On the Dynamics of Periodically Perturbed Quantum Systems · On the dynamics of periodically perturbed quantum systems ... On the dynamics of periodically perturbed quantum systems

On the dynamics of periodically perturbed quantum systems

Consider a system of n ODEs

𝑑

𝑑𝑡𝜓 𝑡 = 𝐴 𝑡 𝜓 𝑡 , 𝜓:ℝ ⟶ 𝑀𝑛×1 ℂ

where 𝐴:ℝ ⟶ 𝑀𝑛×𝑛 ℂ is a continuous, (𝑛 × 𝑛) matrix-valued function of real

parameter t, periodic with period T,

∀ 𝑡 ∈ ℝ ∀ 𝑘 ∈ ℤ ∶ 𝐴 𝑡 + 𝑘𝑇 = 𝐴 𝑡 .

Let Φ 𝑡 to be the fundamental matrix of this system, satisfying the following:

𝑑

𝑑𝑡Φ 𝑡 = 𝐴 𝑡 Φ 𝑡 , detΦ 𝑡 ≠ 0, 𝜓 𝑡 = Φ 𝑡 𝑐

Wrońskian

Any general solution

(c = 𝑐𝑜𝑛𝑠𝑡.) of

system of ODEs


Proposition 1.

a) There exists some 𝑡0 ∈ ℝ such that 𝜓 𝑡 = 𝐸 𝑡, 𝑡0 𝜓 𝑡0 where

𝐸 𝑡, 𝑡0 ≔ Φ 𝑡 Φ 𝑡0−1 is called the resolvent matrix or state transition matrix.

b) Resolvent matrix is a fundamental matrix itself, i.e. it satisfies the same differential

equation

𝑑

𝑑𝑡𝐸 𝑡, 𝑡0 = 𝐴 𝑡 𝐸 𝑡, 𝑡0 .

Proposition 2. Resolvent matrix has some basic properties:

a) Divisibility: 𝐸 𝑡, 𝑡0 = 𝑘=1𝑛 𝐸 𝑡𝑘+1, 𝑡𝑘 for any partition 𝑡𝑘 , 𝑡𝑘+1 of 𝑡0, 𝑡 , iff 𝑡0, 𝑡 =

𝑘=1𝑛 𝑡𝑘 , 𝑡𝑘+1 , and 𝑡𝑘 , 𝑡𝑘+1 ∩ 𝑡𝑘′ , 𝑡𝑘′+1 = ∅ for 𝑘 ≠ 𝑘′

b) 𝐸 𝑡, 𝑡0−1 = 𝐸 𝑡0, 𝑡

c) 𝐸 𝑡0, 𝑡0 = 𝑖𝑑𝑀𝑛 ℂ .


Theorem 1 (Floquet’s).

If Φ 𝑡 is a fundamental matrix of a system of n ODEs and 𝐴 𝑡 is a T-periodic function of

codomain in the 𝑀𝑛 ℂ linear space of n-by-n matrices, then a matrix Φ 𝑡 + 𝑇 is also a

fundamental matrix of this system.

Remark: If Φ 𝑡 + 𝑇 is a fundamental matrix then there exist two constant matrices 𝐶 and 𝐵such that

Φ 𝑡 + 𝑇 = Φ 𝑡 𝐶, 𝐶 = 𝑒𝐵𝑇 .

Assume a spectral decomposition 𝐵 = 𝑘 𝜇𝑘 𝜙𝑘 , .⋅ 𝜙𝑘:

𝑒𝐵𝑇 =

𝑘

𝜆𝑘 𝜙𝑘 , .⋅ 𝜙𝑘 , 𝜆𝑘 = 𝑒𝜇𝑘𝑇.

𝜇𝑘: “Floquet

exponents”


Let 𝜓 𝑡 to be a solution of a system of ODEs, i.e. let it fulfill𝑑

𝑑𝑡𝜓 𝑡 = 𝐴 𝑡 𝜓 𝑡 .

Define 𝜓𝑘 𝑡 ≔ Φ 𝑡 𝜙𝑘 . Then it follows from Floquet’s theorem that

Φ 𝑡 + 𝑇 𝜙𝑘 = Φ 𝑡 𝑒𝐵𝑇𝜙𝑘 = 𝑒𝜇𝑘𝑇𝜓𝑘 𝑡 = 𝜓𝑘 𝑡 + 𝑇

Putting 𝜙𝑘 𝑡 = 𝜓𝑘 𝑡 𝑒−𝜇𝑘𝑡 one gets a set of “base solutions” of system of ODEs,

𝜓𝑘 𝑡 = 𝑒𝜇𝑘𝑡𝜙𝑘 𝑡 , 𝜙𝑘 𝑡 + 𝑇 = 𝜙𝑘 𝑡 .

