(Quantum) chaos theory and statistical physics
far from equilibrium:
Introducing the group for Non-equilibrium quantum and statistical physics
Tomaž Prosen
Department of physics, Faculty of mathematics and physics,
University of Ljubljana
July, 2011
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
People:
dr. Tomaž Prosen, professor, head of the group
dr. Marko Žnudarič, assistant professor, researcher
dr. Martin Horvat, researcher
Bojan Žunkovič, PhD student
Enej Ilievski, PhD student
Simon Jesenko, PhD student
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Research themes
We use methods of theoretical and mathematical physics in the intersection
among the following fields of contemporary physics:
(Hard) condensed matter theory
Non-equilibrium statistical mechanics
Dynamical systems (Nonlinear dynamics, chaos theory)
Quantum information theory
Our group is also a part of the bigger program group (P1-0044) “Condensed
matter theory and statistical physics" shared between Josef Stefan Institute
and the Faculty of Math.& Phys. UL
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Topics of main current research interest:
Fundamental:
Non-equilibrium quantum transport in low dimensional interacting systems
Open quantum many-body system – Lindblad master equation approach:
Its exact, approximate, and numerical solutions
(density-matrix-renormalization group)
Non-equilibrium (quantum) phase transitions
Quantum maps, quantum chaos, random matrix theory:
wave-dynamics, wave-chaos, PT-symmetric Hamiltonians
Quantum chaos in many-body systems
CHAPTER 5. XY
CHAIN FAR FROM
Quantum Information
Theory
andEQUILIBRIUM
Random Matrix Theory
Applied:
Figure 5.1: A schematic representation of a quantum spin chain coupled to two reservoirs with different “temperatures”.
5.1
Lindblad Master equation
Controlling and The
rectifying
heat
flow
lattices
evolution of the System
is given
in termsin
of a quantum/classical
time-independent generator of infinitesimal time translations,
(d/dt)ρ = L̂(ρ)
(5.1)
Thermo-electric,where
thermo-magnetic,
or thermo-chemical
heat engines, and
the generator L̂ must obey usual requirements when applied to a density operator ρ, especially to preserve its trace and positivity.
optimizing their efficiency
fromL̂ isdynamical
systems
perspective
The origin of the generator
illustrated by considering
the Liouville equation
for the density operator R for the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal time translations assuming that the thermal bath
Tomaž Prosen
Non-equilibrium quantum
is in a thermal equilibrium. Finally, for sufficiently large times the generator of time
and statistical physics group
Topics of main current research interest:
Fundamental:
Non-equilibrium quantum transport in low dimensional interacting systems
Open quantum many-body system – Lindblad master equation approach:
Its exact, approximate, and numerical solutions
(density-matrix-renormalization group)
Non-equilibrium (quantum) phase transitions
Quantum maps, quantum chaos, random matrix theory:
wave-dynamics, wave-chaos, PT-symmetric Hamiltonians
Quantum chaos in many-body systems
CHAPTER 5. XY
CHAIN FAR FROM
Quantum Information
Theory
andEQUILIBRIUM
Random Matrix Theory
Applied:
Figure 5.1: A schematic representation of a quantum spin chain coupled to two reservoirs with different “temperatures”.
5.1
Lindblad Master equation
Controlling and The
rectifying
heat
flow
lattices
evolution of the System
is given
in termsin
of a quantum/classical
time-independent generator of infinitesimal time translations,
(d/dt)ρ = L̂(ρ)
(5.1)
Thermo-electric,where
thermo-magnetic,
or thermo-chemical
heat engines, and
the generator L̂ must obey usual requirements when applied to a density operator ρ, especially to preserve its trace and positivity.
optimizing their efficiency
fromL̂ isdynamical
systems
perspective
The origin of the generator
illustrated by considering
the Liouville equation
for the density operator R for the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal time translations assuming that the thermal bath
Tomaž Prosen
Non-equilibrium quantum
is in a thermal equilibrium. Finally, for sufficiently large times the generator of time
and statistical physics group
Topics of main current research interest:
Fundamental:
Non-equilibrium quantum transport in low dimensional interacting systems
Open quantum many-body system – Lindblad master equation approach:
Its exact, approximate, and numerical solutions
(density-matrix-renormalization group)
Non-equilibrium (quantum) phase transitions
Quantum maps, quantum chaos, random matrix theory:
wave-dynamics, wave-chaos, PT-symmetric Hamiltonians
Quantum chaos in many-body systems
CHAPTER 5. XY
CHAIN FAR FROM
Quantum Information
Theory
andEQUILIBRIUM
Random Matrix Theory
Applied:
Figure 5.1: A schematic representation of a quantum spin chain coupled to two reservoirs with different “temperatures”.
