One-dimensional atomic quantum gases Michael Köhl
Quantum condensed matter physics T[K] 1012 100 10-8 n[cm-3] 1038 1023 1013 • extreme conditions • no experiments possible • tunable • precise microscopic understanding • complex systems • technologicallyhighly relevant Characteristic temperature scale:
Condensed matter physics with cold gases Rapidly rotating systems Quantum phases in optical lattices ETH, Hamburg, LENS, Munich/Mainz, MIT, NIST, Yale/Stanford … ENS, JILA, MIT, Oxford,GeorgiaTech, … Low-dimensional systems BEC-BCS crossover 2D 1D Duke, ENS, Innsbruck, JILA, MIT, Rice, … Cambridge, ENS, ETH, LENS, Mainz, MIT, NIST, Penn State, … Fermionic atoms in optical lattices
One-dimensional gases • transverse degrees of freedom are frozen out • asymptotic scattering states are one-dimensional wave functions Conditions for 1D kBT <hw┴ Bosons: m < hw┴ Fermions: EF =Nhwz < hw┴
Regimes of degeneracy in 1D Bose gases weakly interacting Bose gas crossover Tonks-Girardeau gas (“Fermionized Bosons”) 1960: g→ limit solved by Girardeau 1963: Exactly solved for all values of g by Lieb & Liniger 2003: 1D Bose gases first realized by Esslinger et al. (ETH Zürich). 2004: Tonks gas experimentally realized by Weiss et al. (Penn State) & Bloch et al. (Mainz)
Excitations in a 1D Bose gas Two branches of excitations: “particle” excitations and “hole” excitations q q k k -kF kF kF -kF
Generating tight confining potentials Induced electric dipole potential: ac polarizability of the atom electric field of the laser Two options: „red detuned“ „blue detuned“ Optical lattice Energy scale: l/2=380 nm
Optical lattice arrangement Typical experimental parameters: Atoms: 87Rb (bosons) Wavelength of lattice: 764 nm wx= wy ≤ 2p65 kHz (optical lattice) wz= 2p39 Hz (magnetic trap) N ≈ 100per tube Other experiments in 1D: Amsterdam, ENS, ETH, LENS, Mainz, MIT, NIST, Orsay, Penn State, Rice, Vienna …
Lowest Lying Collective Modes „Breathing“ mode(axial compressional oscillation) Dipol mode (center-of-mass oscillation)
Realization of a weakly interacting 1D Gas Measured ratio of frequencies: Bose condensate Dipole oscillation Thermal gas Breathing oscillation H. Moritz, T. Stöferle, MK, T. Esslinger, Phys. Rev. Lett. 91, 250402 (2003)
Tonks-Girardeau gas One route into the Tonks-Girardeau gas: reduce density mean field theory Lieb-Liniger theory T. Kinoshita, T. Wenger, D. Weiss, Science 305, 1125 (2004).
At low density: Lattice Tonks-Girardeau gas Another route into the Tonks-Girardeau gas: tune effective interactions through increasing effective mass slope k-1.9 slope k-0.6 Density n(k) momentum k B. Paredes et al., Nature 429, 277 (2004).
J U Bose-Hubbard Model Expanding the field operator in localized wave functions on each lattice site: on-siteinteraction inhomogeneityof the trap tunneling betweenneighbouring lattice sites Fisher et al. (1989), for atoms: Zoller et al. (1998)
Quantum Phase Transition Quantum phases • U/J < gc: Superfluid state Excitation spectrum is gapless • U/J > gc: Mott-Insulator state Excitation spectrum is gapped First demonstration in 3D: M. Greiner et al., Nature 415, 39 (2002)
Mott-insulator phase diagram in 1D MI(2) MI(2) + SF MI(1) + SF MI (1) SF G. Batrouni et al. PRL 89, 117203 (2002)
Recording the momentum distribution Fourier transform of the onsite (Wannier) wave function -4ħkL -2ħkL 0 2ħkL 4ħkL
Coherence at the transition U/J=5.8, according to mean-field theory U/J≈2, including 1D fluctuations Theory predicts: T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, PRL 92, 130403 (2004). Theory: C. Kollath, U. Schollwöck, J. von Delft, W. Zwerger, PRA 69, 31601 (2004).
Coherent fraction Coherent fraction: T. Stöferle, H. Moritz, C. Schori, MK, and T. Esslinger, PRL 92, 130403 (2004). • F. Gerbier et al., PRL 95, 050404 (2005).
