Strongly-correlated system Quantum phenomena of multi-component bosonic gases in an optical lattice Yongqiang Li 国防科学技术大学

Strongly-correlated system ① Introduction ② Quantum phases of two-component bosons in optical lattices Quantum phases of two-component bosons in optical lattices Quantum phases of two-component bosons in optical lattices ③ Spinor condensates in optical lattices Spinor condensates in optical lattices Spinor condensates in optical lattices ④ Bosonic gases in the presence of long-range interactions ⑤ Spectroscopy of strongly correlated ultracold bosons ① Introduction ② Quantum phases of two-component bosons in optical lattices Quantum phases of two-component bosons in optical lattices Quantum phases of two-component bosons in optical lattices ③ Spinor condensates in optical lattices Spinor condensates in optical lattices Spinor condensates in optical lattices ④ Bosonic gases in the presence of long-range interactions ⑤ Spectroscopy of strongly correlated ultracold bosons Y.-Q. Li, et. al. New J. Phys. 15, (2013); Phys. Rev. A 87, (R) (2013); Phys. Rev. A 86, (2012); Phys. Rev. A 85, (2012); Phys. Rev. B 84, (2011); Phys. Rev. Lett. 106, (2011).

Strongly-correlated system I. Bloch, Les Houches 2011

Strongly-correlated system I. Bloch, Les Houches 2011

Strongly-correlated system Why optical lattice ?

Strongly-correlated system Why optical lattice ? D. Jaksch et. al., PRL 81, 3108 (1998) M. Greiner et. al., Nature 415, 39 (2002)

Strongly-correlated system

Hubbard model Hubbard, Proc. Roy. Soc. London Ser. A 276, 238 (1963). Gutzwiller, Phys. Rev. Lett. 10, 159 (1963) Wannier function

Strongly-correlated system Multi-component bosoic gases M. Lewenstein, Adv. in Phys. 56, 243 (2007)

Many-Body Cooling towards Quantum Magnetism---- Quantum phases of two-component bosons in optical lattices Quantum phases of two-component bosons in optical lattices Quantum phases of two-component bosons in optical lattices Strongly-correlated lattice bosons Y.-Q. Li, et. al. New J. Phys. 15, (2013); Phys. Rev. A 86, (2012); Phys. Rev. A 85, (2012); Phys. Rev. B 84, (2011);

Strongly-correlated lattice bosons First step: observe AF magnetism Second step: observe d-wave SF SF Temperature (pK) Hopping Motivation

Strongly-correlated lattice bosons History of superconductivity H. Kamerlingh Onnes,1911

Strongly-correlated lattice bosons BCS theory: Cooper pair: phonon-mediated pairing mechanism High Tc superconductivity: Spin fluctuation: covered by Hubbard model Multi-layer coupled theory

Quantum magnetism Electrons: Magnetism: Magnetism: 1: Induced magnetism by external magnetic field 1: Induced magnetism by external magnetic field 2: Collective magnetism: exchange effect 2: Collective magnetism: exchange effect Strongly-correlated lattice bosons

Quantum magnetism Electrons: Magnetism: Magnetism: 1: Induced magnetism by external magnetic field 1: Induced magnetism by external magnetic field 2: Collective magnetism: exchange effect 2: Collective magnetism: exchange effect Bohr-van Leeuwen Theorem: Magnetism presents quantum effects 1: F*v=q(E+ v × B)*v =qE*v =qE*v magnetic field does not contribute to the potential energy magnetic field does not contribute to the potential energy Strongly-correlated lattice bosons

Quantum magnetism Electrons: Magnetism: Magnetism: 1: Induced magnetism by external magnetic field 1: Induced magnetism by external magnetic field 2: Collective magnetism: exchange effect 2: Collective magnetism: exchange effect Bohr-van Leeuwen Theorem: Magnetism presents quantum effects 2: Zero for infinite phase space Strongly-correlated lattice bosons

