Dynamical phase diagram of the strongly interacting BoseEinstein

Dynamical phase diagram of the strongly interacting Bose-Einstein Condensate in an optical lattice Bishwajyoti Dey Department of Physics University of Pune, Pune with Galal Al- Akhaly

First unambiguous observation of BEC was reported by Eric Cornell, Carl Wieman (1995) in Colorado (US). BEC was observed cooling a gas of rubidium-87 to a temperature 170 n. K Fig. Velocity distribution. The axes are x and z velocities and third axis is number density of atoms. Macroscopic fraction (~10%) of the atoms are in the ground state.

Optical lattice : an artificial crystal of light – a periodic intensity pattern that is formed by the interference of two or more laser beams. More lasers give 3 D spatial structure. Trapping atom in optical lattice – atoms can be trapped in the bright or dark regions of the optical lattice via Stark shift. Strength of the optical potential confining can be increased by increasing laser intensity. BEC mounted on a optical lattice is like electrons in a periodic potential of ions in conventional solid. Condensate atoms plays the role of electrons and optical lattice the role of ions.

Atoms trapped in an optical lattice move due to quantum tunneling even if the potential depth of the lattice point exceeds the kinetic energy. Strongly interacting limit. However when the well depth is large then the interaction energy of the atoms become more than the hopping energy, then the atoms will be trapped in potential minima and cannot move freely. This phase is called Mott insulator. Atoms in an optical lattice provide an ideal quantum system where all parameters can be controlled. This can be used to observe effects which are difficult to observe in real crystals. Examples: Bloch oscillation, Efimov effect, Superfluid to Mott insulator transition etc.

Transition from superfluid (BEC) to Mott insulator is possible when BEC is placed in a period lattice (optical lattice). Figure 2 (a) Two atoms (dark blue balls) occupy neighboring potential wells. U is the energy cost for them to be in the same well (pale blue balls). (b) Lowering one well relative to the other allows atoms originally in separate wells (light blue) to occupy the same well (dark blue). (Adapted from ref. 1. )

Bloch Oscillation The quantum dynamics of accelerated particles in periodic potential leads to an oscillatory motion instead of a linear increase in velocity. This is termed as Bloch oscillation The periodicity of the potential implies eigenfunctions obey relation In presence of an accelerating force F, the quasimomentum evolves linearly in time In combination with the periodicity of the band structure, this causes an oscillatory motion, the Bloch oscillation. The oscillation period is .

Bloch Oscillation In solid state systems scattering due to impurity of the crystals structure leads to damping of Bloch oscillations on time scales much shorter than the oscillation period itself. Hence difficult to observe experimentally. Optical lattice on the other hand constitute a perfect optical crystal and BEC on optical lattice have enabled the first direct observation of Bloch oscillation. Due to interactions between atoms, Bloch oscillation decays from dynamical instabilities.

Efimov effect Quantum Mechanics of three-body systems : Efimov effect (1970) There exist bound states (Efimov states) of three bosons even if the two-particle attraction is too weak to allow two bosons to form a pair. The sequence of three-body bound states have universal properties, it is insensitive to the details of two-body potential at short distances. Efimov’s theoretical prediction could only be verified experimentally in 2005 in ultra cold gas of cesium atoms. A system of atoms with attractive two-body interactions, is unstable against collapse above certain critical number of atoms Nc. An addition of a repulsive three-body interaction can overcome the collapse and region of stability for the condensate can be extended beyond Nc.

Dynamics of BEC in an optical lattice: order parameter and mean-field theory The many-body Hamiltonian describing N-interacting Bosons confined by an external potential is given by where are boson field operators, potential. The field operators can be written as is the two-body interaction where are the single-particle wave function and are the corresponding annihilation operators defined as, with commutation rules

Using the Heisenberg equation the time evolution of the field operator is given by Bogoliubov first order theory for the excitations of interacting Bose-gas where is a classical field, the order parameter or the wave function of the condensate. The condensate density. Assuming that only binary collisions at low energy are relevant and these collisions are characterized by a single parameter, the s-wave scattering length, independent of the details of the two-body potential, we replace The coupling constant where a is the scattering length.

