Two level systems, geometric phases, and...
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Two level systems, geometric phases,and differential operators of Dunkl type
Alexander Moroz
Wave-scattering.com
Topological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016
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Overview
Motivation
Two-levels systems (TLS) in a classical field
Two-levels systems (TLS) in a quantized field
The Fulton-Gouterman reduction
Differential operators of Dunkl type
Consequences of tridiagonality
“Quantum” integrability
Conclusions
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Overview
Motivation
Two-levels systems (TLS) in a classical field
Two-levels systems (TLS) in a quantized field
The Fulton-Gouterman reduction
Differential operators of Dunkl type
Consequences of tridiagonality
“Quantum” integrability
Conclusions
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 2
/ 86
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Overview
Motivation
Two-levels systems (TLS) in a classical field
Two-levels systems (TLS) in a quantized field
The Fulton-Gouterman reduction
Differential operators of Dunkl type
Consequences of tridiagonality
“Quantum” integrability
Conclusions
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 2
/ 86
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Overview
Motivation
Two-levels systems (TLS) in a classical field
Two-levels systems (TLS) in a quantized field
The Fulton-Gouterman reduction
Differential operators of Dunkl type
Consequences of tridiagonality
“Quantum” integrability
Conclusions
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 2
/ 86
![Page 6: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/6.jpg)
Overview
Motivation
Two-levels systems (TLS) in a classical field
Two-levels systems (TLS) in a quantized field
The Fulton-Gouterman reduction
Differential operators of Dunkl type
Consequences of tridiagonality
“Quantum” integrability
Conclusions
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 2
/ 86
![Page 7: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/7.jpg)
Overview
Motivation
Two-levels systems (TLS) in a classical field
Two-levels systems (TLS) in a quantized field
The Fulton-Gouterman reduction
Differential operators of Dunkl type
Consequences of tridiagonality
“Quantum” integrability
Conclusions
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 2
/ 86
![Page 8: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/8.jpg)
Overview
Motivation
Two-levels systems (TLS) in a classical field
Two-levels systems (TLS) in a quantized field
The Fulton-Gouterman reduction
Differential operators of Dunkl type
Consequences of tridiagonality
“Quantum” integrability
Conclusions
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 2
/ 86
![Page 9: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/9.jpg)
Overview
Motivation
Two-levels systems (TLS) in a classical field
Two-levels systems (TLS) in a quantized field
The Fulton-Gouterman reduction
Differential operators of Dunkl type
Consequences of tridiagonality
“Quantum” integrability
Conclusions
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 2
/ 86
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Motivation
Fundamental physics
Moore’s law - the number of transistors in a dense integrated circuitdoubles approximately every two years
ultimate structural limit of individual atomsholonomic quantum computation using quantum gates implementingthe geometric phasesexperimental advances
A majority of systems discussed at this workshop have beenintrinsically TLS - cf. H = h · σ
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Motivation
Fundamental physics
Moore’s law - the number of transistors in a dense integrated circuitdoubles approximately every two years
ultimate structural limit of individual atomsholonomic quantum computation using quantum gates implementingthe geometric phasesexperimental advances
A majority of systems discussed at this workshop have beenintrinsically TLS - cf. H = h · σ
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 3
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Motivation
Fundamental physics
Moore’s law - the number of transistors in a dense integrated circuitdoubles approximately every two years
ultimate structural limit of individual atoms
holonomic quantum computation using quantum gates implementingthe geometric phasesexperimental advances
A majority of systems discussed at this workshop have beenintrinsically TLS - cf. H = h · σ
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 3
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Motivation
Fundamental physics
Moore’s law - the number of transistors in a dense integrated circuitdoubles approximately every two years
ultimate structural limit of individual atomsholonomic quantum computation using quantum gates implementingthe geometric phases
experimental advances
A majority of systems discussed at this workshop have beenintrinsically TLS - cf. H = h · σ
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 3
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Motivation
Fundamental physics
Moore’s law - the number of transistors in a dense integrated circuitdoubles approximately every two years
ultimate structural limit of individual atomsholonomic quantum computation using quantum gates implementingthe geometric phasesexperimental advances
A majority of systems discussed at this workshop have beenintrinsically TLS - cf. H = h · σ
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 3
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Motivation
Fundamental physics
Moore’s law - the number of transistors in a dense integrated circuitdoubles approximately every two years
ultimate structural limit of individual atomsholonomic quantum computation using quantum gates implementingthe geometric phasesexperimental advances
A majority of systems discussed at this workshop have beenintrinsically TLS - cf. H = h · σ
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 3
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Berry’s observation - Proc. R. Soc. A 429, 61 (1990)
The most general 2× 2 form of the hermitian operator H,
H(τ) = Aσ0 + (B cosφ)σ1 + (B sinφ)σ2 + Dσ3
=
(A(τ) + D(τ) B(τ) exp−iφ(τ)
B(τ) expiφ(τ) A(τ)− D(τ)
)
can be brought into a unitary equivalent real, traceless and symmetricdiabatic form
H(τ) =
(D(τ) B(τ)
B(τ) −D(τ)
)= B(τ)σ1 + D(τ)σ3
B(τ) ... detuning ∆(τ)D(τ) = D(τ)− ~
2 φ(τ) ... Rabi frequency Ω(τ)
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Berry’s observation - Proc. R. Soc. A 429, 61 (1990)
The most general 2× 2 form of the hermitian operator H,
H(τ) = Aσ0 + (B cosφ)σ1 + (B sinφ)σ2 + Dσ3
=
(A(τ) + D(τ) B(τ) exp−iφ(τ)
B(τ) expiφ(τ) A(τ)− D(τ)
)can be brought into a unitary equivalent real, traceless and symmetricdiabatic form
H(τ) =
(D(τ) B(τ)
B(τ) −D(τ)
)= B(τ)σ1 + D(τ)σ3
B(τ) ... detuning ∆(τ)D(τ) = D(τ)− ~
2 φ(τ) ... Rabi frequency Ω(τ)
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A canonical adiabatic form
Given an alternative diabatic form
H(τ) = ρ(τ)
(cos(θ(τ)) sin(θ(τ))sin(θ(τ)) − cos(θ(τ))
)ρ(τ) =
√D2 + B2 ... the length of a field vector, θ(τ) ... the angle it
makes with a z-axis, the adiabatic form is obtained by unitarytransforming with
U(τ) =
(cos(θ/2) − sin(θ/2)sin(θ/2) cos(θ/2)
)
H(τ)→ U−1H(τ)U =
(ρ(τ) ig(τ)−ig(τ) −ρ(τ)
)= −g(τ)σ2 + ρ(τ)σ3
where the so-called adiabatic coupling
g(t) ≡ −〈χ+|χ−(τ)〉 = ∆(τ)Ω(τ)−∆(τ)Ω(τ)2(∆2(τ)+Ω2(τ))
= θ(τ)2
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A canonical adiabatic form
Given an alternative diabatic form
H(τ) = ρ(τ)
(cos(θ(τ)) sin(θ(τ))sin(θ(τ)) − cos(θ(τ))
)ρ(τ) =
√D2 + B2 ... the length of a field vector, θ(τ) ... the angle it
makes with a z-axis, the adiabatic form is obtained by unitarytransforming with
U(τ) =
(cos(θ/2) − sin(θ/2)sin(θ/2) cos(θ/2)
)
H(τ)→ U−1H(τ)U =
(ρ(τ) ig(τ)−ig(τ) −ρ(τ)
)= −g(τ)σ2 + ρ(τ)σ3
where the so-called adiabatic coupling
g(t) ≡ −〈χ+|χ−(τ)〉 = ∆(τ)Ω(τ)−∆(τ)Ω(τ)2(∆2(τ)+Ω2(τ))
= θ(τ)2
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Any TLS makes a cyclic evolution
A TLS Hamiltonian of the general two-level system
H =
(α ββ∗ −α
)generates a cyclic evolution with the period
τp =2π
λ1 − λ2, λ1,2 = ±2
√α2 + |β|2
and total phase
θt = − 2πλ2
λ1 − λ2= − 2πλ1
λ1 − λ2mod 2π
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Any TLS makes a cyclic evolution
The corresponding eigenvectors are
|ψ1〉 =
(cos(ω/2) e iχ/2
sin(ω/2) e−iχ/2
), |ψ2〉 =
(− sin(ω/2) e iχ/2
cos(ω/2) e−iχ/2
)where
χ = arg(β) and
ω = arccos(σ/√
1 + σ2), σ =α
|β|, λ1,2 = ±|β|
√1 + σ2
[Z. Tang and D. Finkelstein, PRL74, 3134 (1995)]
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Any TLS makes a cyclic evolution
The corresponding eigenvectors are
|ψ1〉 =
(cos(ω/2) e iχ/2
sin(ω/2) e−iχ/2
), |ψ2〉 =
(− sin(ω/2) e iχ/2
cos(ω/2) e−iχ/2
)where
χ = arg(β) and
ω = arccos(σ/√
1 + σ2), σ =α
|β|, λ1,2 = ±|β|
√1 + σ2
[Z. Tang and D. Finkelstein, PRL74, 3134 (1995)]
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Geometric phase
adiabatic evolution
closed paths in a parameter space
cyclic quantum evolution
All that is irrelevant!
The geometric phase (GP) is a Hamiltonian-independent, nonintegrablecomponent of the total phase, depending exclusively on the geometryin the ray spaceThe Abelian GP can be defined without reference to the underlyingdynamical mechanism that drives the evolution
[J. Samuel and R. Bhandari, PRL 60, 2339 (1988)]
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Geometric phase
adiabatic evolution
closed paths in a parameter space
cyclic quantum evolution
All that is irrelevant!
The geometric phase (GP) is a Hamiltonian-independent, nonintegrablecomponent of the total phase, depending exclusively on the geometryin the ray spaceThe Abelian GP can be defined without reference to the underlyingdynamical mechanism that drives the evolution
[J. Samuel and R. Bhandari, PRL 60, 2339 (1988)]
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Geometric phase
adiabatic evolution
closed paths in a parameter space
cyclic quantum evolution
All that is irrelevant!
The geometric phase (GP) is a Hamiltonian-independent, nonintegrablecomponent of the total phase, depending exclusively on the geometryin the ray spaceThe Abelian GP can be defined without reference to the underlyingdynamical mechanism that drives the evolution
[J. Samuel and R. Bhandari, PRL 60, 2339 (1988)]
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Geometric phase
adiabatic evolution
closed paths in a parameter space
cyclic quantum evolution
All that is irrelevant!
The geometric phase (GP) is a Hamiltonian-independent, nonintegrablecomponent of the total phase, depending exclusively on the geometryin the ray spaceThe Abelian GP can be defined without reference to the underlyingdynamical mechanism that drives the evolution
[J. Samuel and R. Bhandari, PRL 60, 2339 (1988)]
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Geometric phase
adiabatic evolution
closed paths in a parameter space
cyclic quantum evolution
All that is irrelevant!
The geometric phase (GP) is a Hamiltonian-independent, nonintegrablecomponent of the total phase, depending exclusively on the geometryin the ray space
The Abelian GP can be defined without reference to the underlyingdynamical mechanism that drives the evolution
[J. Samuel and R. Bhandari, PRL 60, 2339 (1988)]
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 8
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Geometric phase
adiabatic evolution
closed paths in a parameter space
cyclic quantum evolution
All that is irrelevant!
The geometric phase (GP) is a Hamiltonian-independent, nonintegrablecomponent of the total phase, depending exclusively on the geometryin the ray spaceThe Abelian GP can be defined without reference to the underlyingdynamical mechanism that drives the evolution
[J. Samuel and R. Bhandari, PRL 60, 2339 (1988)]
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Geometric phase
adiabatic evolution
closed paths in a parameter space
cyclic quantum evolution
All that is irrelevant!