Quantum magnetism Electrons: Z-AF Magnetism: Magnetism: 1: Induced magnetism by external magnetic field 1: Induced magnetism by external magnetic field 2: Collective magnetism: super-exchange effect 2: Collective magnetism: super-exchange effect Strongly-correlated lattice bosons

Quantum simulation via ultracold gases Hyperfine states: Strongly-correlated lattice bosons

Quantum simulation via ultracold gases D. Jaksch et. al., PRL 81, 3108 (1998) W. Hofstetter et. al., PRL 89, (2002) D. McKay and B. DeMarco, Rep. Prog. Phys. 74, (2011) Hyperfine states+optical lattice : Strongly-correlated lattice bosons

Quantum simulation via ultracold gases D. Jaksch et. al., PRL 81, 3108 (1998) W. Hofstetter et. al., PRL 89, (2002) D. McKay and B. DeMarco, Rep. Prog. Phys. 74, (2011) Hyperfine states+optical lattice : Low-T Strongly-correlated lattice bosons

Quantum simulation via ultracold gases D. Jaksch et. al., PRL 81, 3108 (1998) W. Hofstetter et. al., PRL 89, (2002) D. McKay and B. DeMarco, Rep. Prog. Phys. 74, (2011) Hyperfine states+optical lattice : Z-AF Strongly-correlated lattice bosons

Quantum simulation via ultracold gases D. Jaksch et. al., PRL 81, 3108 (1998) W. Hofstetter et. al., PRL 89, (2002) D. McKay and B. DeMarco, Rep. Prog. Phys. 74, (2011) Hyperfine states+optical lattice : XY-ferro Strongly-correlated lattice bosons

Quantum simulation via ultracold gases D. Jaksch et. al., PRL 81, 3108 (1998) W. Hofstetter et. al., PRL 89, (2002) D. McKay and B. DeMarco, Rep. Prog. Phys. 74, (2011) Hyperfine states+optical lattice : Z-ferro Strongly-correlated lattice bosons

Quantum simulation via ultracold gases Low energy scale: 100 pK sufficiently low temperature needed Low energy scale: 100 pK sufficiently low temperature needed S. Trotzky et. al., Science 319, 295 (2008) Hyperfine states+optical lattice : Deep lattices and strong interaction Hyperfine states+optical lattice : Deep lattices and strong interaction Strongly-correlated lattice bosons

Typical temperatue: 1nK Cooling scheme for ultracold gases: spin-gradient cooling scheme A. M. Rey, Physics 2, 103 (2009) D. M. Weld et. al., PRL 103, (2009) P. Medley et. al., PRL 106, (2011) Cool the system down to 350 pK System: trap + field gradient + 3D optical lattice System: trap + field gradient + 3D optical lattice Strongly-correlated lattice bosons

Two-component Bose-Hubbard model : describe a bosonic system in the presence of a harmonic trap and magnetic field gradient Model

describes interacting bosons in a lattice higher-order extension of Gutzwiller method: resolve long-range spin order real-space generalization to include inhomogeneity describes interacting bosons in a lattice higher-order extension of Gutzwiller method: resolve long-range spin order real-space generalization to include inhomogeneity K. Byczuk and D. Vollhardt, PRB 77, (2008) A. Hubener, M. Snoek and WH, PRB 80, (2009) K. Byczuk and D. Vollhardt, PRB 77, (2008) A. Hubener, M. Snoek and WH, PRB 80, (2009) (Real-space) bosonic dynamical mean-field theory Method

describes interacting bosons in a lattice higher-order extension of Gutzwiller method: resolve long-range spin order real-space generalization to include inhomogeneity describes interacting bosons in a lattice higher-order extension of Gutzwiller method: resolve long-range spin order real-space generalization to include inhomogeneity K. Byczuk and D. Vollhardt, PRB 77, (2008) A. Hubener, M. Snoek and WH, PRB 80, (2009) K. Byczuk and D. Vollhardt, PRB 77, (2008) A. Hubener, M. Snoek and WH, PRB 80, (2009) (Real-space) bosonic dynamical mean-field theory Method

