Adiabatic Expansions

Íñigo L. Egusquiza University of the Basque Country, UPV/EHU

Quantum Days in Bilbao IV BCAM, Bilbao, July 2014

OUTLINE

• Adiabatic elimination in physics

• Singular perturbation. Invariant manifold method

• Iteration and convergence

• Hermiticity of the effective hamiltonian

• Examples

Adiabatic Elimination

• Widely separated scales

• Effective evolution of the lowest energy sector?

• Widespread application, specific names and techniques:

• Born-Oppenheimer

• Schrieffer-Wolff

• Bloch wave operator

• Effective lagrangians

• Adiabatic elimination in quantum optics

Singular perturbationExample:

2

ω

∆

g g1

� � !, g1, g2

Choice of units:

! ! 1

� ! 1/✏

gi ! gi/!

/ω

1

g1 21/ε

g /ω

Singular dynamical system

i↵̇ = �1

2↵+

g1!�

i�̇ =1

2� +

g2!�

i✏�̇ = � +✏

!(g1↵+ g2�)

Secular terms

First order perturbation

↵(t) = ↵(0)

✓1 + ✏

ig21t

!2

◆eit/2

+ ✏2ig1g2!2

�(0) sin(t/2) +O(✏2)

Secular Term

Singular perturbation techniques

• Multiple scales • Lindstet - Poincaré • Averaging • Dynamical Renormalisation Group • …

Our choice here:

Invariant manifold

Invariant manifold methodGiven a flow in a manifold, which are the submanifolds that are invariant under the flow….

… and can be determined perturbatively?Additional structure present, is it preserved in the embedding? !

Hamiltonian flow, e.g.

Translation to quantum mechanics

Schrödinger’s equation: i~@t = H

Invariance: existence of an operator (for Bloch) such that

B

HBH ✓ BHHB = BHB

Bloch equation

[H,B]B = 0

B2 = B ; B† 6= B

Further constraints

By itself, it is not very useful: Fully solving Bloch’s equation is equivalent to diagonalising the Hamiltonian

Need additional input for usefulness

Feshbach partitioning

Assume existence of projector P onto an uncoupled low energy sector

/ω

1

g1 21/ε

g /ω

P

Q = 1� P

Feshbach and BlochRestriction on the Bloch operator B

• Size of the subspace:

• Bloch-Feshbach alignment:

rank(B) = rank(P )

BP = B

Feshbach and Bloch: an implementation

If we write P !✓1 00 0

◆then B !

✓1 0B 0

◆

Very general; alternatively:

B = P +QBP

Feshbach and Bloch: an implementation

Same basis,

H =

✓PHP PHQQHP QHQ

◆= ~

✓! ⌦†

⌦ �

◆

Bloch

[H,B]B = 0 , ⌦+�B = B! +B⌦†B

Separated energy scales

H = ~✓! ⌦†

⌦ �

◆with invertible � and

✏ = k��1k k!k ⌧ 1

✏0 = k��1k k⌦k ⌧ 1

Separated scales: equations of motion

Define ↵ = P

� = Q

Equations of motion:

Also ⌧ = k!kt

i@⌧↵ =!

k!k↵+⌦†

k!k�

i✏@⌧� = k��1k⌦↵+ k��1k��

= ✏⌦

k!k↵+ k��1k��Singular perturbation

Bloch operator and separated scales

Equation: ⌦+�B = B! +B⌦†B

Alternatively: B = ���1⌦+��1B! +��1B⌦†B

= T (B)

If scales are separated:

kT (A)k ✏0�1 + kAk2

�+ ✏kAk

Bloch operator and separated scales

kT (A)k ✏0�1 + kAk2

�+ ✏kAk

0.5 1.0 1.5 2.0 2.5 3.0

0.5

1.0

1.5

2.0

2.5

3.0

Iteration

Result (A)

If there is separation of scales there is a (small) solution of Bloch’s equation

which can be built iteratively or perturbatively

(small: )kBk = O(✏0)

Iteration/perturbationB(0) = 0

B(k+1) = ThB(k)

i

= ���1⌦+��1B(k)! +��1B(k)⌦†B(k)

B(0) = 0

B(1) = ���1⌦

B(2) = ���1⌦���2⌦! +��2⌦⌦†��1⌦

Iteration/perturbationB(1) = ���1⌦

B(2) = ���2⌦!