This yields the equation for the order parameter, the Gross-Pitaevskii equation The GP equation can be written as where Mechanism not included in the GP theory are: three-body collisions which become important when density of the system become large.

In presence of three-body interactions the Gross-Pitaevskii equation become which depend on the Hamiltonian of a single trapped atom as well as twoand three-body coupling constants g 2 and g 3. The three-body coupling constant g 3 has been derived from a microscopic theory of three-body collisions in a BEC (Kohler, PRL (2002)). The spatial coordinates are chosen as the vector from atom 1 to atom 2 ( r 12) and the vector from the center of mass of atoms 1 and 2 to atom 3

Dynamics of BEC: Gross- Pitaevskii (GP) equation – treating the condensate as classical field. GP equation is a variant of the Nonlinear Schrodinger equation (NLS) incorporating an external potential used to confine the condensate. Multicomponent GP equation for spinor condensate. Dimensionality reduction possible in the presence of external periodic potential generated by the optical lattices and in the discrete limit.

Deep periodic potential limit – tight binding model The linear Bloch waves exhibit strong localization in the deep potential limit. Condensate wave function is described with localized Wannier states associated with lowest band. where is the condensate wave function localized in trap n with the orthonormal conditions Using in GP equation and integrating using the orthonormal conditions above we get the dynamics of the condensate described by the discrete nonlinear Schrodinger equation as (DNLS)

DNLS (discrete nonlinear Schrödinger) Equation where

The DNLS equation is the equation of motion and can be derived from the Hamiltonian where and the norm are conjugate variable. Both the Hamiltonian are conserved quantity.

Variational Dynamics To study the dynamical regime of a high density BEC in an array, we consider dynamical evolution of a Gaussian profile wave packet and introduce the variational wave function where the variational parameters respectively of the density associated momenta. and are center and width and are their The dynamical evolution of the variational wave packet can be obtained by a variational principle from the Lagrangian

Using Euler-Lagrange equation the variational equations of motion are The pairs Hamiltonian and are conjugate dynamical variables w. r. t. the The variational equations can be solved numerically to obtain the variational dynamics of the system.

The wave packet group velocity is given by and the inverse effective mass is given by where

Numerical solution of DNLS equation is also solved numerically to compare with the variational dynamics results and also to check stability of the dynamics and phase diagrams over long period of time. We write the order parameter components and in terms of two. DNLS then can be written as The coupled nonlinear equations are solved using Runge-Kutta method.

The variational wave function is used as initial condition and The Hamiltonian and the norm are checked at each steps of the integration to look for their constancy over time.

Dynamical instability of Bloch oscillation: We take gravitational force as external force (tilted wash board potential) and obtain the quasimomentum as , where. Linear regime: for zero condensate interactions of condensate oscillate as (exact solution) Similarly, the width of the condensate density No instability (decay) in linear regime. , the center oscillates as (exact solution)

Numerically, for Bloch oscillation, we calculate numerical average position defined as It is easy to show that , the average position of the center of density. Similarly, gives the numerical width of the wavepacket.

Bloch Oscillation: no instability in absence of interactions

Nonlinear regime: In this case the equation for the center of the density is given by Note : even though there is a damping term in the equation, the dynamics is fully Hamiltonian. The apparent damping is due to the divergence of the effective mass with time due to which the Bloch oscillation decays.

The Bloch oscillation decays as Decay of Bloch oscillation: effect of nonlinear Interactions. (GA, BD 2011) Anderson & Kasevich, Science (1998).

Phase diagram of the interacting BEC High density BEC with deep optical lattice potential supports many interesting phases. Consider the accelerating potential to be zero. Phase diagrams can be obtained from the coupled variational equations and the corresponding Hamiltonian.

The trajectories in the plane can be obtained as The condition implies is obtained from the condition For This gives and for . . which implies and The wave packet stops as the effective mass goes to infinity. This corresponds to the self-trapped regime in the phase diagram.