The geometric phase (GP) is a Hamiltonian-independent, nonintegrablecomponent of the total phase, depending exclusively on the geometryin the ray spaceThe Abelian GP can be defined without reference to the underlyingdynamical mechanism that drives the evolution
[J. Samuel and R. Bhandari, PRL 60, 2339 (1988)]
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Geometric phase
The geometric phase is defined as
γ(C) = arg〈ψ(0)|ψ(s)〉+ i
∫C〈ψ(s ′)| d
ds ′|ψ(s ′)〉 ds ′
where C denotes the path (needn’t be closed) that joins the initialstate |ψ(0)〉 with the final state |ψ(s)〉
This definition is independent of the dynamics that governs theevolution of the state |ψ(s ′)〉, as long as this evolution fixes the pathC : s ′ ∈ [0, s]→ |ψ(s ′)〉The geometric phase is the sum of a total Pancharatnam phasearg〈ψ(0)|ψ(s)〉 and a dynamic phase
i
∫C〈ψ(s ′)|dψ(s ′)〉
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Geometric phase
The geometric phase is defined as
γ(C) = arg〈ψ(0)|ψ(s)〉+ i
∫C〈ψ(s ′)| d
ds ′|ψ(s ′)〉 ds ′
where C denotes the path (needn’t be closed) that joins the initialstate |ψ(0)〉 with the final state |ψ(s)〉This definition is independent of the dynamics that governs theevolution of the state |ψ(s ′)〉, as long as this evolution fixes the pathC : s ′ ∈ [0, s]→ |ψ(s ′)〉
The geometric phase is the sum of a total Pancharatnam phasearg〈ψ(0)|ψ(s)〉 and a dynamic phase
i
∫C〈ψ(s ′)|dψ(s ′)〉
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Geometric phase
The geometric phase is defined as
γ(C) = arg〈ψ(0)|ψ(s)〉+ i
∫C〈ψ(s ′)| d
ds ′|ψ(s ′)〉 ds ′
where C denotes the path (needn’t be closed) that joins the initialstate |ψ(0)〉 with the final state |ψ(s)〉This definition is independent of the dynamics that governs theevolution of the state |ψ(s ′)〉, as long as this evolution fixes the pathC : s ′ ∈ [0, s]→ |ψ(s ′)〉The geometric phase is the sum of a total Pancharatnam phasearg〈ψ(0)|ψ(s)〉 and a dynamic phase
i
∫C〈ψ(s ′)|dψ(s ′)〉
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Geometric phase
The geometric phase is
parameter independent, i.e., invariant under s → s ′ = s ′(s)
invariant under local gauge transformations:
|ψ(s)〉 → |ψ(s)〉 = exp[iα(s)]|ψ(s)〉
(can be used to nullify either of the two contributions)
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Geometric phase
The geometric phase is
parameter independent, i.e., invariant under s → s ′ = s ′(s)
invariant under local gauge transformations:
|ψ(s)〉 → |ψ(s)〉 = exp[iα(s)]|ψ(s)〉
(can be used to nullify either of the two contributions)
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Geometric phase for a TLS
For an arbitrary normalized initial state
|ψ0〉 =
(cos(ω0/2) e iχ0/2
sin(ω0/2) e−iχ0/2
)= c |ψ1〉+ d |ψ2〉, |c |2 + |d |2 = 1
the total phase θt = 2πλ2/(λ1 − λ2) is the sum of a dynamic phase
θd = −∫ τp
0〈ψ(t)|H |ψ(t)〉 dt = − 2π
λ1 − λ2(|c |2λ1 + |d |2λ2)
and the energy independent geometric phase
θg = θt − θd
= − 2πλ2
λ1 − λ2+
2π
λ1 − λ2[|c |2λ1 + (1− |c |2)λ2]
= 2π|c |2
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Geometric phase for a TLS
For an arbitrary normalized initial state
|ψ0〉 =
(cos(ω0/2) e iχ0/2
sin(ω0/2) e−iχ0/2
)= c |ψ1〉+ d |ψ2〉, |c |2 + |d |2 = 1
the total phase θt = 2πλ2/(λ1 − λ2) is the sum of a dynamic phase
θd = −∫ τp
0〈ψ(t)|H |ψ(t)〉 dt = − 2π
λ1 − λ2(|c |2λ1 + |d |2λ2)
and the energy independent geometric phase
θg = θt − θd
= − 2πλ2
λ1 − λ2+
2π
λ1 − λ2[|c |2λ1 + (1− |c |2)λ2]
= 2π|c |2
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Geometric phase for a TLS
For an arbitrary normalized initial state
|ψ0〉 =
(cos(ω0/2) e iχ0/2
sin(ω0/2) e−iχ0/2
)= c |ψ1〉+ d |ψ2〉, |c |2 + |d |2 = 1
the total phase θt = 2πλ2/(λ1 − λ2) is the sum of a dynamic phase
θd = −∫ τp
0〈ψ(t)|H |ψ(t)〉 dt = − 2π
λ1 − λ2(|c |2λ1 + |d |2λ2)
and the energy independent geometric phase
θg = θt − θd
= − 2πλ2
λ1 − λ2+
2π
λ1 − λ2[|c |2λ1 + (1− |c |2)λ2]
= 2π|c |2
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Motivation for TLS in a quantized field
The Berry phase is being mostly studied only in a semiclassicalcontext, when field itself has never been quantized
The effects of the vacuum field on the geometric evolution areunknown
Many effects in quantum optics such as quantum jumps, collapsesand revivals of the Rabi oscillations, can only be explained byconsidering a quantum field, showing the importance of fieldquantization in the complete description of physical systems
Many interesting effects are observed due to the interaction ofquantum systems with the vacuum (spontaneous emission, the Lambshift)
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Motivation for TLS in a quantized field
The Berry phase is being mostly studied only in a semiclassicalcontext, when field itself has never been quantized
The effects of the vacuum field on the geometric evolution areunknown
Many effects in quantum optics such as quantum jumps, collapsesand revivals of the Rabi oscillations, can only be explained byconsidering a quantum field, showing the importance of fieldquantization in the complete description of physical systems
Many interesting effects are observed due to the interaction ofquantum systems with the vacuum (spontaneous emission, the Lambshift)
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Motivation for TLS in a quantized field
The Berry phase is being mostly studied only in a semiclassicalcontext, when field itself has never been quantized
The effects of the vacuum field on the geometric evolution areunknown
Many effects in quantum optics such as quantum jumps, collapsesand revivals of the Rabi oscillations, can only be explained byconsidering a quantum field, showing the importance of fieldquantization in the complete description of physical systems
Many interesting effects are observed due to the interaction ofquantum systems with the vacuum (spontaneous emission, the Lambshift)
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Motivation for TLS in a quantized field
The Berry phase is being mostly studied only in a semiclassicalcontext, when field itself has never been quantized
The effects of the vacuum field on the geometric evolution areunknown
Many effects in quantum optics such as quantum jumps, collapsesand revivals of the Rabi oscillations, can only be explained byconsidering a quantum field, showing the importance of fieldquantization in the complete description of physical systems
Many interesting effects are observed due to the interaction ofquantum systems with the vacuum (spontaneous emission, the Lambshift)
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Rabi model Hamiltonian HR
HR = ~ω1a†a + µσ3 + ~gσ1(a† + a)
= ~ω1a†a + µσ3 + ~g(σ+ + σ−)(a† + a)
= ~ω1a†a + µσ3 + ~g(σ+a + σ−a†) + ~g(σ+a
† + σ−a)
HGR = ~ω1a†a + µσ3 + ~g1(σ+a + σ−a†) + ~g2(σ+a
† + σ−a)
HJC = ~ω1a†a + µσ3 + ~g(σ+a + σ−a†)
ω ... cavity mode frequency, [a, a†] = 11 ... the unit matrix, g ... a dipole coupling constantµ = ~ω0/2, ω0 ... TLS resonance frequencyσj ... the Pauli matrices∆ = µ/ω∆′ = (ω0 − ω)/ω ... relative detuning
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Rabi model coupling regimes
strong coupling regime: dimensionless coupling strengthκ = g/ω . 10−2 - the physics of the Rabi model is well captured bythe analytically solvable approximate Jaynes and Cummings model(JCM) [S. Haroche, Nobel prize 2012]
ultrastrong coupling regime: κ & 0.1 - the validity of therotating-wave approximation (RWA) breaks down and the relevantphysics can only be described by the full Rabi model - since lastdecade achievable experimentally
deep strong coupling regime: κ & 1 - the relevance of the Rabimodel becomes even more prominent - expected to be achieved soon
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Rabi model coupling regimes
strong coupling regime: dimensionless coupling strengthκ = g/ω . 10−2 - the physics of the Rabi model is well captured bythe analytically solvable approximate Jaynes and Cummings model(JCM) [S. Haroche, Nobel prize 2012]
ultrastrong coupling regime: κ & 0.1 - the validity of therotating-wave approximation (RWA) breaks down and the relevantphysics can only be described by the full Rabi model - since lastdecade achievable experimentally
deep strong coupling regime: κ & 1 - the relevance of the Rabimodel becomes even more prominent - expected to be achieved soon
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Rabi model coupling regimes
strong coupling regime: dimensionless coupling strengthκ = g/ω . 10−2 - the physics of the Rabi model is well captured bythe analytically solvable approximate Jaynes and Cummings model(JCM) [S. Haroche, Nobel prize 2012]
ultrastrong coupling regime: κ & 0.1 - the validity of therotating-wave approximation (RWA) breaks down and the relevantphysics can only be described by the full Rabi model - since lastdecade achievable experimentally
deep strong coupling regime: κ & 1 - the relevance of the Rabimodel becomes even more prominent - expected to be achieved soon
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Geometric phase
Submit the system to a unitary transformation
U(ϕ(t)) = exp[−iϕ(t)a†a]
A nonvanishing geometric phase
γn = i
∮C〈ψn|U†(ϕ)
d
dϕU(ϕ)|ψn〉 dϕ = 2π〈ψn|a†a|ψn〉
with |ψn〉 being the nth eigenstate of the considered Hamiltonian,appears both in the Jaynes-Cummings model and in the Rabi model
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Geometric phase
Submit the system to a unitary transformation
U(ϕ(t)) = exp[−iϕ(t)a†a]
A nonvanishing geometric phase
γn = i
∮C〈ψn|U†(ϕ)
d
dϕU(ϕ)|ψn〉 dϕ = 2π〈ψn|a†a|ψn〉
with |ψn〉 being the nth eigenstate of the considered Hamiltonian,appears both in the Jaynes-Cummings model and in the Rabi model
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Geometric phase for the Rabi model
Dashed lines correspond to phases associated to the eigenstates of the JCM
[J. Calderon and F. De Zela, PRA 93, 033823 (2015)]
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Fulton-Gouterman form
Any 2× 2 hermitian operator H can be expressed in the form
H =3∑
j=0
hjσj , hj = 12 Tr (Hσj) (1)
where hj ’s are one-dimensional operators in a suitable Hilbert space H.
H is of the Fulton-Gouterman type, and denoted by HFG ,
(*) if the expansion coefficients hj of H in Eq. (1),
HFG = A1 + Bσ1 + Cσ2 + Dσ3,
satisfy[R,A] = [R,B] = 0, R,C = R,D = 0
for some hermitian mirror-like operator R, R2 ≡ 1
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Fulton-Gouterman form
Any 2× 2 hermitian operator H can be expressed in the form
H =3∑
j=0
hjσj , hj = 12 Tr (Hσj) (1)
where hj ’s are one-dimensional operators in a suitable Hilbert space H.
H is of the Fulton-Gouterman type, and denoted by HFG ,
(*) if the expansion coefficients hj of H in Eq. (1),
HFG = A1 + Bσ1 + Cσ2 + Dσ3,
satisfy[R,A] = [R,B] = 0, R,C = R,D = 0
for some hermitian mirror-like operator R, R2 ≡ 1
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Fulton-Gouterman form
Any 2× 2 hermitian operator H can be expressed in the form
H =3∑
j=0
hjσj , hj = 12 Tr (Hσj) (1)
where hj ’s are one-dimensional operators in a suitable Hilbert space H.
H is of the Fulton-Gouterman type, and denoted by HFG ,
(*) if the expansion coefficients hj of H in Eq. (1),
HFG = A1 + Bσ1 + Cσ2 + Dσ3,
satisfy[R,A] = [R,B] = 0, R,C = R,D = 0
for some hermitian mirror-like operator R, R2 ≡ 1
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ABCD theorem for decomposition into invariant paritysubspaces
Parity symmetry ΠFG = Rσ1, [HFG , ΠFG ] = 0 =⇒ an original HFG inthe Hilbert space H⊗ C2 decouples under H → UFG HU−1
FG into two
distinct one-dimensional L± in H
UFG =1√2
(1 1
R −R
)=
1
2
[(1 + R)U13 + (1− R)U−1
2
]where U13 = (σ1 + σ3)/
√2 and U2 = (1 + iσ2)/
√2
L± = A + D ± (B − iC )R
(± sign corresponds to the respective positive and negative parityeigenspaces)
L± remains hermitian, because
(BR)† = RB = BR, (C R)† = RC = −C R
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ABCD theorem for decomposition into invariant paritysubspaces
Parity symmetry ΠFG = Rσ1, [HFG , ΠFG ] = 0 =⇒ an original HFG inthe Hilbert space H⊗ C2 decouples under H → UFG HU−1
FG into two
distinct one-dimensional L± in H
UFG =1√2
(1 1
R −R
)=
1
2
[(1 + R)U13 + (1− R)U−1
2
]where U13 = (σ1 + σ3)/
√2 and U2 = (1 + iσ2)/
√2
L± = A + D ± (B − iC )R
(± sign corresponds to the respective positive and negative parityeigenspaces)
L± remains hermitian, because
(BR)† = RB = BR, (C R)† = RC = −C R
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ABCD theorem for decomposition into invariant paritysubspaces
Parity symmetry ΠFG = Rσ1, [HFG , ΠFG ] = 0 =⇒ an original HFG inthe Hilbert space H⊗ C2 decouples under H → UFG HU−1
FG into two
distinct one-dimensional L± in H
UFG =1√2
(1 1
R −R
)=
1
2
[(1 + R)U13 + (1− R)U−1
2
]where U13 = (σ1 + σ3)/
√2 and U2 = (1 + iσ2)/
√2
L± = A + D ± (B − iC )R
(± sign corresponds to the respective positive and negative parityeigenspaces)
L± remains hermitian, because
(BR)† = RB = BR, (C R)† = RC = −C R
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ABCD theorem for decomposition into invariant paritysubspaces
Parity symmetry ΠFG = Rσ1, [HFG , ΠFG ] = 0 =⇒ an original HFG inthe Hilbert space H⊗ C2 decouples under H → UFG HU−1
FG into two
distinct one-dimensional L± in H
UFG =1√2
(1 1
R −R
)=
1
2
[(1 + R)U13 + (1− R)U−1
2
]where U13 = (σ1 + σ3)/
√2 and U2 = (1 + iσ2)/
√2
L± = A + D ± (B − iC )R
