Bosonic Cold Atoms : from Continuum to Latticecmschool/coldatom/talks/venkat_2.pdf · Venkat Pai!...

Venkat Pai

Bosonic Cold Atoms : ��from Continuum to Lattice

School on Physics of Cold Atoms February 10-14, 2014 @HRI

[email protected]

•  Lecture I : Bose-Einstein Condensation (BEC), BEC of Cold Atoms in Harmonic Traps, Effect of Interactions, and Gross-Pitaevskii Equation

•  Lecture II : More on Gross-Pitaevskii Equation, Optical Lattices, and Mott Insulator - Superfluid Transition

•  Lecture III : More on Lattice Bosons, Long-Range Interactions, Quantum Simulators, and Strong Coupling Theory

Bosonic Cold Atoms : from Continuum to Lattice

•  Slides will be available at : http://www.hri.res.in/~cmschool/coldatom/

Plan of Lectures

Plan of the Talk - II •  Recap : Weakly Interacting Bosons

•  Bosonic Coherent States

•  Description of Condensate : Gross-Pitaevskii Equation (GPE)

•  Applications of GPE

•  A Suprfluidity Primer

•  Bosons in Optical Lattice

•  Gutzwiller Variational Mean Field Theory

•  Ground state Condensate Fraction depletes

Effect of Interactions

•  System develops Superfluidity (flows without viscosity, upto a critical velocity)

•  Energy spectrum changes dramatically; System to be described in terms of quasiparticles with modified dispersion

Theory of Weakly Interacting Bosons

•  Zero momentum condensates and small number of excitations

Theory of Weakly Interacting Bosons (contd.)

Theory of Weakly Interacting Bosons (contd.)

•  New quasiparticle operator (Bogoliubov transformation)

Bosonic Coherent States

•  itself is a coherent state with

•  A condensate (or, its ground state) is a coherent state

•  Coherent states do not have a definite value of quantum number since they are not eigenstates of number operator

•  Probability of observing value in state is a Poissonian distribution

•  Macroscopic coherent states : is essentially infinite and hence, is negligible compared to

•  To a good approximation, many operator expectation values can be replaced by their mean field values by replacing

•  They have a definite phase though it does not have a definite value for number operator

Bosonic Coherent States (Contd.)

•  For any complex

•  Since the term containing depends on

Bosonic Coherent States (Contd.)

•  In the condensed phase with particles,

•  The ground state is a Macroscopic Condensate and has a Macroscopic Wavefunction

•  What would be the effective Schrodinger equation satisfied by this state?

•  This state is a Bosonic coherent state

Towards a description of the Condensate : Gross-Pitaevskii Equation

•  Minimize the energy in the SF state (replace operators by classical field)

Towards a description of the Condensate : Gross-Pitaevskii Equation (Contd.)

•  Chemical potential is introduced to maintain macroscopic, constant normalization of wavefunction

•  Time-dependent Gross-Pitaevskii Equation : Replace the operators by classical fields in the Heisenberg equation of motion

•  Pseudopotential for atom-atom interaction

Towards a description of the Condensate : Gross-Pitaevskii Equation (Contd.)

•  Stationary ground state has

•  Mean Field Approximation; valid in the dilute limit

•  Describes ground state and low lying excitations

•  The ground state could be homogeneous or inhomogeneous

•  Can be generalized to include additional forces (fields), perturbations (including time dependent)

•  Can be used to study coherence properties such as Josephson effect

Applications of Gross-Pitaevskii Equation - I : Ground State Properties in a Trap

•  Size of cloud :

•  Potential energy per particle :

•  Kinetic energy per particle :

•  Size of the cloud (KE and PE are equal) :

•  Stationary Ground state solutions : Nonlinear, solvable only in special case

•  Approximate Methods : Scaling, Variational wavefunctions, Thomas-Fermi

•  Numerical Solution

Scaling Methods : Non-interacting Limit

Applications of Gross-Pitaevskii Equation - I : Ground State Properties in a Trap (Contd.)