Zero-temperature phase diagram (3D cubic lattice) spin-order is observed within BDMFT and pronounced in the phase diagram with hopping as the axes filling influences phase diagram n=1 n=2 U b,d >0, U bd >0

Finite-temperature phase diagram (3D cubic lattice) n=1 n=2 re-entrant SF-MI transition system favors localization upon heating Pomeranchuk effect higher critical temperature for filling n = 2 compared to n = 1 low critical temperature : 100 pKNew cooling scheme required U b,d >0, U bd >0

Finite-temperature phase diagram (3D cubic lattice) n=1 n=2 re-entrant SF-MI transition system favors localization upon heating Pomeranchuk effect higher critical temperature for filling n = 2 compared to n = 1 compare to single-component case S. Trotzky et. al., Nature Physics 6, 998 (2010) low critical temperature : 100 pKNew cooling scheme required U b,d >0, U bd >0

Finite-temperature phase diagram (3D cubic lattice) n=1 n=2 re-entrant SF-MI transition system favors localization upon heating Pomeranchuk effect higher critical temperature for filling n = 2 compared to n = 1 low critical temperature : 100 pKNew cooling scheme required U b,d >0, U bd >0

AF spin-order is observed within BDMFT and pronounced in real 2D system Temperature: ~100 pK n=1 U b,d >0, U bd >0 n=2

Spin-gradient cooling in the presence of magnetic-field gradient s/n x x SF Two-MI High field gradient: large entropy per particle for two-compoent MI High field gradient: large entropy per particle for two-compoent MI Single-MI  entropy is more important than temperature (1) diffcult to measure T, but total entropy is constant (2) phase is controlled by entropy  entropy is more important than temperature (1) diffcult to measure T, but total entropy is constant (2) phase is controlled by entropy Single-MI U b,d >0, U bd >0

Spin-gradient cooling in the presence of magnetic-field gradient s/n x x SF Two-MI Low field gradient: entropy per particle is lowered adiabatically, but the total entropy is fixed Low field gradient: entropy per particle is lowered adiabatically, but the total entropy is fixed Single-MI U b,d >0, U bd >0

Density, magnetism and entropy distribution domain wall forms due to magnetic field density and spin distribution measured in-situ, width of domain wall = thermometer entropy carried by spin-mixed region and mobile particles Density, magnetism and entropy distribution domain wall forms due to magnetic field density and spin distribution measured in-situ, width of domain wall = thermometer entropy carried by spin-mixed region and mobile particles Spin-gradient cooling in the presence of magnetic-field gradient U b,d >0, U bd >0

D. M. Weld et. al., Phys. Rev. A 82, (2010) Spin-gradient cooling in the presence of magnetic-field gradient U b,d >0, U bd >0

D. M. Weld et. al., Phys. Rev. A 82, (2010) Spin-gradient cooling in the presence of magnetic-field gradient Spin-gradient cooling in good agreement with zero-tunneling approximation (deep lattice) quantitative theoretical validation of recent experiments Spin-gradient cooling in good agreement with zero-tunneling approximation (deep lattice) quantitative theoretical validation of recent experiments U b,d >0, U bd >0

U b,d >0, U bd <0  Existence of a PSF for asymmetric hopping amplitudes  Charge density wave  PSF will typically compete with single-species condensations

U b,d >0, U bd <0  Existence of a PSF for asymmetric hopping amplitudes  PSF will typically compete with single-species condensations

U b,d >0, U bd <0 Hard-core bosons U b,d = 2|U bd | P. Cheng, et. al. PRB 82, (R) (2010).