(± sign corresponds to the respective positive and negative parityeigenspaces)
L± remains hermitian, because
(BR)† = RB = BR, (C R)† = RC = −C R
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A comparison with Berry’s diabatic form
The most general 2× 2 form of the hermitian operator H,
H(τ) = Aσ0 + (B cosφ)σ1 + (B sinφ)σ2 + Dσ3
=
(A(τ) + D(τ) B(τ) exp−iφ(τ)
B(τ) expiφ(τ) A(τ)− D(τ)
)can be brought into a unitary equivalent real, traceless and symmetricdiabatic form
H(τ) =
(D(τ) B(τ)
B(τ) −D(τ)
)= B(τ)σ1 + D(τ)σ3, D = D(τ)− ~
2 φ(τ)
[M. V. Berry, Proc. R. Soc. A 429, 61-72 (1990)]
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Examples involving R = e iπa†a
By means of U13 = (σ1 + σ3)/√
2
HR = ω1a†a + gσ1(a† + a) + µσ3 ⇒ ~ω1a†a + µσ1 + gσ3(a† + a)
e iπNa = ae iπ(N−1) = −ae iπN , e iπNa† = a†e iπ(N+1) = −a†e iπN
HgR = ω1a†a+µσ3+g1
(a†σ−+aσ+
)+ g2
(a†σ++aσ−
)The Foulton-Gouterman form:
A = a†a, B = ∆, C = iλ−κ
a†, D = κa +λ+
κa†
∆ :=µ
ω, λ± :=
g21 ± g2
2
2ω2, κ :=
√g1g2
ω
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Examples involving R = e iπa†a
By means of U13 = (σ1 + σ3)/√
2
HR = ω1a†a + gσ1(a† + a) + µσ3 ⇒ ~ω1a†a + µσ1 + gσ3(a† + a)
e iπNa = ae iπ(N−1) = −ae iπN , e iπNa† = a†e iπ(N+1) = −a†e iπN
HgR = ω1a†a+µσ3+g1
(a†σ−+aσ+
)+ g2
(a†σ++aσ−
)
The Foulton-Gouterman form:
A = a†a, B = ∆, C = iλ−κ
a†, D = κa +λ+
κa†
∆ :=µ
ω, λ± :=
g21 ± g2
2
2ω2, κ :=
√g1g2
ω
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Examples involving R = e iπa†a
By means of U13 = (σ1 + σ3)/√
2
HR = ω1a†a + gσ1(a† + a) + µσ3 ⇒ ~ω1a†a + µσ1 + gσ3(a† + a)
e iπNa = ae iπ(N−1) = −ae iπN , e iπNa† = a†e iπ(N+1) = −a†e iπN
HgR = ω1a†a+µσ3+g1
(a†σ−+aσ+
)+ g2
(a†σ++aσ−
)The Foulton-Gouterman form:
A = a†a, B = ∆, C = iλ−κ
a†, D = κa +λ+
κa†
∆ :=µ
ω, λ± :=
g21 ± g2
2
2ω2, κ :=
√g1g2
ω
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Examples involving R = e iπa†a
By means of U13 = (σ1 + σ3)/√
2
HR = ω1a†a + gσ1(a† + a) + µσ3 ⇒ ~ω1a†a + µσ1 + gσ3(a† + a)
e iπNa = ae iπ(N−1) = −ae iπN , e iπNa† = a†e iπ(N+1) = −a†e iπN
HgR = ω1a†a+µσ3+g1
(a†σ−+aσ+
)+ g2
(a†σ++aσ−
)The Foulton-Gouterman form:
A = a†a, B = ∆, C = iλ−κ
a†, D = κa +λ+
κa†
∆ :=µ
ω, λ± :=
g21 ± g2
2
2ω2, κ :=
√g1g2
ω
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An intensity-dependent Rabi Hamiltonian
B. M. Rodrıguez-Lara, J. Opt. Soc. Am. B 31, 1719 (2014):
HiRM = ωn +ω0
2σ3 + g
(√n + 2k a + a†
√n + 2k
)σ1
On introducing su(1, 1) Lie algebra generators
K0 = n + k , K+ = a†√n + 2k σ1, K− =
√n + 2k aσ1
[K+, K−] = −2K0, [K0, K±] = ±K±
the Foulton-Gouterman form relative to R = e iπa†a ∼ e iπK0 :
HiRM = ω(K0 − k) +ω0
2σ1 + g
(K+ + K−
)σ3
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An intensity-dependent Rabi Hamiltonian
B. M. Rodrıguez-Lara, J. Opt. Soc. Am. B 31, 1719 (2014):
HiRM = ωn +ω0
2σ3 + g
(√n + 2k a + a†
√n + 2k
)σ1
On introducing su(1, 1) Lie algebra generators
K0 = n + k , K+ = a†√n + 2k σ1, K− =
√n + 2k aσ1
[K+, K−] = −2K0, [K0, K±] = ±K±
the Foulton-Gouterman form relative to R = e iπa†a ∼ e iπK0 :
HiRM = ω(K0 − k) +ω0
2σ1 + g
(K+ + K−
)σ3
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An intensity-dependent Rabi Hamiltonian
B. M. Rodrıguez-Lara, J. Opt. Soc. Am. B 31, 1719 (2014):
HiRM = ωn +ω0
2σ3 + g
(√n + 2k a + a†
√n + 2k
)σ1
On introducing su(1, 1) Lie algebra generators
K0 = n + k , K+ = a†√n + 2k σ1, K− =
√n + 2k aσ1
[K+, K−] = −2K0, [K0, K±] = ±K±
the Foulton-Gouterman form relative to R = e iπa†a ∼ e iπK0 :
HiRM = ω(K0 − k) +ω0
2σ1 + g
(K+ + K−
)σ3
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Bosonisation of further nonlinear Hamiltonians
Y.-Z. Zhang, JMP 54, 102104 (2013):
H2p = ωa†a + βσ3 + g σ1
[(a†)2 + a2
]H2m = ω(a†1a1 + a†2a2) + βσ3 + g σ1(a†1a
†2 + a1a2)
In terms of suitable su(1, 1) Lie algebra generators theFoulton-Gouterman form relative to R = e iπK0 :
H2FG = γ (K0 − c) + ∆σ1 + σ3(K+ + K−)
where γ = ω/g , ∆ = β/g
The TPRM: c = 1/4,
K+ = 12 (a†)2, K− = 1
2a2, K0 = 1
2
(a†a + 1
2
)The TMRM: c = 1/2,
K+ = a†1a†2, K− = a1a2, K0 = 1
2 (a†1a1 + a†2a2 + 1)
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Bosonisation of further nonlinear Hamiltonians
Y.-Z. Zhang, JMP 54, 102104 (2013):
H2p = ωa†a + βσ3 + g σ1
[(a†)2 + a2
]H2m = ω(a†1a1 + a†2a2) + βσ3 + g σ1(a†1a
†2 + a1a2)
In terms of suitable su(1, 1) Lie algebra generators theFoulton-Gouterman form relative to R = e iπK0 :
H2FG = γ (K0 − c) + ∆σ1 + σ3(K+ + K−)
where γ = ω/g , ∆ = β/g
The TPRM: c = 1/4,
K+ = 12 (a†)2, K− = 1
2a2, K0 = 1
2
(a†a + 1
2
)The TMRM: c = 1/2,
K+ = a†1a†2, K− = a1a2, K0 = 1
2 (a†1a1 + a†2a2 + 1)
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Bosonisation of further nonlinear Hamiltonians
Y.-Z. Zhang, JMP 54, 102104 (2013):
H2p = ωa†a + βσ3 + g σ1
[(a†)2 + a2
]H2m = ω(a†1a1 + a†2a2) + βσ3 + g σ1(a†1a
†2 + a1a2)
In terms of suitable su(1, 1) Lie algebra generators theFoulton-Gouterman form relative to R = e iπK0 :
H2FG = γ (K0 − c) + ∆σ1 + σ3(K+ + K−)
where γ = ω/g , ∆ = β/g
The TPRM: c = 1/4,
K+ = 12 (a†)2, K− = 1
2a2, K0 = 1
2
(a†a + 1
2
)
The TMRM: c = 1/2,
K+ = a†1a†2, K− = a1a2, K0 = 1
2 (a†1a1 + a†2a2 + 1)
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Bosonisation of further nonlinear Hamiltonians
Y.-Z. Zhang, JMP 54, 102104 (2013):
H2p = ωa†a + βσ3 + g σ1
[(a†)2 + a2
]H2m = ω(a†1a1 + a†2a2) + βσ3 + g σ1(a†1a
†2 + a1a2)
In terms of suitable su(1, 1) Lie algebra generators theFoulton-Gouterman form relative to R = e iπK0 :
H2FG = γ (K0 − c) + ∆σ1 + σ3(K+ + K−)
where γ = ω/g , ∆ = β/g
The TPRM: c = 1/4,
K+ = 12 (a†)2, K− = 1
2a2, K0 = 1
2
(a†a + 1
2
)The TMRM: c = 1/2,
K+ = a†1a†2, K− = a1a2, K0 = 1
2 (a†1a1 + a†2a2 + 1)
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Borrowing from other talks
From Prof. Kwon Park talk on Weyl semimetals:
H(k) '[t(k2
1 − k20 ) +
m
2(k2
2 + k23 )]σ1 + 2t(k2σ2 + k3σ3)
is of the Foulton-Gouterman form relative to inversion in momentumspace
R : kj → −kj (2)
Parity symmetry ΠFG = Rσ1
Parity invariant subspaces: interference states involving both k and−k dependent functional dependencies
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Borrowing from other talks
From Prof. Kwon Park talk on Weyl semimetals:
H(k) '[t(k2
1 − k20 ) +
m
2(k2
2 + k23 )]σ1 + 2t(k2σ2 + k3σ3)
is of the Foulton-Gouterman form relative to inversion in momentumspace
R : kj → −kj (2)
Parity symmetry ΠFG = Rσ1
Parity invariant subspaces: interference states involving both k and−k dependent functional dependencies
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Borrowing from other talks
From Prof. Kwon Park talk on Weyl semimetals:
H(k) '[t(k2
1 − k20 ) +
m
2(k2
2 + k23 )]σ1 + 2t(k2σ2 + k3σ3)
is of the Foulton-Gouterman form relative to inversion in momentumspace
R : kj → −kj (2)
Parity symmetry ΠFG = Rσ1
Parity invariant subspaces: interference states involving both k and−k dependent functional dependencies
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 23
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Model Hamiltonians in the Bargmann-Fock representation
Bargmann-Fock representation:
a→ d/dz = dz , a† → z
=⇒HgR =
[(z + κ)dz +
λ+z
κ
]+
(λ−z
κ+ ∆
)σ3R
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Model Hamiltonians in the Bargmann-Fock representation
Bargmann-Fock representation:
a→ d/dz = dz , a† → z
=⇒HgR =
[(z + κ)dz +
λ+z
κ
]+
(λ−z
κ+ ∆
)σ3R
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Linear ordinary differential operators L± of Dunkl type
GRM
L± = (z + κ)dz +λ+z
κ±(λ−z
κ+ ∆
)R
In the limit of the RM
L± = (z + κ)dz + κz ±∆R
iRML± = ω(zdz − k) + g (z + dz)± ω0
2(−1)−kR
For the respective TPRM and the TMRM one finds
L±;2p = 2zd2z + (4q + γz)dz + z
2 + γ(q − 1
4
)±∆R
L±;2m = zd2z + (2q + γz)dz + z + γ
(q − 1
2
)±∆R
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Linear ordinary differential operators L± of Dunkl type
GRM
L± = (z + κ)dz +λ+z
κ±(λ−z
κ+ ∆
)R
In the limit of the RM
L± = (z + κ)dz + κz ±∆R
iRML± = ω(zdz − k) + g (z + dz)± ω0
2(−1)−kR
For the respective TPRM and the TMRM one finds
L±;2p = 2zd2z + (4q + γz)dz + z
2 + γ(q − 1
4
)±∆R
L±;2m = zd2z + (2q + γz)dz + z + γ
(q − 1
2
)±∆R
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Linear ordinary differential operators L± of Dunkl type
GRM
L± = (z + κ)dz +λ+z
κ±(λ−z
κ+ ∆
)R
In the limit of the RM
L± = (z + κ)dz + κz ±∆R
iRML± = ω(zdz − k) + g (z + dz)± ω0
2(−1)−kR
For the respective TPRM and the TMRM one finds
L±;2p = 2zd2z + (4q + γz)dz + z
2 + γ(q − 1
4
)±∆R
L±;2m = zd2z + (2q + γz)dz + z + γ
(q − 1
2
)±∆R
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Linear ordinary differential operators L± of Dunkl type
GRM
L± = (z + κ)dz +λ+z
κ±(λ−z
κ+ ∆
)R
In the limit of the RM
L± = (z + κ)dz + κz ±∆R
iRML± = ω(zdz − k) + g (z + dz)± ω0
2(−1)−kR
For the respective TPRM and the TMRM one finds
L±;2p = 2zd2z + (4q + γz)dz + z
2 + γ(q − 1
4
)±∆R
L±;2m = zd2z + (2q + γz)dz + z + γ
(q − 1
2
)±∆R
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Linear ordinary differential operators L± of Dunkl type
Each term of any of L± does not change the degree of a monomial zn
by more than ±1
=⇒ the resulting eigenvalue problem (L− ε)ψ = 0 in a suitableBargmann-Fock space of analytic functions naturally reduces to athree-term recurrence relation (TTRR)
No level crossing while varying coupling parameters [thenondegeneracy applies to all problems where the Hamiltonianoperator is a self-adjoint extension of a tridiagonal Jacobi matrix ofdeficiency index (1, 1)]
All HFG leading to a TTRR have avoided level crossings
When solving for eigenvalues numerically, the respective eigenvalues,which can alternatively be determined as zeros of an orthogonalpolynomials, decrease monotonically to a corresponding fixed withan increased matrix cutoff
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Linear ordinary differential operators L± of Dunkl type
Each term of any of L± does not change the degree of a monomial zn
by more than ±1
=⇒ the resulting eigenvalue problem (L− ε)ψ = 0 in a suitableBargmann-Fock space of analytic functions naturally reduces to athree-term recurrence relation (TTRR)
No level crossing while varying coupling parameters [thenondegeneracy applies to all problems where the Hamiltonianoperator is a self-adjoint extension of a tridiagonal Jacobi matrix ofdeficiency index (1, 1)]
All HFG leading to a TTRR have avoided level crossings
When solving for eigenvalues numerically, the respective eigenvalues,which can alternatively be determined as zeros of an orthogonalpolynomials, decrease monotonically to a corresponding fixed withan increased matrix cutoff
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Linear ordinary differential operators L± of Dunkl type
Each term of any of L± does not change the degree of a monomial zn
by more than ±1
=⇒ the resulting eigenvalue problem (L− ε)ψ = 0 in a suitableBargmann-Fock space of analytic functions naturally reduces to athree-term recurrence relation (TTRR)
No level crossing while varying coupling parameters [thenondegeneracy applies to all problems where the Hamiltonianoperator is a self-adjoint extension of a tridiagonal Jacobi matrix ofdeficiency index (1, 1)]
All HFG leading to a TTRR have avoided level crossings
When solving for eigenvalues numerically, the respective eigenvalues,which can alternatively be determined as zeros of an orthogonalpolynomials, decrease monotonically to a corresponding fixed withan increased matrix cutoff
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Linear ordinary differential operators L± of Dunkl type
Each term of any of L± does not change the degree of a monomial zn
by more than ±1
=⇒ the resulting eigenvalue problem (L− ε)ψ = 0 in a suitableBargmann-Fock space of analytic functions naturally reduces to athree-term recurrence relation (TTRR)
No level crossing while varying coupling parameters [thenondegeneracy applies to all problems where the Hamiltonianoperator is a self-adjoint extension of a tridiagonal Jacobi matrix ofdeficiency index (1, 1)]
All HFG leading to a TTRR have avoided level crossings
When solving for eigenvalues numerically, the respective eigenvalues,which can alternatively be determined as zeros of an orthogonalpolynomials, decrease monotonically to a corresponding fixed withan increased matrix cutoff
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Linear ordinary differential operators L± of Dunkl type
Each term of any of L± does not change the degree of a monomial zn
by more than ±1
=⇒ the resulting eigenvalue problem (L− ε)ψ = 0 in a suitableBargmann-Fock space of analytic functions naturally reduces to athree-term recurrence relation (TTRR)
No level crossing while varying coupling parameters [thenondegeneracy applies to all problems where the Hamiltonianoperator is a self-adjoint extension of a tridiagonal Jacobi matrix ofdeficiency index (1, 1)]
All HFG leading to a TTRR have avoided level crossings
When solving for eigenvalues numerically, the respective eigenvalues,which can alternatively be determined as zeros of an orthogonalpolynomials, decrease monotonically to a corresponding fixed withan increased matrix cutoff
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Tridiagonality is generic (Lanczos, Haydock, etc)
There always exists an orthonormal basis en∞n=0 such that a givenself-adjoint operator takes on a tridiagonal form
Hen = anen + bn+1en+1 + bnen−1 (3)
with real recurrence coefficients an and bn, where bn ≥ 0, n ≥ 0.
