Open quantum physics:
recent topics in non-Hermitiansystems and beyond
University of Tokyo
Yuto Ashida
Collaborators:
Masahito Ueda (advisor, UT), Shunsuke Furukawa (UT), Sho Higashikawa (UT),
Kazuaki Takasan (Kyoto), Zongping Gong (UT), Kohei Kawabata (UT),
Eugene Demler (Harvard), Richard Shmidt (Harvard→MPQ), Leticia Tarruell (ICFO),
Ignacio Cirac (MPQ), Tao Shi (MPQ→CAS), Mari Carmen Banuls (MPQ)Microscopic observation of Macroscopic Quantum systems
Revolutionary techniques to observe and manipulate many-body systems:•

Observing many-body systems at the single-atom level.

Microscopic observation of Macroscopic Quantum systems
Revolutionary techniques to observe and manipulate many-body systems:•

Observing many-body systems at the single-atom level.
single-shot
imaging

Measurement backaction on many-body systems?
•

Manipulating single quantum particles in many-body systems.

Strongly correlated open systems?

Microscopic observation of Macroscopic Quantum systems
Revolutionary techniques to observe and manipulate many-body systems:•

Observing many-body systems at the single-atom level.
single-shot
imaging•

Measurement
on many-bodysystems?
New frontier pioneered
with backaction
microscopic
observationand manipulation
capabilities
ofsystems.
many-body systems?
single quantum particles
Manipulatingin many-bodyStrongly correlated open systems?

Continuous Monitoring of Quantum Systems: Quantum TrajectoryNon-unitary dynamics due to the measurement backaction.
If one does not know measurement outcomes,→ Lindblad master equation (dissipative dynamics)
measurement
outcome: jump operator

Continuous observation
If one can access the information about measurement outcomes,→ Quantum trajectory dynamics (keeping the purity of the state)(norm preserva<on)unitary
evolution

non-unitary evolution
without quantum jumps*Taking ensemble average E over dNa,

stochastic quantum jumps

obeys Lindblad master eq.
Dalibard, Castin, Mølmer; Carmichael (1993)

Continuous Monitoring of Quantum Systems: Quantum Trajectory
Hilbert space

non-Hermitian time evolution:(effective non-Hermitian Hamiltonian)quantum jumptime

Con1nuous observa1on
If one can access the information about measurement outcomes,→ Quantum trajectory dynamics (keeping the purity of the state)(norm preserva1on)

unitary
evolution

non-unitary evolution
without quantum jumps*Taking ensemble average E over dNa,

stochastic quantum jumps

obeys Lindblad master eq.
Dalibard, Castin, Mølmer; Carmichael (1993)

Continuous Observation: the ability to access quantum jumps
What does differentiate continuous observation from dissipation?
The ability to access the information about quantum jumps.
examples of continuously monitored dynamics:
・complete time record of all quantum jumps
・coarse-grained information: the number of quantum jumps
post-measurement state: subspace of quantum trajectories
determined by a number of jumps

・no-count process: non-Hermitian dynamics
Such continuously observed dynamics is different fromthe dissipative Lindblad dynamics (where the system is easily driven
into, e.g., an infinite temperature state or the vaccum).

Simple example: continuous observation of cavity photons
・・・

Effective Hamiltonian :jump operator :single-mode
photondetector

example 1: Fock stateexample 2: Squeezed state
trajectoryensemblenon-Hermitian dynamics

jump

time

time

Experiments: continuous observation of small quantum systems
Microwave cavity photon

e
or

g

Serge Haroche
2012 Nobel Prizeobservation of wavefunction collapse

observa9on of quantum jumps
Atom
State

e
g
1

CavityPhoton 0

quantum jumpTime
Gleyzes et al., Nature 446, 297 (2007)

Superconducting qubit
Vijay et al., PRL 106, 110502 (2011)

Guerlin et al., Nature 448, 889 (2007)

Quantum dots
Vamivakas et al., Nature 467, 297 (2010)

Experiments: continuous observation of small quantum systems
Microwave cavity photon

e
or

g

Serge Haroche
2012 Nobel Prizeobservation of wavefunction collapse

observa9on of quantum jumps
Atom
State

e
g
1

CavityPhoton 0

quantum jump
Restricted
to quantum systems
Time
with
small
degrees of freedom
Gleyzes et al., Nature 446, 297 (2007)
Guerlin et al., Nature 448, 889 (2007)

Superconducting qubit
Vijay et al., PRL 106, 110502 (2011)

Quantum dots
Vamivakas et al., Nature 467, 297 (2010)

Quantum gas microscopy
Offers a new approach to quantum many-body systems
via in-situ imaging of ultracoldUltracold
atoms atoms trapped in 2D potential.
Measuring quantum gases
at the single-particle level.