T-periodic


Considering 𝜓𝑘 𝑡0 + 𝑇 one obtains an eigenequation of Φ 𝑡0 + 𝑇 Φ−1 𝑡0 :

𝜓𝑘 𝑡0 + 𝑇 = Φ 𝑡0 + 𝑇 Φ−1 𝑡0 𝜙𝑘 𝑡0 = 𝑒𝜇𝑘𝑇𝜙𝑘 𝑡0

Floquet’s operator

𝐹 𝑡0 ≔ 𝐸 𝑡0 + 𝑇, 𝑡0

Floquet’s basis

{𝜙𝑘≔ 𝜙𝑘 𝑡0 }

𝐸 𝑡0 + 𝑇, 𝑡0


• 𝐴 𝑡 = 𝐻 𝑡 – periodic, self-adjoint Hamiltonian of quantum-mechanical system

• ODEs describe an evolution of wavefunction (state) 𝜓 𝑡 – Schrödinger equation:

𝑑

𝑑𝑡𝜓 𝑡 = −

𝑖

ℏ𝐻 𝑡 𝜓 𝑡 , 𝐻 𝑡 + 𝑇 = 𝐻 𝑡 .

• Resolvent matrix – unitary propagator 𝑼 𝒕, 𝒕𝟎 :

𝐸 𝑡, 𝑡0 = 𝑈 𝑡, 𝑡0 = Texp −𝑖

ℏ

𝑡0

𝑡

𝐻 𝑡′ 𝑑𝑡′ ,𝑑𝑈 𝑡, 𝑡0

𝑑𝑡= −

𝑖

ℏ𝐻 𝑡 𝑈 𝑡, 𝑡0 .

• Floquet’s operator 𝐹 𝑡0 = 𝑈 𝑡0 + 𝑇, 𝑡0 = 𝑒−𝑖𝑇

ℏ 𝐻:

𝑭 𝒕𝟎 𝝓𝒌 𝒕𝟎 = 𝒆−𝒊𝑻ℏ𝝐𝒌𝝓𝒌 𝒕𝟎 ,

𝜙𝑘 ≔ 𝜙𝑘 𝑡0 – Floquet basis, 𝜖𝑘 – set of Bohr-Floquet quasienergies.


Floquet Hamiltonian:

𝐻𝐹 𝑟, 𝑡 = 𝐻 𝑟, 𝑡 − 𝑖ℏ𝑑

𝑑𝑡, 𝐻𝐹𝜓 𝑟, 𝑡 = 0

Main analysis based on Schrödinger equation for states 𝜙𝑘 𝑟, 𝑡 = 𝜙𝑘 𝑟, 𝑡 + 𝑇 :

𝐻𝐹𝜙𝑘 𝑟, 𝑡 = 𝜖𝑘𝜙𝑘 𝑟, 𝑡

Solutions are not unique:

They generate the same physical state 𝜓𝑘 𝑟, 𝑡

𝜙𝑘𝑛 𝑟, 𝑡 ≔ 𝜙𝑘 𝑟, 𝑡 𝑒𝑖𝑛Ω𝑡 𝐻𝐹𝜙𝑘𝑛 𝑟, 𝑡 = 𝜖𝑘 + 𝑛ℏΩ 𝜙𝑘 𝑟, 𝑡

𝜖𝑘𝑛Higher Floquet

modes


Extended Hilbert space ℋ⟶ℋ′ = ℛ⊗𝒯 (example: particle in free space)

𝒯 = ℒ2 𝕋𝑇1 , 𝑑𝑡 = span 𝜒𝑛 𝑡 ≔ 𝑒𝑖𝑛Ω𝑡

Space of square-integrable functions

with period T = 2𝜋/Ω, defined over a

circle 𝕋1.

𝜒𝑛, 𝜒𝑛′ =1

𝑇 𝕋1𝜒𝑛 𝑡 𝜒𝑛′ 𝑡 𝑑𝑡 = 𝛿𝑛𝑛′

𝑞

𝜒𝑞∗ ⋅ 𝜒𝑞 = 𝑖𝑑𝒯

ℛ = ℒ2 ℝ3, 𝑑𝑉 = span 𝑓𝑘: ℝ3 → ℂ

Space of square-integrable functions

defined over ℝ3.