•  density :

•  Interaction energy of a particle :

•  This shifts the minimum of total energy to larger values of . Hence as interaction increases, kinetic energy becomes less important. At strong coupling, neglect kinetic energy

•  Minimum when Interaction and Potential energies become equal :

•  Energy per particle

Scaling Methods : Repulsive Interaction


•  At small energies and for small number of particles, minimum determined by the non-interacting value

•  For large number of particles, local minimum becomes shallower and disappers at some critical

Scaling Methods : Attractive Interaction

Pethick and Smith (2002)


•  Use a Gaussian ansatz with zero point amplitude as a variational parameter

Variational Wavefunctions

Thomas - Fermi •  For larger clouds, and in the strong repulsive regime, kinetic energy

negligible

•  Boundary of the cloud at


Variational and Thomas-Fermi

Numerical Gross-Pitaevskii

Pethick and Smith (2002)

Dalfovo et al. (1998)

Applications of Gross-Pitaevskii Equation - II : Excitations

Bogoliubov Excitations

Steinhauer et al. (2002)

Applications of Gross-Pitaevskii Equation - II : Excitations (Contd.)

Dipolar BEC

Rydberg Atoms

Nath and Santos (2010)

Henkel et al. (2010)





•  A Superfluidity Primer



•  Non-classical rotational inertia

•  Persistent flow

•  Quantization of Circulation

•  Second sound

•  Interference and Josephson Effect

Superfluidity

•  A fluid in a torus geometry rotating in a torsion pendulum setup

•  Normal liquid : At rest initially, soon comes to equilibrium with the walls and rotates with the same angular velocity

•  Superfluid : Part of it never comes into rotation (below a critical velocity). Thus there are two components

•  Normal component : moves with wall contributing to rotational inertia

•  Superfluid component : at rest with respect to the walls, experience no frictional drag, and does not contribute to rotational inertia

Superfluidity

Superfluidity : Two-fluid Model •  Rotational momet of inertia appears less than expected : Non-classical

rotational inertia

•  Fraction of fluid associated with normal component where moment of inertia in SF phase, in Normal phase, mass density

•  Superfluid density :

•  Persistent flow : Liquid can exist indefinitely in a metastable state in which it flows persistently around the ring without decaying

•  Momentum density for normal fluid :

•  velocity of walls of the container a normal liquid is dragged along

•  For a superfluid,

•  If ,

•  In general, . Two fluids move with different velocities

Superfluidity : Order Parameter, Landau Theory

•  Suppose, ,

•  Any state which has varying with space must have an energy larger than by ,

Superfluidity : Order Parameter, Landau Theory

•  This is the extra kinetic energy associated with uniform flow of fluid ,

•  Assume to be a quantum mechanical wavefunction

•  Probability density :

•  Current density : ,

•  Superfluid mass density :

•  SF mass current :

•  SF velocity :

•  Kinetic energy is increased, due to motion of superfluid, by

•  Why supercurrents do not decay? ,

Superfluidity : Landau Theory (Contd.)

•  The superflow is a metastable state which cannot decay because of topological constraint ,

•  The line integral is equal to the phase difference between the end points

•  If the end points coincide, line integral around a closed contour is

•  If phase is constant,

•  Metastable states with

•  If the system reaches equilibrium in one of the metastable states, there is a finite current (decays at a rate that is unobservable!).

•  Only way it can decay is through Phase Slips that involve distorting the order parameter, and hence leads to energy barriers!

Quantization of Circulation

•  Topological constraint is usually described in terms of “circulation of fluid”

•  However, this is valid only if the closed contour encloses a region where is nonzero everywhere and hence is well-defined throughout.

•  When this is not the case, we write

•  On going around a closed contour , phase can change by

•  Circulation is quantized in units of

Quantization of Circulation (Contd.)

•  If we can shrink the contour to a point with well-defined phase everywhere (simply connected) , with finite everywhere Stokes’ theorem guarantees since everywhere.

•  For a torus, contour cannot be shrunk to a point (multiply connected). Hence, nonzero circulation

•  In a rotating bucket, fluid can become multiply connected, by having vortex lines running parallel to the axis of rotation. Hence, singularity since at the vortex cores

Quantization of Circulation (Contd.)

•  First sound due to pressure waves. Second sound due to entropy (heat or temperature) waves

•  Total density and the net current is a sum of the superfluid and normal components

•  Neglecting the effect of gravity, irreversible effects such as dissipation, and higher order (in velocity) effects, the (macroscopic) hydrodynamic equations are

Second Sound

Two Fluid Model

Mass conservation

Momentum conservation

Entropy conservation

•  These equations admit two different wave-like solutions

•  First sound : density (pressure) wave with (almost) constant entropy, with both the fluids moving together

•  Second sound : entropy (temperature) wave with (almost) constant total density, the two fluids move in anti-phase in order to keep the net flow of mass to be zero

Second Sound (Contd.)

Wave in the z-direction

•  Problem with the two-fluid model : Since He is a boson, all atoms are identical, and as such cannot be separated into two separate categories. However, the model describes many of the observed features, including the second sound

Second Sound (Contd.)