“Snapshot” of many-body
wavefunction.
W. S. W.
Bakr
al. Nature
462, 74(2009)S.etBakr
et al. Nature462,
74 (2009)

Developments in Quantum Gas Microscopy
Quantum Gas Microscopy

Superfluid-to-Mott insulator transition has
been observed at the single-particle level.
SFMI

2D ultracold atoms
W. S. Bakr et al., Science 329, 547 (2010).

Recent breakthroughs:
Measurement of entanglement entropy:
Islam et al. Nature 528 77 (2015).
Observation of antiferromagnetic fermionic correlations:
Cheuk et al., Parsons et al., Boll et al., Science 353 1253-1260 (2016),
Mazurenko et al., Nature 545 462 (2017).
Observation of many-body localization:
Schreiber et al. Science 349 842 (2015),
J-y. Choi et al. ibid 352 1547 (2016)

Developments in Quantum Gas Microscopy
Superfluid-to-Mott insulator transition has
been observed at the single-particle level.

Quantum Gas Microscopy

SFMI

New direc8on: Con8nuous monitoring of many-body systems by QGM.
2D ultracold atoms

Many-body physics

Interplay
Measurement
W. S. Bakr et al., Science
329, 547 (2010).

Recent breakthroughs:
Measurement of entanglement entropy:
Islam et al. Nature 528 77 (2015).

backaction

Observation of antiferromagnetic fermionic correlations:
Cheuk et al., Parsons et al., Boll et al., Science 353 1253-1260 (2016),
Mazurenko et al., Nature 545 462 (2017).
Observation of many-body localization:
Schreiber et al. Science 349 842 (2015),
J-y. Choi et al. ibid 352 1547 (2016)

1. Quantum many-body systems
under continuous observation
Wavefunction collapse and super-resolved imaging:
YA and M. Ueda, PRL 115, 095301 (2015).
YA and M. Ueda, Opt. Lett. 41, 72-75 (2016).
Quantum critical phenomena:
YA, S. Furukawa and M. Ueda, Nat. Commun. 8, 15791 (2017).
YA, S. Furukawa and M. Ueda, PRA 94, 053615 (2016).
Nonequilibrium dynamics:
YA and M. Ueda, PRL 120, 185301 (2018).
YA and M. Ueda, PRA 95, 022124 (2017).

Quantum critical phenomena under continuous observation
Imagine that we continuously monitor a strongly correlated many-body system.

Hilbert space

non-Hermitian time evolution :stochastic
jump

quantum trajectorytimeeffective Hamiltonian:measurement backaction

Quantum critical phenomena under continuous observation
Q. Does such measurement backaction change
universality class of quantum critical phenomena?

Idea:(sine-Gordon model)(PT-symmetry)

Quantum Phase Transition + Spectral Singularity → New universality classunique to open systems?

Singularity due to non-diagonalizability= unique to non-Hermitian systems

Experimental studies of non-Hermitian systems
Non-hermitian systems exhibit unique features at Exceptional Point (i.e., Spectral singularity)and have attracted considerable attentions. Nat. Phys. 14, 11 (2018); Nat. Photon. 11, 741 (2017).
Op#cs
periodic“Lasing” edge mode
unidirectionality
gain-loss

Regensberger et al. Nature 488 167 (2012)

Thermal atomic gas

Harari et al. Science
10.1126/science.aar4003 (2018)*Open quantum
systems:
ultracold atoms

J. Li et al., arXiv:1608.05061
bifurca#on incollec#ve spin mode
Peng et al. Nature Phys. 12, 1139 (2016)

quantum walks
L. Xiao et al., Nat. Phys. (2017)

Experimental studies of non-Hermitian systems
Non-hermitian systems exhibit unique features at Exceptional Point (i.e., Spectral singularity)and have attracted considerable attentions. Nat. Phys. 14, 11 (2018); Nat. Photon. 11, 741 (2017).
Opticsperiodic“Lasing” edge mode
unidirectionality
gain-loss