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Favard’s theorem (Theorem I-4.4 of Chihara’s book)
For given arbitrary sequences of complex numbers cn and λn in theTTRR
xPn = Pn+1 + cnPn + λnPn−1
there always exists a moment functional, that is a linear functional Lacting in the space of (complex) monic polynomials C[E ], such that thepolynomials Pn defined by the TTRR are orthogonal under L:
L(Pk Pl) = 0 k 6= l ∈ N
The functional L is unique if we impose the normalization conditionL(P0) = L(1) = µ0, where µ0 is a chosen positive constant.
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Trivial and always valid statement
For any linear self-adjoint H there always exists an orthonormal basisen∞n=0 such that eigenstates |E 〉 of H
H|E 〉 = E |E 〉
can be expanded as
|E 〉 =∞∑n=0
pn(E )en
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Haydock’s recursive solution
(H1) the expansion coefficients pn(E ) are polynomials with degreepn = n, orthonormal with respect to the density of states (DOS),n0(E ), ∫ ∞
−∞pn(E )pm(E )n0(E )dE = δnm
(H2) energy eigenstates |E 〉 are the generating function of theorthogonal polynomials
(H3) the orthogonality of the energy eigenstates, 〈E |E ′〉 = δEE ′ yieldsa dual orthogonality relation
n0(E )∞∑n=0
pn(E )pn(E ′) = δEE ′
where E and E ′ are both eigenvalues
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Haydock’s recursive solution
(H1) the expansion coefficients pn(E ) are polynomials with degreepn = n, orthonormal with respect to the density of states (DOS),n0(E ), ∫ ∞
−∞pn(E )pm(E )n0(E )dE = δnm
(H2) energy eigenstates |E 〉 are the generating function of theorthogonal polynomials
(H3) the orthogonality of the energy eigenstates, 〈E |E ′〉 = δEE ′ yieldsa dual orthogonality relation
n0(E )∞∑n=0
pn(E )pn(E ′) = δEE ′
where E and E ′ are both eigenvalues
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Haydock’s recursive solution
(H1) the expansion coefficients pn(E ) are polynomials with degreepn = n, orthonormal with respect to the density of states (DOS),n0(E ), ∫ ∞
−∞pn(E )pm(E )n0(E )dE = δnm
(H2) energy eigenstates |E 〉 are the generating function of theorthogonal polynomials
(H3) the orthogonality of the energy eigenstates, 〈E |E ′〉 = δEE ′ yieldsa dual orthogonality relation
n0(E )∞∑n=0
pn(E )pn(E ′) = δEE ′
where E and E ′ are both eigenvalues
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Braak’s definition of integrability [PRL 107, 100401 (2011)]
If each eigenstate of a quantum system with f1 discrete and f2continuous degrees of freedom can be uniquely labeled by f1 + f2 = fquantum numbers d1, . . . , df1 , c1, . . . , cf2, such that the dj can takeon dim(Hj) different values, where Hj is the state space of the jthdiscrete degree of freedom and the ck range from 0 to infinity, thenthis system is quantum integrable.
The RM has f1 = f2 = 1 and degeneracies take place only betweenlevels of states with different parity, whereas within the paritysubspaces no level crossings occur. The global label for the RM [validfor all values of coupling constant] is two dimensional, with one labelfor the parity and the other being the energy sorting number within agiven parity subspace.
=⇒ the RM is quantum integrable
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Braak’s definition of integrability [PRL 107, 100401 (2011)]
If each eigenstate of a quantum system with f1 discrete and f2continuous degrees of freedom can be uniquely labeled by f1 + f2 = fquantum numbers d1, . . . , df1 , c1, . . . , cf2, such that the dj can takeon dim(Hj) different values, where Hj is the state space of the jthdiscrete degree of freedom and the ck range from 0 to infinity, thenthis system is quantum integrable.
The RM has f1 = f2 = 1 and degeneracies take place only betweenlevels of states with different parity, whereas within the paritysubspaces no level crossings occur. The global label for the RM [validfor all values of coupling constant] is two dimensional, with one labelfor the parity and the other being the energy sorting number within agiven parity subspace.
=⇒ the RM is quantum integrable
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Braak’s definition of integrability [PRL 107, 100401 (2011)]
If each eigenstate of a quantum system with f1 discrete and f2continuous degrees of freedom can be uniquely labeled by f1 + f2 = fquantum numbers d1, . . . , df1 , c1, . . . , cf2, such that the dj can takeon dim(Hj) different values, where Hj is the state space of the jthdiscrete degree of freedom and the ck range from 0 to infinity, thenthis system is quantum integrable.
The RM has f1 = f2 = 1 and degeneracies take place only betweenlevels of states with different parity, whereas within the paritysubspaces no level crossings occur. The global label for the RM [validfor all values of coupling constant] is two dimensional, with one labelfor the parity and the other being the energy sorting number within agiven parity subspace.
=⇒ the RM is quantum integrable
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However in that case:
All physically reasonable HFG are necessarily quantum integrable
Inflation of integrable models, because the avoided level crossingsbetween states of equal parity is generic for the TLS studied here
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However in that case:
All physically reasonable HFG are necessarily quantum integrable
Inflation of integrable models, because the avoided level crossingsbetween states of equal parity is generic for the TLS studied here
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Quantum invariants
(a) A quantum integrability cannot be inferred from quantum invariantsas simply as classical integrability can be inferred from integrals of themotion (analytic invariants)
(b) Commuting operators can always be constructed irrespective ofwhether the model is (classically) integrable or not (A. Peres, PRL53, 1711 (1984))
(c) Any operator T that is not already an invariant, [H,T ] 6= 0, can beturned into an invariant via time average
(d) In the energy representation, the time average strips T of all itsoff-diagonal elements. The resulting operator IT = 〈T 〉 being diagonalin the energy representation thus commutes with H by construction
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Quantum invariants
(a) A quantum integrability cannot be inferred from quantum invariantsas simply as classical integrability can be inferred from integrals of themotion (analytic invariants)
(b) Commuting operators can always be constructed irrespective ofwhether the model is (classically) integrable or not (A. Peres, PRL53, 1711 (1984))
(c) Any operator T that is not already an invariant, [H,T ] 6= 0, can beturned into an invariant via time average
(d) In the energy representation, the time average strips T of all itsoff-diagonal elements. The resulting operator IT = 〈T 〉 being diagonalin the energy representation thus commutes with H by construction
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Quantum invariants
(a) A quantum integrability cannot be inferred from quantum invariantsas simply as classical integrability can be inferred from integrals of themotion (analytic invariants)
(b) Commuting operators can always be constructed irrespective ofwhether the model is (classically) integrable or not (A. Peres, PRL53, 1711 (1984))
(c) Any operator T that is not already an invariant, [H,T ] 6= 0, can beturned into an invariant via time average
(d) In the energy representation, the time average strips T of all itsoff-diagonal elements. The resulting operator IT = 〈T 〉 being diagonalin the energy representation thus commutes with H by construction
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Quantum invariants
(a) A quantum integrability cannot be inferred from quantum invariantsas simply as classical integrability can be inferred from integrals of themotion (analytic invariants)
(b) Commuting operators can always be constructed irrespective ofwhether the model is (classically) integrable or not (A. Peres, PRL53, 1711 (1984))
(c) Any operator T that is not already an invariant, [H,T ] 6= 0, can beturned into an invariant via time average
(d) In the energy representation, the time average strips T of all itsoff-diagonal elements. The resulting operator IT = 〈T 〉 being diagonalin the energy representation thus commutes with H by construction
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A general spin-boson model or nothing but a differentparametrization of the spin-S GRM
H = ~ωa†a + ~ω0Sz + Λ cosα(S+a + S−a
†)
+ Λ sinα(S+a
† + S−a)
[V. V. Stepanov et al, Phys. Rev. E 77, 066202 (2008)]
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Quantum invariant 〈A〉ν = 〈ν|a†(σ− + σ+)|ν〉 versusquantum invariant Eν = 〈ν|H |ν〉
-4
-3
-2
-1
0
1
2
3
4
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(a)-4
-3
-2
-1
0
1
2
3
4
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(a)-4
-3
-2
-1
0
1
2
3
4
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(a)-4
-3
-2
-1
0
1
2
3
4
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(a)-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(b)-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(b)-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(b)-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(b)
-2
-1
0
1
2
0 5 10 15 20 25 30 35 40
⟨A⟩ k
Ek
(c)-2
-1
0
1
2
0 5 10 15 20 25 30 35 40
⟨A⟩ k
Ek
(c)-2
-1
0
1
2
0 5 10 15 20 25 30 35 40
⟨A⟩ k
Ek
(c)-2
-1
0
1
2
0 5 10 15 20 25 30 35 40
⟨A⟩ k
Ek
(c)
For the eigenstates |ν〉 with parity +1 of the spin-boson model with the spin s = 12
, ~ω = 1, λ.= (Λ/~ω)2 = 0.09, and (a)
α = 0, (b) α = π/2, (c) α = π/4
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Quantum invariant 〈A〉ν = 〈ν|a†(σ− + σ+)|ν〉 versusquantum invariant Eν = 〈ν|H |ν〉 - changes upon couplingvariations
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
10 10.5 11 11.5 12 12.5 13
⟨A⟩ m
n
Emn
λ=0.01 λ=0.25
λ=0.01λ=0.25
(a)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
10 10.5 11 11.5 12 12.5 13
⟨A⟩ m
n
Emn
λ=0.01 λ=0.25
λ=0.01λ=0.25
(a)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
10 10.5 11 11.5 12 12.5 13
⟨A⟩ m
n
Emn
λ=0.01 λ=0.25
λ=0.01λ=0.25
(a)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
10 10.5 11 11.5 12 12.5 13
⟨A⟩ m
n
Emn
λ=0.01 λ=0.25
λ=0.01λ=0.25
(a)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
10 10.5 11 11.5 12 12.5 13
⟨A⟩ m
n
Emn
λ=0.01 λ=0.25
λ=0.01λ=0.25
(a)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
10 10.5 11 11.5 12 12.5 13
⟨A⟩ m
n
Emn
λ=0.01 λ=0.25
λ=0.01λ=0.25
(a)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
10 10.5 11 11.5 12 12.5 13
⟨A⟩ m
n
Emn
λ=0.01 λ=0.25
λ=0.01λ=0.25
(a)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
10 10.5 11 11.5 12 12.5 13
⟨A⟩ m
n
Emn
λ=0.01 λ=0.25
λ=0.01λ=0.25
(a)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
16.5 17 17.5 18 18.5
⟨A⟩ k
Ek
λ=0.01
λ=0.25
λ=0.01
λ=0.25
λ=0.09 λ=0.09
(c)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
16.5 17 17.5 18 18.5
⟨A⟩ k
Ek
λ=0.01
λ=0.25
λ=0.01
λ=0.25
λ=0.09 λ=0.09
(c)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
16.5 17 17.5 18 18.5
⟨A⟩ k
Ek
λ=0.01
λ=0.25
λ=0.01
λ=0.25
λ=0.09 λ=0.09
(c)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
16.5 17 17.5 18 18.5
⟨A⟩ k
Ek
λ=0.01
λ=0.25
λ=0.01
λ=0.25
λ=0.09 λ=0.09
(c)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
16.5 17 17.5 18 18.5
⟨A⟩ k
Ek
λ=0.01
λ=0.25
λ=0.01
λ=0.25
λ=0.09 λ=0.09
(c)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
16.5 17 17.5 18 18.5
⟨A⟩ k
Ek
λ=0.01
λ=0.25
λ=0.01
λ=0.25
λ=0.09 λ=0.09
(c)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
16.5 17 17.5 18 18.5
⟨A⟩ k
Ek
λ=0.01
λ=0.25
λ=0.01
λ=0.25
λ=0.09 λ=0.09
(c)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
16.5 17 17.5 18 18.5
⟨A⟩ k
Ek
λ=0.01
λ=0.25
λ=0.01
λ=0.25
λ=0.09 λ=0.09
(c)
-4
-2
0
2
4
32 32.5 33 33.5 34 34.5 35
⟨A⟩ m
n
Emn
λ=0.01
λ=0.25
λ=0.01
λ=0.25
(b)
-4
-2
0
2
4
32 32.5 33 33.5 34 34.5 35
⟨A⟩ m
n
Emn
λ=0.01
λ=0.25
λ=0.01
λ=0.25
(b)
-4
-2
0
2
4
32 32.5 33 33.5 34 34.5 35
⟨A⟩ m
n
Emn
λ=0.01
λ=0.25
λ=0.01
λ=0.25
(b)
-4
-2
0
2
4
32 32.5 33 33.5 34 34.5 35
⟨A⟩ m
n
Emn
λ=0.01
λ=0.25
λ=0.01
λ=0.25
(b)
-4
-2
0
2
4
32 32.5 33 33.5 34 34.5 35
⟨A⟩ m
n
Emn
λ=0.01
λ=0.25
λ=0.01
λ=0.25
(b)
-4
-2
0
2
4
32 32.5 33 33.5 34 34.5 35
⟨A⟩ m
n
Emn
λ=0.01
λ=0.25
λ=0.01
λ=0.25
(b)
-4
-2
0
2
4
32 32.5 33 33.5 34 34.5 35
⟨A⟩ m
n
Emn
λ=0.01
λ=0.25
λ=0.01
λ=0.25
(b)
-4
-2
0
2
4
32 32.5 33 33.5 34 34.5 35
⟨A⟩ m
n
Emn
λ=0.01
λ=0.25
λ=0.01
λ=0.25
(b)
Trace of one pair of eigenstates |ν〉 with parity +1 (identified by full circles) as the interaction parameter λ increases a specifiedamount at constant value (a) α = 0, (b) α = π/2, (c) α = π/4 of the integrability parameter
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 36
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Conclusions on integrability issues
The well known level-statistics criteria which have been applied withgreat success to autonomous particle systems are not applicable tothe generalized Rabi models
The nearest-neighbour distribution of levels is not of the general typeassociated with chaotic systems and does not offer any conclusiveevidence for quantum nonintegrability
Only the analysis of two-dimensional patterns of quantum invariants(εn, 〈T 〉n) yields an unambiguous answer here
Braak’s definition of integrability was shown not only to contradictthe earlier pattern studies by Muller et al. but also to imply that anyphysically reasonable differential operator of Fulton-Gouterman type(i.e. leading to a TTRR) is integrable
This suggests that Braak’s definition of integrability is most probablya faulty one - supported by the conclusions of M. T. Batchelor et al[PRA 91, 053808 (2015)] that the Rabi model is not Yang-Baxterintegrable
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 37
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Conclusions on integrability issues
The well known level-statistics criteria which have been applied withgreat success to autonomous particle systems are not applicable tothe generalized Rabi models
The nearest-neighbour distribution of levels is not of the general typeassociated with chaotic systems and does not offer any conclusiveevidence for quantum nonintegrability
Only the analysis of two-dimensional patterns of quantum invariants(εn, 〈T 〉n) yields an unambiguous answer here
Braak’s definition of integrability was shown not only to contradictthe earlier pattern studies by Muller et al. but also to imply that anyphysically reasonable differential operator of Fulton-Gouterman type(i.e. leading to a TTRR) is integrable
This suggests that Braak’s definition of integrability is most probablya faulty one - supported by the conclusions of M. T. Batchelor et al[PRA 91, 053808 (2015)] that the Rabi model is not Yang-Baxterintegrable
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 37
