on the dynamics of periodically perturbed quantum systems · on the dynamics of periodically...
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On the dynamics of periodically perturbed quantum systems
Consider a system of n ODEs
ð
ðð¡ð ð¡ = ðŽ ð¡ ð ð¡ , ð:â ⶠððÃ1 â
where ðŽ:â ⶠððÃð â is a continuous, (ð à ð) matrix-valued function of real
parameter t, periodic with period T,
â ð¡ â â â ð â †ⶠðŽ ð¡ + ðð = ðŽ ð¡ .
Let Ί ð¡ to be the fundamental matrix of this system, satisfying the following:
ð
ðð¡ÎŠ ð¡ = ðŽ ð¡ Ί ð¡ , detΊ ð¡ â 0, ð ð¡ = Ί ð¡ ð
WroÅskian
Any general solution
(c = ðððð ð¡.) of
system of ODEs
On the dynamics of periodically perturbed quantum systems
Proposition 1.
a) There exists some ð¡0 â â such that ð ð¡ = ðž ð¡, ð¡0 ð ð¡0 where
ðž ð¡, ð¡0 â Ί ð¡ Ί ð¡0â1 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
ð
ðð¡ðž ð¡, ð¡0 = ðŽ ð¡ ðž ð¡, ð¡0 .
Proposition 2. Resolvent matrix has some basic properties:
a) Divisibility: ðž ð¡, ð¡0 = ð=1ð ðž ð¡ð+1, ð¡ð for any partition ð¡ð , ð¡ð+1 of ð¡0, ð¡ , iff ð¡0, ð¡ =
ð=1ð ð¡ð , ð¡ð+1 , and ð¡ð , ð¡ð+1 â© ð¡ðâ² , ð¡ðâ²+1 = â for ð â ðâ²
b) ðž ð¡, ð¡0â1 = ðž ð¡0, ð¡
c) ðž ð¡0, ð¡0 = ðððð â .
On the dynamics of periodically perturbed quantum systems
Theorem 1 (Floquetâs).
If Ί ð¡ is a fundamental matrix of a system of n ODEs and ðŽ ð¡ is a T-periodic function of
codomain in the ðð â linear space of n-by-n matrices, then a matrix Ί ð¡ + ð is also a
fundamental matrix of this system.
Remark: If Ί ð¡ + ð is a fundamental matrix then there exist two constant matrices ð¶ and ðµsuch that
Ί ð¡ + ð = Ί ð¡ ð¶, ð¶ = ððµð .
Assume a spectral decomposition ðµ = ð ðð ðð , .â ðð:
ððµð =
ð
ðð ðð , .â ðð , ðð = ðððð.
ðð: âFloquet
exponentsâ
On the dynamics of periodically perturbed quantum systems
Let ð ð¡ to be a solution of a system of ODEs, i.e. let it fulfillð
ðð¡ð ð¡ = ðŽ ð¡ ð ð¡ .
Define ðð ð¡ â Ί ð¡ ðð . Then it follows from Floquetâs theorem that
Ί ð¡ + ð ðð = Ί ð¡ ððµððð = ðððððð ð¡ = ðð ð¡ + ð
Putting ðð ð¡ = ðð ð¡ ðâððð¡ one gets a set of âbase solutionsâ of system of ODEs,
ðð ð¡ = ðððð¡ðð ð¡ , ðð ð¡ + ð = ðð ð¡ .
T-periodic
On the dynamics of periodically perturbed quantum systems
Considering ðð ð¡0 + ð one obtains an eigenequation of Ί ð¡0 + ð Ίâ1 ð¡0 :
ðð ð¡0 + ð = Ί ð¡0 + ð Ίâ1 ð¡0 ðð ð¡0 = ðððððð ð¡0
Floquetâs operator
ð¹ ð¡0 â ðž ð¡0 + ð, ð¡0
Floquetâs basis
{ððâ ðð ð¡0 }
ðž ð¡0 + ð, ð¡0
On the dynamics of periodically perturbed quantum systems
⢠ðŽ ð¡ = ð» ð¡ â periodic, self-adjoint Hamiltonian of quantum-mechanical system
⢠ODEs describe an evolution of wavefunction (state) ð ð¡ â Schrödinger equation:
ð
ðð¡ð ð¡ = â
ð
âð» ð¡ ð ð¡ , ð» ð¡ + ð = ð» ð¡ .
⢠Resolvent matrix â unitary propagator ðŒ ð, ðð :
ðž ð¡, ð¡0 = ð ð¡, ð¡0 = Texp âð
â
ð¡0
ð¡
ð» ð¡â² ðð¡â² ,ðð ð¡, ð¡0
ðð¡= â
ð
âð» ð¡ ð ð¡, ð¡0 .
⢠Floquetâs operator ð¹ ð¡0 = ð ð¡0 + ð, ð¡0 = ðâðð
â ð»:
ð ðð ðð ðð = ðâðð»âðððð ðð ,
ðð â ðð ð¡0 â Floquet basis, ðð â set of Bohr-Floquet quasienergies.
