Consider a system of n ODEs d ψ t = A t ψ t, dt ψ: R M n 1 C where A: R M n 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 Φ t = A t Φ t, det Φ t 0, ψ t = Φ t c dt Wrońskian Any general solution (c = const.) of system of ODEs
Proposition 1. a) There exists some t 0 R such that ψ t = E t, t 0 ψ t 0 where E t, t 0 Φ t Φ t 1 0 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 dt E t, t 0 = A t E t, t 0. Proposition 2. Resolvent matrix has some basic properties: n a) Divisibility: E t, t 0 = k=1 E t k+1, t k for any partition t k, t k+1 of t 0, t, iff t 0, t = n k=1 t k, t k+1, and t k, t k+1 t k, t k +1 = for k k b) E t, t 0 1 = E t 0, t c) E t 0, t 0 = id Mn 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 M n 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 such that is a fundamental matrix then there exist two constant matrices C and B Φ t + T = Φ t C, C = e BT. Assume a spectral decomposition B = k μ k φ k,. φ k : e BT = λ k φ k,. φ k, λ k = e μkt. k μ k : Floquet exponents
Let ψ t to be a solution of a system of ODEs, i.e. let it fulfill d ψ t = A t ψ t. dt Define ψ k t Φ t φ k. Then it follows from Floquet s theorem that Φ t + T φ k = Φ t e BT φ k = e μ kt ψ k t = ψ k t + T Putting φ k t = ψ k t e μ kt one gets a set of base solutions of system of ODEs, ψ k t = e μ kt φ k t, φ k t + T = φ k t. T-periodic
Considering ψ k t 0 + T one obtains an eigenequation of Φ t 0 + T Φ 1 t 0 : ψ k t 0 + T = Φ t 0 + T Φ 1 t 0 φ k t 0 = e μ kt φ k t 0 E t 0 + T, t 0 Floquet s operator Floquet s basis F t 0 E t 0 + T, t 0 {φ k φ k t 0 }
A t = H t periodic, self-adjoint Hamiltonian of quantum-mechanical system ODEs describe an evolution of wavefunction (state) ψ t Schrödinger equation: d dt ψ t = i H t ψ t, H t + T = H t. ħ Resolvent matrix unitary propagator U t, t 0 : E t, t 0 = U t, t 0 = T exp i ħ t0 t H t dt, du t, t 0 dt = i ħ H t U t, t 0. Floquet s operator F t 0 = U t 0 + T, t 0 = e it ħ H : F t 0 φ k t 0 = e it ħ ε k φ k t 0, φ k φ k t 0 Floquet basis, ε k set of Bohr-Floquet quasienergies.
Floquet Hamiltonian: H F r, t = H r, t iħ d dt, H Fψ r, t = 0 Main analysis based on Schrödinger equation for states φ k r, t = φ k r, t + T : Solutions are not unique: H F φ k r, t = ε k φ k r, t φ kn r, t φ k r, t e inωt H F φ kn r, t = ε k + nħω φ k r, t Higher Floquet modes ε kn They generate the same physical state ψ k r, t
Extended Hilbert space H H = R T (example: particle in free space) R = L 2 R 3, dv = span f k : R 3 C Space of square-integrable functions defined over R 3. f k, f k = n f k R 3 f k r f k r dv r = δ kk f k = id R T = L 2 T T 1, dt = span χ n t e inωt Space of square-integrable functions with period T = 2π/Ω, defined over a circle T 1. χ n, χ n = 1 T T 1χ n t χ n t dt = δ nn q χ q χ q = id T R T = span e kn f k χ n, e kn r, t = f k r e inωt kn e kn e kn = id R T, e kn e k n = δ kk δ nn, e kn R T
Structure of Hamiltonian: H r, t = H 0 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 = H 0 r + V r, θ + θp θ Canonical quantization: θ θ, p θ iħ θ, θ, p θ = iħ H r, θ, θ = H 0 r + V r, θ iħω θ, H r, θ, θ φ kn r, θ = ε kn φ kn r, θ
H r, θ, θ φ kn r, θ = ε kn φ kn r, θ, ε kn = ε k + nħω φ kn R T, T = L 2 T 1 2π, 1 Ω dθ Square-integrable functions of period 2π over a unit circle T 1 = θ = Ωt How to include multi-mode setting? Ansatz: add a sufficient number of new θ i variables, such that H r, θ, θ = H 0 r + V r, θ 1,, θ N iħ N j=1 Ω j θ j, Ω i = 2π T i
New Schrödinger equation: H r, θ 1, θ 2,, θ N φ kn1 n 2 n N r, θ 1, θ 2,, θ N = ε kn1 n 2 n N φ kn1 n 2 n N r, θ 1, θ 2,, θ N Periodicity of φ functions: φ kn1 n 2 n N r, θ 1 + 2π, θ 2 + 2π,, θ N + 2π = φ kn1 n 2 n N r, θ 1, θ 2,, θ N Extension of Hilbert space of φ functions: φ kn1 n 2 n N R T 1 T 2 T N, 2 T j = L 2π T 1, 1 dθ Ω j j N j=1 2 L 2π T 1, 1 dθ Ω j L 2 T 1 T 1 T 1, dτ = L 2 T N, dτ, dτ = j N j=1 dθj Ω j 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 θ 1 : R T 1 R T 1, F θ 1 = X T 2 U T 2, 0, X T 2 φ θ 1 0 = φ θ 1 0 T 2.