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Conclusions on integrability issues
The well known level-statistics criteria which have been applied withgreat success to autonomous particle systems are not applicable tothe generalized Rabi models
The nearest-neighbour distribution of levels is not of the general typeassociated with chaotic systems and does not offer any conclusiveevidence for quantum nonintegrability
Only the analysis of two-dimensional patterns of quantum invariants(εn, 〈T 〉n) yields an unambiguous answer here
Braak’s definition of integrability was shown not only to contradictthe earlier pattern studies by Muller et al. but also to imply that anyphysically reasonable differential operator of Fulton-Gouterman type(i.e. leading to a TTRR) is integrable
This suggests that Braak’s definition of integrability is most probablya faulty one - supported by the conclusions of M. T. Batchelor et al[PRA 91, 053808 (2015)] that the Rabi model is not Yang-Baxterintegrable
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 37
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Conclusions on integrability issues
The well known level-statistics criteria which have been applied withgreat success to autonomous particle systems are not applicable tothe generalized Rabi models
The nearest-neighbour distribution of levels is not of the general typeassociated with chaotic systems and does not offer any conclusiveevidence for quantum nonintegrability
Only the analysis of two-dimensional patterns of quantum invariants(εn, 〈T 〉n) yields an unambiguous answer here
Braak’s definition of integrability was shown not only to contradictthe earlier pattern studies by Muller et al. but also to imply that anyphysically reasonable differential operator of Fulton-Gouterman type(i.e. leading to a TTRR) is integrable
This suggests that Braak’s definition of integrability is most probablya faulty one - supported by the conclusions of M. T. Batchelor et al[PRA 91, 053808 (2015)] that the Rabi model is not Yang-Baxterintegrable
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 37
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Conclusions on integrability issues
The well known level-statistics criteria which have been applied withgreat success to autonomous particle systems are not applicable tothe generalized Rabi models
The nearest-neighbour distribution of levels is not of the general typeassociated with chaotic systems and does not offer any conclusiveevidence for quantum nonintegrability
Only the analysis of two-dimensional patterns of quantum invariants(εn, 〈T 〉n) yields an unambiguous answer here
Braak’s definition of integrability was shown not only to contradictthe earlier pattern studies by Muller et al. but also to imply that anyphysically reasonable differential operator of Fulton-Gouterman type(i.e. leading to a TTRR) is integrable
This suggests that Braak’s definition of integrability is most probablya faulty one - supported by the conclusions of M. T. Batchelor et al[PRA 91, 053808 (2015)] that the Rabi model is not Yang-Baxterintegrable
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 37
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Realizations of linear models
The RM and GRM are presently the focus of intense experimentaland theoretical activity for cavity- and circuit-QED setups,superconducting q-bits, nitrogen vacancy centers
The GRM serves as a non-trivial model in spin resonance, for variousproblems involving the interaction between electronic and vibrationaldegrees of freedom in molecules and solids, and in quantum optics
The RM with a negative sign of its parameters g and µ is used todescribe an excitation hopping between two sites (µ is then atunneling parameter) and is relevant in understanding the transitionbetween untrapped and trapped behavior of an exciton
GRM can be mapped onto the model describing a two-dimensionalelectron gas with Rashba (αR ∼ g1) and Dresselhaus (αD ∼ g2)spin-orbit couplings subject to a perpendicular magnetic field (theZeeman splitting thereby equals 2µ)
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 38
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Realizations of linear models
The RM and GRM are presently the focus of intense experimentaland theoretical activity for cavity- and circuit-QED setups,superconducting q-bits, nitrogen vacancy centers
The GRM serves as a non-trivial model in spin resonance, for variousproblems involving the interaction between electronic and vibrationaldegrees of freedom in molecules and solids, and in quantum optics
The RM with a negative sign of its parameters g and µ is used todescribe an excitation hopping between two sites (µ is then atunneling parameter) and is relevant in understanding the transitionbetween untrapped and trapped behavior of an exciton
GRM can be mapped onto the model describing a two-dimensionalelectron gas with Rashba (αR ∼ g1) and Dresselhaus (αD ∼ g2)spin-orbit couplings subject to a perpendicular magnetic field (theZeeman splitting thereby equals 2µ)
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 38
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Realizations of linear models
The RM and GRM are presently the focus of intense experimentaland theoretical activity for cavity- and circuit-QED setups,superconducting q-bits, nitrogen vacancy centers
The GRM serves as a non-trivial model in spin resonance, for variousproblems involving the interaction between electronic and vibrationaldegrees of freedom in molecules and solids, and in quantum optics
The RM with a negative sign of its parameters g and µ is used todescribe an excitation hopping between two sites (µ is then atunneling parameter) and is relevant in understanding the transitionbetween untrapped and trapped behavior of an exciton
GRM can be mapped onto the model describing a two-dimensionalelectron gas with Rashba (αR ∼ g1) and Dresselhaus (αD ∼ g2)spin-orbit couplings subject to a perpendicular magnetic field (theZeeman splitting thereby equals 2µ)
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 38
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Realizations of linear models
The RM and GRM are presently the focus of intense experimentaland theoretical activity for cavity- and circuit-QED setups,superconducting q-bits, nitrogen vacancy centers
The GRM serves as a non-trivial model in spin resonance, for variousproblems involving the interaction between electronic and vibrationaldegrees of freedom in molecules and solids, and in quantum optics
The RM with a negative sign of its parameters g and µ is used todescribe an excitation hopping between two sites (µ is then atunneling parameter) and is relevant in understanding the transitionbetween untrapped and trapped behavior of an exciton
GRM can be mapped onto the model describing a two-dimensionalelectron gas with Rashba (αR ∼ g1) and Dresselhaus (αD ∼ g2)spin-orbit couplings subject to a perpendicular magnetic field (theZeeman splitting thereby equals 2µ)
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 38
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Realizations of linear models
The Rashba spin-orbit coupling can be tuned by an applied electricfield while the Zeeman term is tuned by an applied magnetic field.This allows to explore the whole parameter space of the model.
A possible realization of tunable Rashba and Dresselhaus SOC withultracold alkali atoms is proposed, where each state is coupled by atwo-photon Raman transition.
Further examples of physical realizations of the GRM include (i)electric-magnetic coupling of light and matter, and (ii) effectiverealization of the model using 3- and 4-level emitters
For a review of different realization see M. Tomka et al, Sci. Rep. 5,13097 (2015)
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 39
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Realizations of linear models
The Rashba spin-orbit coupling can be tuned by an applied electricfield while the Zeeman term is tuned by an applied magnetic field.This allows to explore the whole parameter space of the model.
A possible realization of tunable Rashba and Dresselhaus SOC withultracold alkali atoms is proposed, where each state is coupled by atwo-photon Raman transition.
Further examples of physical realizations of the GRM include (i)electric-magnetic coupling of light and matter, and (ii) effectiverealization of the model using 3- and 4-level emitters
For a review of different realization see M. Tomka et al, Sci. Rep. 5,13097 (2015)
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 39
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Realizations of linear models
The Rashba spin-orbit coupling can be tuned by an applied electricfield while the Zeeman term is tuned by an applied magnetic field.This allows to explore the whole parameter space of the model.
A possible realization of tunable Rashba and Dresselhaus SOC withultracold alkali atoms is proposed, where each state is coupled by atwo-photon Raman transition.
Further examples of physical realizations of the GRM include (i)electric-magnetic coupling of light and matter, and (ii) effectiverealization of the model using 3- and 4-level emitters
For a review of different realization see M. Tomka et al, Sci. Rep. 5,13097 (2015)
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 39
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Realizations of linear models
The Rashba spin-orbit coupling can be tuned by an applied electricfield while the Zeeman term is tuned by an applied magnetic field.This allows to explore the whole parameter space of the model.
A possible realization of tunable Rashba and Dresselhaus SOC withultracold alkali atoms is proposed, where each state is coupled by atwo-photon Raman transition.
Further examples of physical realizations of the GRM include (i)electric-magnetic coupling of light and matter, and (ii) effectiverealization of the model using 3- and 4-level emitters
For a review of different realization see M. Tomka et al, Sci. Rep. 5,13097 (2015)
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 39
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Conclusions and message to take
The Fulton-Gouterman reduction by means of the ABCD theorem fortwo-levels systems in a quantized field is a kind of analogue of Berry’sdiabatic form of a general Hamiltonian for two-levels systems in aclassical field
Calculation can be reduced to one-dimensional ordinary differentialequations involving Dunkl operators
Tridiagonality accompanied by OPS is generic
TLS in quantized field do have nontrivial geometric phases
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 40
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Conclusions and message to take
The Fulton-Gouterman reduction by means of the ABCD theorem fortwo-levels systems in a quantized field is a kind of analogue of Berry’sdiabatic form of a general Hamiltonian for two-levels systems in aclassical field
Calculation can be reduced to one-dimensional ordinary differentialequations involving Dunkl operators
Tridiagonality accompanied by OPS is generic
TLS in quantized field do have nontrivial geometric phases
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 40
/ 86
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Conclusions and message to take
The Fulton-Gouterman reduction by means of the ABCD theorem fortwo-levels systems in a quantized field is a kind of analogue of Berry’sdiabatic form of a general Hamiltonian for two-levels systems in aclassical field
Calculation can be reduced to one-dimensional ordinary differentialequations involving Dunkl operators
Tridiagonality accompanied by OPS is generic
TLS in quantized field do have nontrivial geometric phases
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 40
/ 86
![Page 116: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/116.jpg)
Conclusions and message to take
The Fulton-Gouterman reduction by means of the ABCD theorem fortwo-levels systems in a quantized field is a kind of analogue of Berry’sdiabatic form of a general Hamiltonian for two-levels systems in aclassical field
Calculation can be reduced to one-dimensional ordinary differentialequations involving Dunkl operators
Tridiagonality accompanied by OPS is generic
TLS in quantized field do have nontrivial geometric phases
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 40
/ 86
![Page 117: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/117.jpg)
Thank you
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 41
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Further reading
This talk has been based on excerpts from my arxiv:1205.3139; EPL 100, 60010 (2012);AP 338, 319-340 (2013); AP 340, 252-266 (2014); J. Phys. A: Math. Theor. 47(49),495204 (2014); AP 351, 960-974 (2014); J. Phys. A: Math. Theor. 48(41), 415201(2015); EPL 113, 50004 (2016)
F77 source codes and numerical data files for the Rabi model can be freely downloadedfrom http://www.wave-scattering.com/rabi.html
My F77 scattering, photonic crystals, and plasmonic codes are freely available athttp://www.wave-scattering.com/codes.html
For list of some my projects see http://www.wave-scattering.com/projects.html
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Tridiagonality is generic
There always exists an orthonormal basis en∞n=0 such that a givenself-adjoint operator takes on a tridiagonal form
Hen = anen + bn+1en+1 + bnen−1 (4)
with real recurrence coefficients an and bn, where bn ≥ 0, n ≥ 0.
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 43
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Proof of Hen = anen + bn+1en+1 + bnen−1
n = 0: find a0, b1, e1
He0 = a0e0 + b1e1
a0 = 〈e0,He0〉b2
1 = 〈(H− a0)e0, (H− a0)e0〉e1 = (H− a0)e0/b1
Induction step to determine an, bn+1, en+1
〈en−1,Hen〉 = 〈en,Hen−1〉 ≡ bnan = 〈en,Hen〉bn+1en+1 = (H− an)en − bnen−1
b2n+1 = 〈(H− an)en − bnen−1, (H− an)en − bnen−1〉
en+1 = [(H− an)en − bnen−1]/bn+1
By construction en+1 normalized to one and orthogonal to en and en−1.
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 44
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Proof of Hen = anen + bn+1en+1 + bnen−1
n = 0: find a0, b1, e1
He0 = a0e0 + b1e1
a0 = 〈e0,He0〉b2
1 = 〈(H− a0)e0, (H− a0)e0〉e1 = (H− a0)e0/b1
Induction step to determine an, bn+1, en+1
〈en−1,Hen〉 = 〈en,Hen−1〉 ≡ bnan = 〈en,Hen〉bn+1en+1 = (H− an)en − bnen−1
b2n+1 = 〈(H− an)en − bnen−1, (H− an)en − bnen−1〉
en+1 = [(H− an)en − bnen−1]/bn+1
By construction en+1 normalized to one and orthogonal to en and en−1.
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 44
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Final check: en+1 orthogonal to en−2,. . . ,e0
Tridiagonality of Hen, self-adjointness, and normalizability
bn+1〈em, en+1〉 = 〈em,Hen〉 = 〈Hem, en〉
Tridiagonality for m < n − 1
Hem is a linear combination of em−1, em, and em+1, all of which havezero overlap with en if m + 1 < n.
