Exploring quantum phases by driven dissipation

Exploring quantum phases by driven dissipation

Nicolai Lang

Institute for Theoretical Physics III

University of Stuttgart, Germany

SFB TRR21 Tailored quantum matter

Research group:

Hans Peter Büchler, David Peter, Adam Bühler, Przemek Bienias, Sebastian Weber

RySQ Workshop 2015 Aarhus University, AIAS

1. General concept

Lindblad master equation

Outline

3. Lattice gauge theory

Dissipative implementation

2. Dissipative quantum phase transitions

Paradigmatic model of a purely dissipative system


General concept

1

Dissipation and decoherence

Master equation

• coupling between system and reservoirs

• dephasing and decoherence

• Born-Markov approximation- no-memory of the reservoir - weak coupling between system and bath


• e.g., optical master equation laser cooling

system

bath

⇤t⇥ = L [⇥] =⇤

�

��

�c�⇥c†� �

12c†�c�⇥� 1

2⇥c†�c�

⇥

: jump operator

Dark states

Dark states

• eigenstates of all jump operators with vanishing eigenvalue

• pure state

• decoherence free subspace

• stationary solution of the master equation

c�|D� = 0

⇤t⇥ = L [⇥] =⇤

�

��

�c�|D⌅⇤D|c†� � 1

2c†�c�|D⌅⇤D|� 1

2|D⌅⇤D|c†�c�

⇥= 0

� = |D⇥�D|

Goal: • engineering of jump operator with desired state a dark state • dark state is unique stationary solution

system

bath

subspace of dark states

|D�

Dephasing versus cooling

Dephasing

• hermitian jump operator

• each eigenstate is stationary state • diagonal density matrix

c†� = c�

Cooling

• non-hermitian jump operator

• preparation into the subspace of dark states

• arbitrary initial density matrix evolves into unique pure state

� � |D⇤⇥D|

c†� �= c�

⇤ =�

�,µ

c�µ|�⇤⇥µ| ��

�

p�|�⇤⇥�|

c†�|�� = �|��

loss of coherence

�(t)

L [�]

�(t + �t)

|D� |D�

Implementation

Digital quantum simulation

• Implementation with Rydberg atoms H. Weimer, et al., Nature Physics 6, 382 (2010)

• Implementation with Ion traps Barreiro, et al., Nature 470, 486 (2011)

Dissipative element: spontaneous emission, optical pumping

Paradigmatic model of a purely dissipative system

Dissipative quantum phase transitions

2

Example: Paramagnet

Spin system in dimension d

• spins at lattice sites s

• unique dark state

• parent Hamiltonian

|Di =Y

s

| !is

= 2X

s

[1� �x

s

]

H =X

s

P †sPs

IFTHEN

{{

Ps

=p�z

s

[1� �x

s

]

: frustration free, unique zero energy ground state external magnetic field along x-direction

Example: Ferromagnet

Spin system in dimension d

• spins at lattice sites s

• two dark states: two ferromagnetic states

• parent Hamiltonian: ferromagnetic Ising model

IFTHEN

{{

Fs

= �x

s

"1� 1

q

X

t2s

�z

t

�z

s

#

|Di =Y

s

| "is

|Di =Y

s

| #is

number of nearest neighbors

Exploring quantum phases

Non-equilibrium steady state phase diagram?

• both competing drives

• is there a phase transition?

• does the phase diagram resemble the “blue-print” Hamiltonian system?

• parent Hamiltonian: transverse Ising model

@t⇢ =X

s

Ps⇢P

†s � 1

2P †sPs⇢�

1

2⇢P †

sPs

�+

Fs⇢F

†s � 1

2F †sFs⇢�

1

2⇢F †

sFs

�

H = �X

hs,ti

�z

s

�z

t

� X

s

�x

s

Coherent and dissipative dynamics

Phase transitions and metastability for competing dissipative and coherent drives

Bose-Hubbard, Rydberg atoms, Fermionic systems, conceptional questions

• Diehl, Zoller, Fazio PRL (2010) • Lee, Cross (2011) • Lesanovsky, • Maria Ray, Hazzard (2013) • Eisert (2012) • Fleischhauer, Moos, Höning (2012) • Shirai, Mori, Miyashita (2014) • Immamoglu, Cirac, Lukin (2012)

Here:

Only dissipative dynamics with quantum mechanics encoded in non-commuting jump operators

cf. several talks in WP4 yesterday

• effective master equation for

• self-consistency

Methods

• Wave function Monte Carlo simulation of master equation: only small systems

• DMRG simulations: only in one dimension

• Keldysh path integral formulation

Mean-field theory

Exploring quantum phases

• exact in high dimensions

• ansatz for density matrix:

• homogeneous density matrix

⇢ =Y

s

⇢s

⇢s ⌘ ⇢̂(m)

@t⇢̂(m) = L⇢̂(m)partial trace

m↵ = Tr [�↵⇢̂(m)]