5.1
Lindblad Master equation
Controlling and The
rectifying
heat
flow
lattices
evolution of the System
is given
in termsin
of a quantum/classical
time-independent generator of infinitesimal time translations,
(d/dt)ρ = L̂(ρ)
(5.1)
Thermo-electric,where
thermo-magnetic,
or thermo-chemical
heat engines, and
the generator L̂ must obey usual requirements when applied to a density operator ρ, especially to preserve its trace and positivity.
optimizing their efficiency
fromL̂ isdynamical
systems
perspective
The origin of the generator
illustrated by considering
the Liouville equation
for the density operator R for the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal time translations assuming that the thermal bath
Tomaž Prosen
Non-equilibrium quantum
is in a thermal equilibrium. Finally, for sufficiently large times the generator of time
and statistical physics group
Topics of main current research interest:
Fundamental:
Non-equilibrium quantum transport in low dimensional interacting systems
Open quantum many-body system – Lindblad master equation approach:
Its exact, approximate, and numerical solutions
(density-matrix-renormalization group)
Non-equilibrium (quantum) phase transitions
Quantum maps, quantum chaos, random matrix theory:
wave-dynamics, wave-chaos, PT-symmetric Hamiltonians
Quantum chaos in many-body systems
CHAPTER 5. XY
CHAIN FAR FROM
Quantum Information
Theory
andEQUILIBRIUM
Random Matrix Theory
Applied:
Figure 5.1: A schematic representation of a quantum spin chain coupled to two reservoirs with different “temperatures”.
5.1
Lindblad Master equation
Controlling and The
rectifying
heat
flow
lattices
evolution of the System
is given
in termsin
of a quantum/classical
time-independent generator of infinitesimal time translations,
(d/dt)ρ = L̂(ρ)
(5.1)
Thermo-electric,where
thermo-magnetic,
or thermo-chemical
heat engines, and
the generator L̂ must obey usual requirements when applied to a density operator ρ, especially to preserve its trace and positivity.
optimizing their efficiency
fromL̂ isdynamical
systems
perspective
The origin of the generator
illustrated by considering
the Liouville equation
for the density operator R for the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal time translations assuming that the thermal bath
Tomaž Prosen
Non-equilibrium quantum
is in a thermal equilibrium. Finally, for sufficiently large times the generator of time
and statistical physics group
Topics of main current research interest:
Fundamental:
Non-equilibrium quantum transport in low dimensional interacting systems
Open quantum many-body system – Lindblad master equation approach:
Its exact, approximate, and numerical solutions
(density-matrix-renormalization group)
Non-equilibrium (quantum) phase transitions
Quantum maps, quantum chaos, random matrix theory:
wave-dynamics, wave-chaos, PT-symmetric Hamiltonians
Quantum chaos in many-body systems
CHAPTER 5. XY
CHAIN FAR FROM
Quantum Information
Theory
andEQUILIBRIUM
Random Matrix Theory
Applied:
Figure 5.1: A schematic representation of a quantum spin chain coupled to two reservoirs with different “temperatures”.
5.1
Lindblad Master equation
Controlling and The
rectifying
heat
flow
lattices
evolution of the System
is given
in termsin
of a quantum/classical
time-independent generator of infinitesimal time translations,
(d/dt)ρ = L̂(ρ)
(5.1)
Thermo-electric,where
thermo-magnetic,
or thermo-chemical
heat engines, and
the generator L̂ must obey usual requirements when applied to a density operator ρ, especially to preserve its trace and positivity.
optimizing their efficiency
fromL̂ isdynamical
systems
perspective
The origin of the generator
illustrated by considering
the Liouville equation
for the density operator R for the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal time translations assuming that the thermal bath
Tomaž Prosen
Non-equilibrium quantum
is in a thermal equilibrium. Finally, for sufficiently large times the generator of time
and statistical physics group
Topics of main current research interest:
Fundamental:
Non-equilibrium quantum transport in low dimensional interacting systems
Open quantum many-body system – Lindblad master equation approach:
Its exact, approximate, and numerical solutions
(density-matrix-renormalization group)
Non-equilibrium (quantum) phase transitions
Quantum maps, quantum chaos, random matrix theory:
wave-dynamics, wave-chaos, PT-symmetric Hamiltonians
Quantum chaos in many-body systems
CHAPTER 5. XY
CHAIN FAR FROM
Quantum Information
Theory
andEQUILIBRIUM
Random Matrix Theory
Applied:
Figure 5.1: A schematic representation of a quantum spin chain coupled to two reservoirs with different “temperatures”.