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Final check: en+1 orthogonal to en−2,. . . ,e0
Tridiagonality of Hen, self-adjointness, and normalizability
bn+1〈em, en+1〉 = 〈em,Hen〉 = 〈Hem, en〉
Tridiagonality for m < n − 1
Hem is a linear combination of em−1, em, and em+1, all of which havezero overlap with en if m + 1 < n.
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Relation to orthogonal polynomials I
Wave function expansion
|E 〉 =∞∑n=0
pn(E )en
Three-term recurrence relation (TTRR)
Expansion coefficients satisfy
Epn(E ) = anpn(E ) + bn+1pn+1(E ) + bnpn−1(E )
with bk ≥ 0 and an initial condition p0 = 1 and p−1 = 0.pn(E ) are by the very definition orthogonal polynomials.
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Relation to orthogonal polynomials I
Wave function expansion
|E 〉 =∞∑n=0
pn(E )en
Three-term recurrence relation (TTRR)
Expansion coefficients satisfy
Epn(E ) = anpn(E ) + bn+1pn+1(E ) + bnpn−1(E )
with bk ≥ 0 and an initial condition p0 = 1 and p−1 = 0.pn(E ) are by the very definition orthogonal polynomials.
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Relation to orthogonal polynomials II
Shorthand for basis
en = pn(H)e0
Orthogonality relations
〈em, en〉 = 〈pm(H)e0, pn(H)e0〉 = δmn
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Relation to orthogonal polynomials II
Shorthand for basis
en = pn(H)e0
Orthogonality relations
〈em, en〉 = 〈pm(H)e0, pn(H)e0〉 = δmn
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Relation to orthogonal polynomials III
Change to the orthonormal basis of eigenstates
e0 =∑k
ωkψ(Ek)
Orthogonal polynomials of a discrete variable
〈em, en〉 =∑k
|ωk |2pm(Ek)pn(Ek) = δmn
Weight function is the local density of function (DOS)
n(E ) =∑k
|ωk |2δ(E − Ek)
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Relation to orthogonal polynomials III
Change to the orthonormal basis of eigenstates
e0 =∑k
ωkψ(Ek)
Orthogonal polynomials of a discrete variable
〈em, en〉 =∑k
|ωk |2pm(Ek)pn(Ek) = δmn
Weight function is the local density of function (DOS)
n(E ) =∑k
|ωk |2δ(E − Ek)
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Relation to orthogonal polynomials III
Change to the orthonormal basis of eigenstates
e0 =∑k
ωkψ(Ek)
Orthogonal polynomials of a discrete variable
〈em, en〉 =∑k
|ωk |2pm(Ek)pn(Ek) = δmn
Weight function is the local density of function (DOS)
n(E ) =∑k
|ωk |2δ(E − Ek)
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Weight function more rigorously (pp. 62-63 of Chihara’sbook)
limn→∞ xnl = ξl σ ≡ limj→∞ ξj
(a) ξl = σ = −∞ (l ≥ 1)
(b) −∞ < ξ1 < ξ2 < . . . < ξl = σ for some l ≥ 1
(c) −∞ < ξ1 < ξ2 < . . . < ξl < . . . < σ =∞
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Weight function more rigorously (pp. 62-63 of Chihara’sbook)
limn→∞ xnl = ξl σ ≡ limj→∞ ξj
(a) ξl = σ = −∞ (l ≥ 1)
(b) −∞ < ξ1 < ξ2 < . . . < ξl = σ for some l ≥ 1
(c) −∞ < ξ1 < ξ2 < . . . < ξl < . . . < σ =∞
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Weight function more rigorously (pp. 62-63 of Chihara’sbook)
limn→∞ xnl = ξl σ ≡ limj→∞ ξj
(a) ξl = σ = −∞ (l ≥ 1)
(b) −∞ < ξ1 < ξ2 < . . . < ξl = σ for some l ≥ 1
(c) −∞ < ξ1 < ξ2 < . . . < ξl < . . . < σ =∞
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Weight function more rigorously (pp. 62-63 of Chihara’sbook)
limn→∞ xnl = ξl σ ≡ limj→∞ ξj
(a) ξl = σ = −∞ (l ≥ 1)
(b) −∞ < ξ1 < ξ2 < . . . < ξl = σ for some l ≥ 1
(c) −∞ < ξ1 < ξ2 < . . . < ξl < . . . < σ =∞
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Weight function is the set of limit points of flows of zeros
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Numerical recipe
Choose Nc ≥ N0 and determine the first N0 zeros xNc l , l ≤ N0, ofPNc (x). Usually a good starting point is to take Nc ≈ N0 + 20.Because PNc (x) has Nc simple zeros, any omission of a zero could beeasily identified.
Gradually increase the cut-off value of Nc . The latter is what drivesthe incessant flows of polynomial zeros xNc l , wherein each flow ischaracterized by the parameter l .
Monitor convergence of the respective flows induced by the very firstn zeros of PNc (x). Each flow is a monotonically decreasing sequencehaving necessary a fixed limit point. Terminate your calculationswhen the N0-th zero of PNc (x) converged to ξN0 withinpredetermined accuracy. Then as a rule all other flows xNc l withl < N0 have converged, too.
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Numerical recipe
Choose Nc ≥ N0 and determine the first N0 zeros xNc l , l ≤ N0, ofPNc (x). Usually a good starting point is to take Nc ≈ N0 + 20.Because PNc (x) has Nc simple zeros, any omission of a zero could beeasily identified.
Gradually increase the cut-off value of Nc . The latter is what drivesthe incessant flows of polynomial zeros xNc l , wherein each flow ischaracterized by the parameter l .
Monitor convergence of the respective flows induced by the very firstn zeros of PNc (x). Each flow is a monotonically decreasing sequencehaving necessary a fixed limit point. Terminate your calculationswhen the N0-th zero of PNc (x) converged to ξN0 withinpredetermined accuracy. Then as a rule all other flows xNc l withl < N0 have converged, too.
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Numerical recipe
Choose Nc ≥ N0 and determine the first N0 zeros xNc l , l ≤ N0, ofPNc (x). Usually a good starting point is to take Nc ≈ N0 + 20.Because PNc (x) has Nc simple zeros, any omission of a zero could beeasily identified.
Gradually increase the cut-off value of Nc . The latter is what drivesthe incessant flows of polynomial zeros xNc l , wherein each flow ischaracterized by the parameter l .
Monitor convergence of the respective flows induced by the very firstn zeros of PNc (x). Each flow is a monotonically decreasing sequencehaving necessary a fixed limit point. Terminate your calculationswhen the N0-th zero of PNc (x) converged to ξN0 withinpredetermined accuracy. Then as a rule all other flows xNc l withl < N0 have converged, too.
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Convergence toward the 1000th eigenvalue of the Rabimodel in positive parity eigenspace
0 50 100 1501E-13
1E-11
1E-9
1E-7
1E-5
1E-3
0,1
10
n,99
9 - 99
9
(n-1000)/
κ=0.2, ∆=0.4 κ=1.0, ∆=0.7
ε999 = 998.907883759510, 997.950425260357, 973.989087026621 for(κ,∆) = (0.2, 0.4), (1, 0.7), (5, 0.4), respectively
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An approximation of the spectrum εl = l − κ2 of displacedharmonic oscillator by the zeros xnl
1 2 50 60 70 80 90 1001E-71E-51E-30,110
1 2 400 420 440 460 480 5001E-71E-51E-30,110
1 2 860 880 900 920 940 960 980 10001E-71E-51E-30,110
l
κ=2n=100
nl -
l
n=500
n=1000
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Table of the classical OPS
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Table of the classical OPS
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From the classical OPS to SE
From E. C. Titchmarsh, Eigenfunction Expansions, Part I, 2nd Edition, Oxford UniversityPress, Oxford, 1962 (its first edition appeared in 1946)
Usually attributed to A. Bhattacharjie and E. C. G. Sudarshan, A class of solvablepotentials, Nuovo Cimento 25(4), 864-879 (1962).
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From the classical OPS to SE
From E. C. Titchmarsh, Eigenfunction Expansions, Part I, 2nd Edition, Oxford UniversityPress, Oxford, 1962 (its first edition appeared in 1946)
Usually attributed to A. Bhattacharjie and E. C. G. Sudarshan, A class of solvablepotentials, Nuovo Cimento 25(4), 864-879 (1962).
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The world map of solvable potentials
(From G. L’evai, Spontaneous breakdown of PT symmetry in exactly,semi-analytically and numerically solvable potentials, 2015)
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The world map of solvable potentials
(From G. L’evai, Spontaneous breakdown of PT symmetry in exactly,semi-analytically and numerically solvable potentials, 2015)
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An approximation of the spectrum εl = l − κ2 of displacedharmonic oscillator by the zeros xnl
1 2 50 60 70 80 90 1001E-71E-51E-30,110
1 2 400 420 440 460 480 5001E-71E-51E-30,110
1 2 860 880 900 920 940 960 980 10001E-71E-51E-30,110
l
κ=2n=100
nl -
l
n=500
n=1000
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The message
Tridiagonality is fundamental
Three-term recurrence relations are indispensable
Orthogonal polynomial systems are unavoidable
an efficient numerical method for energy levelscharacterization of integrable and QES models in terms of intrinsicpolynomial propertiesconfine integrable spectra to four basic classescharacterization of chaos?
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 60
/ 86
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The message
Tridiagonality is fundamental
Three-term recurrence relations are indispensable
Orthogonal polynomial systems are unavoidable
an efficient numerical method for energy levelscharacterization of integrable and QES models in terms of intrinsicpolynomial propertiesconfine integrable spectra to four basic classescharacterization of chaos?
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 60
/ 86
![Page 150: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/150.jpg)
The message
Tridiagonality is fundamental
Three-term recurrence relations are indispensable
Orthogonal polynomial systems are unavoidable
an efficient numerical method for energy levelscharacterization of integrable and QES models in terms of intrinsicpolynomial propertiesconfine integrable spectra to four basic classescharacterization of chaos?
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 60
/ 86
![Page 151: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/151.jpg)
The message
Tridiagonality is fundamental
Three-term recurrence relations are indispensable
Orthogonal polynomial systems are unavoidable
an efficient numerical method for energy levels
characterization of integrable and QES models in terms of intrinsicpolynomial propertiesconfine integrable spectra to four basic classescharacterization of chaos?
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 60
/ 86
![Page 152: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/152.jpg)
The message
Tridiagonality is fundamental
Three-term recurrence relations are indispensable
Orthogonal polynomial systems are unavoidable
an efficient numerical method for energy levelscharacterization of integrable and QES models in terms of intrinsicpolynomial properties
confine integrable spectra to four basic classescharacterization of chaos?
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 60
/ 86
![Page 153: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/153.jpg)
The message
Tridiagonality is fundamental
Three-term recurrence relations are indispensable
Orthogonal polynomial systems are unavoidable
an efficient numerical method for energy levelscharacterization of integrable and QES models in terms of intrinsicpolynomial propertiesconfine integrable spectra to four basic classes
characterization of chaos?
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 60
/ 86
![Page 154: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/154.jpg)
The message
Tridiagonality is fundamental
Three-term recurrence relations are indispensable
Orthogonal polynomial systems are unavoidable
an efficient numerical method for energy levelscharacterization of integrable and QES models in terms of intrinsicpolynomial propertiesconfine integrable spectra to four basic classescharacterization of chaos?
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 60
/ 86
![Page 155: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/155.jpg)
Structure relation and bispectral condition
Divided-difference operator or discrete derivative
Dx f (x) =f (ι+(x))− f (ι−(x))
ι+(x)− ι−(x)
Structure relation
Dxpn(x) = −Bn(x)pn(x) + An(x)pn−1(x)
where Dx : Πn[x ]→ Πn−1[x ], Πn[x ] being the linear space of polynomialsin x over C with degree at most n ∈ Z≥0
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Structure relation and bispectral condition
Divided-difference operator or discrete derivative
Dx f (x) =f (ι+(x))− f (ι−(x))
ι+(x)− ι−(x)
Structure relation
Dxpn(x) = −Bn(x)pn(x) + An(x)pn−1(x)
where Dx : Πn[x ]→ Πn−1[x ], Πn[x ] being the linear space of polynomialsin x over C with degree at most n ∈ Z≥0
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Consequences of a structure relation
if there is one structure relation there is another - simply apply thefundamental TTRR
a pair of mutually adjoint raising and lowering ladder operators
orthogonal polynomials satisfy in general a second-order differenceequation = bispectral property
allows one to introduce a discrete analogue of the Bethe Ansatzequations
the polynomials are hypergeometric of Askey-Wilson type
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 62
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Consequences of a structure relation
if there is one structure relation there is another - simply apply thefundamental TTRR
a pair of mutually adjoint raising and lowering ladder operators
orthogonal polynomials satisfy in general a second-order differenceequation = bispectral property
allows one to introduce a discrete analogue of the Bethe Ansatzequations
the polynomials are hypergeometric of Askey-Wilson type
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 62
/ 86
![Page 159: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/159.jpg)
Consequences of a structure relation
if there is one structure relation there is another - simply apply thefundamental TTRR
a pair of mutually adjoint raising and lowering ladder operators
orthogonal polynomials satisfy in general a second-order differenceequation = bispectral property
allows one to introduce a discrete analogue of the Bethe Ansatzequations
the polynomials are hypergeometric of Askey-Wilson type
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 62
/ 86
![Page 160: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/160.jpg)
Consequences of a structure relation
if there is one structure relation there is another - simply apply thefundamental TTRR
a pair of mutually adjoint raising and lowering ladder operators
orthogonal polynomials satisfy in general a second-order differenceequation = bispectral property
allows one to introduce a discrete analogue of the Bethe Ansatzequations
the polynomials are hypergeometric of Askey-Wilson type
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 62
/ 86
![Page 161: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/161.jpg)
Consequences of a structure relation
if there is one structure relation there is another - simply apply thefundamental TTRR
a pair of mutually adjoint raising and lowering ladder operators
orthogonal polynomials satisfy in general a second-order differenceequation = bispectral property
allows one to introduce a discrete analogue of the Bethe Ansatzequations
the polynomials are hypergeometric of Askey-Wilson type
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 62
/ 86
![Page 162: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/162.jpg)
Dx requires four primary classes of special non-uniformlattices
1 the linear lattice
2 the linear q-lattice
3 the quadratic lattice
4 the q-quadratic lattice
EitherΛ = x | x = u2n
2 + u1n + u0, n ∈ N
orΛ = x | x = u2q
−n + u1qn + u0, n ∈ N
where uj are real constants and the real parameter q satisfies 0 < q < 1
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Dx requires four primary classes of special non-uniformlattices
1 the linear lattice
2 the linear q-lattice
3 the quadratic lattice
4 the q-quadratic lattice
EitherΛ = x | x = u2n
2 + u1n + u0, n ∈ N
orΛ = x | x = u2q
−n + u1qn + u0, n ∈ N
where uj are real constants and the real parameter q satisfies 0 < q < 1
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 63
/ 86
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Dx requires four primary classes of special non-uniformlattices
1 the linear lattice
2 the linear q-lattice
3 the quadratic lattice
4 the q-quadratic lattice
EitherΛ = x | x = u2n
2 + u1n + u0, n ∈ N
orΛ = x | x = u2q
−n + u1qn + u0, n ∈ N
where uj are real constants and the real parameter q satisfies 0 < q < 1
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 63
/ 86
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Dx requires four primary classes of special non-uniformlattices
1 the linear lattice
2 the linear q-lattice
3 the quadratic lattice
4 the q-quadratic lattice
EitherΛ = x | x = u2n
2 + u1n + u0, n ∈ N
orΛ = x | x = u2q
−n + u1qn + u0, n ∈ N
where uj are real constants and the real parameter q satisfies 0 < q < 1
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 63
/ 86
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Dx requires four primary classes of special non-uniformlattices
1 the linear lattice
2 the linear q-lattice
3 the quadratic lattice
4 the q-quadratic lattice
EitherΛ = x | x = u2n
2 + u1n + u0, n ∈ N
orΛ = x | x = u2q
−n + u1qn + u0, n ∈ N
where uj are real constants and the real parameter q satisfies 0 < q < 1
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 63
/ 86
![Page 167: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/167.jpg)
Dx requires four primary classes of special non-uniformlattices
1 the linear lattice
2 the linear q-lattice
3 the quadratic lattice
4 the q-quadratic lattice
EitherΛ = x | x = u2n
2 + u1n + u0, n ∈ N
orΛ = x | x = u2q
−n + u1qn + u0, n ∈ N
where uj are real constants and the real parameter q satisfies 0 < q < 1
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 63
/ 86
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Is Rabi model integrable or solvable?