𝑓𝑘 , 𝑓𝑘′ =

ℝ3

𝑓𝑘 𝑟 𝑓𝑘′ 𝑟 𝑑𝑉 𝑟 = 𝛿𝑘𝑘′

𝑛

𝑓𝑘∗ ⋅ 𝑓𝑘 = 𝑖𝑑ℛ

ℛ⊗𝒯 = span 𝑒𝑘𝑛 ≔ 𝑓𝑘 ⊗𝜒𝑛 , 𝑒𝑘𝑛 𝑟, 𝑡 = 𝑓𝑘 𝑟 𝑒𝑖𝑛Ω𝑡

𝑘𝑛

𝑒𝑘𝑛∗ ⋅ 𝑒𝑘𝑛 = 𝑖𝑑ℛ⊗𝒯 , 𝑒𝑘𝑛

∗ 𝑒𝑘′𝑛′ = 𝛿𝑘𝑘′𝛿𝑛𝑛′ , 𝑒𝑘𝑛∗ ∈ ℛ ⊗𝒯 ∗


Structure of Hamiltonian: 𝐻 𝑟, 𝑡 = 𝐻0 𝑟 + 𝑉 𝑟, 𝑡 , 𝑉 𝑟, 𝑡 + 𝑇 = 𝑉 𝑟, 𝑡 .

Idea: We are applying a transformation of variables:

𝜃 = Ω𝑡, 𝜃 = Ω

𝐻 𝑟, 𝜃, 𝜃 = 𝐻0 𝑟 + 𝑉 𝑟, 𝜃 + 𝜃𝑝𝜃

Canonical

quantization:

𝜃 → 𝜃,

𝑝𝜃 → −𝑖ℏ𝜕

𝜕𝜃,

𝜃, 𝑝𝜃 = 𝑖ℏ

𝐻 𝑟, 𝜃, 𝜃 = 𝐻0 𝑟 + 𝑉 𝑟, 𝜃 − 𝑖ℏΩ𝜕

𝜕𝜃, 𝐻 𝑟, 𝜃, 𝜃 𝜙𝑘𝑛 𝑟, 𝜃 = 𝜖𝑘𝑛𝜙𝑘𝑛 𝑟, 𝜃


𝐻 𝑟, 𝜃, 𝜃 𝜙𝑘𝑛 𝑟, 𝜃 = 𝜖𝑘𝑛𝜙𝑘𝑛 𝑟, 𝜃 , 𝜖𝑘𝑛 = 𝜖𝑘 + 𝑛ℏΩ

𝜙𝑘𝑛 ∈ ℛ ⊗𝒯, 𝒯 = ℒ2 𝕋2𝜋1 ,

1

Ω𝑑𝜃

Square-integrable functions of period

2𝜋 over a unit circle 𝕋1 = 𝜃 = 𝛺𝑡

How to include multi-mode setting?

Ansatz: add a sufficient number of new 𝜃𝑖 variables, such that

𝐻 𝑟, 𝜃, 𝜃 = 𝐻0 𝑟 + 𝑉 𝑟, 𝜃1, … , 𝜃𝑁 − 𝑖ℏ

𝑗=1

𝑁

Ω𝑗𝜕

𝜕𝜃𝑗, Ω𝑖 =

2𝜋

𝑇𝑖


New Schrödinger equation:

𝐻 𝑟, 𝜃1, 𝜃2, … , 𝜃𝑁 𝜙𝑘𝑛1𝑛2…𝑛𝑁 𝑟, 𝜃1, 𝜃2, … , 𝜃𝑁 = 𝜖𝑘𝑛1𝑛2…𝑛𝑁𝜙𝑘𝑛1𝑛2…𝑛𝑁 𝑟, 𝜃1, 𝜃2, … , 𝜃𝑁

Periodicity of 𝜙 functions:

𝜙𝑘𝑛1𝑛2…𝑛𝑁 𝑟, 𝜃1 + 2𝜋, 𝜃2 + 2𝜋,… , 𝜃𝑁 + 2𝜋 = 𝜙𝑘𝑛1𝑛2…𝑛𝑁 𝑟, 𝜃1, 𝜃2, … , 𝜃𝑁

Extension of Hilbert space of 𝜙 functions:

𝜙𝑘𝑛1𝑛2…𝑛𝑁 ∈ ℛ ⊗𝒯1⊗𝒯2⊗⋯⊗𝒯𝑁, 𝒯𝑗 = ℒ2𝜋2 𝕋1,

1

𝛺𝑗𝑑𝜃𝑗

𝑗=1

𝑁

ℒ2𝜋2 𝕋1,

1

𝛺𝑗𝑑𝜃𝑗 ≡ ℒ2 𝕋1 × 𝕋1 ×⋯× 𝕋1, 𝑑𝜏 = ℒ2 𝕋𝑁, 𝑑𝜏 , 𝑑𝜏 =

𝑗=1

𝑁𝑑𝜃𝑗

𝛺𝑗

Product measure


Qpen problem:

How to incorporate the multi-mode Floquet theory into Open Quantum Systems

realm?