5.1
Lindblad Master equation
Controlling and The
rectifying
heat
flow
lattices
evolution of the System
is given
in termsin
of a quantum/classical
time-independent generator of infinitesimal time translations,
(d/dt)ρ = L̂(ρ)
(5.1)
Thermo-electric,where
thermo-magnetic,
or thermo-chemical
heat engines, and
the generator L̂ must obey usual requirements when applied to a density operator ρ, especially to preserve its trace and positivity.
optimizing their efficiency
fromL̂ isdynamical
systems
perspective
The origin of the generator
illustrated by considering
the Liouville equation
for the density operator R for the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal time translations assuming that the thermal bath
Tomaž Prosen
Non-equilibrium quantum
is in a thermal equilibrium. Finally, for sufficiently large times the generator of time
and statistical physics group
Topics of main current research interest:
Fundamental:
Non-equilibrium quantum transport in low dimensional interacting systems
Open quantum many-body system – Lindblad master equation approach:
Its exact, approximate, and numerical solutions
(density-matrix-renormalization group)
Non-equilibrium (quantum) phase transitions
Quantum maps, quantum chaos, random matrix theory:
wave-dynamics, wave-chaos, PT-symmetric Hamiltonians
Quantum chaos in many-body systems
CHAPTER 5. XY
CHAIN FAR FROM
Quantum Information
Theory
andEQUILIBRIUM
Random Matrix Theory
Applied:
Figure 5.1: A schematic representation of a quantum spin chain coupled to two reservoirs with different “temperatures”.
5.1
Lindblad Master equation
Controlling and The
rectifying
heat
flow
lattices
evolution of the System
is given
in termsin
of a quantum/classical
time-independent generator of infinitesimal time translations,
(d/dt)ρ = L̂(ρ)
(5.1)
Thermo-electric,where
thermo-magnetic,
or thermo-chemical
heat engines, and
the generator L̂ must obey usual requirements when applied to a density operator ρ, especially to preserve its trace and positivity.
optimizing their efficiency
fromL̂ isdynamical
systems
perspective
The origin of the generator
illustrated by considering
the Liouville equation
for the density operator R for the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal time translations assuming that the thermal bath
Tomaž Prosen
Non-equilibrium quantum
is in a thermal equilibrium. Finally, for sufficiently large times the generator of time
and statistical physics group
Topics of main current research interest:
Fundamental:
Non-equilibrium quantum transport in low dimensional interacting systems
Open quantum many-body system – Lindblad master equation approach:
Its exact, approximate, and numerical solutions
(density-matrix-renormalization group)
Non-equilibrium (quantum) phase transitions
Quantum maps, quantum chaos, random matrix theory:
wave-dynamics, wave-chaos, PT-symmetric Hamiltonians
Quantum chaos in many-body systems
CHAPTER 5. XY
CHAIN FAR FROM
Quantum Information
Theory
andEQUILIBRIUM
Random Matrix Theory
Applied:
Figure 5.1: A schematic representation of a quantum spin chain coupled to two reservoirs with different “temperatures”.
5.1
Lindblad Master equation
Controlling and The
rectifying
heat
flow
lattices
evolution of the System
is given
in termsin
of a quantum/classical
time-independent generator of infinitesimal time translations,
(d/dt)ρ = L̂(ρ)
(5.1)
Thermo-electric,where
thermo-magnetic,
or thermo-chemical
heat engines, and
the generator L̂ must obey usual requirements when applied to a density operator ρ, especially to preserve its trace and positivity.
optimizing their efficiency
fromL̂ isdynamical
systems
perspective
The origin of the generator
illustrated by considering
the Liouville equation
for the density operator R for the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal time translations assuming that the thermal bath
Tomaž Prosen
Non-equilibrium quantum
is in a thermal equilibrium. Finally, for sufficiently large times the generator of time
and statistical physics group
Topics of main current research interest:
Fundamental:
Non-equilibrium quantum transport in low dimensional interacting systems
Open quantum many-body system – Lindblad master equation approach:
Its exact, approximate, and numerical solutions
(density-matrix-renormalization group)
Non-equilibrium (quantum) phase transitions
Quantum maps, quantum chaos, random matrix theory:
wave-dynamics, wave-chaos, PT-symmetric Hamiltonians
Quantum chaos in many-body systems
CHAPTER 5. XY
CHAIN FAR FROM
Quantum Information
Theory
andEQUILIBRIUM
Random Matrix Theory
Applied:
Figure 5.1: A schematic representation of a quantum spin chain coupled to two reservoirs with different “temperatures”.
5.1
Lindblad Master equation
Controlling and The
rectifying
heat
flow
lattices
evolution of the System
is given
in termsin
of a quantum/classical
time-independent generator of infinitesimal time translations,
(d/dt)ρ = L̂(ρ)
(5.1)
Thermo-electric,where
thermo-magnetic,
or thermo-chemical
heat engines, and
the generator L̂ must obey usual requirements when applied to a density operator ρ, especially to preserve its trace and positivity.
