Quantum critical states and phase transitions in the presence of non equilibrium noise Emanuele G. Dalla Torre – Weizmann Institute of Science, Israel.

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