U b,d >0, U bd <0 3D real system with unbalanced mixtures

 Quantum magnetism of two-component bosons  New cooling scheme: 1.Pomeranchuk cooling scheme due to spin physics 2.spin-gradient cooling scheme Strongly-correlated lattice bosons Summary for the first part

Quantum phases of spinor bosons in optical lattices Strongly-correlated lattice bosons D. Stamper-Kurn

Strongly-correlated lattice bosons J. Stenger, Nature 396, 345 (1998)

Strongly-correlated lattice bosons Coherent dynamics of spinor condensates Widera et al., New J. Phys. 8:152 (2006)

Strongly-correlated lattice bosons Model : At sufficiently high lattice depth, the atoms with spin symmetric tunneling are well described by the generated Bose-Hubbard model Method: bosonic dynamical mean-field theory

Strongly-correlated lattice bosons a.U 2 =0 b.U 0 >U 2 >0 c.U 2 >U 0 >0 Real system: U 2 =0.04U 0 for 23 Na E. Delmer, PRL 88, (2002)

Strongly-correlated lattice bosons How to classify spinor states  Spin ½ atoms (two component Bose mixture) Local spin expectation value is in terms of creation operator:  Spin 1 atoms Nematic order parameter. Needed to characterize e.g. F=1,F z =0 state

Strongly-correlated lattice bosons  left: U 2 /U 0 =0.04, right: U 2 /U 0 =0.01  Three different phases in this diagram: polar superfluid (SF), nematic insulator (NI) and spin-singlet insulator (SSI).  Coexistence regions (Coex) are found between SF and SSI (left) and NI (right)  MF: S. Tsuchiya, PRA 70, (2004) Finite temperature phase diagram

Strongly-correlated lattice bosons  left: U 2 /U 0 =0.04, right: U 2 /U 0 =0.01  Three different phases in this diagram: polar superfluid (SF), nematic insulator (NI) and spin-singlet insulator (SSI).  Coexistence regions (Coex) are found between SF and SSI (left) and NI (right)  MF: S. Tsuchiya, PRA 70, (2004) Finite temperature phase diagram

Strongly-correlated lattice bosons  left: U 2 /U 0 =0.04, right: U 2 /U 0 =0.01  Three different phases in this diagram: polar superfluid (SF), nematic insulator (NI) and spin-singlet insulator (SSI).  Coexistence regions (Coex) are found between SF and SSI (left) and NI (right)  MF: S. Tsuchiya, PRA 70, (2004) Finite temperature phase diagram

Strongly-correlated lattice bosons

Finite temperature phase diagram Five phases in this diagram : polar superfluid (SF), nematic insulator (NI), spin-singlet insulator (SSI), unordered insulator (UI) and normal state (NS). S

 Quantum phases of spinor bosons in optical lattices  In contrast to the mean-field theory, we find that SF and NI demonstrates non-zero local magnetism, and SSI is a spin-dependent-interaction sensitive pair state. Strongly-correlated lattice bosons Summary for the second part

Strongly-correlated lattice Bosons Overview ① Quantum phases of two-component bosons in optical lattices a)Quantum magnetism b)Many-body cooling scheme Many-body cooling schemeMany-body cooling scheme c)Pair-superfluidity ② Quantum phases of bosonic gases in the presence of long-range interactions a)Quantum phases of bosonic gases coupled to an optical cavity Quantum phases of bosonic gases coupled to an optical cavityQuantum phases of bosonic gases coupled to an optical cavity b)Ultracold dipolar bosonic gases ③ Spectroscopy of strongly correlated ultracold bosons Spectroscopy of strongly correlated ultracold bosons Spectroscopy of strongly correlated ultracold bosons ① Quantum phases of two-component bosons in optical lattices a)Quantum magnetism b)Many-body cooling scheme Many-body cooling schemeMany-body cooling scheme c)Pair-superfluidity ② Quantum phases of bosonic gases in the presence of long-range interactions a)Quantum phases of bosonic gases coupled to an optical cavity Quantum phases of bosonic gases coupled to an optical cavityQuantum phases of bosonic gases coupled to an optical cavity b)Ultracold dipolar bosonic gases ③ Spectroscopy of strongly correlated ultracold bosons Spectroscopy of strongly correlated ultracold bosons Spectroscopy of strongly correlated ultracold bosons

