Out-of-equilibrium quantum matter
Successes, challenges, and opportunities

Nano, Quantum and Materials Physics (NQMP) community day

Nijmegen, 24 June 2025

Jean-Sébastien Caux

Institute of Physics
University of Amsterdam

Plan of the talk


Out-of-equilibrium: why?

Why NOT?

  • Real systems equilibrate very rapidly
  • Experimentally: tough to control
  • Theoretically: can't do it - toolbox too limited

Why INDEED?

  • Real isolated systems can equilibrate slowly
  • Novel states & properties
  • Experimentally: frontier of current capabilities
  • Theoretically: the toughest challenge around?

Mechanical
motivation

Out of equilibrium: terminology

pat • pet • stroke
to touch or handle in a tender manner
jumble • muss • rumple
to undo the proper order or arrangement of
jiggle • joggle • jolt • wobble
to make a series of small irregular or violent movements
jerk • twitch • yank
to move or cause to move with a sharp quick motion
beat • pulse • throb
to expand and contract in a rhythmic manner
swish • wag • whisk
a quick jerky movement from side to side or up and down
bump • knock • ram
to come into usually forceful contact with something
slam • smack • thump • whack • bludgeon • clobber • thwack
to deliver a blow usually in a strong vigorous manner
batter • pelt • pound • pummel • wallop
to strike repeatedly

Out of equilibrium: what? how?

Probed
  • Neutron scattering
  • Bragg spectroscopy
  • Transport
  • Linear response
  • Traditional methods
  • CFT, Bosonization, ...
Pulsed
  • Magnetic resonance
  • Photoemission
  • Ultrafast spectr.
  • Free models
  • Numerics
  • Nonlinear response theory
Quenched
  • Feshbach resonances
  • Integrability
  • Hydrodynamics ($t\rightarrow \infty$)
  • GGE ($t = \infty$)
  • Quench Action ($\forall t$)
Driven
  • Masers/lasers
  • Resonators
  • Free models
  • Floquet theory
  • Numerics
  • Despair?

Out-of-equilibrium using Integrability

Pulsed
  • Split Fermi seas (Moses states)
  • Spin echo in quantum dots
  • Quasisolitons
Quenched
  • The super Tonks-Girardeau gas
  • Interaction quench in Richardson
  • Domain wall release in Heisenberg
  • Geometric quench
  • Interaction cutoff in Lieb-Liniger
  • Release of trapped Lieb-Liniger
  • BEC to Lieb-Liniger quench
  • Quantum Newton’s Cradle in TG
  • Néel to XXZ quench
  • Generalized hydrodynamics
Driven
  • Floquet driven spin chains

Quantum Gases

The Lieb-Liniger Model

$ H = \int_0^L dx \left[ \partial_x \Psi^{\dagger}(x) \partial_x \Psi(x) + c \Psi^{\dagger} (x) \Psi^{\dagger}(x) \Psi(x) \Psi(x) \right] $

Eigenstates: Bethe Ansatz

\[ \begin{align} \Psi_N(\{ x \} | \{ \lambda \}) &= \prod_{N \geq j_1 > j_2 \geq 1} sgn(x_{j_1} - x_{j_2}) sgn (\lambda_{j_1} - \lambda_{j_2}) \times \\ &\times \sum_{P \in \pi_N} (-1)^{[P]} e^{i \sum_{j=1}^N \lambda_{P_j} x_j + \frac{i}{2} \sum_{N \geq j_1 > j_2 \geq 1} sgn(x_{j_1} - x_{j_2}) \phi (\lambda_{P_{j_1}} - \lambda_{P_{j_2}})} \end{align} \]

in which $\phi(\lambda) \equiv 2\mbox{atan}(\lambda/c)$

The Lieb-Liniger Model

(Dynamical) Correlation functions

\[ {\cal S}^{a \bar{a}} (k, \omega) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k(j-j')} \int_{-\infty}^\infty dt e^{i\omega t} \langle \frac{1}{2} \left[ {\cal O}^a_j (t), ({\cal O}^a_{j'} (0))^\dagger \right] \rangle \]

Simplest examples:

  • Dynamical structure factor: ${\cal O}_j = \rho_j \equiv \Psi^\dagger(x_j) \Psi(x_j)$
  • One-body function (Green's): ${\cal O}_j = \Psi(x_j)$ or $\Psi^\dagger(x_j)$

Dynamical structure factor
from (algebraic) Bethe Ansatz

Abacus (post Achilles)

Abacus_Explorer.png

Dynamical structure factor
from (algebraic) Bethe Ansatz



The ABACUS algorithm


Dynamical structure factor from Bethe Ansatz

extension to finite T

M. Panfil and J-SC, PRA 89 (2014)

Lieb-Liniger using cold atoms (I)

Lieb-Liniger using cold atoms (II)

Let's

Interaction quench in Lieb-Liniger

We will consider a sudden switch of the interaction:

$ c (t) = \left\{ \begin{array}{ll} c_i & t < 0, \\ c_f & t > 0 \end{array} \right. $

starting in ground/(any eigen)state of $H(c_i)$ at $t < 0$

For many reasons, this is a complicated problem...