One-body (mean-field) phenomena have been well explored.
Harari et al. Science

Regensberger
al. Nature(2012)
We
extendetthe
idea488
of167
PT-symmetry
to many-bodysystems.(2018)10.1126/science.aar4003

Thermal atomic gas*Open quantum
systems:
ultracold atoms

J. Li et al., arXiv:1608.05061
bifurcation incollective spin mode
Peng et al. Nature Phys. 12, 1139 (2016)

quantum walks
L. Xiao et al., Nat. Phys. (2017)

Many-body paradigm: BKT transition and sine-Gordon model
1D Bose gas in a real periodic potential
shallow periodic potential(far-detuned light)
1D Bose gas

Low-energy Hamiltonian (sine-Gordon model):,depth of real potential

RG flows & Phase diagram:
gapped phase(Mott insulator (MI))
Berezinskii-Kosterlitz-Thoulesstransition

gr

gapless quantum critical phase(Tomonaga-Luttinger liquid (TLL))
TLL parameter2016 Nobel PrizeUnder continuous observation: Generalized sine-Gordon model
1D Bose gas subject to spatially modulated one-body lossa far-detuned light(real potential)
a weak near-resonant light(imaginary potential)
periodic gain-loss
structureLow-energy Hamiltonian (Generalized sine-Gordon model):Re V(φ) Im V(φ): depth of real potential: depth of imaginary potential→

PT symmetry
Bender & Boettcher PRL (1998)

Phase diagram of PT-symmetric sine-Gordon model
depth of imaginary potential

PT-symmetric sine-Gordon*c.f.) conventional sine-Gordon

grPhase diagram of PT-symmetric sine-Gordon model
depth of imaginary potential

PT-symmetric sine-Gordon*c.f.) conventional sine-Gordon

gr

2D phase boundary:• MI phase (below) and TLL phase (above)

Phase diagram of PT-symmetric sine-Gordon model
depth of imaginary potential

PT-symmetric sine-Gordon*c.f.) conventional sine-Gordon

gr

2D phase boundary:• MI phase (below) and TLL phase (above)

Phase diagram of PT-symmetric sine-Gordon model
depth of imaginary potential

PT-symmetric sine-Gordon

2D phase boundary:• MI phase (below) and TLL phase (above)
Two phase transitions:• BKT transition (K>2)• PT transition (K<2)*c.f.) conventional sine-Gordon

grPhase diagram of PT-symmetric sine-Gordon model
depth of imaginary potential

PT-symmetric sine-Gordon

2D phase boundary:• MI phase (below) and TLL phase (above)
Two phase transitions:• BKT transition (K>2)• PT transition (K<2)
Measurement-induced QPTSpectral Singularity
Re E*c.f.) conventional sine-Gordon

grcomplexconjugate
pair

real
<latexit sha1_base64="/uzIqAyBbpxBp7q2pozHb8EXR+8=">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</latexit>Phase diagram of PT-symmetric sine-Gordon model
depth of imaginary potential

PT-symmetric sine-Gordon

2D phase boundary:• MI phase (below) and TLL phase (above)
Two phase transitions:• BKT transition (K>2)• PT transition (K<2)
Measurement-induced QPTSpectral Singularity*c.f.) conventional sine-Gordon

gr

Merging line:• Each point = RG fixed point.
• Scale invariance at K=2, gs=gc

Spectral singularity+
quantum cri=cal point→ a nontrivial RG ﬁxed point.
New universality class.
cf. non-unitary CFT [Ikhlef et al., PRL 116, 130601 (2016)].

New type of RG flows: Violation of c-theorem
PT-symmetric sine-Gordon
depth of imaginary potential

PT broken (gi>gr)*c.f.) conventional sine-Gordon• Semicircular RG flows appear:→ anomalous increase of TLL parameter K= varying critical exponent:gr• Violation of c-theorem(Zamorodchikov, 1984)
YA, S. Furukawa and M. Ueda, Nat. Commun. 8, 15791 (2017).

