Nonequilibrium quantum dynamics in condensed matter:

excitons, chaos, and quantum walk

Takashi Oka (U-Tokyo)

previous talk

“Meson” and their “turbulent higher mode condensation”

Hashimoto, Kinoshita, Murata, TO 2014

Strong field physics in Condensed matter and Nuclear physics

hole density

T

superconductor

phase diagram of hadron (Fukushima-Hatsuda)

phase diagram of Hi Tc

Strong field physics in Condensed matter and Nuclear physics

ion collision pump probe exp.

Hirori, Tanaka et al. Nat. Com. 2011

THz laser pulses (now stronger than the Schwinger limit) strong E and B fields

Difference Movie

Wang et al. … N. Gedik Phys. Rev. Lett. 109, 127401 (2012)

I(E, kx, ky, t<0) data from N. Gedik (MIT)

Pump-probe technique

Time resolved ARPES (angle resolved photo emission spectroscopy)

Gedik@MIT group

Part I: Condensation of Excitons

Question: How do you obtain

from quantum mechanics?

Part I: Condensation of Excitons

Proposed phase diagram of the extended Hubbard model at half filling

Jeckelmann PRB 67 (2003)

Mott insulator

J: Hopping between lattice sites

extended Hubbard model

ij: nearest neighbor site

U: On-site Coulomb interaction V: long-range Coulomb interaction (although it is just neighbor site)

+U +2V

competition between U and V

Mott insulator (spin density wave)

Mott insulator (spin density wave)

Excitations from the Mott insulator

Mott insulator (spin density wave) doublon (- charge) hole (+ charge)

Exciton = doublon-hole boundstate

Exciton string (~QCD string?)

CDW droplet with a fractural structure (in higher dim.)

: arbitrary complex number

higher ``exciton” condensate ~

?

Lu, Sota, Matsueda, Bonča, Tohyama PRL 2012

U=10

Pulse Laser induced CDW (exciton condensation)

short pulse laser

charge-charge correlation

Maybe related to the higher meson condensation

Question: How do you obtain

from quantum dynamics?

Part II: chaos and quantum walk

Part II: chaos and quantum walk

Time independent Hamiltonian

fixed

How are cn determined?

Time dependent Hamiltonian

Spec H(B)

Question: Which are quantum chaotic?

Nakamura Thomas PRL61 ‘88

Spec H(B)

Question: Which are quantum chaotic?

Chaotic non-chaotic

Nakamura Thomas PRL61 ‘88

cf) classical chaos is distinguished by the Lyapunov exponent

Level repulsion (Wigner distribution)

Level crossing (Poisson distribution)

Classic case

Quench problem

Classic case

Quench problem

fluctuation growth

Production rate of B

Quantum case

slow Quench problem

Quantum case

Quench problem

Quantum case

Quench problem

distribution ~ “thermal state”?

If the system were non-chaotic …

Quench problem

pure state

②

③

vacuum decay rate/ Euler-Heisenberg Lagrangian (fidelity, Loschmidt echo)

② Oka, Aoki PRL 2005 Hashimoto, Oka 2013

Hashimoto, Kinoshita, Murata, Oka 2014

① Full dynamics

Oka, Arita, Aoki PRL 2003

(Gauge/gravity)

①

③

Oka, Aoki 2010, Oka 2012

production rate

φ two level approximation

Semenoff, Zarembo 2011 Sato, Yoshida 2013,..

Schwinger mechanism

②

③ Hashimoto, Kinoshita, Murata, Oka 2014

① Full dynamics

Oka, Arita, Aoki PRL 2003

(Gauge/gravity)

①

2. Quantum walk gives a rough sketch of the level dynamics

1. Excited states are multiple doublon-hole pairs ~ exciton string (~ higher meson state)

In the Hubbard model,

Long time behavior!

creation

creation

annihilation

The dynamics can be decomposed into 2✖2 unitary evolution

The tunneling probability is given by

Schwinger mechanism

Oka, Konno, Aoki PRL 2005

1D Quantum walk with a reflecting boundary!

introductory reviews; J. Kempe, Contemporary Physics 44, 307 (2003). Nayak et.al　quant-ph/0010117. 今野紀雄,「数理科学」 2004 年 6 月号

"量子ウォークの極限定理" in Japanese

[evolution rule]

two state (up, down) at each site n and time τ.

at the boundary

unitary matrix

evolution matrices

Properties of the distribution!

classical stochastic system!

quantum walk!

quantum interference …. leads to Anderson localization

main difference from classical system

mapping to a quantum walk!

evolution matrix

p is the Landau-Zener tunneling rate

quantum walk

creation

creation

annihilation

dielectric breakdown of Mott insulator

similar

Oka, Konno, Aoki PRL 2005

Path integral in energy space!

contribution from each path = product of P，Q matrices!

initial vector generally,!

transition amplitude (2×2 matrix)

Oka, Konno, Aoki PRL 2005

PQRS method and recurrence formula !recurrence formula

multiplication rule

＋

generating function of the wave function

Oka, Konno, Aoki PRL 2005

time evolution of the wave function!expand in z

asymptotic distribution

ground state

phase interference from various paths

Anderson localization in energy space =dynamical localization

Oka, Konno, Aoki PRL 2005

localization-delocalization transition !

p=0.01 p=0.2 p=0.4

electric field

δ関数 core

adiabatic evolution!（δfunction）

delocalized state!localized state!

Oka, Konno, Aoki PRL 2005

Higher meson condensation??!

Conclusion “Meson” and their “higher mode condensation”

Exciton string

1. Similarity with the exciton string/ CDW droplets

2. Localization-delocalization transition in the quantum walk may explain the dynamical higher mode condensation