Lattice modulation spectroscopy Periodic modulation of lattice height • explicitly time dependent J(t) and U(t) • experimental modulation ~20% • measure energy absorption
1D Mott-Insulator: Excitation Spectrum superfluid Excitation rate [a. u.] Mott-Insulator Excitation frequency [kHz] T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, PRL 92, 130403 (2004).
Structure of the spectrum Peak at U: particle-hole excitation spectral weight very small Peak at 2U: two particle-hole excitations single particle-hole excitation 2U peak only present in inhomogeneous systems or at finite temperature C.Kollath et al., PRL 97, 050402 (2006)
Superfluid to a Mott insulator in 1D T. Stöferle, H. Moritz, C. Schori, MK, and T. Esslinger, PRL 92, 130403 (2004).
Transport experiments in cold atoms A simple transport experiment k -k Three dimensions: onset of superfluidity Ketterle et al. PRL 2000 -2ħk 0 +2ħk Scattered atoms with spherical (s-wave) symmetry Other transport experiments: ETH, LENS, NIST, Orsay, Pisa, Stanford,, ...
Quantum Newton’s cradle • Two density wave packets with momentum ±2ħk • Reflection probability per collision: 4% D. Weiss, Penn. State Univ.
Long time behaviour g = 18, Ncoll=600 g = 3, Ncoll=2750 g = 1, Ncoll=6000 Steady state but no thermalization. (in 3D it takes 2.7 collisions to thermalize) T. Kinoshita et al., Nature 440, 900 (2006).
Transport in the solid state 1D quantum wire containing impurities from which electrons scatter I Open quantum system: energy is continuously transferred into the system U potential difference U continuously accelerates electrons
atom laser Tomographic in-situ measurements const. B-field shells hn= µ B radio frequency coherently spinflips atoms gravitational sag trapped atoms F=2 untrapped atoms F=1 Airy function gravity mF -2 -1 0 1 2 Hyperfine ground state in a weak magnetic field
Spatial addressing 1D Bose gases |F=1, mF=-1> Radio frequency resonance: |F=1, mF=-1> → |F=1, mF=0> at ħnRF= gFmBB(x,y,z) ≈ mBB(z)/2
Generation of spin impurities • width of impurity wave packet: 2.5 mm (≈ 3 atoms) • same transverse confinement:propagation of impurities is purely one-dimensional • same scattering lengths: a-1,-1 ≈ a-1,0 ≈ a0,0 • accelerated impurity breaks integrability of the 1D Bose gas → interesting dynamics |F=1, mF=-1> V= m/2 wz2 z2 - m g z |F=1, mF=0> V=- m g z quick transfer (RF p-pulse, 200 ms)
In situ detection of the spin wave packet F=2 F=1 Distance [mm]
Time evolution • strong interaction-induced dynamics • significant back action of the impurity onto the majority component • open quantum system: impurity atoms can transfer continuously energy into trapped component by collisions S. Palzer, C. Zipkes, C. Sias, M.K., arXiv:0903.4823
Dynamic structure factor Scattering rate of an impurity (Fermi’s golden rule): S(q,w): dynamic structure factor ki, kf : initial and final momentum of the impurity w(q): excitation spectrum of the gas
Dynamic structure factor in 1D Dynamic structure factor calculation: Brand & Cherny, PRA (2004); Caux & Calabrese, PRA (2006).
Collision rate and energy dissipation For equal masses: impurities move collisionless through a superfluidand a Tonks-Girardeau gases for v<c. Collision rate (v>c) Energy dissipation (v>c) constant force 50% of gravity for heavy impurities: Astrakharchik & Pitaevskii, Davis et al., ...
Center-of-mass motion of the impurities part of the impurities have already left the gas Tonks gas weakly interacting Bose gas ballistic S. Palzer, C. Zipkes, C. Sias, M.K., arXiv:0903.4823
Release measurement Simple model Every collision resets the impurity’s velocity to 0, then gravity accelerates again Mean time between collision events: Time delay accumulated: total number of collisions x time delay per collision tdelay S. Palzer, C. Zipkes, C. Sias, M.K., arXiv:0903.4823
“Far field” distribution of the impurities unscattered scattered g =1.5 g =2.5 g =3.8 • Fermionization? • Enhancement of multiple collision events due to strong interactions? • ? S. Palzer, C. Zipkes, C. Sias, M.K., arXiv:0903.4823
Thanks! Carlo Sias (Postdoc), Christoph Zipkes (PhD), Stefan Palzer (PhD), Michael Feld (PhD), Bernd Fröhlich (PhD), M.K. www.quantumoptics.eu £££: EPSRC, University of Cambridge, Herchel-Smith Fund