optimizing their efficiency
fromL̂ isdynamical
systems
perspective
The origin of the generator
illustrated by considering
the Liouville equation
for the density operator R for the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal time translations assuming that the thermal bath
Tomaž Prosen
Non-equilibrium quantum
is in a thermal equilibrium. Finally, for sufficiently large times the generator of time
and statistical physics group
A pedestrian path:
from one, two, to (infinitely) many dynamical degrees of freedom
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Chaotic versus integrable maps: trajectories versus ensembles
(qt+1 , pt+1 ) = F (qt , pt )
Standard map
Suris map
Triangle map
t!7
t!5
t!3
Perturbed cat
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Chaos and complexity: transport in Fourier space
Standard map
Suris map
Triangle map
Ky!154
Ky!164
Ky!410
Kx!2867
Kx!164
Ky!164
Kx!8192
Kx!205
Kx!205
Ky!410
Kx!1587
Ky!4096
Ky!1720
t!7
Kx!429
Kx!164
Ky!410
Kx!308
Ky!794
Ky!258
t!5
Kx!64
Ky!164
t!3
Perturbed cat
Ky!39
Chaotic maps have densities which diffuse exponentially fast in Fourier space.
Kx!164
Kx!205
T.P., Complexity and nonseparability of classical Liouvillian dynamics, Phys. Rev. E 83, 031124 (2011).
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Open problem: Deterministic diffusion and mixing in non-chaotic maps
Triangle map
qt+1
=
qt + pt (mod 2)
pt+1
=
pt + α sgnqt+1 + β (mod 2)
on (q, p) ∈ [−1, 1] × [−1, 1]
M. Horvat, M. Degli Esposti, S. Isola, T. Prosen and L.
Bunimovich, Physica D 238, 395 (2009).
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Quantum chaos: playing billiards
Integrable dynamics:
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Chaos and double-slit experiment
Numerical experiment (Giulio Casati and T.P. Phys. Rev. A 72, 032111 (2005))
“Leaking of quantum particles/waves through two slits inside a regular or
chaotic billiard.
a
a
s
s
Λ
l
Λ
l
screen
screen
absorber
absorber
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
HaL
HbL
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
What about changing gears and going to many-body systems?
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Universality in spectral statistics of quantum chaotic many body systems
Quantum Chaos Conjecture (Berry 1977, Casati, Guarneri, Vals-Griz 1980,
Bohigas, Giannoni, Schmit 1984):
Spectral correlations (and some other statistical properties of spectra and
eigenfunctions) of - even very simple - quantum systems, which are chaotic
in the classical limit, can be described by universal (no free parameter)
ensembles of Gausssian random matrices
0.8
0.6
p(s)
1.6
non-integrable
0.4
0.8
0.2
0.4
0
0
1
2
s
3
integrable
1.2
4
0
0
1
2
s
3
4
Is there a "quantum chaos conjecture" for many body quantum systems which
do not possess a classical limit?
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Case study: Kicked Ising Chain
There were many results reporting random matrix statistics on non-integrable
strongly correlated quantum systems (e.g. Montambaux et al 1993).
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Case study: Kicked Ising Chain
There were many results reporting random matrix statistics on non-integrable
strongly correlated quantum systems (e.g. Montambaux et al 1993).
Here we describe a recent detailed study (C. Pineda and T.P. PRE 2007) of
quasi-energy level statistics in a non-integrable regime of
Kicked Ising spin 1/2 chain (T.P. PTPS 2000, T.P. PRE 2002):
(
)
L−1
X
X
z z
x
z
H(t) =
Jσj σj+1 + (hx σj + hz σj )
δ(t − m)
m∈Z
j=0
UFloquet
« Y
„ Z 1+
`
´
`
z ´
exp −i(hx σjx + hz σjz ) exp −iJσjz σj+1
= T exp −i dt 0 H(t 0 ) =
0+
j
where [σjα , σkβ ] = 2iεαβγ σjγ δjk .
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Case study: Kicked Ising Chain
There were many results reporting random matrix statistics on non-integrable
strongly correlated quantum systems (e.g. Montambaux et al 1993).
Here we describe a recent detailed study (C. Pineda and T.P. PRE 2007) of
quasi-energy level statistics in a non-integrable regime of
Kicked Ising spin 1/2 chain (T.P. PTPS 2000, T.P. PRE 2002):
(
)
L−1
X
X
z z
x
z
H(t) =
Jσj σj+1 + (hx σj + hz σj )
δ(t − m)
m∈Z
j=0
UFloquet
« Y
„ Z 1+
`
´
`
z ´
exp −i(hx σjx + hz σjz ) exp −iJσjz σj+1
= T exp −i dt 0 H(t 0 ) =
0+
j
where [σjα , σkβ ] = 2iεαβγ σjγ δjk .
The model is completely integrable in terms of Jordan-Wigner transformation if
hx = 0 (longitudinal field)
hz = 0 (transverse field)
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Case study: Kicked Ising Chain
There were many results reporting random matrix statistics on non-integrable
strongly correlated quantum systems (e.g. Montambaux et al 1993).