: central charge: RG scale

Numerical test of RG phase diagram (exact diagonalization)
PT-symmetric spin-chain model:

Correspondence to the effective field theory:
Physical implementation in ultracold atoms:
PT symmetricshallow potential

deep optical lattice(irrelevant tocritical behavior)
hard-core bosons↑：presence↓：absence⇔Numerical test of RG phase diagram (exact diagonalization)
PT-symmetric spin-chain model:⇔

Correspondence to the effective field theory:
Typical low-energy excitation spectrum

BKT transition point:crossing of appropriate energy levels(level spectroscopy)
K. Nomura, J. Phys. A 28, 5451 (1995)

PT transi/on point:
merging of two energy levels
with square-root scaling(characteristic of exceptional point)
T. Kato, Perturbation theory for linear
operators (1966)

YA, S. Furukawa and M. Ueda, Nat. Commun. 8, 15791 (2017).

Numerical test of RG phase diagram (exact diagonalization)
PT-symmetric spin-chain model:

Correspondence to the effective field theory:
Phase diagram (fixed hs)⇔
fixed gc

Numerical supports for RG phase diagram.

YA, S. Furukawa and M. Ueda, Nat. Commun. 8, 15791 (2017).

Numerical test of anomalous RG flows (iTEBD)
PT-symmetric spin-chain model:

Correspondence to the effective field theory:⇔

PT broken (gi>gr)
Semicircular RG flows:
large-distance • Varying critical exponent in a larger distance• Violation of c-theorem
behavior
Method:infinite time-evolving block decimation (iTEBD)
algorithm (G. Vidal, PRL 98, 070201 (2007))
imaginary-time evolution → ground state

YA, S. Furukawa and M. Ueda, Nat. Commun. 8, 15791 (2017).

Numerical test of anomalous RG flows (iTEBD)
PT-symmetric spin-chain model:⇔

Correspondence to the effective field theory:

Varying TLL parameter K

TLL parametercorrelation function

Critical decay in PT broken phase

-1

10

χ=60
χ=100
χ=900
101

102
distance r

YA, S. Furukawa and M. Ueda, Nat. Commun. 8, 15791 (2017).

Numerical test of anomalous RG flows (iTEBD)
PT-symmetric spin-chain model:⇔

Correspondence to the effective field theory:
RG analysis

Varying TLL parameter K

Anomalouslarge-distance
RG flows violating the c-theorem
behavior
have
been numerically verified!
* Violation of the g-theorem has been also found in
the non-Hermitian Kondo model:
M. Nakagawa et al., arXiv:1806.04039

Semicircular RG flows= Enhancement of TLL parameterin a larger distance

YA, S. Furukawa and M. Ueda, Nat. Commun. 8, 15791 (2017).

1. Quantum many-body systems
under continuous observation
Wavefunction collapse and super-resolved imaging:
YA and M. Ueda, PRL 115, 095301 (2015).
YA and M. Ueda, Opt. Lett. 41, 72-75 (2016).
Quantum critical phenomena:
YA, S. Furukawa and M. Ueda, Nat. Commun. 8, 15791 (2017).
YA, S. Furukawa and M. Ueda, PRA 94, 053615 (2016).
Nonequilibrium dynamics (short summary):
YA and M. Ueda, PRL 120, 185301 (2018).
YA and M. Ueda, PRA 95, 022124 (2017).

Full-counting “dynamics” under continuous observation

Hilbert spacen quantum jumps

・・・
jump

0

t

time

Post-measurement density matrix conditioned on the number of jumps n :: subspace of quantum trajectories
having n jumps*different from the unconditional dynamics (= Lindblad dynamics)

Full-counting “dynamics” under continuous observation

Hilbert spacen quantum jumps

・・・
jump

0

New types of many-particle phenomena
unique to conditional dynamics?
ttime

Post-measurement density matrix conditioned on the number of jumps n :: subspace of quantum trajectories
having n jumps*different from the unconditional dynamics (= Lindblad dynamics)

Propagation beyond the Lieb-Robinson bound (summary)
Setup
time

0

N atoms
continuous observation
jump : one-body lossn quantum jumps
N - n atomst

Measuring total atom number by QGM

Conditional dynamics
Ensemble average over trajectories
including n jumps during [0,t]YA and M. Ueda, PRL 120, 185301 (2018).

Propagation beyond the Lieb-Robinson bound (summary)
Exact calculations of quench dynamics for solvable model
Lindblad dynamics:

Lieb-Robinson (LR) bound for Lindblad dynamics
Poulin PRL 104, 190401 2010

YA and M. Ueda, PRL 120, 185301 (2018).