Mean-field theory

Paramagnetic jump operators

• local on each lattice site and remain the same within mean-field theory

Ferromagnetic jump operators

• ferromagnetic drive

• dephasing terms

f1 = �x [1�mz

�z]

f2 =1p2d

p1�m2

z�y

f3 =1p2d

�z

f0 =p�z [1� �x]

@t⇢̂ =3X

i=0

h2fi⇢̂f

†i � f†

i fi⇢̂� ⇢̂f†i fi

i

m↵ = Tr [�↵⇢̂]

three coupled non-linear equations

@tm = F(m)

⇢̂ =1 +m�

2

Mean-field theory

Second order phase transition

• critical value:

• continuous behavior of the order parameter

• ferromagnetic to paramagnetic phase transition

• in general: mixed state, with finite purity

• purity is minimal at phase transition point

• behavior resembles the thermal phase diagram for the parent Hamiltonian

• critical exponents in analogy to mean-field exponents for the Hamiltonian system

κ = 2.9/3.1

δmx

κ = 2.9/3.1

δmy

κ = 2.9/3.1

δmz

κ = 3.0

δmx

κ = 3.0

δmy

κ = 3.0

δmz

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

Mag

netization/C

orrelation

0 20 40 60 80 100

Time t

1 2 3

-1.0

-0.5

0.0

0.5

1.0

0 1 2 3 ∞

0.0

0.4

0.8

1.2

Mag

netization

mz

Purity

|m̂|

Ratio κ

ferromagnetic paramagnetic

≀ ≀

c

d

b

a

c = 4

✓1� 1

2d

◆

Dissipative Transverse Ising model

Hamiltonian system • second order phase transition

• two-parameter phase diagram: temperature and transverse field

• identical critical exponents

• gapped system with gapless point at critical point

Dissipative model • second order phase transition

• critical value for drive; minimal purity at phase transition

• mean-field critical exponent

• gap in Lindblad spectrum with gapless point for critical drive

Can a dissipatively driven system explore the full richness of the

Hamiltonian “blue print” model?

Dissipative implementation

Lattice gauge theory

3


Z2 lattice gauge Higgs model

• simplest model of gauge field and charged particles

Ie = �zs⌧

ze �

zs0

: minimal coupling between matter and gauge field

Bp =Y

e2p

⌧ze

Bp

�s ⌧e

H = �X

s

�x

s

� �X

e

Ie

�X

e

⌧xe

� !X

p

Bp

chemical potential

electric field

kinetic energy

magnetic fieldcharges gauge field

: magnetic flux

Gauge symmetry:

[H,Gs] = 0

Wegner, F. J. Journal of Mathematical Physics 12, 2259 (1971) Fradkin, Susskind, Physical Review D 17, 2637 (1978) Fradkin, Shenker, Physical Review D 19, 3682 (1979)

Gs

⌘ �x

s

Y

e2s

⌧xe


Condensed matter approach

• terms in the Hamiltonian enforce the gauge constraint

High energy approach

• all physical observable are gauge invariant

[A,Gs] = 0

Gs| i = | i

physical states are equivalence classes of states in different gauges

emergent gauge theory at low energies



H = �X

s

�x

s

� �X

e

Ie

�X

e

⌧xe

� !X

p

Bp

chemical potential

electric field

kinetic energy

magnetic field

e.g. Karl Jansen’s talk yesterday

e.g. Alex Glätzle’s talk yesterday


H = �X

s

�x

s

� �X

e

Ie

�X

e

⌧xe

� !X

p

Bp

Ie = �zs⌧

ze �

zs0 Bp =

Y

e2p

⌧ze

Implementation of this model by dissipation?

• three corners of the phase diagram can be dissipatively prepared




Confining phase:

• Hamiltonian:

• gauge invariance:

� = ! = 0

H = �X

s

�x

s

�X

e

⌧xe

charges connected by a string of electric field

mesons gauge loop

Design jump operator to prepare into the ground state:

• “naive approach” breaks gauge invariance

Fundamental excitations:

require gauge invariant jump operators

�z

s

[1� �x

s

] ⌧ze

[1� ⌧xe

]

Gs

⌘ �x

s

Y

e2s

⌧xe

Lattice gauge theoryConfining phase:� = ! = 0

Ie

[1� ⌧xe

]

Removing gauge loops and deformation of loops:

Removing confined charges and hopping of charges:

