Exploring quantum phases by driven dissipation
Transcript of 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
Lindblad master equation
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
Lindblad master equation
• 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
Lattice gauge theory
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
Lattice gauge theory
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
Z2 lattice gauge Higgs model
• simplest model of gauge field and charged particles
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
Lattice gauge theory
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
Z2 lattice gauge Higgs model
• simplest model of gauge field and charged particles
Lattice gauge theory
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.
Lattice gauge theory
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
Lattice gauge theory
• 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?