Natan Andrei (Rutgers): Quench dynamics of the interacting Bose gas in one dimension complementary to Aditi's talk

Quenching and time evolution

Start in H0, turn on interaction at t = 0, propagate with full Hamiltonian.  New experiments allow us to do exactly this in cold atoms, nano-devices etc, since the time scales are much longer than in typical condensed matter systems.

Interesting questions:
- how do the observables evolve?  correlations functions are now functions of two times (time difference and waiting time).

Example:
Quantum dot with tunneling turned on suddenly, measure appearance of the Kondo effect, how does the resonance evolve with time?

Bosons in one dimension - what are the effects of interactions on the dynamics?
One dimension because strong quantum fluctuations enhance all interactions, plus powerful mathematical methods - RG, bosonization, CFT and *Bethe Ansatz* (BA), which allows one to diagonalize an integrable Hamiltonian and find all its eigenvalues.  Many integral models, like Hubbard, Heisenberg etc are experimentably realizable.  So how can one use the BA to solve hte quench dynamics of many body systems?

Setting it up:

A given state can be time evolved knowing the energy eigenstates.  These are known via the BA.
Standard approach - impose periodic boundary conditions, generate BA equations with quantized momenta, get spectrum and reconstruct thermo.  Nonequalibrium requires overlaps, summing over the whole basis and take infinite volume limits, but typically has required heavy numerics, while we want to be elegant.

BA review:

General N-particle state generally very complicated, but BA-wave-function is much simpler - it is a product of single particle wave-functions and S-matrices (describing scattering phase shifts of two particles).  Illustrating on Lieb-Linnigar model - bosons on line with delta-function interaction.

Divide configuration space into N! regions (labeled by Q), within each region there is no interaction, wave-function is product of exponentials.  Regions are related by S-matrices - interaction on the boundaries.  Assign amplitude to each region, A(Q), and S(ij) relating the regions.  However, we have to be careful to do this consistently, if say we are exchanging between three regions there are two ways to do it and they must be the same.  This consistency relationship is called the Yang-Baxter equation: S(12)S(13)S(23) = S(23)S(13)S(12), which is satisfied only when the system has enough conservation laws (eg - integrable).

Calculating overlaps is still difficult.  Directly going to the infinite system can simplify things since momenta are unconstrained, and so can be integrated on a contour in the momenta space - which may be complicated, but allows a certain simplification using the non-interacting overlap. Dominated by the poles in the full-fledged Bethe eigenstates. Unfortunately, implementing the formula is very system-specific, really a guide for how to go about doing the problem.

Piers Q: New way of writing completeness? (yes) Does it only work for integrable systems?
A: Not really - would be similar formulas if we have a set of lambda where we know the poles.

Applications to bosons in optical traps:
(can make predictions for experiments - noise etc)

Lieb-Linniger model - standard model for bosons in 1 or 2 dimensions.  Short range interaction - kinetic energy + local interaction when bosons are on top of one another.  Coupling constant can be either attractive or repulsive, can be controlled experimentally via a Feshbach resonance.
Conventional BA results: Eigenstates labeled by momenta.  Dynamic factor - normally 1, negative if they cross.  Momenta real for repulsive, c > 0, complex pairs for attractive, c < 0, forming bound states.

Initial condition 1 - relax periodic lattice
Intial condition 2 - relax trap and allow bosons to expand and interact - interested in how they interact.

Central theorem: Expand state in term of eigenstates in contour representation.  Must use real contours for repulsive case, complex strings for attractive case.  Using contour represntation, can easily write down time-evolved state analytically.

Q: number of particles? A: any number of particles.

Keldysh: Time-evolving operator, O is non-perturbative Keldysh (forward and backwards in time on contour C) - not expanding in interaction, just solving exactly.

Okay - let's calculate.
1. Evolution of density -> time of flight experiment.
2. Evolution of noise correlation (more physics than in average density) -> time-dependent Hanbury-Brown Twiss effect.
C2 = <rho(1) rho(2)>/<rho(1)><rho(2)>

HBT - vanishing correlations between two sources (stars) - measure photons find bunching (C2 ~ cos x), fermions find anti-bunching (-cosx)
Many free particles gives more structure, time-dependent - curious about effects of interactions -> repulsive bosons evolve into fermions, attractve bosons evolve to a condensate.

Prepare initial state with two bosons, weakly overlapping.  Evolve in time.  Can carry out all integrals for two particles.  Calculate B\dg b.  Results: no interaction, then bosons quantum broaden and nothing interesting happens, density just decays in time.  Repulsive interaction - pretty much like free bosons.  Attractive interaction competes with the diffusion, as it favors more of an overlap, increasing the effect of interactions, and get much more interesting, oscillatory structure of the density(t).

For simplicity look at the long time limit - bosons turn into fermions - expression develops antisymmetric correlations and can rewrite original bosons in terms of fermions interacted on by free fermion Hamiltonian (Tonks-Girardeau Hamiltonian is c = \infinity limit, where hard-core bosons are effectively free fermions - it flows to this fixed point for any c > 0).
Scaling interaction fails for attracive bosons.

Q: How does time flow on the contour?
A: Time flows only forwards - back is just taking bra.

Looking at corrections to long time - can use stationary phase approximation at large times. Attractive has saddle point + poles and  bound states.  xi = x/2t only t-dependence.

C2(x1,x2,t) -> C2(xi1,xi2).  Develop antipeaks in the C2(x1-x2) at zero, indicating fermionic behavior (bosons develop peak that sharpens in time).

Can do for any number of particles - always fermionic physics, with antidip, and then more structure at longer distances.

Q: Piers - origin of periodicity in 10 particle? A: Starting from periodic initial state (lattice)
Q: Bosons as fermions - wavefunction is antisymmetric?  A: Really hardcore bosons, just changing variables into fermions. (okay because 1D)

Time-evolution Renormalization Group:
time flow is RG flow - increasing time plays role of increasing bandwidth, since flows to c = +/- infinity in long-time limits.  Can argue that there are basins of attraction, so adding perturbations (short range interactions beyond delta-function), dominated by same fixed points - eg - Bose-Hubbard model (putting LL on the lattice) -> hard core bosons hopping on lattice in long time limit.

Conclusions:
Don't need spectrum of H or normalized eigenstates.  Just need theorem on initial state.  Takes into account bound states without summing over strings, and is mysterious.  Can completely calculate all asymptotics, and can compare to experiments, especially as results are universal.

Working on applying to more models, adding impurities, finite volume, density, temperatures.  Approach to steady state, testing dynamical RG hypothesis.

Big Questions:

- what drives thermalization of pure states? is it canonical typicality, entanglement entropy?
- general priciples out of equilibrium?
- what is univeral?

Questions:
- HBT results recovered for bosons and fermions?
A - not yet compared to experiments, but should be recovered.
Chubukov: large time asymptotics means what time scales compared to experiments?
A: time scales are milliseconds.  t > 1/coupling (and coupling is known)

Comment: von Neumann did work on thermalization - considered quantum ergodic thm, related to quantum central limit thm - system in long time limit goes to thermal limit.
A: not everyone accepts that this has been answered, but this is even more general and we have full control - not all states thermalize - they remember their initial conditions.

Piers: Vacuum is absence of bosons, and is eigenstate.  Could do for magnons against FM vacuum.  But what about nontrivial background, like AFM?
A: Can start with Neel state (not eigenstate), act with H and propagate - where does it go?  Or can take XXZ model, has phase transition so prepare as eigenstate of Ising model, propagate with XY model.