Exact solution using Quench Action

$ S^Q = L\int_0^\infty d\lambda \rho(\lambda) \log\left(\frac{\lambda^2}{c^2} \left( \frac{1}{4} + \frac{ \lambda^2}{c^2} \right) \right) + S^{YY} $

Saddle-point equation:

$ \ln \eta(\lambda) = g(\lambda) - h - \int_{-\infty}^\infty \frac{d\lambda'}{2\pi} K (\lambda - \lambda') \ln \left[ 1 + \eta^{-1}(\lambda') \right], $

with driving term $ g(\lambda) = \ln \left[ \frac{\lambda^2}{c^2} \left( \frac{\lambda^2}{c^2} + \frac{1}{4} \right) \right] $

Analytic solution for steady state:

$ \begin{align} \rho_{sp} (\lambda) &= -\frac{\gamma}{4\pi} \frac{1}{1 + a(\lambda)} \frac{\partial a(\lambda)}{\partial \gamma},\\ a(\lambda) &\equiv \frac{2\pi}{\frac{\lambda}{n} \sinh \frac{2\pi \lambda}{c}} I_{1 - 2i\frac{\lambda}{c}} \left(\frac{4}{\sqrt{\gamma}}\right) I_{1 + 2i\frac{\lambda}{c}} \left(\frac{4}{\sqrt{\gamma}}\right) \end{align} $

Exact solution using Quench Action

Steady state distribution: non-thermal
J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014

Generalized Hydrodynamics

Quenches from spatially inhomogeneous states

  • B. Bertini, M. Collura, J. De Nardis and M. Fagotti, PRL 117, 207201 (2016)
  • O. A. Castro-Alvaredo, B. Doyon and T. Yoshimura, PRX 6, 041065 (2016)
  • B. Doyon and T. Yoshimura, SciPost Phys. 2, 014 (2017)


After initial dephasings: 'hydrodynamic' evolution described by local GGE

$\partial_t \rho (\lambda) + \partial_x (v^{\tiny \mbox{eff}}(\lambda) \rho (\lambda)) = 0$

with effective velocities determined by the dressing equation

$v^{\tiny \mbox{eff}}(\lambda) = v^{\tiny \mbox{gr}}(\lambda) + \int d\lambda' \frac{\varphi(\lambda, \lambda')}{p'(\lambda)} \rho(\lambda') \left(v^{\tiny \mbox{eff}}(\lambda') - v^{\tiny \mbox{eff}}(\lambda)\right)$

Effective dynamics:
"flea gas" of scattering quasiparticles

Flea gas GHD
for the Quantum Newton's Cradle

JSC, B. Doyon, J. Dubail, R. Konik and T. Yoshimura, SciPost Phys. 6, 070 (2019)

Short time scales:

Flea gas GHD
for the Quantum Newton's Cradle

JSC, B. Doyon, J. Dubail, R. Konik and T. Yoshimura, SciPost Phys. 6, 070 (2019)

Longer time scales:

Generalized Hydrodynamics with space/time-dependent potentials

Phys. Rev. Lett. 123, 130602 (2019)

Adiabatic formation of bound states in the 1d Bose gas

Phys. Rev. B 103, 165121 (2021)

Observing Bethe strings in an attractive Bose gas far from equilibrium

Milena Horvath, Alvise Bastianello, Sudipta Dhar, Rebekka Koch, Yanliang Guo,
Jean-Sébastien Caux, Manuele Landini, and Hanns-Christoph Nägerl, arXiv:2505.10550

Observing Bethe strings in an attractive Bose gas far from equilibrium

Milena Horvath, Alvise Bastianello, Sudipta Dhar, Rebekka Koch, Yanliang Guo,
Jean-Sébastien Caux, Manuele Landini, and Hanns-Christoph Nägerl, arXiv:2505.10550

Quantum Spin Chains

The structure factor
of the Heisenberg chain

Quantum wake dynamics in Heisenberg

A. Scheie, P. Laurell, B. Lake, S. E. Nagler, M. B. Stone, JSC, D. A. Tennant
Nature Communications 13, 5796 (2022)

  • Inelastic neutron scattering on $K Cu F_3$
  • Really accurate data in momentum/frequency space enabling...
  • Fourier transform to real space / time
  • Look specifically at high-energy features

Quantum wake dynamics in Heisenberg

A. Scheie, P. Laurell, B. Lake, S. E. Nagler, M. B. Stone, JSC, D. A. Tennant
Nature Communications 13, 5796 (2022)

  • Quantum wake:
    a local, coherent, long-lived, quasiperiodically oscillating magnetic state emerging out of the distillation of propagating excitations following a local quantum quench

Quantum wake dynamics in Heisenberg

Quantum wake dynamics in Heisenberg

Quantum wake dynamics in Heisenberg

Selectively (patch-wise) Fourier transforming: $k=\pi/2$ (mod $2\pi$) states are responsible for the quantum wake dynamics

Beyond the reach of any low-energy method

Not mentioned here

  • Renormalization from integrability
  • (Nonlinear) Luttinger liquids
  • Lessons for field theory
  • "Bethe liquid theory"
  • Quasisolitons in quantum gases

Challenges and Opportunities

  • Detailed phenomenology for quantum gases
  • Transport: (sub/super)diffusion
  • Ultrafast magnetism
  • Central spin / NV centers
  • Floquet dynamics
  • Driven dissipative systems
  • Quantum circuits

Thanks!