Breaking of topological gauge loops:

pure state as steady state

Bp

"1� 1

q

X

e2p

⌧xe

#

Ie

"1� 1

2

X

s2e

�x

s

#

Lattice gauge theoryFull set of gauge invariant jump operators

• requires 6 jump operators

• three edges of the phase diagram can be prepared

5

Figure 3. Conceptual foundation of the dissipative Z2-Gauge-Higgs model. (a) illustrates qualitatively the well-known phasediagram of the Hamiltonian Z2-Gauge-Higgs theory in the !-�-plane. There are three characteristic phases: The (I) confinedcharge, (II) free charge, and (III) Higgs phase. In order to drive the system dissipatively in a distinct phase, combinations ofthe baths adjacent to the labels (I), (II), and (III) are employed. (b) depicts the e↵ects of the six types of jump operators(characterizing the baths) on elementary excitations in two spatial dimensions. Asymmetric arrows denote asymmetric quantumjump probabilities. The symbols read as follows: Yellow site , �x = �1 (electric charge); Red edge , ⌧x = �1 (gauge string);Blue site+edge , I

e

= �1 (Higgs excitation); Blue face , Bp

= �1 (magnetic flux). The formal definitions are given inTable I.

spin-1/2 representations attached to sites s (the matterfield, denoted by �k

s ) and edges e (the gauge field, denotedby ⌧ke ). Here, �k

s and ⌧ke (k = x, y, z) denote Pauli ma-trices. Then the Hamiltonian of the Z2GH model reads

HZ2GH = �Xs

�xs � �

Xe

Ie �Xe

⌧xe � !Xp

Bp (6)

where s, e and p denote sites, edges and faces of the(hyper-)cubic lattice, respectively; ! and � are non-negative real parameters. The plaquette operators Bp ⌘Q

e2p ⌧ze describe a four-body interaction of gauge spins

on the perimeter of face p and Ie ⌘ �zs1⌧

ze �

zs2 (where

e = {s1, s2}) realizes a gauged Ising interaction betweenadjacent matter spins. Note that HZ2GH features thelocal gauge symmetry Gs ⌘ �x

s

Qe:s2e ⌧

xe = �x

sAs, i.e.[H,Gs] = 0 for all sites s. Here As ⌘

Qe:s2e ⌧

xe denotes

a 2D-body interaction of gauge spins located on the edgesadjacent to site s.

The expected quantum phase diagram in 2+ 1 dimen-sions is sketched in Fig. 3 (a) and features three distinctphases [33, 37]: The (I) confined charge, (II) free charge,and (III) Higgs phase, respectively. To contrive a familyof baths that explore these three phases and give rise to anon-equilibrium analogy of Fig. 3 (a), it proves advanta-geous to analyse the elementary excitations of HZ2GH inthe three parameter regimes: We aim at jump operatorsthat remove the elementary excitations of each phase andthereby drive the system towards the latter. In addition,this scheme leads inevitably to gauge invariant jump op-erators L, i.e. [L,Gs] = 0 for all sites s — which is

a necessary condition for the intended gauge-symmetryconstrained dynamics. We stress that any realistic imple-mentation would have to deal with gauge-symmetry vio-lating imperfections, demanding additional mechanismsto enforce gauge-invariance [34, 35].

For the sake of brevity, we label localised excita-tions (“quasiparticles”) by the corresponding operator inHamiltonian (6) and its eigenvalue. E.g. �x

s = �1 refers

Bath Jump operator

Gauge string tension F (1)p

= ⌘1 Bp

�1� ⌧x

e2p

�

Gauge string fragility F (2)e

= ⌘2 Ie (1� ⌧x

e

)

Higgs brane tension D(1)s

= ⌘3 �x

s

(1� Ie2s

)

Higgs brane fragility D(2)e

= ⌘4 ⌧x

e

(1� Ie

)

Charge hopping Te

= ⌘5 Ie (1� �x

s2e

)

Flux string tension Be

= ⌘6 ⌧x

e

(1�Bp2e

)

Table I. Jump operators for the dissipative Z2-Gauge-Higgsmodel. Their action is described in the text. Pictorial descrip-tions can be found in Fig. 3. s, e and p denote sites, edges andfaces, respectively. The short-hand notation e 2 p denotes thenormalized sum over all edges e adjacent to face p. The freeparameters of the theory are labeled ⌘

i

for i = 1, . . . , 6. Thesecond column lists the jump operators of the gauge theorywith non-trivial gauge condition �x

s

As

= 1.


Mean-field theory for lattice gauge model

• two mean-fields:

• all three phases are predicted within mean-field theory

• well known artifacts of MF for lattice gauge theories

Exploring the full phase diagram by competing dissipative drives

⇢ =Y

e

⇢eY

s

⇢sDissipative MF phase diagram

parallels the well-known MF phase diagram of

the Hamiltonian theory

Drouffe, Zuber, Phys. Rep. 102,1 (1983)

ConclusionExploring quantum phases by driven dissipation• paradigmatic transverse Ising model

• reveals the Hamiltonian phase diagram in high dimensions


• first demonstration how to implement a lattice gauge theory by dissipation

• resembles the phase diagram of the Hamiltonian system

Do dissipative systems in general reveal the different ground state phases of the “blue-print” Hamiltonian system?

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Exploring quantum phases by driven dissipation