Herwig Ott

University of Kaiserslautern

Driven-dissipative Bose-Einstein condensates

Outline

The quantum gas and its

environment

Experimental approach Quantum Zeno dynamics in a BEC

Negative differential conductivity

Bistable tunneling transport

The quantum gas and its environment

Quantum gases are typically well isloated from the environment => unitary dynamics

Residual photon scattering and trap shaking => heating

Background gas collisions => global losses

The master equation General interaction with the environment: Master equation in Lindblad form

System density operator unitary time evolution

non-unitary time evolution

Coupling rates Lindblad operators

Experimental platforms

Fluorescence measurement e.g. Quantum Zeno Effect Itano et al. PRA 41, 2295 (1990)

Coupling atoms to cavities e.g. atom laser in a cavity Öttl et al. PRL 94, 090404 (2005)

Coupling to other particles e.g. Digital Open-System Quantum Simulator with Ions Barreiro et al. Nature 470, 486 (2011)

Local losses and decoherence Dissipation in a many-body quantum system

Coupling photons to atoms e.g. Cavity field decay Brune et al. PRL 101, 240402

Coupling to solid state system e.g.in superconducting qubits Katz et al. Science 312, 1498 (2005)

Particle loss as dissipation

loss rate at each lattice site

A one dimensional system: leaky optical lattice

P. Barmettler and C. Kollath PRA 84, 041606 (2011)

D. Witthaut,…, S. Wimberger, PRA 83, 063608 (2011)

anihilation and creation operators at each lattice site

Continuous losses in a bulk system

Gross-Pitaevskii equation with imaginary potential

In mean field approximation for a BEC, the total master equation is equivalent to a time-dependent Gross-Pitaevskii equation:

What is the generic effect of an imaginary potential?

Simple toy model in 1D: an imaginary potential barrier in one dimension

Increasing dissipation inhibits losses

The extreme limit

Misra and Sudarshan J. Math. Phys. 18, 756 (1977) Fachi et al. Phys. Lett. A 275, 12 (2000) Fachi and Pascazio J. Phys. A 41, 493001 (2008)

Projection to a unitary time evolution in the subspace with no losses:

x

0V0V

m

pH

2

2

Phase noise as decoherence

Fluctuating potential in a lattice site

number operator effective decoherence rate

V(t)

The working principle

2D imaging

Manipulation and preparation tool

Spatial resolution = Beam diameter > 100 nm

Single atom sensitivity

In situ technique

In vivo technique

Magneto-optical trap

MOT: 3s, 1 x 109 atoms

Optical dipole trap

CO2 Laser on

10 W CO2 laser

Weak probing limit

Small beam current and fast imaging sequence: In situ image of a Bose-Einstein condensate

High precision density probing

Increase statistics by many repetitions of the experiment

Gericke et al. Nat. Phys (2008) Würtz et al. PRL (2009)

Optical lattices

k k

keff keff

Image gallery

Local particle loss in a thermal gas

ultracold atoms

beam locally removes atoms, collect created ions

ion

s/µ

s

time (ms)

fast depletion

steady-state

generic behaviour: small beam current (I=15 nA)

Data acquisition and analysis

time

event Dt

Fluctuations

Electron beam locally probes quantum gas

Mean count rate 𝜂𝛾(𝑟 )𝑛(𝑟 )

𝛾(𝑟 ) =𝜎𝑡𝑜𝑡

𝑒

𝐼

2𝜋𝜔2 𝑒− 𝜚− 𝜚02/(2𝜔2)

𝑔 2 (𝜏) temporal pair correlation function

local loss rate atom density

detection efficiency

Gaussian beam profile

Temporal g(2) - correlations

thermal gas

T=120 nK

)2(Exp1)(2

2)2(

th

rrg

nm600200)/(2 B

2

th Tmk

Probability of finding two atoms at a distance r

Probability of finding two atoms at a distance r with time delay t

)/1

12(Exp

)/1(

11),(

222

2

2/322

)2(

cthc t

r

ttrg

Tkc

B

Thermal bunching

BEC

45 nK

100 nK

Guerrera et al. PRL (2012)

Dissipative defect in a BEC Time-dependent GPE with imaginary potential

),(2

)(),()()(

2),(

222

trr

itrtgNrVm

tri extt

Quantum Zeno dynamics I measurement protocol

Quantum Zeno dynamics II

Barontini et al PRL 110, 035302 (2013)

Numerical simulation via GPE

Imaginary potential is a good description for our loss processes

From unitary to non-unitary dynamics

Unitary dynamics dominate

Non-unitary dynamics dominate 𝜇𝐵𝐸𝐶

Hole drilling mechanism in a BEC

Image of the hole:

Linescan through the BEC after 1 ms if dissipatoin

Microscopic hole drilling Toy model: two atoms in double well potential:

Quantum jump: apply the anihilation operator to the left well, you get

Probability for a quantum jump in time intervall dt

L R

aL rate

0 2

Microscopic hole drilling Non-detection: no atom is detected in an infinite time intervall