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Characterization of tridiagonalHen−1 = an−1en−1 + bnen + bn−1en−2
(A) bn = 0 for some n = N > 0
(B) bn 6= 0 for all n ≥ 0
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Characterization of tridiagonalHen−1 = an−1en−1 + bnen + bn−1en−2
(A) bn = 0 for some n = N > 0
(B) bn 6= 0 for all n ≥ 0
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 65
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Hen−1 = an−1en−1 + bnen + bn−1en−2 with bN = 0 forsome N > 0
pN (E ) enters the TTRR first for n = N . One has
pn+N (E ) = pN (E )Qn(E ) (n ≥ 0)
i.e. pN+1(E ) is proportional to pN (E ). The quotient polynomialsQn(E ) form a new orthogonal sequence of polynomials.
The polynomials pn(E )N−1n=0 decouple from pn(E )∞n=N .
The operator H has necessary a finite dimensional invariant subspaceVN = enN−1
n=0 .
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 66
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Hen−1 = an−1en−1 + bnen + bn−1en−2 with bN = 0 forsome N > 0
pN (E ) enters the TTRR first for n = N . One has
pn+N (E ) = pN (E )Qn(E ) (n ≥ 0)
i.e. pN+1(E ) is proportional to pN (E ). The quotient polynomialsQn(E ) form a new orthogonal sequence of polynomials.
The polynomials pn(E )N−1n=0 decouple from pn(E )∞n=N .
The operator H has necessary a finite dimensional invariant subspaceVN = enN−1
n=0 .
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 66
/ 86
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Hen−1 = an−1en−1 + bnen + bn−1en−2 with bN = 0 forsome N > 0
pN (E ) enters the TTRR first for n = N . One has
pn+N (E ) = pN (E )Qn(E ) (n ≥ 0)
i.e. pN+1(E ) is proportional to pN (E ). The quotient polynomialsQn(E ) form a new orthogonal sequence of polynomials.
The polynomials pn(E )N−1n=0 decouple from pn(E )∞n=N .
The operator H has necessary a finite dimensional invariant subspaceVN = enN−1
n=0 .
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 66
/ 86
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Quasi-exact solvability of a linear differential operator H
H preserves a finite dimensional subspace VN of a Hilbert spaceL2(S) (S ⊂ R), HVN ⊂ VN , dimVN = N <∞, on which theoperator H is naturally defined
The basis of VN admits an explicit analytic formVN = span e0(x), . . . , eN−1(x)
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Quasi-exact solvability of a linear differential operator H
H preserves a finite dimensional subspace VN of a Hilbert spaceL2(S) (S ⊂ R), HVN ⊂ VN , dimVN = N <∞, on which theoperator H is naturally defined
The basis of VN admits an explicit analytic formVN = span e0(x), . . . , eN−1(x)
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 67
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Basic properties of finite OPS defined byxPn = Pn+1 + cnPn + λnPn−1
If λk > 0 and ck are real for k = 0, 1, . . . ,N − 1 then:
the polynomials of the resulting finite orthogonal polynomial sequencePkNk=1 have real and simple zeros
the zeros of PkNk=1 interlace (the Sturm property)
νP(E ) =∑N−1
k=0 wk θ(E − Ek), where θ(x) is Heaviside’s stepfunction, Ek are the N zeros of PN , and the coefficients wk can befound by solving
N−1∑l=0
Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.
wk > 0 for 0 ≤ k < NdνP(E ) is unique
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 68
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Basic properties of finite OPS defined byxPn = Pn+1 + cnPn + λnPn−1
If λk > 0 and ck are real for k = 0, 1, . . . ,N − 1 then:
the polynomials of the resulting finite orthogonal polynomial sequencePkNk=1 have real and simple zeros
the zeros of PkNk=1 interlace (the Sturm property)
νP(E ) =∑N−1
k=0 wk θ(E − Ek), where θ(x) is Heaviside’s stepfunction, Ek are the N zeros of PN , and the coefficients wk can befound by solving
N−1∑l=0
Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.
wk > 0 for 0 ≤ k < NdνP(E ) is unique
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 68
/ 86
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Basic properties of finite OPS defined byxPn = Pn+1 + cnPn + λnPn−1
If λk > 0 and ck are real for k = 0, 1, . . . ,N − 1 then:
the polynomials of the resulting finite orthogonal polynomial sequencePkNk=1 have real and simple zeros
the zeros of PkNk=1 interlace (the Sturm property)
νP(E ) =∑N−1
k=0 wk θ(E − Ek), where θ(x) is Heaviside’s stepfunction, Ek are the N zeros of PN , and the coefficients wk can befound by solving
N−1∑l=0
Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.
wk > 0 for 0 ≤ k < NdνP(E ) is unique
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 68
/ 86
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Basic properties of finite OPS defined byxPn = Pn+1 + cnPn + λnPn−1
If λk > 0 and ck are real for k = 0, 1, . . . ,N − 1 then:
the polynomials of the resulting finite orthogonal polynomial sequencePkNk=1 have real and simple zeros
the zeros of PkNk=1 interlace (the Sturm property)
νP(E ) =∑N−1
k=0 wk θ(E − Ek), where θ(x) is Heaviside’s stepfunction, Ek are the N zeros of PN , and the coefficients wk can befound by solving
N−1∑l=0
Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.
wk > 0 for 0 ≤ k < NdνP(E ) is unique
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 68
/ 86
![Page 180: Two level systems, geometric phases, and …pcs.ibs.re.kr/PCS_Workshops/TSLB_Talks_files/Moroz.pdfTwo level systems, geometric phases, and di erential operators of Dunkl type Alexander](https://reader035.fdocuments.us/reader035/viewer/2022070917/5fb799c35312cd5f2b528417/html5/thumbnails/180.jpg)
Basic properties of finite OPS defined byxPn = Pn+1 + cnPn + λnPn−1
If λk > 0 and ck are real for k = 0, 1, . . . ,N − 1 then:
the polynomials of the resulting finite orthogonal polynomial sequencePkNk=1 have real and simple zeros
the zeros of PkNk=1 interlace (the Sturm property)
νP(E ) =∑N−1
k=0 wk θ(E − Ek), where θ(x) is Heaviside’s stepfunction, Ek are the N zeros of PN , and the coefficients wk can befound by solving
N−1∑l=0
Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.
wk > 0 for 0 ≤ k < N
dνP(E ) is unique
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 68
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Basic properties of finite OPS defined byxPn = Pn+1 + cnPn + λnPn−1
If λk > 0 and ck are real for k = 0, 1, . . . ,N − 1 then:
the polynomials of the resulting finite orthogonal polynomial sequencePkNk=1 have real and simple zeros
the zeros of PkNk=1 interlace (the Sturm property)
νP(E ) =∑N−1
k=0 wk θ(E − Ek), where θ(x) is Heaviside’s stepfunction, Ek are the N zeros of PN , and the coefficients wk can befound by solving
N−1∑l=0
Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.
wk > 0 for 0 ≤ k < NdνP(E ) is unique
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 68
/ 86
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Normalization paradox
Orthogonality
L[π(x)pn(x)] = 0
for every polynomial π(x) of degree k < n. Hence the norm of pn(E ) forn ≥ N vanishes, i.e.
0 ≡ 1
bN+1L[ENpN (E )]
Paradox
One cannot satisfy (H1), i.e. that pn(E ) are orthonormal with respectto the density of states (DOS), n0(E ),∫ ∞
−∞pn(E )pm(E )n0(E )dE = δnm
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Resolution
Assume that the respective spectra obtained in VN and L2(S)\VN formtwo disjoint intervals on the real axis, S = Sqes ∪ S∞, sup Sqes < inf S∞.
On Sqes is n0(E )dE represented by the discrete measure dνP(E )
On S∞ is n0(E )dE represented by
dνPQ(E ) =1
p2N (E )
dνQ(E ) (E ∈ S∞)
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Resolution
Assume that the respective spectra obtained in VN and L2(S)\VN formtwo disjoint intervals on the real axis, S = Sqes ∪ S∞, sup Sqes < inf S∞.
On Sqes is n0(E )dE represented by the discrete measure dνP(E )
On S∞ is n0(E )dE represented by
dνPQ(E ) =1
p2N (E )
dνQ(E ) (E ∈ S∞)
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Resolution
Assume that the respective spectra obtained in VN and L2(S)\VN formtwo disjoint intervals on the real axis, S = Sqes ∪ S∞, sup Sqes < inf S∞.
On Sqes is n0(E )dE represented by the discrete measure dνP(E )
On S∞ is n0(E )dE represented by
dνPQ(E ) =1
p2N (E )
dνQ(E ) (E ∈ S∞)
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R. Haydock, p. 217 of his review:
“The generality of this result suggests a new way of looking at quantummechanics. Since any quantum system can be transformed into a chainmodel, we need only investigate such chain models in order to see thevarieties of quantum phenomena that are possible. To understand aparticular physical system, we need only find the chain model appropriateto it.”
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Comparison with Braak’s PRL 107, 100401 (2011)
A transcendental function is constructed
G±(ζ) =∞∑n=0
Kn(ζ, κ)
[1∓ ∆
ζ − n
]κn
and one determines eigenvalues as zeros of G±(ζ).
The expansion coefficients Kn are determined by a TTRR
φn+1 −fn(ζ)
(n + 1)φn +
1
n + 1φn−1 = 0
where
fn(ζ) = 2κ+1
2κ
(n − ζ − ∆2
n − ζ
).
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Comparison with Braak’s PRL 107, 100401 (2011)
A transcendental function is constructed
G±(ζ) =∞∑n=0
Kn(ζ, κ)
[1∓ ∆
ζ − n
]κn
and one determines eigenvalues as zeros of G±(ζ).
The expansion coefficients Kn are determined by a TTRR
φn+1 −fn(ζ)
(n + 1)φn +
1
n + 1φn−1 = 0
where
fn(ζ) = 2κ+1
2κ
(n − ζ − ∆2
n − ζ
).
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Shortcomings of Braak’s approach
Relies heavily on unproven hypotheses that G±(ζ) takes on a zero valuebetween its subsequent poles at ζ = n and ζ = n + 1
once - by implicitly presuming that at one of the poles G±(ζ) goes to+∞ and at the neighboring pole goes to −∞, with a monotonicbehavior from +∞ to −∞ between the poles;
twice - implicitly presuming that G±(ζ) goes to one of ±∞ at bothsubsequent poles, and in between the poles it has rather a featurelessbehavior, e.g., similar to a cord hanging on two posts;
none - occurs under the similar circumstances as described in theprevious item, if the “cord is too short”, e.g., it does not stretchsufficiently up or down so as to cross the abscissa.
Numerically no advantage over the classical Schweber’s solution [Ann.Phys. (N.Y.) 41, 205 (1967)].
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Shortcomings of Braak’s approach
Relies heavily on unproven hypotheses that G±(ζ) takes on a zero valuebetween its subsequent poles at ζ = n and ζ = n + 1
once - by implicitly presuming that at one of the poles G±(ζ) goes to+∞ and at the neighboring pole goes to −∞, with a monotonicbehavior from +∞ to −∞ between the poles;
twice - implicitly presuming that G±(ζ) goes to one of ±∞ at bothsubsequent poles, and in between the poles it has rather a featurelessbehavior, e.g., similar to a cord hanging on two posts;
none - occurs under the similar circumstances as described in theprevious item, if the “cord is too short”, e.g., it does not stretchsufficiently up or down so as to cross the abscissa.
Numerically no advantage over the classical Schweber’s solution [Ann.Phys. (N.Y.) 41, 205 (1967)].
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Shortcomings of Braak’s approach
Relies heavily on unproven hypotheses that G±(ζ) takes on a zero valuebetween its subsequent poles at ζ = n and ζ = n + 1
once - by implicitly presuming that at one of the poles G±(ζ) goes to+∞ and at the neighboring pole goes to −∞, with a monotonicbehavior from +∞ to −∞ between the poles;
twice - implicitly presuming that G±(ζ) goes to one of ±∞ at bothsubsequent poles, and in between the poles it has rather a featurelessbehavior, e.g., similar to a cord hanging on two posts;
none - occurs under the similar circumstances as described in theprevious item, if the “cord is too short”, e.g., it does not stretchsufficiently up or down so as to cross the abscissa.