Propagation beyond the Lieb-Robinson bound (summary)
Exact calculations of quench dynamics for solvable model
Lindblad dynamics:
conditional dynamics:
Lieb-Robinson (LR) bound for Lindblad dynamics
Poulin PRL 104, 190401 2010

YA and M. Ueda, PRL 120, 185301 (2018).

Propagation beyond the Lieb-Robinson bound (summary)
Exact calculations of quench dynamics for solvable model
Lindblad dynamics:
conditional
dynamics:
Propagation
beyond the LR bound can (probabilistically)

appear as a result of the global postselection.

Lieb-Robinson (LR) bound for Lindblad dynamics
Poulin PRL 104, 190401 2010

YA and M. Ueda, PRL 120, 185301 (2018).

Classical non-Hermitian systems: when continuous
observation reduces to one-body dissipative dynamics
Continuous observation(non-Hermitian):
Markovian dissipative dynamics:

Sometimes, continuous observation and dissipation can be equivalent and reduce toclassical (one-body) dynamics:
Case I: Loss process in the single-particle sector
annihilates the particle → Jump term is trivial<latexit sha1_base64="jAjH4QC5YEpRf3IJiueL8afmddg=">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</latexit><latexit sha1_base64="/6PHmd1gexyNK75D9sP9ioh0Q0E=">AAACtnichVG7btRAFD0xj4TlkQUaJJoVqyCq1TVCSkQVQZOCIpuwSaQ4WY2didfK2DMaz64UHP8AP5CCCiSEEB/AB9DwAxQpaEGIMkg0FNz1WkEQAdfy3DNn7rlzZiY0Kskd0dGUd+bsufPTMxcaFy9dvjLbvHptLddDG8lepJW2G6HIpUoy2XOJU3LDWCnSUMn1cO/heH19JG2e6Oyx2zdyKxVxluwmkXBM9ZvdYCBc8ajsi1aFAjvQ5Qm3XQQ7Io6lLQNjtXH6gAvS1khEZWBFFisZqCo1TviDfrNNHaqidRr4NWijjmXdfIUAO9CIMEQKiQyOsYJAzt8mfBAMc1somLOMkmpdokSDtUOuklwhmN3jMebZZs1mPB/3zCt1xLso/i0rW5ijD/Sajuk9vaEv9OOvvYqqx9jLPudwopWmP/v0xur3/6pSzg6DX6p/enbYxULlNWHvpmLGp4gm+tGTw+PV+ytzxW16QV/Z/3M6ond8gmz0LXrZlSvP0OAH8P+87tNg7W7Hp47fvddefFA/xQxu4hbu8H3PYxFLWEaP932Lj/iEz96Ct+1JL56UelO15jp+C8/8BCcArD4=</latexit>

Case II: Loss process with a coherent state(e.g., classical waves of photons and phonons, or BEC)
is an eigenstate of<latexit sha1_base64="jAjH4QC5YEpRf3IJiueL8afmddg=">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</latexit>→

Jump term just modifies the normalization.

<latexit sha1_base64="7zRU4dGanDfRspXpxKXUprLUt48=">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</latexit>(Essentially) classical phenomena, yet still interesting anddirectly relevant to experiments in classical optics & mechanics.
Topological aspects of non-Hermitian systems?

Topological aspects of non-Hermitian systems (short summary)Definition of allowed continuous deformations
Band gap not closed= EF ( ) not touched.

⇨ EB ( ) not touched.

Two non-Hermitian Hamiltonians are equivalent ! " ~ !′ "iff ∃ !& " (' ∈ [0,1]) s.t.
(i) !. " = ! " , !0 " = !′ " and(ii) det !& " − 56 ≠ 0 ∀ ' ∈ [0,1] and " ∈ B. Z.
Consistent with the definition in Hermitian systems.
Parameter fine tuning not required → “Generic” nonunitary operation allowed.
Gong, YA, Kawabata, Takasan, Higashikawa and Ueda, arXiv:1802.07964 [to appear in PRX]

Topological aspects of non-Hermitian systems (short summary)
Minimal model:
Asymmetric hopping of a single particle on 1D ring.

