On the Dynamics of Periodically Perturbed Quantum SystemsΒ Β· On the dynamics of periodically...
Transcript of On the Dynamics of Periodically Perturbed Quantum SystemsΒ Β· On the dynamics of periodically...
On the dynamics of periodically perturbed quantum systems
Consider a system of n ODEs
π
ππ‘π π‘ = π΄ π‘ π π‘ , π:β βΆ ππΓ1 β
where π΄:β βΆ ππΓπ β is a continuous, (π Γ π) matrix-valued function of real
parameter t, periodic with period T,
β π‘ β β β π β β€ βΆ π΄ π‘ + ππ = π΄ π‘ .
Let Ξ¦ π‘ to be the fundamental matrix of this system, satisfying the following:
π
ππ‘Ξ¦ π‘ = π΄ π‘ Ξ¦ π‘ , detΞ¦ π‘ β 0, π π‘ = Ξ¦ π‘ π
WroΕskian
Any general solution
(c = ππππ π‘.) of
system of ODEs
On the dynamics of periodically perturbed quantum systems
Proposition 1.
a) There exists some π‘0 β β such that π π‘ = πΈ π‘, π‘0 π π‘0 where
πΈ π‘, π‘0 β Ξ¦ π‘ Ξ¦ π‘0β1 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
π
ππ‘πΈ π‘, π‘0 = π΄ π‘ πΈ π‘, π‘0 .
Proposition 2. Resolvent matrix has some basic properties:
a) Divisibility: πΈ π‘, π‘0 = π=1π πΈ π‘π+1, π‘π for any partition π‘π , π‘π+1 of π‘0, π‘ , iff π‘0, π‘ =
π=1π π‘π , π‘π+1 , and π‘π , π‘π+1 β© π‘πβ² , π‘πβ²+1 = β for π β πβ²
b) πΈ π‘, π‘0β1 = πΈ π‘0, π‘
c) πΈ π‘0, π‘0 = ππππ β .
On the dynamics of periodically perturbed quantum systems
Theorem 1 (Floquetβs).
If Ξ¦ π‘ is a fundamental matrix of a system of n ODEs and π΄ π‘ is a T-periodic function of
codomain in the ππ β linear space of n-by-n matrices, then a matrix Ξ¦ π‘ + π is also a
fundamental matrix of this system.
Remark: If Ξ¦ π‘ + π is a fundamental matrix then there exist two constant matrices πΆ and π΅such that
Ξ¦ π‘ + π = Ξ¦ π‘ πΆ, πΆ = ππ΅π .
Assume a spectral decomposition π΅ = π ππ ππ , .β ππ:
ππ΅π =
π
ππ ππ , .β ππ , ππ = ππππ.
ππ: βFloquet
exponentsβ
On the dynamics of periodically perturbed quantum systems
Let π π‘ to be a solution of a system of ODEs, i.e. let it fulfillπ
ππ‘π π‘ = π΄ π‘ π π‘ .
Define ππ π‘ β Ξ¦ π‘ ππ . Then it follows from Floquetβs theorem that
Ξ¦ π‘ + π ππ = Ξ¦ π‘ ππ΅πππ = ππππππ π‘ = ππ π‘ + π
Putting ππ π‘ = ππ π‘ πβπππ‘ one gets a set of βbase solutionsβ of system of ODEs,
ππ π‘ = ππππ‘ππ π‘ , ππ π‘ + π = ππ π‘ .
T-periodic
On the dynamics of periodically perturbed quantum systems
Considering ππ π‘0 + π one obtains an eigenequation of Ξ¦ π‘0 + π Ξ¦β1 π‘0 :
ππ π‘0 + π = Ξ¦ π‘0 + π Ξ¦β1 π‘0 ππ π‘0 = ππππππ π‘0
Floquetβs operator
πΉ π‘0 β πΈ π‘0 + π, π‘0
Floquetβs basis
{ππβ ππ π‘0 }
πΈ π‘0 + π, π‘0
On the dynamics of periodically perturbed quantum systems
β’ π΄ π‘ = π» π‘ β periodic, self-adjoint Hamiltonian of quantum-mechanical system
β’ ODEs describe an evolution of wavefunction (state) π π‘ β SchrΓΆdinger equation:
π
ππ‘π π‘ = β
π
βπ» π‘ π π‘ , π» π‘ + π = π» π‘ .
β’ Resolvent matrix β unitary propagator πΌ π, ππ :
πΈ π‘, π‘0 = π π‘, π‘0 = Texp βπ
β
π‘0
π‘
π» π‘β² ππ‘β² ,ππ π‘, π‘0
ππ‘= β
π
βπ» π‘ π π‘, π‘0 .
