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

Two level systems, geometric phases,and differential operators of Dunkl type

Alexander Moroz

Wave-scattering.com

[email protected]

Topological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016

Alexander Moroz Two level systemsTopological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 1

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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







Conclusions


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Overview

Motivation







Conclusions


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Overview

Motivation







Conclusions


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Overview

Motivation







Conclusions


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Overview

Motivation







Conclusions


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Overview

Motivation







Conclusions


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Overview

Motivation







Conclusions


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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





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Motivation

Fundamental physics


ultimate structural limit of individual atoms

holonomic quantum computation using quantum gates implementingthe geometric phasesexperimental advances



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Motivation

Fundamental physics


ultimate structural limit of individual atomsholonomic quantum computation using quantum gates implementingthe geometric phases

experimental advances



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Motivation

Fundamental physics





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Motivation

Fundamental physics





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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)



=



)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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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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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







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Geometric phase

adiabatic evolution







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Geometric phase

adiabatic evolution







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Geometric phase

adiabatic evolution




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



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Geometric phase

adiabatic evolution







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Geometric phase

adiabatic evolution







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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


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


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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|ψ0〉 =

(cos(ω0/2) e iχ0/2


)= c |ψ1〉+ d |ψ2〉, |c |2 + |d |2 = 1


θd = −∫ τp

0〈ψ(t)|H |ψ(t)〉 dt = − 2π

λ1 − λ2(|c |2λ1 + |d |2λ2)


θg = θt − θd

= − 2πλ2

λ1 − λ2+

2π

λ1 − λ2[|c |2λ1 + (1− |c |2)λ2]

= 2π|c |2


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|ψ0〉 =

(cos(ω0/2) e iχ0/2


)= c |ψ1〉+ d |ψ2〉, |c |2 + |d |2 = 1


θd = −∫ τp

0〈ψ(t)|H |ψ(t)〉 dt = − 2π

λ1 − λ2(|c |2λ1 + |d |2λ2)


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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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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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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H =3∑

j=0





HFG = A1 + Bσ1 + Cσ2 + Dσ3,




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H =3∑

j=0





HFG = A1 + Bσ1 + Cσ2 + Dσ3,




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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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FG into two


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



(BR)† = RB = BR, (C R)† = RC = −C R


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FG into two


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



(BR)† = RB = BR, (C R)† = RC = −C R


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FG into two


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



(BR)† = RB = BR, (C R)† = RC = −C R


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A comparison with Berry’s diabatic form



=



)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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2




(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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2




(a†σ−+aσ+

)+ g2

(a†σ++aσ−


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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2




(a†σ−+aσ+

)+ g2

(a†σ++aσ−


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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HiRM = ωn +ω0

2σ3 + g

(√n + 2k a + a†

√n + 2k

)σ1


K0 = n + k , K+ = a†√n + 2k σ1, K− =

√n + 2k aσ1

[K+, K−] = −2K0, [K0, K±] = ±K±



2σ1 + g

(K+ + K−

)σ3


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HiRM = ωn +ω0

2σ3 + g

(√n + 2k a + a†

√n + 2k

)σ1


K0 = n + k , K+ = a†√n + 2k σ1, K− =

√n + 2k aσ1

[K+, K−] = −2K0, [K0, K±] = ±K±



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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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)


H2FG = γ (K0 − c) + ∆σ1 + σ3(K+ + K−)


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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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)


H2FG = γ (K0 − c) + ∆σ1 + σ3(K+ + K−)


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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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)


H2FG = γ (K0 − c) + ∆σ1 + σ3(K+ + K−)


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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H(k) '[t(k2

1 − k20 ) +

m

2(k2

2 + k23 )]σ1 + 2t(k2σ2 + k3σ3)


R : kj → −kj (2)




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H(k) '[t(k2

1 − k20 ) +

m

2(k2

2 + k23 )]σ1 + 2t(k2σ2 + k3σ3)


R : kj → −kj (2)




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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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GRM

L± = (z + κ)dz +λ+z

κ±(λ−z

κ+ ∆

)R


L± = (z + κ)dz + κz ±∆R


2(−1)−kR


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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GRM

L± = (z + κ)dz +λ+z

κ±(λ−z

κ+ ∆

)R


L± = (z + κ)dz + κz ±∆R


2(−1)−kR


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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GRM

L± = (z + κ)dz +λ+z

κ±(λ−z

κ+ ∆

)R


L± = (z + κ)dz + κz ±∆R


2(−1)−kR


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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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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by more than ±1






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by more than ±1






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by more than ±1






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by more than ±1






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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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n0(E )∞∑n=0




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n0(E )∞∑n=0




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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






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Quantum invariants






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Quantum invariants






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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


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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


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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)


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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)


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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


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Thank you


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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.