[c.f. Hatano and Nelson, PRL 77, 570 (1996)]

Z ⇡dk
Tr[H 1 (k)@k H(k)]
⇡ 2⇡i
Z ⇡
dk E 0 (k)w=⇡ 2⇡i E(k)w=<latexit sha1_base64="dGvz2I7mXzwUrHWMNQJ9o9F3tkY=">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</latexit><latexit sha1_base64="U6A1WizZHlw8B4M5uAD+DNJBx+A=">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</latexit>*Robust against on-site disorder*General classification possible(in parallel with Floquet systems
c.f. Roy & Harper, PRB 96, 155118 (2017))
Gong, YA, Kawabata, Takasan, Higashikawa and Ueda, arXiv:1802.07964 [to appear in PRX]

Title:Open quantum physics: non-Hermitian systems and beyond
So far, we focused on open quantum systems (weakly) coupled toa Markovian environment.
What happens if strong system-environment entanglement is present?
(Theoretical analysis beyond the Born-Markov approximation needed).

2. Strongly correlated open systems:
Nonequilibrium quantum impurities
YA, T. Shi, M. C. Bañuls, J. I. Cirac and E. Demler, PRL 121, 026805 (2018).
YA, T. Shi, M. C. Bañuls, J. I. Cirac and E. Demler, PRB 98, 024103 (2018).
M. Kanász-Nagy, YA, T. Shi, C. P. Moca, T. N. Ikeda, S. Foelling, J. I. Cirac,
G. Zarand and E. Demler, PRB 97, 155156 (2018).
YA, R. Schmidt, L. Tarruell and E. Demler, PRB 97, 060302(R) (2018).

Quantum impurity: a paradigm in many-body physics
・Relevance to a variety of physical systems:
quantum dots

Goldhaber-Gordon et al.,Nature 391, 156 (1998).

heavy electron materials

ultracold atoms

Monthoux et al.,Nature 450, 1177 (2007).

Physics 9, 86 (2016).

・Prototypical open system ・Numerical method
decoherence

Shin et al., PRB 88, 161412 (2013).

DMFT solver

From Web page of LMU
theoretical nanophysics.

・Universal quantum
computation

M. C. Tran & J. M. Taylor,
arXiv:1801.04006

Quantum impurity: a paradigm in many-body physics
・Relevance to a variety of physical systems:
quantum dotsheavy electron materials

ultracold atoms

New theoretical approach to solve generic quantum spin
Goldhaber-Gordon et al.,Monthoux et al.,impurity
problems(including
Kondo or centralPhysics
spin9, models)?
86 (2016).
Nature 391, 156 (1998).
Nature 450, 1177 (2007).
・Prototypical open system ・Numerical method
decoherence

Shin et al., PRB 88, 161412 (2013).

DMFT solver

From Web page of LMU
theoretical nanophysics.

・Universal quantum
computation

M. C. Tran & J. M. Taylor,
arXiv:1801.04006

New “disentangling” canonical transformation
Essential feature of spin-impurity systems:
strong entanglement between impurity and bath(*this strong correlation invalidates the Markov approximation).
Kondo singlet

impurityspin 1/2

itinerant fermionsNat. Commun. 8, 15210 (2017)

New “disentangling” canonical transformation
Construct a “disentangling” canonical transformation U
Original frame
impurityspin 1/2
itinerant
fermions

Strong entanglement“Corotating” frame
canonical
transformation

Impurity spin is decoupled from bath!

Efficient variational states:
Gaussian states
YA, T. Shi, M. C. Bañuls, J. I. Cirac and E. Demler, PRL 121, 026805 (2018).
YA, T. Shi, M. C. Bañuls, J. I. Cirac and E. Demler, PRB 98, 024103 (2018).

New “disentangling” canonical transformation
1. Notice parity symmetry in a Hamiltonian of a generic spin-impurity system:

invariance under π rotation of impurity and bath spins around z axis:

spin operatorof bath mode n:parity operator:
bath parity:

2. Find the unitary transformation U mapping the parity to the impurity spin-1/2:
impurity is decoupled !
Transformed Hamiltonian
effective bath-bath interaction
mediated by the impurity

is c-number.