On the dynamics of periodically perturbed quantum systems
Floquet Hamiltonian:
ð»ð¹ ð, ð¡ = ð» ð, ð¡ â ðâð
ðð¡, ð»ð¹ð ð, ð¡ = 0
Main analysis based on Schrödinger equation for states ðð ð, ð¡ = ðð ð, ð¡ + ð :
ð»ð¹ðð ð, ð¡ = ðððð ð, ð¡
Solutions are not unique:
They generate the same physical state ðð ð, ð¡
ððð ð, ð¡ â ðð ð, ð¡ ðððΩð¡ ð»ð¹ððð ð, ð¡ = ðð + ðâΩ ðð ð, ð¡
ðððHigher Floquet
modes
On the dynamics of periodically perturbed quantum systems
Extended Hilbert space ââ¶ââ² = ââð¯ (example: particle in free space)
ð¯ = â2 ðð1 , ðð¡ = span ðð ð¡ â ðððΩð¡
Space of square-integrable functions
with period T = 2ð/Ω, defined over a
circle ð1.
ðð, ððâ² =1
ð ð1ðð ð¡ ððâ² ð¡ ðð¡ = ð¿ððâ²
ð
ððâ â ðð = ððð¯
â = â2 â3, ðð = span ðð: â3 â â
Space of square-integrable functions
defined over â3.
ðð , ððâ² =
â3
ðð ð ððâ² ð ðð ð = ð¿ððâ²
ð
ððâ â ðð = ððâ
ââð¯ = span ððð â ðð âðð , ððð ð, ð¡ = ðð ð ðððΩð¡
ðð
ðððâ â ððð = ððââð¯ , ððð
â ððâ²ðâ² = ð¿ððâ²ð¿ððâ² , ðððâ â â âð¯ â
On the dynamics of periodically perturbed quantum systems
Structure of Hamiltonian: ð» ð, ð¡ = ð»0 ð + ð ð, ð¡ , ð ð, ð¡ + ð = ð ð, ð¡ .
Idea: We are applying a transformation of variables:
ð = Ωð¡, ð = Ω
ð» ð, ð, ð = ð»0 ð + ð ð, ð + ððð
Canonical
quantization:
ð â ð,
ðð â âðâð
ðð,
ð, ðð = ðâ
ð» ð, ð, ð = ð»0 ð + ð ð, ð â ðâΩð
ðð, ð» ð, ð, ð ððð ð, ð = ðððððð ð, ð
On the dynamics of periodically perturbed quantum systems
ð» ð, ð, ð ððð ð, ð = ðððððð ð, ð , ððð = ðð + ðâΩ
ððð â â âð¯, ð¯ = â2 ð2ð1 ,
1
Ωðð
Square-integrable functions of period
2ð over a unit circle ð1 = ð = ðºð¡
How to include multi-mode setting?
Ansatz: add a sufficient number of new ðð variables, such that
ð» ð, ð, ð = ð»0 ð + ð ð, ð1, ⊠, ðð â ðâ
ð=1
ð
Ωðð
ððð, Ωð =
2ð
ðð
On the dynamics of periodically perturbed quantum systems
New Schrödinger equation:
ð» ð, ð1, ð2, ⊠, ðð ððð1ð2âŠðð ð, ð1, ð2, ⊠, ðð = ððð1ð2âŠððððð1ð2âŠðð ð, ð1, ð2, ⊠, ðð
Periodicity of ð functions:
ððð1ð2âŠðð ð, ð1 + 2ð, ð2 + 2ð,⊠, ðð + 2ð = ððð1ð2âŠðð ð, ð1, ð2, ⊠, ðð
Extension of Hilbert space of ð functions:
ððð1ð2âŠðð â â âð¯1âð¯2ââ¯âð¯ð, ð¯ð = â2ð2 ð1,
1
ðºðððð
ð=1
ð
â2ð2 ð1,
1
ðºðððð â¡ â2 ð1 à ð1 Ãâ¯Ã ð1, ðð = â2 ðð, ðð , ðð =
ð=1
ðððð
ðºð
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 ð = 2 (2-dimensional torus)
(H. R. Jauslin and J. L. Lebowitz, Chaos 1, 114 (1991))
Generalized Floquet operator ð¹ ð1 : â â ð¯1 â¶ââð¯1,
ð¹ ð1 = ð âð2 ð ð2, 0 , ð âð2 ð ð1 0 = ð ð1 0 â ð2 .
On the dynamics of periodically perturbed quantum systems
Theorem 2.
If ð â ââð¯1 is an eigenfunction of Floquet operator, ð¹ð = ðâððð2ð, then the
function ð â ââð¯1âð¯2 defined
ð ð1, ð2 = ððð2ðð 0,âð2 ð ð1 â ð2
is an eigenfunction of ð» ð, ð1, ð2, ð1, ð2 with eigenvalue (quasienergy) ð.
