Two level systems, geometric phases, and...

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

Alexander Moroz

Wave-scattering.com

wavescattering@yahoo.com

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

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

Overview

Motivation

Conclusions

Overview

Motivation

Conclusions

Overview

Motivation

Conclusions

Overview

Motivation

Conclusions

Overview

Motivation

Conclusions

Overview

Motivation

Conclusions

Overview

Motivation

Conclusions

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

Motivation

Fundamental physics

Motivation

Fundamental physics

ultimate structural limit of individual atoms

holonomic quantum computation using quantum gates implementingthe geometric phasesexperimental advances

Motivation

Fundamental physics

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

experimental advances

Motivation

Fundamental physics

Motivation

Fundamental physics

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)

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)

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

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

Any TLS makes a cyclic evolution

A TLS Hamiltonian of the general two-level system

(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

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)]

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)]

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)]

Geometric phase

adiabatic evolution

Geometric phase

adiabatic evolution

Geometric phase

adiabatic evolution

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

Geometric phase

adiabatic evolution

Geometric phase

adiabatic evolution

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

∫C〈ψ(s ′)|dψ(s ′)〉

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

∫C〈ψ(s ′)|dψ(s ′)〉

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

∫C〈ψ(s ′)|dψ(s ′)〉

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)

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)

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+

λ1 − λ2[|c |2λ1 + (1− |c |2)λ2]

= 2π|c |2

|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+

λ1 − λ2[|c |2λ1 + (1− |c |2)λ2]

= 2π|c |2

|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+

λ1 − λ2[|c |2λ1 + (1− |c |2)λ2]

= 2π|c |2

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)

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

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

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)

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

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)

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

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)]

Fulton-Gouterman form

Any 2× 2 hermitian operator H can be expressed in the form

H =3∑

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

H =3∑

HFG = A1 + Bσ1 + Cσ2 + Dσ3,

H =3∑

HFG = A1 + Bσ1 + Cσ2 + Dσ3,

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

R −R

[(1 + R)U13 + (1− R)U−1

]where U13 = (σ1 + σ3)/

√2 and U2 = (1 + iσ2)/

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

FG into two

UFG =1√2

R −R

[(1 + R)U13 + (1− R)U−1

]where U13 = (σ1 + σ3)/

√2 and U2 = (1 + iσ2)/

L± = A + D ± (B − iC )R

(BR)† = RB = BR, (C R)† = RC = −C R

FG into two

UFG =1√2

R −R

[(1 + R)U13 + (1− R)U−1

]where U13 = (σ1 + σ3)/

√2 and U2 = (1 + iσ2)/

L± = A + D ± (B − iC )R

(BR)† = RB = BR, (C R)† = RC = −C R

FG into two

UFG =1√2

R −R

[(1 + R)U13 + (1− R)U−1

]where U13 = (σ1 + σ3)/

√2 and U2 = (1 + iσ2)/

L± = A + D ± (B − iC )R

(BR)† = RB = BR, (C R)† = RC = −C R

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)]

Examples involving R = e iπa†a

By means of U13 = (σ1 + σ3)/√

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+

(a†σ++aσ−

)The Foulton-Gouterman form:

A = a†a, B = ∆, C = iλ−κ

a†, D = κa +λ+

κa†

∆ :=µ

ω, λ± :=

g21 ± g2

2ω2, κ :=

√g1g2

(a†σ−+aσ+

(a†σ++aσ−

The Foulton-Gouterman form:

A = a†a, B = ∆, C = iλ−κ

a†, D = κa +λ+

κa†

∆ :=µ

ω, λ± :=

g21 ± g2

2ω2, κ :=

√g1g2

(a†σ−+aσ+

(a†σ++aσ−

A = a†a, B = ∆, C = iλ−κ

a†, D = κa +λ+

κa†

∆ :=µ

ω, λ± :=

g21 ± g2

2ω2, κ :=

√g1g2

(a†σ−+aσ+

(a†σ++aσ−

A = a†a, B = ∆, C = iλ−κ

a†, D = κa +λ+

κa†

∆ :=µ

ω, λ± :=

g21 ± g2

2ω2, κ :=

√g1g2

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

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−

HiRM = ωn +ω0

2σ3 + g

(√n + 2k a + a†

√n + 2k

K0 = n + k , K+ = a†√n + 2k σ1, K− =

√n + 2k aσ1

[K+, K−] = −2K0, [K0, K±] = ±K±

2σ1 + g

(K+ + K−

HiRM = ωn +ω0

2σ3 + g

(√n + 2k a + a†

√n + 2k

K0 = n + k , K+ = a†√n + 2k σ1, K− =

√n + 2k aσ1

[K+, K−] = −2K0, [K0, K±] = ±K±

2σ1 + g

(K+ + K−

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

(a†a + 1

)The TMRM: c = 1/2,

K+ = a†1a†2, K− = a1a2, K0 = 1

2 (a†1a1 + a†2a2 + 1)

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

(a†a + 1

)The TMRM: c = 1/2,

K+ = a†1a†2, K− = a1a2, K0 = 1

2 (a†1a1 + a†2a2 + 1)

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

(a†a + 1

The TMRM: c = 1/2,

K+ = a†1a†2, K− = a1a2, K0 = 1

2 (a†1a1 + a†2a2 + 1)

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

(a†a + 1

)The TMRM: c = 1/2,

K+ = a†1a†2, K− = a1a2, K0 = 1

2 (a†1a1 + a†2a2 + 1)

Borrowing from other talks

From Prof. Kwon Park talk on Weyl semimetals:

H(k) '[t(k2

1 − k20 ) +

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

H(k) '[t(k2

1 − k20 ) +

2 + k23 )]σ1 + 2t(k2σ2 + k3σ3)

R : kj → −kj (2)

H(k) '[t(k2

1 − k20 ) +

2 + k23 )]σ1 + 2t(k2σ2 + k3σ3)

R : kj → −kj (2)

Model Hamiltonians in the Bargmann-Fock representation

Bargmann-Fock representation:

a→ d/dz = dz , a† → z

=⇒HgR =

[(z + κ)dz +

(λ−z

κ+ ∆

Model Hamiltonians in the Bargmann-Fock representation

Bargmann-Fock representation:

a→ d/dz = dz , a† → z

=⇒HgR =

[(z + κ)dz +

(λ−z

κ+ ∆

Linear ordinary differential operators L± of Dunkl type

L± = (z + κ)dz +λ+z

κ±(λ−z

κ+ ∆

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

)±∆R

L±;2m = zd2z + (2q + γz)dz + z + γ

(q − 1

)±∆R

L± = (z + κ)dz +λ+z

κ±(λ−z

κ+ ∆

L± = (z + κ)dz + κz ±∆R

2(−1)−kR

L±;2p = 2zd2z + (4q + γz)dz + z

2 + γ(q − 1

)±∆R

L±;2m = zd2z + (2q + γz)dz + z + γ

(q − 1

)±∆R

L± = (z + κ)dz +λ+z

κ±(λ−z

κ+ ∆

L± = (z + κ)dz + κz ±∆R

2(−1)−kR

L±;2p = 2zd2z + (4q + γz)dz + z

2 + γ(q − 1

)±∆R

L±;2m = zd2z + (2q + γz)dz + z + γ

(q − 1

)±∆R

L± = (z + κ)dz +λ+z

κ±(λ−z

κ+ ∆

L± = (z + κ)dz + κz ±∆R

2(−1)−kR

L±;2p = 2zd2z + (4q + γz)dz + z

2 + γ(q − 1

)±∆R

L±;2m = zd2z + (2q + γz)dz + z + γ

(q − 1

)±∆R

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

by more than ±1

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.

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.

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

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

n0(E )∞∑n=0

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

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

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

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

Quantum invariants

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)]

Quantum invariant 〈A〉ν = 〈ν|a†(σ− + σ+)|ν〉 versusquantum invariant Eν = 〈ν|H |ν〉

0 5 10 15 20 25 30 35 40

⟨A⟩ m

0 5 10 15 20 25 30 35 40

⟨A⟩ m

0 5 10 15 20 25 30 35 40

⟨A⟩ m

0 5 10 15 20 25 30 35 40

⟨A⟩ m

(a)-3.5

0 5 10 15 20 25 30 35 40

⟨A⟩ m

(b)-3.5

0 5 10 15 20 25 30 35 40

⟨A⟩ m

(b)-3.5

0 5 10 15 20 25 30 35 40

⟨A⟩ m

(b)-3.5

0 5 10 15 20 25 30 35 40

⟨A⟩ m

0 5 10 15 20 25 30 35 40

⟨A⟩ k

0 5 10 15 20 25 30 35 40

⟨A⟩ k

0 5 10 15 20 25 30 35 40

⟨A⟩ k

0 5 10 15 20 25 30 35 40

⟨A⟩ k

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

Quantum invariant 〈A〉ν = 〈ν|a†(σ− + σ+)|ν〉 versusquantum invariant Eν = 〈ν|H |ν〉 - changes upon couplingvariations