Supersolid Phase of Strongly Correlated Bosons Coupled to an Optical Cavity Strongly-correlated lattice bosons

Motivation K. Baumann, et.al., Nature 464, 1301(2010) P < P c P > P c Superfluid Coexistence: -- non-trivial diagonal long-range order -- off-diagonal long-range order Supersolid

Motivation BEC is excited. Cavity mode is excited Dynamical process K. Baumann, et.al., Nature 464, 1301(2010)

How can strongly-correlated bosons be described? Jaynes-Cummings model + Bose-Hubbard model 1: Conventional optical lattice  coupling acts as external potential 2: Optical cavity  Strong coupling  Strong onsite interactions 1: Conventional optical lattice  coupling acts as external potential 2: Optical cavity  Strong coupling  Strong onsite interactions

1.Hamiltonian: Extended Bose-Hubbard model 1.Hamiltonian: Extended Bose-Hubbard model What does the Hamiltonian look like? C. Maschler et.al., Phys. Rev. Lett. 95, (2005). J. Larson, et.al., Phys. Rev. Lett. 100, (2008). Assumptions: Excited states are adiabatically eliminated Coherent states for the cavity mode 2. Method: Real-space bosonic dynamical mean field theory Effective staggered potential due to induced long-range interactions:

Results : Checker-board order vs. filling Experimentally relevant parameters: 2D square lattice Depth of pump laser kept fixed : n(r) n(k) Density

Results : Checker-board order vs. filling Superfluid n(r) n(k) Density Experimentally relevant parameters: 2D square lattice Depth of pump laser kept fixed :

Results : Checker-board order vs. filling Supersolid n(r) n(k) Density Experimentally relevant parameters: 2D square lattice Depth of pump laser kept fixed :

Results : Checker-board order vs. filling Mott insulator n(r) n(k) Density Experimentally relevant parameters: 2D square lattice Depth of pump laser kept fixed :

Results : Checker-board order vs. filling Checker-board solid n(r) n(k) Density Experimentally relevant parameters: 2D square lattice Depth of pump laser kept fixed :

Results : Checker-board order vs. filling Superfluid Supersolid Mott insulator Long-range interactions Contact interactions Long-range interactions Contact interactions Checker-board order Mott insulator Checker-board solid Experimentally relevant parameters: 2D square lattice Depth of pump laser kept fixed :

Total particle number is 139, 167, 184 and 220 for panels (a), (b), (c) and (d) respectively. Parameters same as in homogeneous case. Harmonic trap -> Density distribution -> strength of long- range interactions. Total particle number is 139, 167, 184 and 220 for panels (a), (b), (c) and (d) respectively. Parameters same as in homogeneous case. Harmonic trap -> Density distribution -> strength of long- range interactions. Effects of inhomogeneity

In panels (a) No checker-board pattern. No excitation in cavity mode. In panels (a) No checker-board pattern. No excitation in cavity mode. Effects of inhomogeneity N_tot= 139

In panels (b), Supersolid phase appears Cavity mode is excited In panels (b), Supersolid phase appears Cavity mode is excited Cut of panels (b) along the center of the trap Effects of inhomogeneity N_tot= 167

In panels (c), Checker-board phase disappears Mott-insulating core appears around trap center In panels (c), Checker-board phase disappears Mott-insulating core appears around trap center Effects of inhomogeneity N_tot= 184

In panels (d), Coexistence of phases: Core: Checker-board solid Inner ring: Mott-insulator Outer ring: Supersolid In panels (d), Coexistence of phases: Core: Checker-board solid Inner ring: Mott-insulator Outer ring: Supersolid Effects of inhomogeneity N_tot= 220