Evolution for small dt

rate projective

measurement

Action of both mechanism together: Hole is made by the non-detection of an atom

Experimental signature

Measure the probability to detect one atom right after the detection of another one! = probe the system after a quantum jump

),()(ˆ)(ˆ)(ˆ)(ˆ21

)2(

1221 rrGrrrr

BEC thermal cloud

Negative differential conductivity

Superimpose a one-dimensional optical lattice along the BEC Lattice constant 600 nm Discretization of axial coordinate

Experimental settings:

Prepare initial non-equilibrium

One lattice site is emptied by electron beam axial motion is frozen (s=30)

Each lattice site is an independent 2D codensate with about 800 atoms Mean-field with chemical potential µ

minimum instance

Microscopic level structure

Dµ = h x 1500 Hz nr = 180 Hz

mean field ~ 800 atoms

radially excited single particle

states

Energy conservation requires tunneling into radially excited state!

Effective tunneling coupling

Renormalization of J upon refilling, the effective tunneling coupling recovers its normal value J

𝐽𝑒𝑓𝑓 = 𝜂 × 𝐽

𝜂 = 𝜓𝑒𝑥𝑝 𝜓𝑛 ≈ 0.14

Tunneling becomes density dependent mean field version of correlated hopping

𝐽 = 𝐽(Δ𝜇)

Refilling dynamics and NDC

No oscilllations visible Timescale depends on tunneling coupling Pronounced ‚s-shape‘ visible

1. take derivate of

experimental data current

2. convert atom number difference in difference in chemical potential voltage

Refilling dynamics and NDC

No oscilllations visible Timescale depends on tunneling coupling Pronounced ‚s-shape‘ visible

Microscopic modelling

Basic phenomenology can be understood from effective tunneling What happens in the central site?

Steady state Short after the beginning of tunneling

Conversion of interaction energy into thermal energy Joule-Thompson effect

𝜇 = 2𝑘𝐵𝑇

Collisions and thermalization!

Collisional decoherence

The collision rate in the central site can be estimated to a few hundred collisions per second.

Tunneling rate J/h is slower than collision rate incoherent dynamics

Transport is a true steady-state transport and we observe a DC current

𝜏 ∝ Γ𝑑𝑒𝑐/𝐽2

Quantum Zeno result

Theoretical modelling

Full problem not treatable due to many spatial modes

Use effective single particle mode with modified tunneling coupling at the central site and phase noise at the central site (phase noise is a fit parameter)

G

Jeff Jeff J J

Theoretical modelling

Master equation for one spatial mode

Effective model for reduced single particle density matrix

Theoretical modelling

Fits with theoretical model (decoherence rate is fit parameter)

Fitted decoherence rate is of similar magnitude as the collisional rate in the lattice site (few hundred Hz)

Incoherent hopping transport!

Remarks on NDC

NDC is the basis of tunneling diodes in electronics possible applications in atomtronics (sustained Josephson oscillations)

Single particle model works surprisingly well

Internal decoherence can be treated as Markovian, even though the whole system is closed

Strong influence of intrinsic dissipative channels on many-body dynamics

Future investigation of NDC in strongly correlated systems

Bistable tunneling transport

Same settings as before BUT: continuously probing with the electron beam for different starting conditions

Bistable tunneling transport

Electron beam is scanned over one lattice site with variable rate

Typical behaviour

Initial dynamics Stationary state

Stationary states

(iii) Site is almost empty

(ii) Two stable solutions depending on initial conditions

(i) Site is completely full

(i) Why is the site full?

Same current voltage characteristics as a superconductor!

(i) Why is the site full?

Consider Josephson junction array Meanfield treatment via discrete nonlinear Schrödinger equation Losses appear as imaginary potential

Stationary state with unit filling and finite phase difference between adjacent site -> superfluid state

(ii) Why is there bistability?

Consider red points (initialy empty site)

Due to the nonlinear tunneling coupling J‘(N), the site is kept empty

Finite difference in chemical potential

Resistive transport!

Initial dynamics for resistive branch

Critical slowing down

Non-equilibrium phase transition?

Critical slowing down

In equilibrium: condensation happens already at a filling of 10 percent

Here: critical slowing down happens at a filling of 30 - 40 percent

Final phase diagram

Outlook

Dissipation and decoherence in strongly correlated systems

Atomtronics with dissipatively engineered quantum states Non-equilibrium phase transitions

Preparation of dissipative attractor states (dark soliton)

The team Andreas

Vogler Thorsten Manthey

Peter Würtz

Tobias Weber

Ralf Labouvie

Thomas Niederprüm

Former group members: Vera Guarrera, Giovanni Barontini, Matthias Scholl Arne Ewerbeck, Felix Stubenrauch, Philipp Langer, Tatjana Gericke

Oliver Thomas

Simon Heun

Bodhaditya Santra