Proof in: H. R. Jauslin and J. L. Lebowitz, Chaos 1, 114 (1991)
Problem:
Spectrum of ð» may become very complex (p.p., a.c. or s.c.), even in finite
dimensional case.
On the dynamics of periodically perturbed quantum systems
ð, âð
ð 1, âð 1
ð 2, âð 2
ð 3, âð 3ð 4, âð 4
ð ð,
âð ð
â = âð ââð 1 ââ¯ââð ð
ð» = ð»ð +
ð=1
ð
ð»ð ð +
ð=1
ð
ðð
ð»ð â¡ ð»ð â ðŒð 1 ââ¯â ðŒð ð
ð»ð ð â¡ ðŒð ââ¯âð»ð ð ââ¯â ðŒð ð
ðð = ðð
ðŒ
ðð,ðŒ âð ð,ðŒ
ðð,ðŒ:âð â¶âð, ð ð,ðŒ:âð ð â¶âð ð
ð1
ð2
ð3
ð4
ðð
âð ð¡ =ðð ð¡
ðð¡=
ð,ð
ð
ðââ€
ðºððð + ðΩ ðð,ð ð, ð ð ð¡ ðð,ð
â ð, ð â1
2ðð,ðâ ð, ð ðð,ð ð, ð , ð ð¡ ,
ðºððð =
ââ
â
ðððð¡ ð ð,ð ð¡ ð ð,ð ðð¡ , ðº âð = ðââðððµððº ð
On the dynamics of periodically perturbed quantum systems
KMS condition
(in equilibrium)
Floquet Theorem â±ðð = ðâðâððððð
ðð quasienergies
ðð Floquet basis
Fourier transform of
ðð,ð ð¡
Bohr frequencies
ð =1
âðð â ðð
ð + ðΩ , ð â â€
Bohr â Floquet
quasifrequencies
ðð ð¡ = ðâ ð¡ ððð ð¡ = ðð
ð
ðð,ð ð¡ â ð ð,ð ð¡ , ð ð¡ = ΀ exp âð
â
0
ð¡
ð»ð ð¡â² ðð¡â²
On the dynamics of periodically perturbed quantum systems
Îð¡,ð¡0 = ΀exp
ð¡0
ð¡
â ð¡â² ðð¡â² â¡ ð° ð¡, ð¡0 ðð¡âð¡0 â
Dynamical map reconstructed from its interaction picture:
ð° ð¡, ð¡0 â one-parameter unitary map defined on C*-algebra of operators ð,
ð°:ð à 0,â ⶠð defined as ð° ð¡ ðŽ = ð ð¡ ðŽðâ ð¡ .
ð ð¡ = Îð¡ ð0 = ð ð¡ ðð¡âð0 ðâ ð¡
ð ð¡ in interaction
picture
ð ð¡ in Schrödinger
picture
On the dynamics of periodically perturbed quantum systems
9 Floquet quasifrequencies: 0,±Ωð , ±Ω,± Ω â Ωð , ± Ω + Ωð
Interaction with molecular gas Interaction with
electromagnetic field
ð ðð,
âðð
ð ð, âð
Two-level system
âð â¡ â2
Bosonic heat bath
(EM field)
â±+ âðâ =
ð=0
â1
ð!âðâ
âð
+
ðð
laser, Ωð0
Dephasing bath
(molecular gas),
âð â¡ â2 â3, ðð ð
ðð
ðð = ð3âð¹ð
ð¹ð:âð â¶âð
ðð = ð1â ðâ ð + ð ð
On the dynamics of periodically perturbed quantum systems
Markovian master equation
in interaction picture:
ð
ðð¡ð ð¡ = âððð ðððð + âðððâðð ððð ð ð¡
ðð11 ð¡
ðð¡= â ðŒ + ð
âΩâΩð ðð ð¿â + ð¿+ ð11 ð¡ + ð
âΩð ðððŒ + ð¿â + ð
âΩ+Ωð ðð ð¿+ ð22 ð¡
ðŸ =1
2ðŒ0 + ðŒ 1 + ð
âΩð ðð + ð¿0 1 + ð
âΩðð + ð¿â 1 + ð
âΩâΩð ðð + ð¿+ 1 + ð
âΩ+Ωð ðð
ðð22 ð¡
ðð¡= â
ðð11 ð¡
ðð¡,
ðð21 ð¡
ðð¡= âðŸð21 ð¡ ,
ðð12 ð¡
ðð¡= âðŸð12 ð¡ .
ð¿Â± =Ωð ± Î
2Ωð
2
ðºð Ω ± Ωð , ðŒ0 =2Î
Ωð
2
ðºð 0 , ðŒ =2ð
Ωð
2
ðºð Ωð ,
On the dynamics of periodically perturbed quantum systems
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