B(k+1) = ��1B(k) +��1k�1X

l=1

B(k�l)⌦†B(l)

B(1) = ���1⌦

B(2) = ���2⌦!

B(3) = ���3⌦! +��2⌦⌦†��1⌦

B(k) �kX

l=1

B(l) = O⇣��(k+1)

⌘

Result (B)

Resummation of the secular terms!

he↵ = ! + ⌦†B

i@t↵ = he↵↵

Effective hamiltoniansh(k)e↵ = ! + ⌦†B(k)

h(k)e↵ = ! +kX

l=1

⌦†B(l)

h(1)e↵ = ! � ⌦†��1⌦

h(2)e↵ = h(1)

e↵ + ⌦†��2⌦⌦†��1⌦

� ⌦†��2⌦!

Not hermitian!

Why not hermitian?Essentially, he↵ ⇠ PHB

Physically, =

✓↵B↵

◆

Carries a fraction of the norm

Total norm: † = ↵† �1 +B†B

�↵

Hermitian hamiltonian

hB =�1 +B†B

�1/2he↵

�1 +B†B

��1/2

is hermitian if a solution of Bloch’s equationB

• Similarity transformation • Isospectral • Induces approximations • Hermitian approximations?

Hermitian Hamiltonian If a solution of Bloch’s equationB

�1 +B†B

� �! + ⌦†B

�= ! + ⌦†B +B†⌦+B†�B

sohB =

�1 +B†B

�1/2he↵

�1 +B†B

��1/2

=1p

1 +B†B

�! + ⌦†B +B†⌦+B†�B

� 1p1 +B†B

amenable to hermitian approximations

Connection to other approaches

Equivalence to unitary transformations with perturbative exponents

X =

✓1 �B†

B 1

◆0

@1p

1+B†B0

0

1p1+BB†

1

A

= exp

arctanh

✓0 �B†

B 0

◆�

block diagonalises the hamiltonian, if solution of Bloch

Connection to other approaches

Interleaves projectors

XP = PXXwith exact projector onto the dressed low

energy sector

PX

Examples: Lambda system

2

ω

∆

g g1

20 40 60 80 100 120D t

0.2

0.4

0.6

0.8

1.0

p

Exacth(4)e↵

!

Examples: Lambda system

2

ω

∆

g g1

20 40 60 80 100 120D t

0.2

0.4

0.6

0.8

1.0

p

Exacth(4)e↵

h(10)B

Example: phenomenological time of arrival

H =p2

2m+

~⌦(x)2

�x

� ~ (2�(x) + i�)

4(1� �

z

)

Question: what is the effective evolution of the spatial wave function of the (1 0)T

component?

Example: phenomenological time of arrival

Bloch equation:

B =⌦

2�+ i�+

1

m~1

2�+ i�

⇥B, p2

⇤� 1

2�+ i�B⌦B

Conditions:• Small kinetic energy • Either detuning or

dissipation dominates over Rabi

Example: phenomenological time of arrival

Effective potential W =~2⌦B

Adiabatic approximations:

W

(1)(x) =~2

⌦2(x)

2�(x) + i�

W

(2)(x, p) = W

(1)(x)� ~2

⌦4(x)

(2�(x) + i�)3

� ~2m

⌦

2�+ i�

"~✓

⌦

2�+ i�

◆00+ 2i

✓⌦

2�+ i�

◆0p

#

Non local term

Conclusions

• Adiabatic elimination in quantum optics is another instance of a general structure

• Separation of energy scales leads to singular perturbation dynamical systems

• The invariant manifold method resums secular terms • Guaranteed existence of scale separated Bloch

operator • Explicit construction of hermitian effective

hamiltonians

Thank you!