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

On the dynamics of periodically perturbed quantum systems

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 .

On the dynamics of periodically perturbed quantum systems

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”

On the dynamics of periodically perturbed quantum systems

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

On the dynamics of periodically perturbed quantum systems

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

On the dynamics of periodically perturbed quantum systems

• 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.

On the dynamics of periodically perturbed quantum systems

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

On the dynamics of periodically perturbed quantum systems

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 ∗

On the dynamics of periodically perturbed quantum systems

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

On the dynamics of periodically perturbed quantum systems

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

On the dynamics of periodically perturbed quantum systems

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

On the dynamics of periodically perturbed quantum systems

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 .

On the dynamics of periodically perturbed quantum systems

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.

On the dynamics of periodically perturbed quantum systems

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

On the dynamics of periodically perturbed quantum systems

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′

On the dynamics of periodically perturbed quantum systems

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

On the dynamics of periodically perturbed quantum systems

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

On the dynamics of periodically perturbed quantum systems

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 ,

On the dynamics of periodically perturbed quantum systems

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