Here we describe a recent detailed study (C. Pineda and T.P. PRE 2007) of
quasi-energy level statistics in a non-integrable regime of
Kicked Ising spin 1/2 chain (T.P. PTPS 2000, T.P. PRE 2002):
(
)
L−1
X
X
z z
x
z
H(t) =
Jσj σj+1 + (hx σj + hz σj )
δ(t − m)
m∈Z
j=0
UFloquet
« Y
„ Z 1+
`
´
`
z ´
exp −i(hx σjx + hz σjz ) exp −iJσjz σj+1
= T exp −i dt 0 H(t 0 ) =
0+
j
where [σjα , σkβ ] = 2iεαβγ σjγ δjk .
The model is completely integrable in terms of Jordan-Wigner transformation if
hx = 0 (longitudinal field)
hz = 0 (transverse field)
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Case study: Kicked Ising Chain
There were many results reporting random matrix statistics on non-integrable
strongly correlated quantum systems (e.g. Montambaux et al 1993).
Here we describe a recent detailed study (C. Pineda and T.P. PRE 2007) of
quasi-energy level statistics in a non-integrable regime of
Kicked Ising spin 1/2 chain (T.P. PTPS 2000, T.P. PRE 2002):
(
)
L−1
X
X
z z
x
z
H(t) =
Jσj σj+1 + (hx σj + hz σj )
δ(t − m)
m∈Z
j=0
UFloquet
« Y
„ Z 1+
`
´
`
z ´
exp −i(hx σjx + hz σjz ) exp −iJσjz σj+1
= T exp −i dt 0 H(t 0 ) =
0+
j
where [σjα , σkβ ] = 2iεαβγ σjγ δjk .
The model is completely integrable in terms of Jordan-Wigner transformation if
hx = 0 (longitudinal field)
hz = 0 (transverse field)
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Quasi-energy level statistics
Fix J = 0.7, hx = 0.9, hz = 0.9, s.t. KI is (strongly) non-integrable.
Diagonalize UFloquet |ni = exp(−iϕn )|ni. For each conserved total momentum
K quantum number, we find N ∼ 2L /L levels, normalized to mean level
spacing as sn = (N /2π)ϕn .
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Quasi-energy level statistics
Fix J = 0.7, hx = 0.9, hz = 0.9, s.t. KI is (strongly) non-integrable.
Diagonalize UFloquet |ni = exp(−iϕn )|ni. For each conserved total momentum
K quantum number, we find N ∼ 2L /L levels, normalized to mean level
spacing as sn = (N /2π)ϕn .
N(s) = #{sn < s} = Nsmooth (s) + Nfluct (s)
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Quasi-energy level statistics
Fix J = 0.7, hx = 0.9, hz = 0.9, s.t. KI is (strongly) non-integrable.
Diagonalize UFloquet |ni = exp(−iϕn )|ni. For each conserved total momentum
K quantum number, we find N ∼ 2L /L levels, normalized to mean level
spacing as sn = (N /2π)ϕn .
N(s) = #{sn < s} = Nsmooth (s) + Nfluct (s)
For kicked quantum quantum systems spectra are expected to be statistically
uniformly dense
Nsmooth (s) = s
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Quasi-energy level statistics
Fix J = 0.7, hx = 0.9, hz = 0.9, s.t. KI is (strongly) non-integrable.
Diagonalize UFloquet |ni = exp(−iϕn )|ni. For each conserved total momentum
K quantum number, we find N ∼ 2L /L levels, normalized to mean level
spacing as sn = (N /2π)ϕn .
N(s) = #{sn < s} = Nsmooth (s) + Nfluct (s)
For kicked quantum quantum systems spectra are expected to be statistically
uniformly dense
Nsmooth (s) = s
For classically chaotic systems, statistical distribution of mode fluctuations
Nfluct (s) has been predicted to be Gaussian (Aurich, Bäcker, Steiner 1997).
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Mode fluctuations
We find perfect agreement with Gaussian mode fluctuations for KI chain.
n
dP/dNfluct
Nfluct
0
6000
12000
2
0
!2
0.4
0.3
0.2
0.1
0.0
!3
!2
!1
0
1
2
3
Nfluct
We plot the mode fluctuation Nfluct as a function of the level number n (upper
panel) and its normalized distribution (lower panel) for an example of a KI
spectrum with L = 18 and K = 6 (N=14599). Gaussian fit: χ2 = 102.46 and
100 equal size bins.
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Long-range statistics: spectral form factor
Spectral form factor K2 (τ ) is for nonzero integer t defined as
˛2
˛
˛
˛
˛
1 ˛˛
1 ˛X −iϕn t ˛
t ˛2
K2 (t/N ) =
tr U
=
e
˛ .
˛
˛
N
N ˛
n
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Long-range statistics: spectral form factor
Spectral form factor K2 (τ ) is for nonzero integer t defined as
˛2
˛
˛
˛
˛
1 ˛˛
1 ˛X −iϕn t ˛
t ˛2
K2 (t/N ) =
tr U
=
e
˛ .