Numerically no advantage over the classical Schweber’s solution [Ann.Phys. (N.Y.) 41, 205 (1967)].
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Shortcomings of Braak’s approach
Relies heavily on unproven hypotheses that G±(ζ) takes on a zero valuebetween its subsequent poles at ζ = n and ζ = n + 1
once - by implicitly presuming that at one of the poles G±(ζ) goes to+∞ and at the neighboring pole goes to −∞, with a monotonicbehavior from +∞ to −∞ between the poles;
twice - implicitly presuming that G±(ζ) goes to one of ±∞ at bothsubsequent poles, and in between the poles it has rather a featurelessbehavior, e.g., similar to a cord hanging on two posts;
none - occurs under the similar circumstances as described in theprevious item, if the “cord is too short”, e.g., it does not stretchsufficiently up or down so as to cross the abscissa.
Numerically no advantage over the classical Schweber’s solution [Ann.Phys. (N.Y.) 41, 205 (1967)].
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Comparison with Braak’s PRL 107, 100401 (2011) - II
The TTRR for the coefficients Kn is not a fundamental one, i.e. notof Haydock’s type, because of fn(ζ) is not a linear function of energy.
Any relation with the orthogonal polynomials pn is eluded.
Instead at looking straight at the zeros of pN of sufficiently largedegree, a complicated composite object, G±(ζ), is formed.
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Comparison with Braak’s PRL 107, 100401 (2011) - II
The TTRR for the coefficients Kn is not a fundamental one, i.e. notof Haydock’s type, because of fn(ζ) is not a linear function of energy.
Any relation with the orthogonal polynomials pn is eluded.
Instead at looking straight at the zeros of pN of sufficiently largedegree, a complicated composite object, G±(ζ), is formed.
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Comparison with Braak’s PRL 107, 100401 (2011) - II
The TTRR for the coefficients Kn is not a fundamental one, i.e. notof Haydock’s type, because of fn(ζ) is not a linear function of energy.
Any relation with the orthogonal polynomials pn is eluded.
Instead at looking straight at the zeros of pN of sufficiently largedegree, a complicated composite object, G±(ζ), is formed.
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Is Rabi model integrable or solvable?
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Proof of Berry’s observation - I
(1) Write the solution of
i~|ψ(τ)〉 = H(τ)|ψ(τ)〉
as|ψ(τ)〉 = U(τ)|ψ(τ)〉
Then |ψ(τ)〉 = U(τ)|ψ(τ)〉+ U(τ)| ˙ψ(τ)〉(2) |ψ(τ)〉 can be shown to satisfy
i~| ˙ψ(τ)〉 = H(τ)|ψ(τ)〉,H(τ) = U−1(τ) [H(τ)− i~dτ ]U(τ)
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Proof of Berry’s observation - I
(1) Write the solution of
i~|ψ(τ)〉 = H(τ)|ψ(τ)〉
as|ψ(τ)〉 = U(τ)|ψ(τ)〉
Then |ψ(τ)〉 = U(τ)|ψ(τ)〉+ U(τ)| ˙ψ(τ)〉
(2) |ψ(τ)〉 can be shown to satisfy
i~| ˙ψ(τ)〉 = H(τ)|ψ(τ)〉,H(τ) = U−1(τ) [H(τ)− i~dτ ]U(τ)
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Proof of Berry’s observation - I
(1) Write the solution of
i~|ψ(τ)〉 = H(τ)|ψ(τ)〉
as|ψ(τ)〉 = U(τ)|ψ(τ)〉
Then |ψ(τ)〉 = U(τ)|ψ(τ)〉+ U(τ)| ˙ψ(τ)〉(2) |ψ(τ)〉 can be shown to satisfy
i~| ˙ψ(τ)〉 = H(τ)|ψ(τ)〉,H(τ) = U−1(τ) [H(τ)− i~dτ ]U(τ)
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Proof of Berry’s observation - II
(3) One can get rid of the phase factors e±iφ in H(τ) by means of theunitary transformation
Uφ =
(exp− i
2 φ(τ) 0
0 exp i2 φ(τ)
).
Because [Uφ, σ0] = [Uφ, σ3] = 0
U−1φ H(τ)Uφ = diag H(τ)
+U−1φ
(0 B(τ) exp−iφ(τ)
B(τ) expiφ(τ) 0
)Uφ
=
(A(τ) + D(τ) B(τ)
B(τ) A(τ)− D(τ)
)
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Proof of Berry’s observation - II
(3) One can get rid of the phase factors e±iφ in H(τ) by means of theunitary transformation
Uφ =
(exp− i
2 φ(τ) 0
0 exp i2 φ(τ)
).
Because [Uφ, σ0] = [Uφ, σ3] = 0
U−1φ H(τ)Uφ = diag H(τ)
+U−1φ
(0 B(τ) exp−iφ(τ)
B(τ) expiφ(τ) 0
)Uφ
=
(A(τ) + D(τ) B(τ)
B(τ) A(τ)− D(τ)
)
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Proof of Berry’s observation - II
(3) One can get rid of the phase factors e±iφ in H(τ) by means of theunitary transformation
Uφ =
(exp− i
2 φ(τ) 0
0 exp i2 φ(τ)
).
Because [Uφ, σ0] = [Uφ, σ3] = 0
U−1φ H(τ)Uφ = diag H(τ)
+U−1φ
(0 B(τ) exp−iφ(τ)
B(τ) expiφ(τ) 0
)Uφ
=
(A(τ) + D(τ) B(τ)
B(τ) A(τ)− D(τ)
)
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Proof of Berry’s observation - III
(4) The contribution of
U−1φ (τ) (−i~dτ )Uφ(τ) = −~
2 φ(τ)σ3
induces the following change in H(τ) = U−1(τ) [H(τ)− i~dτ ]U(τ):
D(τ)→ D(τ)− ~2 φ(τ)
(5) One can get rid of the diagonal term A(τ) by amending Uφ with afactor proportional to σ0:
U = exp
− i
~
∫ τ
0A(τ ′) dτ ′
Uφ (5)
Then obviously U−1H(τ)U = U−1φ H(τ)Uφ, whereas
U−1(τ) (−i~dτ )U(τ) = −A(τ)σ0 − ~2 φ(τ)σ3 (6)
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Proof of Berry’s observation - III
(4) The contribution of
U−1φ (τ) (−i~dτ )Uφ(τ) = −~
2 φ(τ)σ3
induces the following change in H(τ) = U−1(τ) [H(τ)− i~dτ ]U(τ):
D(τ)→ D(τ)− ~2 φ(τ)
(5) One can get rid of the diagonal term A(τ) by amending Uφ with afactor proportional to σ0:
U = exp
− i
~
∫ τ
0A(τ ′) dτ ′
Uφ (5)
Then obviously U−1H(τ)U = U−1φ H(τ)Uφ, whereas
U−1(τ) (−i~dτ )U(τ) = −A(τ)σ0 − ~2 φ(τ)σ3 (6)
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d · E coupling
The atom is coupled with the field via polarization operator
S = σ+ + σ− = σx
Atomic inversion operator
σ3 = |e〉〈e| − |g〉〈g |
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d · E coupling
The atom is coupled with the field via polarization operator
S = σ+ + σ− = σx
Atomic inversion operator
σ3 = |e〉〈e| − |g〉〈g |
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Dipole operator
By parity consideration
〈e|d|e〉 = 〈g |d|g〉 = 0
In general,
d = d|g〉〈e|+ d∗|e〉〈g |= dσ− + d∗σ+
σ+ = |e〉〈g |, σ− = |g〉〈e|
When the states |e〉 and |g〉 are connected by ∆m = 0 transition, it ispossible to adjust the arbitrary phases associated with the states sothat Im d vanishes, d is real, and
d = d(σ− + σ+), 〈e|d|g〉 = d
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Dipole operator
By parity consideration
〈e|d|e〉 = 〈g |d|g〉 = 0
In general,
d = d|g〉〈e|+ d∗|e〉〈g |= dσ− + d∗σ+
σ+ = |e〉〈g |, σ− = |g〉〈e|
When the states |e〉 and |g〉 are connected by ∆m = 0 transition, it ispossible to adjust the arbitrary phases associated with the states sothat Im d vanishes, d is real, and
d = d(σ− + σ+), 〈e|d|g〉 = d
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Dipole operator
By parity consideration
〈e|d|e〉 = 〈g |d|g〉 = 0
In general,
d = d|g〉〈e|+ d∗|e〉〈g |= dσ− + d∗σ+
σ+ = |e〉〈g |, σ− = |g〉〈e|
When the states |e〉 and |g〉 are connected by ∆m = 0 transition, it ispossible to adjust the arbitrary phases associated with the states sothat Im d vanishes, d is real, and
d = d(σ− + σ+), 〈e|d|g〉 = d
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Dipole operator
By parity consideration
〈e|d|e〉 = 〈g |d|g〉 = 0
In general,
d = d|g〉〈e|+ d∗|e〉〈g |= dσ− + d∗σ+
σ+ = |e〉〈g |, σ− = |g〉〈e|
When the states |e〉 and |g〉 are connected by ∆m = 0 transition, it ispossible to adjust the arbitrary phases associated with the states sothat Im d vanishes, d is real, and
d = d(σ− + σ+), 〈e|d|g〉 = d
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d · E coupling
A single-mode cavity field in between two plane parallel mirrors
E(r, t) = e
(~ωε0V
)1/2 [a(t) + a†(t)
]sin(kz),
where e is an arbitrary oriented polarization field.
The dipole interaction Hamiltonian with a single-mode cavity field is
Hint = −d · E = dg(a† + a)
Here d = d · e and
κ = −(
~ωε0V
)1/2
sin(kz)
On using the atomic transition operators
Hint = ~g(σ− + σ+)(a† + a) = ~gσx(a† + a), g =dκ
~·
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d · E coupling
A single-mode cavity field in between two plane parallel mirrors
E(r, t) = e
(~ωε0V
)1/2 [a(t) + a†(t)
]sin(kz),
where e is an arbitrary oriented polarization field.
The dipole interaction Hamiltonian with a single-mode cavity field is
Hint = −d · E = dg(a† + a)
Here d = d · e and
κ = −(
~ωε0V
)1/2
sin(kz)
On using the atomic transition operators
Hint = ~g(σ− + σ+)(a† + a) = ~gσx(a† + a), g =dκ
~·
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/ 86
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d · E coupling
A single-mode cavity field in between two plane parallel mirrors
E(r, t) = e
(~ωε0V
)1/2 [a(t) + a†(t)
]sin(kz),
where e is an arbitrary oriented polarization field.
The dipole interaction Hamiltonian with a single-mode cavity field is
Hint = −d · E = dg(a† + a)
Here d = d · e and
κ = −(
~ωε0V
)1/2
sin(kz)
On using the atomic transition operators
Hint = ~g(σ− + σ+)(a† + a) = ~gσx(a† + a), g =dκ
~·
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 81
/ 86
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d · E coupling
A single-mode cavity field in between two plane parallel mirrors
E(r, t) = e
(~ωε0V
)1/2 [a(t) + a†(t)
]sin(kz),
where e is an arbitrary oriented polarization field.
The dipole interaction Hamiltonian with a single-mode cavity field is
Hint = −d · E = dg(a† + a)
Here d = d · e and
κ = −(
~ωε0V
)1/2
sin(kz)
On using the atomic transition operators
Hint = ~g(σ− + σ+)(a† + a) = ~gσx(a† + a), g =dκ
~·
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 81
/ 86
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Rabi model
In the Schrodinger picture
HR = ~ω1a†a + ~gσ1(a† + a) + µσ3,
where 1 is the unit matrix, a and a† are the conventional bosonannihilation and creation operators satisfying commutation relation[a, a†] = 1, g is a coupling constant, and µ = ~ω0/2. Here and elsewherethe standard representation of the Pauli matrices σj , j = 1, 2, 3, isassumed and the reduced Planck constant in what follows ~ = 1.
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Generalized Rabi model
HgR = γa†a+µσ3+g1
(a†σ−+aσ+
)+ g2
(a†σ++aσ−
)Define for any real parameter λ ∈ R
a = (2~)−1/2(λq + iλ−1p),
a† = (2~)−1/2(λq − iλ−1p).
Then
q =
√~√
2λ(a + a†).
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Quantum invariant 〈A〉ν = 〈ν|a†(σ− + σ+)|ν〉 versusquantum invariant Eν = 〈ν|H |ν〉 for α = 0
-4
-3
-2
-1
0
1
2
3
4
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(a)-4
-3
-2
-1
0
1
2
3
4
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(a)-4
-3
-2
-1
0
1
2
3
4
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(a)-4
-3
-2
-1
0
1
2
3
4
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(a)
For the eigenstates |ν〉 with parity P = +1 of the spin-boson model with σ = 12 ,
~ω = 1, λ.
= (Λ/~ω)2 = 0.09.Alexander Moroz Two level systems
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Quantum invariant 〈A〉ν = 〈ν|a†(S− + S+)|ν〉 versusquantum invariant Eν = 〈ν|H |ν〉 for α = π/2
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(b)-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(b)-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(b)-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 5 10 15 20 25 30 35 40
⟨A⟩ m
n
Emn
(b)
For the eigenstates |ν〉 with parity P = +1 of the spin-boson model with σ = 12 ,
~ω = 1, λ.
= (Λ/~ω)2 = 0.09.[V. V. Stepanov et al, Phys. Rev. E 77, 066202 (2008)]
Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 85
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Quantum invariant 〈A〉ν = 〈ν|a†(S− + S+)|ν〉 versusquantum invariant Eν = 〈ν|H |ν〉 for α = π/4
-2
-1
0
1
2
0 5 10 15 20 25 30 35 40
⟨A⟩ k
Ek
(c)-2
-1
0
1
2
0 5 10 15 20 25 30 35 40
⟨A⟩ k
Ek
(c)-2
-1
0
1
2
0 5 10 15 20 25 30 35 40
⟨A⟩ k
Ek
(c)-2
-1
0
1
2
0 5 10 15 20 25 30 35 40
⟨A⟩ k
Ek
(c)
For the eigenstates |ν〉 with parity P = +1 of the spin-boson model with σ = 12 ,
~ω = 1, λ.
= (Λ/~ω)2 = 0.09.Alexander Moroz Two level systems
Topological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 86/ 86