Donnelly (2009)

First Sound

Second Sound

Superfluid component

Normal component

•  Coherenet atomic tunneling between two Bose Einstein condensates (BEC) confined in a double well magnetic trap

•  Two Gross-Pitaevskii equations (GPE) coupled by a transfer matrix element describes the dynamics in terms of inter-well phase difference and fractional population imbalance

•  In addition to anharmonic generalization of the familiar ac Josephson effect and plasma oscillations occuring in superconducting systems, nonlinear BEC tunneling dynamics sustains a self-maintained population imbalance due to the fact that atoms are neutral : “macroscopic self-trapping”

Josephson Effect

•  Condensate atoms described by a “macroscopic” wavefunction (or order parameter) with , the condensate density

Number of particles in traps 1 and 2

Josephson Effect (Contd.)

Zero point energies in in traps 1 and 2

Tunneling matrix element

•  In terms of the inter-well phase difference and fractional population difference with

•  and are canonically conjugate variables for a “momentum shortened pendulum”


•  Mechanical Analogy : A nonrigid pendulum of tilt angle and length proportional to that decreases with “angular momentum”

•  Bose Josephson junction (BJJ) intertrap tunneling current is

•  Intrinsically nonlinear system unlike a superconductor (which does not support charge imbalance). Hence anharmonic oscillations and sustainable population imbalance expected


Conventional Josephson oscillations


Dynamics Phase Portrait

Anharmonic Josephson oscillations Macroscopic Self-trapping

Smerzi et al. (1997)

•  Towards strongly interacting bosons : Confine bosons in an optical lattice

•  Standing wave formed by three counter-propagating laser beams

•  Atoms experience a sinusoidal potential with

•  Depth of the potential controlled by intensity of the beams

Bosons in Optical Lattice

Superfluid Mott Insulator

Bosons in Optical Lattice : SF and MI •  Small , Large hopping, Kinetic energy dominates, Large number

fluctuations : Superfluidity

•  Large , Small hopping, Interaction dominates, Negligible number fluctuations, Commensurate filling : Mott Insulator

•  A quantum phase transition from a phase coherent condensate to a phase incoherent Mott insulator

Superfluid Mott Insulator

Superfluid-Mott Insulator Transition : Experiments (Coherence)

Greiner et al. (2002)

Superfluid-Mott Insulator Transition : Experiments (Excitations)

Greiner et al. (2002)

Towards an Effective Model : Bose-Hubbard Hamiltonian

•  Wannier Representation

•  Leads to Bose-Hubbard Model

Fisher et al. (1989)

Jaksch et al. (1998)

Double Well Potential : Superfluid State

for M wells and N particles

Double Well Potential : Mott Insulator

for M wells and N particles :

commensurate filling N=gM

Variational Mean Field Theory : Gutzwiller

•  Condensate order parameter

•  Mean field Hamiltonian

•  Variational ground state

•  Variational procedure : minimize

Sheshadri et al. (1993)

Rokhsar and Kotliar (1991)

Gutzwiller Mean Field Theory : Phase Diagram

SF MI-3

MI-2

MI-1

•  Bose-Einstein Condensation in Dilute Gases : C. J. Pethick and H. Smith

•  Superconductivity, Superfluids, and Condensates : J. F. Annett

•  Bose-Einstein Condensation : L. P. Pitaevskii and S. Stringari

•  Quantum Liquids : A. J. Leggett

References Books

Reviews

•  F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999)

•  I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)

•  A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett. 79, 4950 (1997)

•  M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1989)

•  D. S. Rokhsar and B. G. Kotliar, Phys. Rev. B 44, 10328 (1991)

•  K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, Europhys. Lett. 22, 257 (1993)

•  D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998)

•  M. Greiner, O. Mandel, T. Esslinger, T. W. Haensch, and I. Bloch, Nature 415, 39 (2002)

•  D. L. Kovrizhin, G. Venketeswara Pai, and S. Sinha, Europhys. Lett. 72, 162 (2006)

References Papers

•  Show that two different coherent states are not orthogonal. Why?

•  Using the GPE and scaling arguments, calculate the size of a cloud of repulsive atoms and the energy per particle

•  Calculate the size of a cloud of repulsive atoms in a harmonic trap using the Thomas-Fermi approximation

•  Use the time dependent GPE to calculate the Bogoliubov excitations in absence of a trap and show that it has the same behavior as in the case of weakly interacting Bosons

•  In the atomic limit, calculate the values of scaled chamical potential at which various Mott insulator transitions occur

Problems

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