˛
˛
N
N ˛
n
In non-integrable systems with a chaotic classical lomit, form factor has two
regimes:
universal described by RMT,
non-universal described by short classical periodic orbits.
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Long-range statistics: spectral form factor
Note that for kicked systems, Heisenberg integer time τH = N
1
K2
0.8
0.6
0.02
0.4
0.2
0.25
0.5
!0.02
0.5
1
1.5
0.75
1
1.25 1.5 1.75
t/τH
2
We show the behavior of the form factor for L = 18 qubits. We perform
averaging over short ranges of time (τH /25). The results for each of the
K -spaces are shown in colors. The average over the different spaces as well as
the theoretical COE(N) curve is plotted as a black and red curve, respectively.
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Quantum chaos and non-equilibrium statistical mechanics:
Decay of time correlations
Temporal correlation of an extensive traceless observable A:
CA (t) = lim
L→∞
1
tr A(0)A(t),
L2L
A(t) = U −t AU t
Average correlator
DA = lim
T →∞
T −1
1 X
CA (t)
T t=0
signals quantum ergodicity if DA = 0
Quantum chaos regime in KI chain is compatible with exponential decay of
correlations. For integrable, and weakly non-integrable cases, though, we
find saturation of temporal correlations D 6= 0.
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Quantum chaos and non-equilibrium statistical mechanics:
Decay of time correlations
Temporal correlation of an extensive traceless observable A:
CA (t) = lim
L→∞
1
tr A(0)A(t),
L2L
A(t) = U −t AU t
Average correlator
DA = lim
T →∞
T −1
1 X
CA (t)
T t=0
signals quantum ergodicity if DA = 0
Quantum chaos regime in KI chain is compatible with exponential decay of
correlations. For integrable, and weakly non-integrable cases, though, we
find saturation of temporal correlations D 6= 0.
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Quantum chaos and non-equilibrium statistical mechanics:
Decay of time correlations
Temporal correlation of an extensive traceless observable A:
CA (t) = lim
L→∞
1
tr A(0)A(t),
L2L
A(t) = U −t AU t
Average correlator
DA = lim
T →∞
T −1
1 X
CA (t)
T t=0
signals quantum ergodicity if DA = 0
Quantum chaos regime in KI chain is compatible with exponential decay of
correlations. For integrable, and weakly non-integrable cases, though, we
find saturation of temporal correlations D 6= 0.
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
<M(t)M>/L
Decay of time correlatons in KI chain
Three typical cases of parameters:
(c) J = 1, hx = 1.4, hz = 1.4
("quantum chaotic").
0.4
0.2
(b)
10-1
10-2
0.6
10-3
0.4
10
20
30
t
DM/L=0.293
0.2
0
L=20
L=16
L=12
0.25exp(-t/6)
(c)
0.1
-2
10
10-3
0
Tomaž Prosen
|<M(t)M>-DM|/L
0
<M(t)M>/L
(b) J = 1, hx = 1.4, hz = 0.4
(intermediate).
0.6
0.8
|<M(t)M>|/L
(a) J = 1, hx = 1.4, hz = 0.0
(completely integrable).
DM/L=0.485
(a)
0.8
5
10
15
20
25
t
30
35
40
45
50
Non-equilibrium quantum and statistical physics group
Loschmidt echo and decay of fidelity
0.1
|F(t)|
Decay of correlations is closely related to
fidelity decay F (t) = hU −t Uδt (t)i due to
perturbed evolution Uδ = U exp(−iδA)
(Prosen PRE 2002) e.g. in a linear response approximation:
-2
10
(a)
!’=0.0025
!’=0.005
!’=0.01
10-3
δ2
2
C (t 0 − t 00 )
t 0 ,t 00 =1
0.1
|F(t)|
F (t) = 1 −
t
X
!’=0.0025
(b)
!’=0.005
!’=0.01
10-2
-3
10
(a) J = 1, hx = 1.4, hz = 0.0
(completely integrable).
(c) J = 1, hx = 1.4, hz = 1.4
("quantum chaotic").
|F(t)|
(b) J = 1, hx = 1.4, hz = 0.4
(intermediate).
0.1
!’=0.02
(c)
-2
!’=0.01
!’=0.04
10
L=20
L=16
10-3 L=12
theory
-4
10
0
50
100
150
200
t
250
300
350
400
REVIEWED IN: T. Gorin, T. P. , T H. Seligman and
M. Žnidarič: Physics Reports 435, 33-156 (2006)
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Exact (analytical and numerical) treatment of large open quantum systems
CHAPTER 5. XY CHAIN FAR FROM EQUILIBRIUM
Figure
5.1: A
representation
a quantum
spin chain coupled to two reserToy
models
of schematic
interacting
Heisenbergofspin
1/2 chains:
voirs with different “temperatures”.