10 10.5 11 11.5 12 12.5 13

⟨A⟩ m

λ=0.01 λ=0.25

λ=0.01λ=0.25

10 10.5 11 11.5 12 12.5 13

⟨A⟩ m

λ=0.01 λ=0.25

λ=0.01λ=0.25

10 10.5 11 11.5 12 12.5 13

⟨A⟩ m

λ=0.01 λ=0.25

λ=0.01λ=0.25

10 10.5 11 11.5 12 12.5 13

⟨A⟩ m

λ=0.01 λ=0.25

λ=0.01λ=0.25

10 10.5 11 11.5 12 12.5 13

⟨A⟩ m

λ=0.01 λ=0.25

λ=0.01λ=0.25

10 10.5 11 11.5 12 12.5 13

⟨A⟩ m

λ=0.01 λ=0.25

λ=0.01λ=0.25

10 10.5 11 11.5 12 12.5 13

⟨A⟩ m

λ=0.01 λ=0.25

λ=0.01λ=0.25

10 10.5 11 11.5 12 12.5 13

⟨A⟩ m

λ=0.01 λ=0.25

λ=0.01λ=0.25

16.5 17 17.5 18 18.5

⟨A⟩ k

λ=0.01

λ=0.25

λ=0.01

λ=0.25

λ=0.09 λ=0.09

16.5 17 17.5 18 18.5

⟨A⟩ k

λ=0.01

λ=0.25

λ=0.01

λ=0.25

λ=0.09 λ=0.09

16.5 17 17.5 18 18.5

⟨A⟩ k

λ=0.01

λ=0.25

λ=0.01

λ=0.25

λ=0.09 λ=0.09

16.5 17 17.5 18 18.5

⟨A⟩ k

λ=0.01

λ=0.25

λ=0.01

λ=0.25

λ=0.09 λ=0.09

16.5 17 17.5 18 18.5

⟨A⟩ k

λ=0.01

λ=0.25

λ=0.01

λ=0.25

λ=0.09 λ=0.09

16.5 17 17.5 18 18.5

⟨A⟩ k

λ=0.01

λ=0.25

λ=0.01

λ=0.25

λ=0.09 λ=0.09

16.5 17 17.5 18 18.5

⟨A⟩ k

λ=0.01

λ=0.25

λ=0.01

λ=0.25

λ=0.09 λ=0.09

16.5 17 17.5 18 18.5

⟨A⟩ k

λ=0.01

λ=0.25

λ=0.01

λ=0.25

λ=0.09 λ=0.09

32 32.5 33 33.5 34 34.5 35

⟨A⟩ m

λ=0.01

λ=0.25

λ=0.01

λ=0.25

32 32.5 33 33.5 34 34.5 35

⟨A⟩ m

λ=0.01

λ=0.25

λ=0.01

λ=0.25

32 32.5 33 33.5 34 34.5 35

⟨A⟩ m

λ=0.01

λ=0.25

λ=0.01

λ=0.25

32 32.5 33 33.5 34 34.5 35

⟨A⟩ m

λ=0.01

λ=0.25

λ=0.01

λ=0.25

32 32.5 33 33.5 34 34.5 35

⟨A⟩ m

λ=0.01

λ=0.25

λ=0.01

λ=0.25

32 32.5 33 33.5 34 34.5 35

⟨A⟩ m

λ=0.01

λ=0.25

λ=0.01

λ=0.25

32 32.5 33 33.5 34 34.5 35

⟨A⟩ m

λ=0.01

λ=0.25

λ=0.01

λ=0.25

32 32.5 33 33.5 34 34.5 35

⟨A⟩ m

λ=0.01

λ=0.25

λ=0.01

λ=0.25

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

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

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)

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)

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

Thank you

Two level systems, geometric phases, and...

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