Metastable states in a model of spin dependent point interactions. claudio cacciapuoti, raffaele carlone, rodolfo figari

Metastable states in a model of spin dependent point interactions claudio cacciapuoti, raffaele carlone, rodolfo figari

Metastable states in a model of spin dependent point interactions

Metastable states in a model of spin dependent point interactions Resonances, metastable states and exponential decay laws in perturbation theory W.Hunzinker Comm.Math.Phys. 132, 177-188 (1990) Resonance theory for Schrödinger operators O.Costin, A.Soffer Comm.Math.Phys. 224, 133-152 (2001) La regola d'oro di Fermi P.Facchi, S.Pascazio

Metastable states in a model of spin dependent point interactions Spectral analysis for system of atoms and molecules coupled to the quantizied radiation field V.Bach, J.Fröhlich, I.M.Sigal Comm.Math.Phys. 207, 249-290 (1999) Resonances, metastable states and exponential decay laws in perturbation theory W.Hunzinker Comm.Math.Phys. 132, 177-188 (1990) Resonance theory for Schrödinger operators O.Costin, A.Soffer Comm.Math.Phys. 224, 133-152 (2001) La regola d'oro di Fermi P.Facchi, S.Pascazio

Metastable states in a model of spin dependent point interactions Spectral analysis for system of atoms and molecules coupled to the quantizied radiation field V.Bach, J.Fröhlich, I.M.Sigal Comm.Math.Phys. 207, 249-290 (1999) y x Resonances, metastable states and exponential decay laws in perturbation theory W.Hunzinker Comm.Math.Phys. 132, 177-188 (1990) Resonance theory for Schrödinger operators O.Costin, A.Soffer Comm.Math.Phys. 224, 133-152 (2001) La regola d'oro di Fermi P.Facchi, S.Pascazio Resonances in twisted quantum waveguides H. Kovarik, A. Sacchetti J. Phys. A: Math. Theor. 40 8371-8384 (2007) FIG. 1: The Hilbert space of the system: an effective superbecomes diagonal with respect to H meas, [U(t), H meas ] = 0, an effective superselection rule arises and the total Hilbert space is split into subspaces H Pn that are invariant under the evolution. These subspaces are defined by the P n s, i.e., they are eigenspaces belonging to distinct eigenvalues η n : in other words, subspaces that the apparatus is able to distinguish. On the other hand, due to (22), the dynamics within each Zeno subspace H Pn is governed by the diagonal part P n HP n of the system Hamiltonian H. This bridges the gap with the description (1)-(9) and clarifies the role of the detection apparatus. In Fig. 1 we endeavored to give a pictorial representation of the decomposition of the Hilbert space as K is increased. It is worth noticing that the superselection rules discussed here are de facto equivalent to the celebrated W 3 ones [23], but turn out to be a mere consequence of the Zeno dynamics. Four examples will prove useful. First example: reconsider H 3lev in Eq. (10). As K is increased, the Hilbert Quantum Zeno subspaces P. Facchi, S. Pascazio Phys. Rev. Lett. 89, 080401 (2002) 3

Metastable states in a model of spin dependent point interactions The analysis of time decay of resonances will proceed as follows: (1) define an unperturbed Hamiltonian Ĥ0 in a way such that the spectrum has one eigenvalue embedded in the continuous spectrum (2) define Hamiltonian sense, of Ĥ0 Ĥ as a self-adjoint perturbation, in some suitable (3) show that the embedded eigenvalue turnes in a resonance (4) estimate the decay times of such a metastable

Point Interaction: a very short introduction

Point Interaction: a very short introduction (H α k 2 ) 1 = (H k 2 ) 1 4π + 4πα i k (G k( y), )G k ( y) { Being D(H α ) = Ran[(H α k 2 ) 1 ] it is easily seen that { } D(H α ) = ψ L 2 (R 3 ) : ψ = ψ k + qg k ( y); ψ k H 2 (R 3 ), q = 4πψk (y) } 4πα i k, k2 ρ(h α ), k > 0, < α Function ψ k (x) is called regular part and often constant q is referred to as charge. G k (x y) = eik x y 4π x y k > 0 satisfies, in the sense of distributions, the equation ( z)g z = δ y, where δ y is the three dimensional Dirac delta centered in y.

