. Kotochigova, Nature 507, 475 (2014) 12

Feshbach Spectroscopy in Ln Er: A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn, C. Makrides, A. Petrov, and S. Kotochigova, Nature 507, 475 (2014) Er 168 extensive high-resolution trap loss spectroscopy measurements: 13 200 points Very dense spectra (compare to alkali): ~200 resonances in [0,70]G!! Not easy to encompass from the theory side: traditional methods fail... 12

14 1950 s Wigner, 1960 s Dyson, Mehta, and Gaudin. Originally designed for heavy nuclei Handling complex spectra How to characterize some properties of systems with complex spectra? Bohigas and Giannoni (1984)

14 1950 s Wigner, 1960 s Dyson, Mehta, and Gaudin. Originally designed for heavy nuclei Handling complex spectra Bohigas and Giannoni (1984)

Handling complex spectra Wigner (1950): Can we identify / forecast global spectral properties of a system with complex interactions without integrating the microscopic Hamiltonian? statistical approach Random Matrix Theory (RMT) Characterised by few statistical properties of the energy spectrum = correlations Bohigas and Giannoni (1984) 14 1950 s Wigner, 1960 s Dyson, Mehta, and Gaudin. Originally designed for heavy nuclei

15 1950 s Wigner, 1960 s Dyson, Mehta, and Gaudin. Originally designed for heavy nuclei Random Matrix Theory Can predict global spectrum properties: Without integrating the real Hamiltonian. Statistical analysis of a set of random Hamiltonians.

15 1950 s Wigner, 1960 s Dyson, Mehta, and Gaudin. Originally designed for heavy nuclei Random Matrix Theory Ĥ real Can predict global spectrum properties: Without integrating the real Hamiltonian. Statistical analysis of a set of random Hamiltonians. 8 >< >: Ĥ random i = Condition on the random matrix: Respect the same symmetries as the real H! Reproduce the correlations of the complex spectrum of a real system!! 0 B @ H 1,1 H 1,n......... random coupling terms Gaussian distributed H n,1 H n,n 1 C A i 9 >= >; i2[1,n]

16 Level correlations & RMT P (s) 1 0.8 0.6 0.4 P(s): Nearest-neighbor distribution of energy levels Poisson distribution Wigner-Dyson distribution 0.2 0 0 1 2 3 s s = normalized nearest-neighbor level spacing (in units of the mean level spacing)

16 Level correlations & RMT P (s) 1 0.8 0.6 0.4 P(s): Nearest-neighbor distribution of energy levels Poisson distribution Wigner-Dyson distribution Poisson law uncorrelated classical dynamics P (P) (s) =exp( s) 0.2 0 0 1 2 3 s s = normalized nearest-neighbor level spacing (in units of the mean level spacing)

16 Level correlations & RMT P (s) 1 0.8 0.6 0.4 0.2 P(s): Nearest-neighbor distribution of energy levels Poisson distribution Wigner-Dyson distribution 0 0 1 2 3 s s = normalized nearest-neighbor level spacing (in units of the mean level spacing) Poisson law uncorrelated classical dynamics P (P) (s) =exp( s) Wigner-Dyson distr. Dynamics of a quantum system with time-reversal and rotational symmetry predicted within Random Matrix Theory (Gaussian Orthogonal ensemble) P (WD) (s) = s 2 exp( s 2 4 )

16 Level correlations & RMT P (s) 1 0.8 0.6 0.4 0.2 P(s): Nearest-neighbor distribution of energy levels Poisson distribution Wigner-Dyson distribution 0 0 1 2 3 s s = normalized nearest-neighbor level spacing (in units of the mean level spacing) Wigner surmise excludes degeneracies, P (WD) (0) = 0 Poisson law uncorrelated classical dynamics P (P) (s) =exp( s) Wigner-Dyson distr. Dynamics of a quantum system with time-reversal and rotational symmetry predicted within Random Matrix Theory (Gaussian Orthogonal ensemble) P (WD) (s) = s 2 exp( s 2 4 ) Wigner-Dyson distr. expresses level repulsions and spectral rigidity.

16 Level correlations & RMT P (s) 1 0.8 0.6 0.4 0.2 P(s): Nearest-neighbor distribution of energy levels Poisson distribution Wigner-Dyson distribution 0 0 1 2 3 s s = normalized nearest-neighbor level spacing (in units of the mean level spacing) Wigner surmise excludes degeneracies, P (WD) (0) = 0 Poisson law uncorrelated classical dynamics P (P) (s) =exp( s) Wigner-Dyson distr. Dynamics of a quantum system with time-reversal and rotational symmetry predicted within Random Matrix Theory (Gaussian Orthogonal ensemble) P (WD) (s) = s 2 exp( s 2 4 ) Same mean value, different variance. Wigner-Dyson distr. expresses level repulsions and spectral rigidity.

