Download Quantum steady states and phase transitions in the presence of non

NICE-BEC, June 4th - Session on “Non equilibrium dynamics”
Quantum critical states and phase transitions in the
presence of non equilibrium noise
Emanuele G. Dalla Torre – Weizmann Institute of Science, Israel
Collaborators:
Ehud Altman – Weizmann Inst.
Eugene Demler – Harvard Univ.
Thierry Giamarchi – Geneve Univ.
Quantum systems coupled to the environment
External noise from the environment (classical)
System
Zero temperature
thermal bath (quantum)
The systems reaches a non-equilibrium steady state:
Criticality? Phase transitions?
Specific realization in zero dimensions
Iext
Shunted Josephson Junction
Charge Noise with 1/f spectrum
J
R
Bath: Zero temperature resistance
C
~ VN(t)
In the absence of noise this system undergoes a superconductor-insulator
quantum phase transition at the universal value of the resistor
The external noise shift the quantum phase transition
away from its universal value
R/RQ
1
insulator
0.5
arxiv/0908.0868
superconductor
noise
Specific realizations in one dimension
Trapped ions
Noise: Charge fluctuations on the electrodes
Bath: Laser cooling
Dipolar atoms in a cigar shape potential
Noise: fluctuations of the polarizing field
Bath: immersion in a condensate
Cigar shape potential: Bloch group (2004) - BEC immersion: Daley, Fedichev, Zoller (2004)
Outline
1. Review of the equilibrium physics in 1D (no noise)
2. Non equilibrium quantum critical states in one dim.
A. Dynamical response
B. Phase transitions
3. Extension to higher dimensions
4. Outlook and summary
Review of equilibrium physics in 1D:
continuum limit
a : average
distance
(x) :
displacement field
Low-energy effective action: phonons
Luttinger parameter
(controls the quantum fluctuations)
Haldane (1980)
Review : density correlations in 1D
Two types of low-lying density fluctuations
Long-wavelength fluctuations
Crystal fluctuations
Crystalline correlation decay as a power law:
Scale invariant, critical state
Review: effects of a lattice in 1D
Add a static periodic potential
(“lattice”) at integer filling
When does the lattice induce a quantum phase transition
to a Mott insulator?
Effective action:
phonons
lattice potential
Review: Mott transition in 1D
The quadratic term is scale invariant.
How does the lattice change under rescaling ?
K>2
K<2
lattice decays
lattice grows
critical
Mott insulator
Quantum phase transition at K = 2
Buchler, Blatter, Zwerger, PRL (2002)
What are the effects of the external noise?
Can we have non-equilibrium quantum critical states?
Non-equilibrium quantum phase transitions?
Effects of non-equilibrium noise
Immersion in a BEC (or
laser cooling) behaves as
a zero temperature bath
The external noise
couples linearly to
the density
If we assume that the noise is smooth on an inter-particle scale,
we can neglect the cosine term and retain a quadratic action!
Effects of non-equilibrium noise
We can cast the quadratic action into
a linear quantum Langevin equation:
Zero temperature bath induces
both dissipation and fluctuations
(satisfies FDT)
External noise induces only
fluctuations (breaks FDT)
The measured noise spectrum in ion traps
Monroe group, PRL (06), Chuang group, PRL (08)
• Time correlations:
1/f spectrum
• Indications for short range
spatial correlations
Crystalline correlations in the presence of 1/f noise
Using the Langevin equation we can compute correlation functions:
crystal correlations remain power-law,
with a tunable power
noise
dissipation
Non equilibrium quantum critical state!
(Note: exact only in the scale invariant limit , F00 with F0 /  = const.)
Non equilibrium critical state: Bragg spectroscopy
Add a periodic potential
which modulates with time
Goal: compute the energy transferred into the system
In linear response, we have to compute density-density correlations
in the absence of the potential (V=0)
Absorption spectrum in the non equilibrium critical state
Long wavelength limit:
Equilibrium (F0=0)
Luther&Peschel(1973)
Unaffected by noise
Near q0=2π/a:
Non equilibrium (F0/η=2)
Strongly affected by the noise
Absorption spectrum in the non equilibrium critical state
Near q0=2π/a:
The energy loss can be negative
 critical gain spectrum
Non equilibrium quantum phase transitions
Add a static periodic potential
(“lattice”) at integer filling
Does the lattice induce a quantum phase transition?
or
What are the effects of the lattice on the correlation function?
The Hamiltonian is not quadratic and we cannot cast into a Langevin equation
Instead we use a double path integral formalism (Keldysh) and expand in small g
Non equilibrium Mott transition: scaling analysis
2x2 Keldysh action (non equilibrium quantum critical state)
How does the lattice change under rescaling ?
K
critical
Non equilibrium phase
transition at
pinned
F0 /
Extension: General noise source
We develop a real-time Renormalization Group procedure
 > -1
irrelevant: doesn’t affect the phase transition
 < -1
relevant: destroys the phase transition (thermal noise)
 = -1
marginal: non-equilibrium phase transition
Summary: Quantum systems coupled to the environment show
non equilibrium critical steady states and phase transitions
E.G. Dalla Torre, E. Demler, T. Giamarchi, E. Altman - arxiv/0908.0868 (v2)
1. Critical steady state with power-law
correlations (faster decay)
2. Negative response to external probes
(“critical amplifier”)
F0 /
3. Non equilibrium quantum phase
transitions: a real-time RG approach
4. High dimensions: novel phase transitions
tuned by a competition of classical noise
and quantum fluctuations
K
2D superfluid
critical
2D crystal
F0 /
Non equilibrium phase transitions - coupled tubes
Inter-tube tunneling:
K
Phase transition at
2D superfluid
1D critical
F0 /
Non equilibrium phase transitions - coupled tubes
Inter-tube tunneling
K
Inter-tube repulsion
2D superfluid
K
1D critical
1D critical
2D crystal
F0 /
F0 /
Both perturbations (actual situation)
K
2D superfluid
critical
2D crystal
F0 /
Outlook : reintroduce backscattering
In the presence of backscattering, the Hamiltonian is not quadratic
Keldysh path integral enables to treat the cosine perturbatively (relevant/irrelevant)
How to go beyond?
We introduce a new variational approach for many body physics
The idea: substitute the original Hamiltonian by a quadratic variational one
Time dependent variational approach
Variational
Original Hamiltonian
Hamiltonian
The variational parameter fV(t) is determined self consistently by
requiring a vanishing response of
to any variation of fV (t).
We show that this approach is equivalent to Dirac-Frenkel (using a
variational Hamiltonian instead that a variational wavefunction)
We successfully use it to compute the non linear I-V characteristic of
a resistively shunted Josephson Junction