β’ Floquetβs operator πΉ π‘0 = π π‘0 + π, π‘0 = πβππ
β π»:
π ππ ππ ππ = πβππ»βππππ ππ ,
ππ β ππ π‘0 β Floquet basis, ππ β set of Bohr-Floquet quasienergies.
On the dynamics of periodically perturbed quantum systems
Floquet Hamiltonian:
π»πΉ π, π‘ = π» π, π‘ β πβπ
ππ‘, π»πΉπ π, π‘ = 0
Main analysis based on SchrΓΆdinger equation for states ππ π, π‘ = ππ π, π‘ + π :
π»πΉππ π, π‘ = ππππ π, π‘
Solutions are not unique:
They generate the same physical state ππ π, π‘
πππ π, π‘ β ππ π, π‘ πππΞ©π‘ π»πΉπππ π, π‘ = ππ + πβΞ© ππ π, π‘
πππHigher Floquet
modes
On the dynamics of periodically perturbed quantum systems
Extended Hilbert space ββΆββ² = ββπ― (example: particle in free space)
π― = β2 ππ1 , ππ‘ = span ππ π‘ β πππΞ©π‘
Space of square-integrable functions
with period T = 2π/Ξ©, defined over a
circle π1.
ππ, ππβ² =1
π π1ππ π‘ ππβ² π‘ ππ‘ = πΏππβ²
π
ππβ β ππ = πππ―
β = β2 β3, ππ = span ππ: β3 β β
Space of square-integrable functions
defined over β3.
ππ , ππβ² =
β3
ππ π ππβ² π ππ π = πΏππβ²
π
ππβ β ππ = ππβ
ββπ― = span πππ β ππ βππ , πππ π, π‘ = ππ π πππΞ©π‘
ππ
πππβ β πππ = ππββπ― , πππ
β ππβ²πβ² = πΏππβ²πΏππβ² , πππβ β β βπ― β
On the dynamics of periodically perturbed quantum systems
Structure of Hamiltonian: π» π, π‘ = π»0 π + π π, π‘ , π π, π‘ + π = π π, π‘ .
Idea: We are applying a transformation of variables:
π = Ξ©π‘, π = Ξ©
π» π, π, π = π»0 π + π π, π + πππ
Canonical
quantization:
π β π,
ππ β βπβπ
ππ,
π, ππ = πβ
π» π, π, π = π»0 π + π π, π β πβΞ©π
ππ, π» π, π, π πππ π, π = ππππππ π, π
On the dynamics of periodically perturbed quantum systems
π» π, π, π πππ π, π = ππππππ π, π , πππ = ππ + πβΞ©
πππ β β βπ―, π― = β2 π2π1 ,
1
Ξ©ππ
Square-integrable functions of period
2π over a unit circle π1 = π = πΊπ‘
How to include multi-mode setting?
Ansatz: add a sufficient number of new ππ variables, such that
π» π, π, π = π»0 π + π π, π1, β¦ , ππ β πβ
π=1
π
Ξ©ππ
πππ, Ξ©π =
2π
ππ
On the dynamics of periodically perturbed quantum systems
New SchrΓΆdinger equation:
π» π, π1, π2, β¦ , ππ πππ1π2β¦ππ π, π1, π2, β¦ , ππ = πππ1π2β¦πππππ1π2β¦ππ π, π1, π2, β¦ , ππ
Periodicity of π functions:
πππ1π2β¦ππ π, π1 + 2π, π2 + 2π,β¦ , ππ + 2π = πππ1π2β¦ππ π, π1, π2, β¦ , ππ
Extension of Hilbert space of π functions:
πππ1π2β¦ππ β β βπ―1βπ―2ββ―βπ―π, π―π = β2π2 π1,
1
πΊππππ
π=1
π
β2π2 π1,
1
πΊππππ β‘ β2 π1 Γ π1 Γβ―Γ π1, ππ = β2 ππ, ππ , ππ =
π=1
ππππ
πΊπ
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 π = 2 (2-dimensional torus)
(H. R. Jauslin and J. L. Lebowitz, Chaos 1, 114 (1991))
Generalized Floquet operator πΉ π1 : β β π―1 βΆββπ―1,
πΉ π1 = π βπ2 π π2, 0 , π βπ2 π π1 0 = π π1 0 β π2 .
On the dynamics of periodically perturbed quantum systems
Theorem 2.
If π β ββπ―1 is an eigenfunction of Floquet operator, πΉπ = πβπππ2π, then the
function π β ββπ―1βπ―2 defined
π π1, π2 = πππ2ππ 0,βπ2 π π1 β π2
is an eigenfunction of π» π, π1, π2, π1, π2 with eigenvalue (quasienergy) π.