On the other hand, for But , and the effective mass There is complete spreading of the wave packet giving rise to the diffusive regime. The critical line separating these two regime (the self-trapped and the diffusive) is obtained from the condition as

(GA, BD 2011) Self -trapping Diffusion

SOLITON For negative effective mass, i. e. for we get another interesting phase from the fixed point of the trajectory. This gives a regime in the phase diagram where soliton solutions are allowed. The center of mass moves with constant velocity and the shape of the wavepacket do not change with time. Soliton solutions are allowed for the parameter values For there are no soliton solutions, as in this case the trajectory do not have fixed points.

Soliton solution from direct numerical Integration of the Gross-Pitaevskii Equation. (GA, BD 2011)

DISCRETE BREATHER Another interesting phase is the discrete breather which is a spatially localized and time-periodic solution. In this case oscillate with time. The trajectories in the plane are closed. We have discrete breather solution with center of mass travelling with nearly constant velocity and with oscillating width. oscillate around constant value.

Phase space trajectories

Phase space trajectories in the Efimov region : Numerical results shows that the Soliton exist only for large value of For large value of , the soliton line approach the critical line . .

Phase space trajectories in the Efimov region – no discrete breather. In this region and when the trajectory shrink to zero. , the area enclosed by However, for addition of a small two-body interaction in the Efimov region, the discrete breather solution reappears.

Phase diagram of high density scalar BEC (GA, BD 2011) Parameters: Inset:

When the two- and three-body interactions have opposite sign, then Solitons as well as discrete breathers are not allowed. In this case, the soliton as well as the breather lines lies much below the critical line (deep inside the diffusion region) and it is not possible by increasing the width to get these lines approach the critical line.

Future work: 1. Dynamics of BEC in graphene optical lattice. 2. Nonlinear localized solutions in the gap region of the spectrum of BEC on optical lattice. BEC in a Honeycomb optical lattice (Chen and Wu, PRL, August 2011). Dirac point is changed completely by atomic interaction. Dirac point is extended into a closed curve and an intersecting tube structure arises at the original Dirac point. The tube structure is caused by the superfluidity of the system. This implies application of tight-binding model is not the correct one to describe the interacting BEC around Dirac point. May be a correct choice of the Wannier function is necessary?

Chen, Wu PRL, 2011 Tubed structure due to superfluidity of BEC.

2. Localized solutions in the spectrum gap in the Localized solutions can exist in the spectrum gaps forbidden for linear waves. Such solutions are highly stable as they cannot decay by interacting with linear waves. Gap Solitons in BEC in optical lattice was confirmed experimentally (PRL, 2004). General problem of linearly coupled K-d. V equations with nonlinear dispersion – localized excitations in the gap region of the spectrum. GA, BD (PRE, 2011)

Spectrum can also occur in multi-component BEC – the spinor BEC, due to coupling between components. Spectrum gap can also open due to interplay of lattice periodicity and nonlinearity.

Interest in BEC was lost since then, as it was believed that the conditions for BEC can never be produced in a real system. ---Ideal gas ---Density and temperature relation ---Most of the gases will be solid at such low temperature Tc = 34 n. K from the formula of Tc for rubidium vapor (Theory). Experimental value 170 n. K ! Super fluidity (1937) and superconductivity (1957 - BCS theory) were believed to due to Bose-Einstein condensation.

Super fluid Helium-4 (Boson) (1937) - transferring of fluid mass without transferring energy. Landau phenomenological theory (Noble Prize 1962): -- super fluidity is destroyed above a critical velocity. Above this velocity, excitations are created. -- existence of roton (high momentum version of sound). Required to explain specific heat data. Microscopic theory of roton by Feynman. -- existence of quantized vortex lines --existence of second sound (application of heat to a spot in liquid helium results in a heat wave conduction). --super fluidity also observed in He-3 (fermions). Cooper pairing between the atoms (rather than the electrons as in BCS theory of superconductivity). Attractive interaction between the atoms is mediated by spin fluctuations rather than phonons.