Theorem 2. If φ R T 1 is an eigenfunction of Floquet operator, Fφ = e iλt 2φ, then the function ψ R T 1 T 2 defined ψ θ 1, θ 2 = e iθ 2λ U 0, θ 2 φ θ 1 θ 2 is an eigenfunction of H r, θ 1, θ 2, θ 1, θ 2 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.
H = H S H R1 H RN N N H = H S + H Rj + j=1 j=1 V j R 1, H R1 R N, H RN V N V 1 R 2, H R2 H S H S I R1 I RN H Rj I S H Rj I RN S, H S V 2 V 4 V 3 V j = λ j S j,α R j,α α R 4, H R4 R 3, H R3 S j,α : H S H S, R j,α : H Rj H Rj
V j t = U t V j U t = λ j k S j,k t R j,k t, U t = Τ exp i ħ 0 t H S t dt Floquet Theorem Fφ k = e i ħ εkt ε k quasienergies φ k φ k Floquet basis Bohr frequencies ω = 1 ħ ε k ε l ω + qω, q Z Bohr Floquet quasifrequencies Fourier transform of S j,k t dρ t Lρ t = dt = j,k ω q Z G k j ω + qω S j,k q, ω ρ t S j,k q, ω 1 2 S j,k q, ω S j,k q, ω, ρ t, G k j ω = e iωt R j,k t R j,k dt, G ω = e ħω k BT G ω KMS condition (in equilibrium)
Dynamical map reconstructed from its interaction picture: Λ t,t0 = Τ exp t L t dt U t, t 0 e t t 0 L t 0 U t, t 0 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 = Λ t ρ 0 = U t e tl ρ 0 U t ρ t in Schrödinger picture ρ t in interaction picture
Bosonic heat bath (EM field) F + H ph = 1 N=0 N! H ph N + R em, H em V e V e = σ 1 a f + a f Two-level system H S C 2 ω 0 laser, Ω Dephasing bath (molecular gas), H g L 2 R 3, dv r R g, H g V g V g = σ 3 F g F g : H g H g 9 Floquet quasifrequencies: 0, ±Ω R, ±Ω, ± Ω Ω R, ± Ω + Ω R Interaction with molecular gas Interaction with electromagnetic field
Markovian master equation in interaction picture: d dt ρ t = L bosonic + L dephasing ρ t dρ 11 t dt dρ 22 t dt = α + e Ω Ω R T e δ + δ + ρ 11 t + e Ω R T d α + δ + e Ω+Ω R T e δ + ρ 22 t = dρ 11 t dt, dρ 21 t dt = γρ 21 t, dρ 12 t dt = γρ 12 t. γ = 1 2 α 0 + α 1 + e Ω R T d + δ 0 1 + e Ω T e + δ 1 + e Ω Ω R T e + δ + 1 + e Ω+Ω R T e δ ± = Ω R ± Δ 2Ω R 2 G e Ω ± Ω R, α 0 = 2Δ Ω R 2 G g 0, α = 2g Ω R 2 G g Ω R,
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