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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.


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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.


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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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e0 =∑k

ωkψ(Ek)


〈em, en〉 =∑k



n(E ) =∑k

|ωk |2δ(E − Ek)


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e0 =∑k

ωkψ(Ek)


〈em, en〉 =∑k



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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(a) ξl = σ = −∞ (l ≥ 1)

(b) −∞ < ξ1 < ξ2 < . . . < ξl = σ for some l ≥ 1

(c) −∞ < ξ1 < ξ2 < . . . < ξl < . . . < σ =∞


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(a) ξl = σ = −∞ (l ≥ 1)

(b) −∞ < ξ1 < ξ2 < . . . < ξl = σ for some l ≥ 1

(c) −∞ < ξ1 < ξ2 < . . . < ξl < . . . < σ =∞


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(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





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Numerical recipe





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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?


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The message






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The message






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The message




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?


/ 86

The message




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?


/ 86

The message




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?


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The message






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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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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


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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


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2 + u1n + u0, n ∈ N

orΛ = x | x = u2q

−n + u1qn + u0, n ∈ N



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2 + u1n + u0, n ∈ N

orΛ = x | x = u2q

−n + u1qn + u0, n ∈ N



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2 + u1n + u0, n ∈ N

orΛ = x | x = u2q

−n + u1qn + u0, n ∈ N



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2 + u1n + u0, n ∈ N

orΛ = x | x = u2q

−n + u1qn + u0, n ∈ N



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2 + u1n + u0, n ∈ N

orΛ = x | x = u2q

−n + u1qn + u0, n ∈ N



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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


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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 .


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pn+N (E ) = pN (E )Qn(E ) (n ≥ 0)




n=0 .


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pn+N (E ) = pN (E )Qn(E ) (n ≥ 0)




n=0 .


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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)


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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


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νP(E ) =∑N−1


N−1∑l=0

Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.



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νP(E ) =∑N−1


N−1∑l=0

Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.



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νP(E ) =∑N−1


N−1∑l=0

Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.



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νP(E ) =∑N−1


N−1∑l=0

Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.

wk > 0 for 0 ≤ k < N

dνP(E ) is unique


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νP(E ) =∑N−1


N−1∑l=0

Pk(El)wl = δk0, k = 0, 1, . . . ,N − 1.



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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 ),∫ ∞



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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




dνPQ(E ) =1

p2N (E )

dνQ(E ) (E ∈ S∞)


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Resolution




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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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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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



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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



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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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Uφ =

(exp− i

2 φ(τ) 0

0 exp i2 φ(τ)

).



+U−1φ


B(τ) expiφ(τ) 0

)Uφ

=

(A(τ) + D(τ) B(τ)


)


/ 86



Uφ =

(exp− i

2 φ(τ) 0

0 exp i2 φ(τ)

).



+U−1φ


B(τ) expiφ(τ) 0

)Uφ

=

(A(τ) + D(τ) B(τ)


)


/ 86

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


〈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|


d = d(σ− + σ+), 〈e|d|g〉 = d


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Dipole operator


〈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|


d = d(σ− + σ+), 〈e|d|g〉 = d


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Dipole operator


〈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|


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


E(r, t) = e

(~ωε0V

)1/2 [a(t) + a†(t)

]sin(kz),



Hint = −d · E = dg(a† + a)

Here d = d · e and

κ = −(

~ωε0V

)1/2

sin(kz)



~·


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d · E coupling


E(r, t) = e

(~ωε0V

)1/2 [a(t) + a†(t)

]sin(kz),



Hint = −d · E = dg(a† + a)

Here d = d · e and

κ = −(

~ωε0V

)1/2

sin(kz)



~·


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d · E coupling


E(r, t) = e

(~ωε0V

)1/2 [a(t) + a†(t)

]sin(kz),



Hint = −d · E = dg(a† + a)

Here d = d · e and

κ = −(

~ωε0V

)1/2

sin(kz)



~·


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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

Topological states of light and beyond, PCS, Daejeon, Korea, June 24, 2016 84/ 86

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)


~ω = 1, λ.

= (Λ/~ω)2 = 0.09.[V. V. Stepanov et al, Phys. Rev. E 77, 066202 (2008)]


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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)


~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

Documents

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