BEC is obtained with a two-step process of cooling and trapping. The first stage uses laser light for the cooling and trapping. The second stage uses magnetic fields for trapping and cooling by evaporation (allowing high energy particles to escape). Trapping provides a “thermos bottle” that keeps the very cold atoms from coming into contact with hot wall only 1 cm away! No complex cryostats, dilution refrigerators required! Only atom cloud is cooler than the room temperature where the apparatus is! Apparatus – a small glass cell with some coils of wire around it. Now undergraduate lab. Expts!

Getting around the laws of thermodynamics in experiment. Rubidium is a metallic solid in room temperature which is its true thermodynamic equilibrium. Requirement is vapor state of rubidium. The idea is to avoid reaching a true thermodynamic equilibrium. Produce conditions (low temperature and low density) so that the gas remains in its metastable supersaturated-vapor state for a long time. During this time BEC is produced and studied. It takes long time to reach its true equilibrium (solid) state. The condensate lives for 15 -20 seconds. This concept of needing to produce a sample with two very different time scales for equilibration is a critical step for achieving BEC.

BEC (both bosonic and fermionic) is expected to show all properties exhibited by superfluid helium. Experiments on rotating BEC has already shown the existence of quantized vortices. Experiments are on to determine the excitation spectra of BEC (rotons), second sound etc. BEC -BCS crossover: transition from the condensate formed by pair of fermionic atoms forming pair through formation of molecules (BEC condensate) to BCS condensate where atoms form pair through “Cooper pairing” like mechanism. The ability to realize ultra cold gases in periodic potential has enabled detailed studies of many effects in solid state physics,

6. Study quark-gluon plasma (QGP)( state containing free quarks and gluons that existed a fraction of second after Big Bang). When BEC is released from a cigar shaped trap, it expands more rapidly in narrow direction than in the long direction. Similar behavior seen in experiment built to produce quark-gluon plasma. Connection between QGP and strongly interacting Fermi gas is their perfect hydrodynamics – no damping and no viscosity. 7. Strongly interacting Fermi gas can be used as means of testing predictions in other branches of physics where strong interaction dominate. String theory. Theorists at Utrecht Univ. , Netherlands proposed that superstrings could be made in the laboratory by trapping ultracold cloud of fermionic atoms inside a vortex in a BEC (A recipe for making strings in the lab). 8. Condensate to make quantum computers. Condensates have lot of potential quantum bits.

Atom Laser : Laser that emits atoms rather than photons. Atoms that can be collimated to travel long distance or brought to a tiny focus. In BEC all atoms have same energy, same de Broglie wavelength, same phase and is described by the same wave function. BEC therefore can be highly monochromatic source of atoms when released from the condensate. In conventional laser optical cavity is formed by mirror. In BEC the inhomogeneous magnetic field provided a confining potential around the atoms which plays the role of optical cavity. To extract coherent beam from the cavity in conventional laser, partially transmitting mirror is used (output coupling). In atom laser the output coupling is achieved by changing the magnetic state of atom. It is changed from the state that are confined to the state that are not. This is done applying short radio frequency pulse to flip the spin of the atom and therefore release from the trap. The extracted atoms are accelerated away from the trap under gravity.

Application of atom laser The discover of conventional laser was dubbed as “ a solution in search of a problem”. Today laser is billions of dollars industry! Atom laser is expected to spark similar revolution in the field of atomic optics. Lens, mirrors, beam splitters have already been developed to control the atomic beam. Application in atom holography – the de-Broglie wavelength of atoms is much smaller than the wavelength of the conventional lasers. This means atom laser can create much higher resolution holographic images. It can be used to project complex integrated-circuit patterns, just a few nanometer in scale, onto semiconductors. Atom interferometry - atom beams can be split to create interference patterns – more sensitive than optical interferometer. Useful in condensed matter physics and quantum mechanics. Problems with atom laser– getting continuous output. BEC has to be replenished continuously. Steady state condensation formation required. Also output energy is fixed as condensate is in ground state. BEC in excited state requires to be created.