Possible answer for 𝑁 = 2 (2-dimensional torus)

(H. R. Jauslin and J. L. Lebowitz, Chaos 1, 114 (1991))

Generalized Floquet operator 𝐹 𝜃1 : ℛ ⊗ 𝒯1 ⟶ℛ⊗𝒯1,

𝐹 𝜃1 = 𝑋 −𝑇2 𝑈 𝑇2, 0 , 𝑋 −𝑇2 𝜙 𝜃1 0 = 𝜙 𝜃1 0 − 𝑇2 .


Theorem 2.

If 𝜙 ∈ ℛ⊗𝒯1 is an eigenfunction of Floquet operator, 𝐹𝜙 = 𝑒−𝑖𝜆𝑇2𝜙, then the

function 𝜓 ∈ ℛ⊗𝒯1⊗𝒯2 defined

𝜓 𝜃1, 𝜃2 = 𝑒𝑖𝜃2𝜆𝑈 0,−𝜃2 𝜙 𝜃1 − 𝜃2

is an eigenfunction of 𝐻 𝑟, 𝜃1, 𝜃2, 𝜃1, 𝜃2 with eigenvalue (quasienergy) 𝜆.

Proof in: H. R. Jauslin and J. L. Lebowitz, Chaos 1, 114 (1991)

Problem:

Spectrum of 𝐻 may become very complex (p.p., a.c. or s.c.), even in finite

dimensional case.


𝑆, ℋ𝑆

𝑅1, ℋ𝑅1

𝑅2, ℋ𝑅2

𝑅3, ℋ𝑅3𝑅4, ℋ𝑅4

𝑅𝑁,

ℋ𝑅𝑁

ℋ = ℋ𝑆 ⊗ℋ𝑅1 ⊗⋯⊗ℋ𝑅𝑁

𝐻 = 𝐻𝑆 +

𝑗=1

𝑁

𝐻𝑅𝑗 +

𝑗=1

𝑁

𝑉𝑗

𝐻𝑆 ≡ 𝐻𝑆 ⊗ 𝐼𝑅1 ⊗⋯⊗ 𝐼𝑅𝑁

𝐻𝑅𝑗 ≡ 𝐼𝑆 ⊗⋯⊗𝐻𝑅𝑗 ⊗⋯⊗ 𝐼𝑅𝑁

𝑉𝑗 = 𝜆𝑗

𝛼

𝑆𝑗,𝛼 ⊗𝑅𝑗,𝛼

𝑆𝑗,𝛼:ℋ𝑆 ⟶ℋ𝑆, 𝑅𝑗,𝛼:ℋ𝑅𝑗 ⟶ℋ𝑅𝑗

𝑉1

𝑉2

𝑉3

𝑉4

𝑉𝑁

ℒ𝜌 𝑡 =𝑑𝜌 𝑡

𝑑𝑡=

𝑗,𝑘

𝜔

𝑞∈ℤ

𝐺𝑘𝑗𝜔 + 𝑞Ω 𝑆𝑗,𝑘 𝑞, 𝜔 𝜌 𝑡 𝑆𝑗,𝑘

∗ 𝑞, 𝜔 −1

2𝑆𝑗,𝑘∗ 𝑞, 𝜔 𝑆𝑗,𝑘 𝑞, 𝜔 , 𝜌 𝑡 ,

𝐺𝑘𝑗𝜔 =

−∞

∞

𝑒𝑖𝜔𝑡 𝑅𝑗,𝑘 𝑡 𝑅𝑗,𝑘 𝑑𝑡 , 𝐺 −𝜔 = 𝑒−ℏ𝜔𝑘𝐵𝑇𝐺 𝜔


KMS condition

(in equilibrium)