XY spin chain with transverse magnetic field
5.1
n−1 „ equation
Lindblad Master
X
1+γ
« X
n
1−γ y y
σj σj+1 +
hσjz
The evolution of the System is2given in terms of2a time-independent generator of inx
σjx σj+1
+
H=
j=1
finitesimal time translations,
Anisotropic XXZ spin chain (d/dt)ρ = L̂(ρ)
j=1
(5.1)
where the generator L̂ must obey usual requirements when applied to a density operan−1
X
` and
tor ρ, especially to preserve its
trace
x x positivity.
y
z ´
H
=
σj+1 + σjyby
σj+1
+ ∆σjz σj+1
The origin of the generator L̂ σ
isj illustrated
considering
the Liouville equation
for the density operator R forj=1
the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal
time translations
assuming
the thermal
Tomaž Prosen
Non-equilibrium
quantumthat
and statistical
physicsbath
group
Exact (analytical and numerical) treatment of large open quantum systems
CHAPTER 5. XY CHAIN FAR FROM EQUILIBRIUM
Figure
5.1: A
representation
a quantum
spin chain coupled to two reserToy
models
of schematic
interacting
Heisenbergofspin
1/2 chains:
voirs with different “temperatures”.
XY spin chain with transverse magnetic field
5.1
n−1 „ equation
Lindblad Master
X
1+γ
« X
n
1−γ y y
σj σj+1 +
hσjz
The evolution of the System is2given in terms of2a time-independent generator of inx
σjx σj+1
+
H=
j=1
finitesimal time translations,
Anisotropic XXZ spin chain (d/dt)ρ = L̂(ρ)
j=1
(5.1)
where the generator L̂ must obey usual requirements when applied to a density operan−1
X
` and
tor ρ, especially to preserve its
trace
x x positivity.
y
z ´
H
=
σj+1 + σjyby
σj+1
+ ∆σjz σj+1
The origin of the generator L̂ σ
isj illustrated
considering
the Liouville equation
for the density operator R forj=1
the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal
time translations
assuming
the thermal
Tomaž Prosen
Non-equilibrium
quantumthat
and statistical
physicsbath
group
Exact (analytical and numerical) treatment of large open quantum systems
CHAPTER 5. XY CHAIN FAR FROM EQUILIBRIUM
Figure
5.1: A
representation
a quantum
spin chain coupled to two reserToy
models
of schematic
interacting
Heisenbergofspin
1/2 chains:
voirs with different “temperatures”.
XY spin chain with transverse magnetic field
5.1
n−1 „ equation
Lindblad Master
X
1+γ
« X
n
1−γ y y
σj σj+1 +
hσjz
The evolution of the System is2given in terms of2a time-independent generator of inx
σjx σj+1
+
H=
j=1
finitesimal time translations,
Anisotropic XXZ spin chain (d/dt)ρ = L̂(ρ)
j=1
(5.1)
where the generator L̂ must obey usual requirements when applied to a density operan−1
X
` and
tor ρ, especially to preserve its
trace
x x positivity.
y
z ´
H
=
σj+1 + σjyby
σj+1
+ ∆σjz σj+1
The origin of the generator L̂ σ
isj illustrated
considering
the Liouville equation
for the density operator R forj=1
the Universe, (d/dt) R = i[ R, HU ] = L̂U R from which
the state of the System itself may be obtained by tracing over all degrees of freedom
!
"
in the environment, ρ = tr E R and thus ρ(t) = tr E etL̂U R(0) . From there, one can
derive the generator of infinitesimal
time translations
assuming
the thermal
Tomaž Prosen
Non-equilibrium
quantumthat
and statistical
physicsbath
group
Open Many-Body Quantum Systems, method I:
Quantization in the Liouville-Fock space of density operators
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Analytical solution for quasi-free fermionic systems
Consider a general solution of the Lindblad equation:
”
X“
dρ
2Lµ ρL†µ − {L†µ Lµ , ρ}
= L̂ρ := −i[H, ρ] +
dt
µ
for a general quadratic system of n fermions, or n qubits (spins 1/2)
H=
2n
X
wj Hjk wk = w · H w
Lµ =
2n
X
lµ,j wj = l µ · w
j=1
j,k=1
where wj , j = 1, 2, . . . , 2n, are abstract Hermitian Majorana operators
{wj , wk } = 2δj,k
Tomaž Prosen
j, k = 1, 2, . . . , 2n
Non-equilibrium quantum and statistical physics group
Analytical solution for quasi-free fermionic systems
Consider a general solution of the Lindblad equation:
”
X“
dρ
2Lµ ρL†µ − {L†µ Lµ , ρ}
= L̂ρ := −i[H, ρ] +
dt
µ
for a general quadratic system of n fermions, or n qubits (spins 1/2)
H=
2n
X
wj Hjk wk = w · H w
Lµ =
2n
X
lµ,j wj = l µ · w
j=1
j,k=1
where wj , j = 1, 2, . . . , 2n, are abstract Hermitian Majorana operators
{wj , wk } = 2δj,k
j, k = 1, 2, . . . , 2n
Two physical realizations:
†
†
canonical fermions cm , w2m−1 = cm + cm
, w2m = i(cm − cm
), m = 1, . . . , n.