Proof in: H. R. Jauslin and J. L. Lebowitz, Chaos 1, 114 (1991)
Problem:
Spectrum of π» may become very complex (p.p., a.c. or s.c.), even in finite
dimensional case.
On the dynamics of periodically perturbed quantum systems
π, βπ
π 1, βπ 1
π 2, βπ 2
π 3, βπ 3π 4, βπ 4
π π,
βπ π
β = βπ ββπ 1 ββ―ββπ π
π» = π»π +
π=1
π
π»π π +
π=1
π
ππ
π»π β‘ π»π β πΌπ 1 ββ―β πΌπ π
π»π π β‘ πΌπ ββ―βπ»π π ββ―β πΌπ π
ππ = ππ
πΌ
ππ,πΌ βπ π,πΌ
ππ,πΌ:βπ βΆβπ, π π,πΌ:βπ π βΆβπ π
π1
π2
π3
π4
ππ
βπ π‘ =ππ π‘
ππ‘=
π,π
π
πββ€
πΊπππ + πΞ© ππ,π π, π π π‘ ππ,π
β π, π β1
2ππ,πβ π, π ππ,π π, π , π π‘ ,
πΊπππ =
ββ
β
ππππ‘ π π,π π‘ π π,π ππ‘ , πΊ βπ = πββπππ΅ππΊ π
On the dynamics of periodically perturbed quantum systems
KMS condition
(in equilibrium)
Floquet Theorem β±ππ = πβπβπππππ
ππ quasienergies
ππ Floquet basis
Fourier transform of
ππ,π π‘
Bohr frequencies
π =1
βππ β ππ
π + πΞ© , π β β€
Bohr β Floquet
quasifrequencies
ππ π‘ = πβ π‘ πππ π‘ = ππ
π
ππ,π π‘ β π π,π π‘ , π π‘ = Ξ€ exp βπ
β
0
π‘
π»π π‘β² ππ‘β²
On the dynamics of periodically perturbed quantum systems
Ξπ‘,π‘0 = Ξ€exp
π‘0
π‘
β π‘β² ππ‘β² β‘ π° π‘, π‘0 ππ‘βπ‘0 β
Dynamical map reconstructed from its interaction picture:
π° π‘, π‘0 β one-parameter unitary map defined on C*-algebra of operators π,
π°:π Γ 0,β βΆ π defined as π° π‘ π΄ = π π‘ π΄πβ π‘ .
π π‘ = Ξπ‘ π0 = π π‘ ππ‘βπ0 πβ π‘
π π‘ in interaction
picture
π π‘ in SchrΓΆdinger
picture
On the dynamics of periodically perturbed quantum systems
9 Floquet quasifrequencies: 0,Β±Ξ©π , Β±Ξ©,Β± Ξ© β Ξ©π , Β± Ξ© + Ξ©π
Interaction with molecular gas Interaction with
electromagnetic field
π ππ,
βππ
π π, βπ
Two-level system
βπ β‘ β2
Bosonic heat bath
(EM field)
β±+ βπβ =
π=0
β1
π!βπβ
βπ
+
ππ
laser, Ξ©π0
Dephasing bath
(molecular gas),
βπ β‘ β2 β3, ππ π
ππ
ππ = π3βπΉπ
πΉπ:βπ βΆβπ
ππ = π1β πβ π + π π
On the dynamics of periodically perturbed quantum systems
Markovian master equation
in interaction picture:
π
ππ‘π π‘ = βπππ ππππ + βπππβππ πππ π π‘
ππ11 π‘
ππ‘= β πΌ + π
βΞ©βΞ©π ππ πΏβ + πΏ+ π11 π‘ + π
βΞ©π πππΌ + πΏβ + π
βΞ©+Ξ©π ππ πΏ+ π22 π‘
πΎ =1
2πΌ0 + πΌ 1 + π
βΞ©π ππ + πΏ0 1 + π
βΞ©ππ + πΏβ 1 + π
βΞ©βΞ©π ππ + πΏ+ 1 + π
βΞ©+Ξ©π ππ
ππ22 π‘
ππ‘= β
ππ11 π‘
ππ‘,
ππ21 π‘
ππ‘= βπΎπ21 π‘ ,
ππ12 π‘
ππ‘= βπΎπ12 π‘ .
πΏΒ± =Ξ©π Β± Ξ
2Ξ©π
2
πΊπ Ξ© Β± Ξ©π , πΌ0 =2Ξ
Ξ©π
2
πΊπ 0 , πΌ =2π
Ξ©π
2
πΊπ Ξ©π ,
On the dynamics of periodically perturbed quantum systems
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