Floquet Theorem ℱ𝜙𝑘 = 𝑒−𝑖ℏ𝜖𝑘𝑇𝜙𝑘

𝜖𝑘 quasienergies

𝜙𝑘 Floquet basis

Fourier transform of

𝑆𝑗,𝑘 𝑡

Bohr frequencies

𝜔 =1

ℏ𝜖𝑘 − 𝜖𝑙

𝜔 + 𝑞Ω , 𝑞 ∈ ℤ

Bohr – Floquet

quasifrequencies

𝑉𝑗 𝑡 = 𝑈∗ 𝑡 𝑉𝑗𝑈 𝑡 = 𝜆𝑗

𝑘

𝑆𝑗,𝑘 𝑡 ⊗ 𝑅𝑗,𝑘 𝑡 , 𝑈 𝑡 = Τ exp −𝑖

ℏ

0

𝑡

𝐻𝑆 𝑡′ 𝑑𝑡′


Λ𝑡,𝑡0 = Τexp

𝑡0

𝑡

ℒ 𝑡′ 𝑑𝑡′ ≡ 𝒰 𝑡, 𝑡0 𝑒𝑡−𝑡0 ℒ

Dynamical map reconstructed from its interaction picture:

𝒰 𝑡, 𝑡0 – one-parameter unitary map defined on C*-algebra of operators 𝔄,

𝒰:𝔄 × 0,∞ ⟶ 𝔄 defined as 𝒰 𝑡 𝐴 = 𝑈 𝑡 𝐴𝑈∗ 𝑡 .

𝜌 𝑡 = Λ𝑡 𝜌0 = 𝑈 𝑡 𝑒𝑡ℒ𝜌0 𝑈∗ 𝑡

𝜌 𝑡 in interaction

picture

𝜌 𝑡 in Schrödinger

picture


9 Floquet quasifrequencies: 0,±Ω𝑅 , ±Ω,± Ω − Ω𝑅 , ± Ω + Ω𝑅

Interaction with molecular gas Interaction with

electromagnetic field

𝑅𝑒𝑚,

ℋ𝑒𝑚

𝑅𝑔, ℋ𝑔

Two-level system

ℋ𝑆 ≡ ℂ2

Bosonic heat bath

(EM field)

ℱ+ ℋ𝑝ℎ =

𝑁=0

∞1

𝑁!ℋ𝑝ℎ

⊗𝑁

+

𝑉𝑒

laser, Ω𝜔0

Dephasing bath

(molecular gas),

ℋ𝑔 ≡ ℒ2 ℝ3, 𝑑𝑉 𝑟

𝑉𝑔

𝑉𝑔 = 𝜎3⊗𝐹𝑔

𝐹𝑔:ℋ𝑔 ⟶ℋ𝑔

𝑉𝑒 = 𝜎1⊗ 𝑎∗ 𝑓 + 𝑎 𝑓


Markovian master equation

in interaction picture:

𝑑

𝑑𝑡𝜌 𝑡 = ℒ𝑏𝑜𝑠𝑜𝑛𝑖𝑐 + ℒ𝑑𝑒𝑝ℎ𝑎𝑠𝑖𝑛𝑔 𝜌 𝑡

𝑑𝜌11 𝑡

𝑑𝑡= − 𝛼 + 𝑒

−Ω−Ω𝑅𝑇𝑒 𝛿− + 𝛿+ 𝜌11 𝑡 + 𝑒

−Ω𝑅𝑇𝑑𝛼 + 𝛿− + 𝑒

−Ω+Ω𝑅𝑇𝑒 𝛿+ 𝜌22 𝑡

𝛾 =1

2𝛼0 + 𝛼 1 + 𝑒

−Ω𝑅𝑇𝑑 + 𝛿0 1 + 𝑒

−Ω𝑇𝑒 + 𝛿− 1 + 𝑒

−Ω−Ω𝑅𝑇𝑒 + 𝛿+ 1 + 𝑒

−Ω+Ω𝑅𝑇𝑒

𝑑𝜌22 𝑡

𝑑𝑡= −

𝑑𝜌11 𝑡

𝑑𝑡,

𝑑𝜌21 𝑡

𝑑𝑡= −𝛾𝜌21 𝑡 ,

𝑑𝜌12 𝑡

𝑑𝑡= −𝛾𝜌12 𝑡 .

𝛿± =Ω𝑅 ± Δ

2Ω𝑅

2

𝐺𝑒 Ω ± Ω𝑅 , 𝛼0 =2Δ

Ω𝑅

2

𝐺𝑔 0 , 𝛼 =2𝑔

Ω𝑅

2

𝐺𝑔 Ω𝑅 ,


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Nature 461, 70-73 (3 Sep. 2009).

2. R. Alicki, K. Lendi: “Quantum Dynamical Semigroups and Applications”,

Lecture Notes in Physics; Springer 2006.

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