In panels (d), Critical number of particles for checker-board solid: Harmonic trap: 220 Homogeneous: 384 Hence the trap enhances the checkerboard solid order. In panels (d), Critical number of particles for checker-board solid: Harmonic trap: 220 Homogeneous: 384 Hence the trap enhances the checkerboard solid order. N_tot=220 Effects of inhomogeneity 384 N_tot= 220

 Novel quantum phase of BEC-cavity system  Harmonic trap strongly influences properties of the system Strongly-correlated lattice bosons Summary for the second part

Strongly-correlated lattice Bosons Content of table ① Quantum phases of two-component bosons in optical lattices a)Quantum magnetism b)Many-body cooling scheme Many-body cooling schemeMany-body cooling scheme c)Pair-superfluidity ② Quantum phases of bosonic gases in the presence of long-range interactions a)Quantum phases of bosonic gases coupled to an optical cavity Quantum phases of bosonic gases coupled to an optical cavityQuantum phases of bosonic gases coupled to an optical cavity b)Ultracold dipolar bosonic gases ③ Spectroscopy of strongly correlated ultracold bosons Spectroscopy of strongly correlated ultracold bosons Spectroscopy of strongly correlated ultracold bosons ① Quantum phases of two-component bosons in optical lattices a)Quantum magnetism b)Many-body cooling scheme Many-body cooling schemeMany-body cooling scheme c)Pair-superfluidity ② Quantum phases of bosonic gases in the presence of long-range interactions a)Quantum phases of bosonic gases coupled to an optical cavity Quantum phases of bosonic gases coupled to an optical cavityQuantum phases of bosonic gases coupled to an optical cavity b)Ultracold dipolar bosonic gases ③ Spectroscopy of strongly correlated ultracold bosons Spectroscopy of strongly correlated ultracold bosons Spectroscopy of strongly correlated ultracold bosons

Spectroscopy of strongly correlated bosons Strongly-correlated lattice bosons

Two laser beams at coincident angle small frequency shift transferred to system Momentum- and frequency-resolved excitations of the many-particle system Probes quantities not accessible by direct time of flight imaging Method : Dynamic bosonic Gutzwiller

Strongly-correlated lattice bosons Finite intensity of Bragg beam Pulse shape Inhomogeneous trapping Effect beyond linear response Comparison between theory and experiment: Comparison between theory and experiment: experiment theory sound mode mainly broadened by intensity Amplitude mode mainly broadened by trap Clear identification of amplitude mode (Along 45° line in Brillouin zone) Good quantitative agreement only achieved when taking all experimental effects into account U. Bissbort et. al, PRL106, (2011)

Strongly-correlated lattice bosons Dispersion relation For high intensity Bragg beams, a broadening of the sound mode signal occurs, and effects beyond the linear response regime must be included. s=9 The grey squares : Gutzwiller method, Shaded region: uncertainty black dots: quasi-particle energy in the linear response regime. The blue line: the Bogoliubov result. the green lines: variational method for deep lattice. The grey squares : Gutzwiller method, Shaded region: uncertainty black dots: quasi-particle energy in the linear response regime. The blue line: the Bogoliubov result. the green lines: variational method for deep lattice.

 Detect excitations of strongly-correlated bosonic gases  Two mode excitations: sound mode and gapped mode Strongly-correlated lattice bosons Summary for the third part

 zero- and finite-temperature phase diagrams, including magnetic ordering  many-body cooling, such as Pomeranchuk cooling and spin gradient cooling  novel quantum phases of BEC-cavity system  spectroscopy of strongly correlated bosonic systems via Bragg scattering Strongly-correlated lattice bosons Summary

 Spin-orbit coupling  Three-component bosonic gases  High-orbital physics  Dynamics of strongly correlated system Strongly-correlated lattice bosons Outlook

Thanks