Experiment for showing superfluid (BEC) – Mott insulator quantum phase transition. Experiment based on phase of the BEC and Mott insulator. Number-phase uncertainty relation in the condensate and its phase. of the number of atoms All atoms in BEC has same phase (same wave function, that of condensate). Two BEC can show interference pattern. But number of particles in the lattice site is not defined. Mott insulator on the other hand have number of particle on the lattice sites fixed but no phase coherence between the particles. This phase coherence and decoherence during the transition from the superfluid (BEC) to Mott insulator phase is seen in experiment as presence or absence of interference pattern.

Fig. Interference patterns in absorption images (gauged by scale on right) result when a gas of cold atoms in a three-dimensional optical lattice is in its superfluid phase; no interference is seen in a Mott insulating phase. The depth of the potential wells in the lattice is systematically increased from 0 at (a) to 20 Er at (h), where Er is a reference energy. The phase transition occurs somewhere between (f) and (g).

Dynamics of Bose-Einstein Condensate (Dey, Galal) DST project sanctioned (July 2010) entitled “Dynamics and phase diagram of scalar and spinor Bose-Einstein Condensates” Spinor Condensate – In spinor BEC the atoms can be in an internal quantum state that is a superposition of spin states. The BEC of spin-1 particle, like the F=1 ground state, is a single condensate with the atoms in a superposition of the three spin projection, m. F =1, 0 and -1, or the superposition of three coupled BECs with the same spatial wave function, one in each of these spin states. F-state of an atom F=J+I. J=L+S (total electron angular momentum) and I= nuclear spin 87 Rb has nuclear spin I=3/2. Electron configuration n=5, L=0, s=1/2, J=1/2 F=J+I =2 and F=J-I =1.

Spinor condensate is expected to possess a host of quantum phenomena absent in scalar condensates. These include magnetization, spin wave mode, spin domains, ferromagnetic, antiferromagnetic and cyclic phase of ground state etc. Cyclic phase has close analogy with d-wave BCS superfluids.

Problems to be studied (a) Dynamics of the system (b)Effect of two- and three body interactions on some model properties of the system (c) Examine nature of collapse of BEC (d) Dynamic phase diagram of the system (e) Phase transition (f) Dynamics of coupled spinor condensates (g) Magnetic states of the spinor condensates, in particular for F=2 hyperfine states.

Bose-Einstein Condensation – Macroscopic occupation of a single particle state. Peculiar to systems with integer spin (bosons). Total number of neutrons in the atom decide whether the atom is fermion or boson. S. N. Bose (1924) brought to the attention of Einstein a simple derivation of the Planck radiation law that treated photons as particles that obeyed the counting rule that are today called Bose-Einstein statistics. Einstein extended the idea to atoms (bosons). In a gas of atoms obeying Maxwell-Boltzmann statistics, states are microscopically populated (number of particles occupying any one state is very small fraction of total number of particles). Only at T=0, the ground state (E=0) macroscopically populated. Einstein showed that for a gas of identical bosons, macroscopic occupation of the ground state, comparable to the total number of particles in the gas, can occur at a finite nonzero temperature. This phase change is called Bose-Einstein condensation (BEC).

Bose-Einstein statistics – thermal occupation of a state with energy and temp. T Since the occupation number should be +ve, . For a system held at constant T and V, when more particles are added adjusts by increasing towards zero. For the second term in N can be evaluated as and cannot increase further. (convert to 3 D integral over phase space volume). At this point T represent critical value Tc.

Suppose we now add more particles to the system. How can adjust further? To see this expand Thus as becomes macroscopically occupied in a very controlled way. As more particles are added below Tc, they all go into

For low density gas and high temperature, the de Broglie wavelength , which is a measure of delocalization of a particle, is small compared to interparticle separation. For particles in the zero momentum state, the delocalization is infinite. BEC occurs when the interparticle separation is of the order of delocalization. Thermal de Broglie ( a measure of thermodynamics uncertainty in the localization of particle) wavelength From the expression of Tc, one can show that the interparticle spacing is BEC does not occur in a 2 -D gas (in absence of external potential). The second term in N diverges faster than , as , and the occupation of is not favored.