Quantum critical states and phase transitions in the presence of non equilibrium noise Emanuele G....
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Transcript of Quantum critical states and phase transitions in the presence of non equilibrium noise Emanuele G....
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.
NICE-BEC, June 4th - Session on “Non equilibrium dynamics”
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?
Q U A N T U M
NOISE
SYSTEM
CLASSIC
B A T H
1
0.5
Specific realization in zero dimensions
Shunted Josephson Junction
Charge Noise with 1/f spectrum
Bath: Zero temperature resistance~
RJ C
VN(t)
Iext
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
arxiv/0908.0868
R/RQ
superconductor
noise
insulator
Specific realizations in one dimension
Dipolar atoms in a cigar shape potential
Noise: fluctuations of the polarizing field
Bath: immersion in a condensate
Trapped ions
Noise: Charge fluctuations on the electrodes
Bath: Laser cooling
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
(controls the quantum fluctuations)
Luttinger parameter
Haldane (1980)
Review : density correlations in 1D
Crystalline correlation decay as a power law:
Long-wavelength fluctuations Crystal fluctuations
Scale invariant, critical state
Two types of low-lying density fluctuations
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?
lattice potentialphonons
Effective action:
Review: Mott transition in 1D
The quadratic term is scale invariant.
How does the lattice change under rescaling ?
Buchler, Blatter, Zwerger, PRL (2002)
Quantum phase transition at K = 2
K > 2 lattice decays critical
K < 2 lattice grows Mott insulator
Can we have non-equilibrium quantum critical states?
Non-equilibrium quantum phase transitions?
What are the effects of the external noise?
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
Zero temperature bath induces both dissipation and fluctuations (satisfies FDT)
External noise induces only fluctuations (breaks FDT)
We can cast the quadratic action into
a linear quantum Langevin equation:
Monroe group, PRL (06), Chuang group, PRL (08)
• Indications for short range
spatial correlations
• Time correlations:
1/f spectrum
The measured noise spectrum in ion traps
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 powernoise
dissipation
Non equilibrium quantum critical state!
(Note: exact only in the scale invariant limit , F00 with F0 / = const.)
Non equilibrium critical state: Bragg spectroscopy
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)
Add a periodic potential
which modulates with time
Absorption spectrum in the non equilibrium critical state
Equilibrium (F0=0)
Non equilibrium (F0/η=2)
Luther&Peschel(1973)
Unaffected by noise
Long wavelength limit:
Near q0=2π/a:
Strongly affected by the noise
Absorption spectrum in the non equilibrium critical state
The energy loss can be negative
critical gain spectrum
Near q0=2π/a:
Non equilibrium quantum phase transitions
Add a static periodic potential
(“lattice”) at integer filling
Does the lattice induce a quantum phase transition?
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
What are the effects of the lattice on the correlation function?
or
Non equilibrium Mott transition: scaling analysis
Non equilibrium phase
transition at
How does the lattice change under rescaling ?
2x2 Keldysh action (non equilibrium quantum critical state)
K
F0 /
pinned
critical
Extension: General noise source
We develop a real-time Renormalization Group procedure
α > -1 irrelevant: doesn’t affect the phase transition
α < -1 relevant: destroys the phase transition (thermal noise)
α = -1 marginal: non-equilibrium phase transition
Summary: Quantum systems coupled to the environment show non equilibrium critical steady states and phase transitions
F0 /
2D superfluid
2D crystal
critical
K
F0 /
1. Critical steady state with power-law correlations (faster decay)
2. Negative response to external probes (“critical amplifier”)
4. High dimensions: novel phase transitions tuned by a competition of classical noise and quantum fluctuations
E.G. Dalla Torre, E. Demler, T. Giamarchi, E. Altman - arxiv/0908.0868 (v2)
3. Non equilibrium quantum phase transitions: a real-time RG approach
Non equilibrium phase transitions - coupled tubes
Inter-tube tunneling:
Phase transition atK
1D critical
2D superfluid
F0 /
Non equilibrium phase transitions - coupled tubes
F0 /
Inter-tube repulsion
K 1D critical
2D crystal
Inter-tube tunnelingK
1D critical
2D superfluid
F0 /
Both perturbations (actual situation)
F0 /
K2D superfluid
critical
2D crystal
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 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
Original Hamiltonian