17 O. Bohigas, M.J. Giannoni, C. Schmit, PRL (1984) RMT universality & chaos

17 O. Bohigas, M.J. Giannoni, C. Schmit, PRL (1984) RMT universality & chaos BOHIGAS-GIANNONI-SCHMIT CONJECTURE (1984): The spectral fluctuation properties of a quantum system that is fully chaotic in the classical limit follows the WD distribution Random Matrix Theory Chaos

18 RMT universal applications P(s): Nearest-neighbor distribution of energy levels P (s) 1 0.8 0.6 0.4 Poisson distribution Wigner-Dyson distribution - 0.2 0 0 1 2 3 s

RMT universal applications P(s): Nearest-neighbor distribution of energy levels P (s) 1 0.8 0.6 0.4 0.2 Poisson distribution Wigner-Dyson distribution - Neutron scattering cross section of 238 U Sinai billiard Biological networks of yeast O. Bohigas, R.U. Haq, A. Pandey, Nuclear Data for Science and Technology, 1983 O. Bohigas, M.J. Giannoni, C. Schmit, PRL (1984) 18 0 0 1 2 3 s Powerful Tool: Nuclear physics, Disordered and Mesoscopic Systems, Number theory, Quantum Chromodynamics, Quantum Chaos, Biological systems, Quantum gravity and string theory, risk management in finance, Luo et al., Physics Letters A 357 (2006)

Wigner Surmise in Erbium Applying RMT to Er Feshbach resonances 19 A. Frisch et al. Nature 507, 475 (2014)

Wigner Surmise in Erbium P (WD) : NO FREE-PARAMETERS Applying RMT to Er Feshbach resonances 19 A. Frisch et al. Nature 507, 475 (2014)

Wigner Surmise in Erbium P (WD) : NO FREE-PARAMETERS Applying RMT to Er Feshbach resonances First observation of the chaotic nature of ultracold scattering 19 A. Frisch et al. Nature 507, 475 (2014)

Why a new work? A joint effort to better understand the chaotic scattering of Lanthanide atoms. Er Group (Ferlaino et al.) + Dy Group (Pfau et al.) + Theory Group (Kotochigova/Tiesinga) 21

Why a new work? A joint effort to better understand the chaotic scattering of Lanthanide atoms. Er Group (Ferlaino et al.) + Dy Group (Pfau et al.) + Theory Group (Kotochigova/Tiesinga) Is this generalizable to other atomic species? Thorough comparison of the behavior of 2 Lanthanides: Er and Dy (+isotopes). What is the origin of chaotic scattering? Effort toward a better theoretical understanding. From where come some specific behavior? Investigate temperature and magnetic field dependence Development of more accurate numerical simulations for both species. 21

New measurements. D QRUPDOL]HGDWRPQXPEHU Dy 164 E PDJQHWLF HOG * F 22 Q. VVR 3RL. Q (U Q Er 168 @ higher T = 1400 nk. Q. (U Ȉ PDJQHWLF HOG * ' Dy 164 @ 600 nk (Pfau group, Stuttgart) (U QXPEHURIUHVRQDQFHV Additional extensive high-resolution trap loss spectroscopy measurements:. Q \ :LJQHU '\VRQ 1

New measurements. D QRUPDOL]HGDWRPQXPEHU Dy 164 E PDJQHWLF HOG * F Q. VVR 3RL. Q (U Er 168 @ higher T = 1400 nk. Q. (U Ȉ Q PDJQHWLF HOG * very dense spectra Nres @ 400 nk 309 @ 1400 nk 238 164Dy 168Er 168Er 22 @ 350 nk 189 ' Dy 164 @ 600 nk (Pfau group, Stuttgart) (U QXPEHURIUHVRQDQFHV Additional extensive high-resolution trap loss spectroscopy measurements:. Q \ :LJQHU '\VRQ 1

New measurements. D QRUPDOL]HGDWRPQXPEHU Dy 164 E PDJQHWLF HOG * F Q. VVR 3RL. Q (U Er 168 @ higher T = 1400 nk. Q. (U Ȉ Q RMT-based analysis: PDJQHWLF HOG * very dense spectra Nres @ 400 nk 309 @ 1400 nk 238 164Dy 168Er 168Er 22 @ 350 nk 189 ' Dy 164 @ 600 nk (Pfau group, Stuttgart) (U QXPEHURIUHVRQDQFHV Additional extensive high-resolution trap loss spectroscopy measurements:. Q \ :LJQHU '\VRQ Global/statistical properties of the spectra. 1 information on chaoticity / interaction features