New “disentangling” canonical transformation
1. Notice parity symmetry in a Hamiltonian of a generic spin-impurity system:

invariance under π rotation of impurity and bath spins around z axis:

Time-dependent variational principle
spin operatorof bath mode n:↓bath parity:
Efficient variational imaginaryand real-time evolutions
2. Find the unitary transformation U mapping the parity to the impurity spin-1/2:
applicable to in- and out-of-equilibrium problems!
parity operator:

impurity is decoupled !
Transformed Hamiltonian
effective bath-bath interaction
mediated by the impurity

is c-number.

Benchmark tests with MPS results in and out of equilibrium
Ground-state energies (L=200)
red: MPS (D=280)
blue: our variational states

FM

AFM

Quench dynamics
of the impurity

Benchmark tests with MPS results in and out of equilibrium
Ground-state energies (L=200)
red: MPS (D=280)
blue: our variational states

Quench dynamics
of the impurity

State-of-the-art accuracies withorders of magnitude fewer variational parameters:
FM

AFM# of parameters
MPS: LD^2〜10^6~7
Our approach: L^2〜10^4machine timea few hours-daysa few minutes-hours

Spatiotemporal dynamics of correlations after quench
initial state:
Quench the impurity-bath interaction at site x=0.
Real-time formation and spread of
the impurity-bath correlation?

x=0
time

Spatiotemporal dynamics of correlations after quench(i) FM phase: formation of triplet correlation(ii) AFM phase: formation of singlet correlation(iii) FM→AFM crossover: mimicking RG flow in time.

impurity-bath spin correlationFMSpatiotemporal dynamics of correlations after quench(i) FM phase: formation of triplet correlation(ii) AFM phase: formation of singlet correlation(iii) FM→AFM crossover: mimicking RG flow in time.

impurity-bath spin correlation
FM

AFM

Spatiotemporal dynamics of correlations after quench(iii) FM→AFM crossover : mimicking RG flow in timeshort-time = high-energy physics → FM (Jz<0)long-time = low-energy physics → AFM (J>0)
Effective temperature
after quench:
Nordlander et al., PRL 83, 808 (1999)

impurity-bath spin correlation
FM

AFM

AFM
crossoverSpatiotemporal dynamics of correlations after quench(iii) FM→AFM crossover : mimicking RG flow in timeshort-time = high-energy physics → FM (Jz<0)long-time = low-energy physics → AFM (J>0)
Effective temperature
after quench:
Nordlander et al., PRL 83, 808 (1999)

impurity-bath spin correlation
FM

AFM

AFM
crossoverSpatiotemporal dynamics of correlations after quench(iii) FM→AFM crossover : mimicking RG flow in timeshort-time = high-energy physics → FM (Jz<0)long-time = low-energy physics → AFM (J>0)
Effective temperature
after quench:
Nordlander et al., PRL 83, 808 (1999)

impurity-bathcorrelation dynamics
Long-&mespin
many-body

regime.
AFMunexplored
AFM
FM in previously
crossoverSpatiotemporal dynamics of correlations after quench
Experimental proposal
with Yb atoms:
M. Kanász-Nagy, YA, T. Shi, C. P. Moca,
T. N. Ikeda, S. Foelling, J. I. Cirac,
G. Zarand and E. Demler, arXiv:1801.01132✔ Spatiotemporal dynamics can be measured with QGM.
✔ Both FM and AFM regimes can be reached by Floquet engineering.
FM

AFM

AFM
crossoverSummary: Quantum Many-Body Physics in Open Systems
I: Many-body physics under continuous observation & Non-Hermitian physics- New types of critical phenomena, nonequilibrium dynamics and topology.
YA and M. Ueda, PRL 115, 095301 (2015).
YA, S. Furukawa and M. Ueda, Nat. Commun. 8, 15791 (2017).
YA and M. Ueda, PRL 120, 185301 (2018).
Z. Gong, YA, K. Kawabata, K. Takasan, S. Higashikawa
and M. Ueda, arXiv:1802.07964 [to appear in PRX].

II: Strongly correlated open quantum systems: nonequilibrium quantum impurity
Versatile and efficient theoretical approach to generic spin-impurity systems.
- Applicable to in- and out-of-equilibrium problems.
YA, T. Shi, M. C. Bañuls, J. I. Cirac and E. Demler, PRL 121, 026805 (2018).
YA, T. Shi, M. C. Bañuls, J. I. Cirac and E. Demler, PRB 98, 024103 (2018).