Qiang Gu （顾 强） Cold atoms in the synthetic magnetic field Department of Physics, University of Science and Technology Beijing （北京科技大学 物理系） KITPC, Beijing, August 24, 2012

Trapped Bose gas BKT transition of 2D Bose gas Charged Bose gas in magnetic field Cold atoms in the synthetic magnetic field

Charged Bose gas in magnetic field 1949 Possibly earliest study on charged bosons Phys. Rev. 76, 400(1949) Landau diamagnetism: due to the quantization of orbital motions charged of charged particles

1954 Schafroth-Blatt-Butler superconductor Phys. Rev. 96, 1149(1954), Letters to the Editor Charged Bose gas in magnetic field

1955 Meissner-Ochsenfeld effect Phys. Rev. 100, 463 (1955) Charged Bose gas in magnetic field

Phys. Rev. 100, 463 (1955) Charged Bose gas in magnetic field where infinity Phys. Rev. E 60, 5275 (1999)

PRL96, (2006) Charged Bose gas in magnetic field

Ideal fermi gas: Pauli paramagnetism 1927 where is the density of state at Fermi surface is the Bohr magneton due to the intrinsic magnetic moment of electrons Ideal Fermi gas: Landau diamagnetism 1930 due to the quantization of orbital motions of charged particles PM vs DM in Fermi gas

PM vs DM in spin-1 Bose gas where : chemical potential; The Hamiltonian ： : degeneracy; : Zeeman energy : cyclotron frequency : Lande factor : Landau level J. Phy.:Condens. Matt. 23, (2011)

PM vs DM in spin-1 Bose gas where Grand thermodynamical potential ： particle number and magnetization ：

PM vs DM in spin-1 Bose gas = 10 (dash-dot-dotted) ， 3 (dotted) ， 0.3 (dashed) ， 0.05(solid)

PM vs DM in spin-1 Bose gas For S=1 particles: Bose gas, dotted line Fermi gas, dashed line Boltzmann gas, solid line Phy. Lett. A 374, 2580 (2010) J. Phys.: Condens. Matter 23, (2011)

PM vs DM in spin-1 Bose gas Grand thermal potential of Maxwell-Boltzmann gas ： is determined by In the limit cases,

Trapped Bose gas BKT transition of 2D Bose gas Charged Bose gas in magnetic field Cold atoms in the synthetic magnetic field

Berezinskii, JETP 34, 610 (1972) Kosterlitz,Thouless, J. Phys. C 6, 1181 (1973) BKT transition of 2D Bose gas

Experimental systems: liquid 4 He, SC films, SC Josephson array… BKT transition of 2D Bose gas Phys. Rev. Lett. 40, 1727 (1978). Phys. Rev. Lett. 42, 1165 (1979). Phys. Rev. Lett. 47, 1542 (1981).

The transition occurs when T c =πJ/2 BKT transition of 2D Bose gas From SC Josephson array to lattice of Bose-Einstein condensates

Experimental system in cold atoms: Nature 441, 1118 (2006) ENS(2006): Phys. Rev. A 81, (2010) BKT transition of 2D Bose gas At low-T, the interference fringes are straight. Just below the transition temperature, the fringes become wavy due to decreased spatial phase coherence. Phase dislocations become common at temperatures above the transition. These “zipper patterns” indicate the presence of free vortices.

JILA(2007): Phys. Rev. Lett. 99, (2007) BKT transition of 2D Bose gas

Phys. Rev. Lett. 102, (2009) NIST(2009): Thermal —— Quasicondensate —— BKT Superfluid BKT transition of 2D Bose gas

Rotating frame Synthetic magnetic field: Easier to construct optical lattices BKT transition of 2D Bose gas

The frustrated XY model: BKT transition of 2D Bose gas Phys. Rev. A 82, (2010) Phys. Rev. B 14, 2239 (1976). Hofstadter butterfly

The origin of frustration: BKT transition of 2D Bose gas The frustrated XY model was used to describe the superconducting Josephson arrays in transverse magnetic field by Teitel and Jayaprakash, the model can be mapped into the “frustrated” quantum phase model: Using the relationship: Phys. Rev. B 27, 598 (1983); Phys. Rev. Lett. 51, 1999 (1983).

 The physical quantities of a system described by the frustrated XY model can be gauge dependent, although observable quantities are usually gauge invariant in conventional systems.  The imaging of density of the expanding condensates in cold-atom experiments in fact measures the canonical momentum of the original model.  That is why the vector gauge potentials can be detected in the momentum distribution of the density matrix. U(1) gauge symmetry breaking BKT transition of 2D Bose gas

Rev. Mod. Phys. 80, 885 (2008) gauge dependent We can get the density profile by solving the frustrated XY model using the standard Metropolis Monte Carlo method. BKT transition of 2D Bose gas

Metropolis Monte Carlo step 1.Pick a random site i from the lattices 2.Choose a random phase 3.Calculate the energy shift with 4.Accept the random walk with the probability 5.Calculate the new energy of the system and pick a new site j, go to step 1. First of all, we choose a initial state that each site holds a phase zero. Every Monte Carlo step can be performed as follows: (over-relaxation, cluster algorithm, etc.) BKT transition of 2D Bose gas

Illustration of the expansion image of the system at different temperatures T for the fully frustrated case (f =1/2). The BKT transition takes place at about T = 0.5J/kB where the peaks decay to nearly zero in (b). The color represents the relative magnitude of the density which increases from purple to white.

BKT transition of 2D Bose gas The central peak G0 (G(kx = 0, ky = 0)) of expansion image as a function of temperature T for four different fractional frustration f =1/2, 2/5, 1/3, 1/5. The square points are numerical results with the error bar obtained using the standard deviation. The circle in each figure guides the estimated critical transition temperature. The insert shows the corresponding expansion image close to the ground state.

We are searching for experimental possibilities to identify the transition diagram. For example, to find the T C at f=1/3 or 1/2. BKT transition of 2D Bose gas The superconducting Josephson arrays in transverse magnetic field. Phys. Rev. B 27, 598 (1983); Phys. Rev. Lett. 51, 1999 (1983).

the Raman detuning gradient Realized detuning in experiments: For the case that So we get: BKT transition of 2D Bose gas Nature 462, 628 (2009).

BKT transition of 2D Bose gas The transition temperature For: Higher transition temperature can be get by increasing the average particle number at each site N 0 or reducing depth of the lattice potential V 0 。

Trapped Bose gas BKT transition of 2D Bose gas Charged Bose gas in magnetic field Cold atoms in the synthetic magnetic field

Neutral atoms in rotating frame Charged bosons in magnetic field / Neutral bosons in synthetic field Trapped Bose gas Hamiltonian in direction: where and denote the frequencies of harmonic potential in the x, y plane and in the z direction.

The energy spectrum: Trapped Bose gas with where The grand thermodynamic potential, with

结果与分析 Trapped Bose gas (above ) (bellow ) The BEC temperature: Chin. Phys. Lett. 28, (2011) Homogeneous gas in magnetic field To determine the BEC temperature: The Landau diamagnetization: Kling and Pelster, Phys. Rev. A 76, (2007)

Trapped Bose gas in in the synthetic magnetic field Beyond SCA:

Trapped Bose gas in rotating frame Beyond SCA:

Thanks for your attention! Beijing, August 24, 2012