spins 1/2 with canonical Pauli operators ~σm , m = 1, . . . , n,
Y z
Y z
y
w2m = σm
σm0
w2m−1 = σjx
σm 0
m0 <m
Tomaž Prosen
m0 <m
Non-equilibrium quantum and statistical physics group
Analytical solution for quasi-free fermionic systems
Consider a general solution of the Lindblad equation:
”
X“
dρ
2Lµ ρL†µ − {L†µ Lµ , ρ}
= L̂ρ := −i[H, ρ] +
dt
µ
for a general quadratic system of n fermions, or n qubits (spins 1/2)
H=
2n
X
wj Hjk wk = w · H w
Lµ =
2n
X
lµ,j wj = l µ · w
j=1
j,k=1
where wj , j = 1, 2, . . . , 2n, are abstract Hermitian Majorana operators
{wj , wk } = 2δj,k
j, k = 1, 2, . . . , 2n
Two physical realizations:
†
†
canonical fermions cm , w2m−1 = cm + cm
, w2m = i(cm − cm
), m = 1, . . . , n.
spins 1/2 with canonical Pauli operators ~σm , m = 1, . . . , n,
Y z
Y z
y
w2m = σm
σm0
w2m−1 = σjx
σm 0
m0 <m
Tomaž Prosen
m0 <m
Non-equilibrium quantum and statistical physics group
Analytical solution for quasi-free fermionic systems
Consider a general solution of the Lindblad equation:
”
X“
dρ
2Lµ ρL†µ − {L†µ Lµ , ρ}
= L̂ρ := −i[H, ρ] +
dt
µ
for a general quadratic system of n fermions, or n qubits (spins 1/2)
H=
2n
X
wj Hjk wk = w · H w
Lµ =
2n
X
lµ,j wj = l µ · w
j=1
j,k=1
where wj , j = 1, 2, . . . , 2n, are abstract Hermitian Majorana operators
{wj , wk } = 2δj,k
j, k = 1, 2, . . . , 2n
Two physical realizations:
†
†
canonical fermions cm , w2m−1 = cm + cm
, w2m = i(cm − cm
), m = 1, . . . , n.
spins 1/2 with canonical Pauli operators ~σm , m = 1, . . . , n,
Y z
Y z
y
w2m = σm
σm0
w2m−1 = σjx
σm 0
m0 <m
Tomaž Prosen
m0 <m
Non-equilibrium quantum and statistical physics group
Quantum phase transition far from equilibrium in XY spin-1/2 chain
hcritical = 1 − γ 2 .
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Open Many-Body Quantum Systems, method II:
time-dependent density-matrix-renormalization group in operator-space
Non-equlibrium steady state as a fixed point of Liouville equation LρNESS = 0
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Spin Diffusion in Heisenberg chains
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Long-range correlations far from equilibrium
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Solving a long standing problem: Proof of ballistic spin transport in
easy-plane (|∆| < 1) anisotropic Heisenberg spin-1/2 chain
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
ρNESS = 2−n
„
1 + Γµ(Z − Z † ) + Γ2
„
««
µ2
µ
(Z − Z † )2 − [Z , Z † ]
+ O(Γ3 )
2
2
Z is a non-hermitian matrix product operator
Z =
X
(s1 ,...,sn
hL|As1 As2 · · · Asn |Ri
)∈{+,−,0}n
n
Y
s
σj j ,
j=1
where σ 0 ≡ 1 and A0 , A± is a triple of near-diagonal matrix operators acting
on an auxiliary Hilbert space H spanned by {|Li, |Ri, |1i, |2i, . . .}:
A0
=
|LihL| + |RihR| +
∞
X
cos (r λ) |r ihr |,
r =1
∞
X
„ —
«
r +1
sin 2
λ |r ihr +1|,
2
r =1
A+
=
|Lih1| + c
A−
=
|1ihR| − c −1
∞
X
r =1
sin
““ j r k ” ”
2
+1 λ |r +1ihr |,
2
where λ = arccos ∆ ∈ < ∪ i< and bxc is the largest integer not larger than x.
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Towards application:
Improving efficiency of thermo-electric (thermo-chemical) heat-to-power
conversion
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
Improving thermoelectric figure of merit using dynamical systems’ approach
Tomaž Prosen
Non-equilibrium quantum and statistical physics group
A simple dynamical model of thermoelectric engine
Tomaž Prosen
Non-equilibrium quantum and statistical physics group