Point Interaction: a very short introduction (H α k 2 ) 1 = (H k 2 ) 1 4π + 4πα i k (G k( y), )G k ( y) { Being D(H α ) = Ran[(H α k 2 ) 1 ] it is easily seen that { } D(H α ) = ψ L 2 (R 3 ) : ψ = ψ k + qg k ( y); ψ k H 2 (R 3 ), q = 4πψk (y) } 4πα i k, k2 ρ(h α ), k > 0, < α Function ψ k (x) is called regular part and often constant q is referred to as charge. G k (x y) = eik x y 4π x y k > 0 satisfies, in the sense of distributions, the equation ( z)g z = δ y, where δ y is the three dimensional Dirac delta centered in y. this operator matches up with the point Hamiltonian defined formally in the sense that if y / supp[ψ k ] functions ψ and ψ k coincide and H α acts on ψ as.

Point Interaction: a very short introduction (H α k 2 ) 1 = (H k 2 ) 1 4π + 4πα i k (G k( y), )G k ( y) { Being D(H α ) = Ran[(H α k 2 ) 1 ] it is easily seen that { } D(H α ) = ψ L 2 (R 3 ) : ψ = ψ k + qg k ( y); ψ k H 2 (R 3 ), q = 4πψk (y) } 4πα i k, k2 ρ(h α ), k > 0, < α Function ψ k (x) is called regular part and often constant q is referred to as charge. G k (x y) = eik x y 4π x y k > 0 satisfies, in the sense of distributions, the equation ( z)g z = δ y, where δ y is the three dimensional Dirac delta centered in y. this operator matches up with the point Hamiltonian defined formally in the sense that if y / supp[ψ k ] functions ψ and ψ k coincide and H α acts on ψ as. H α ψ = ψ ψ C 0 (R 3 \y)

Point Interaction: a very short introduction

Point Interaction: a very short introduction The spectrum of H α : If α < 0, H α has one eigenvalue σ ac (H α ) = [0, ) σ pp (H α ) = { (4πα) 2 } < α < 0 the corresponding normalized eigenfunction is If α 0, then σ pp =. φ 0 = 2 α e4πα x y x y

Point Interaction: a very short introduction The spectrum of H α : If α < 0, H α has one eigenvalue σ ac (H α ) = [0, ) σ pp (H α ) = { (4πα) 2 } < α < 0 the corresponding normalized eigenfunction is If α 0, then σ pp =. φ 0 = 2 α e4πα x y x y For every k R 3 the generalized eigenfunction of H α corresponding to the energy E = k 2 in the continuous spectrum is given in closed form by Φ y ±(x, k) = e ikx + e iky 4πα ± i k e i k x y x y

Spin dependent point interaction

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction The Hilbert space for a system made up of one particle in dimension d and one spin 1/2 can be written H := L 2 (R d ) C 2 d = 1, 2, 3.

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction The Hilbert space for a system made up of one particle in dimension d and one spin 1/2 can be written Ψ H can be written in the following form Ψ = σ ψ σ χ σ, where the sum runs over σ = ± H := L 2 (R d ) C 2 d = 1, 2, 3.

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction The Hilbert space for a system made up of one particle in dimension d and one spin 1/2 can be written H := L 2 (R d ) C 2 d = 1, 2, 3. Ψ H can be written in the following form Ψ = σ ψ σ χ σ, where the sum runs over σ = ± Ψ, Φ := σ (ψ σ, φ σ ) L 2 Ψ, Φ H.

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction The Hilbert space for a system made up of one particle in dimension d and one spin 1/2 can be written H := L 2 (R d ) C 2 d = 1, 2, 3. Ψ H can be written in the following form Ψ = σ ψ σ χ σ, where the sum runs over σ = ± Ψ, Φ := σ (ψ σ, φ σ ) L 2 Ψ, Φ H. D(S) := C 0 (R d \0) C 2 d = 1, 2, 3 S := I C 2 + I L 2 βˆσ (1) β R +,

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction The Hilbert space for a system made up of one particle in dimension d and one spin 1/2 can be written H := L 2 (R d ) C 2 d = 1, 2, 3. Ψ H can be written in the following form Ψ = σ ψ σ χ σ, where the sum runs over σ = ± Ψ, Φ := σ (ψ σ, φ σ ) L 2 Ψ, Φ H. D(S) := C 0 (R d \0) C 2 d = 1, 2, 3 S := I C 2 + I L 2 βˆσ (1) β R +, D(H) := H 2 (R d ) C 2 d = 1, 2, 3 H := I C 2 + I L 2 βˆσ (1) β R +,

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction The Hilbert space for a system made up of one particle in dimension d and one spin 1/2 can be written H := L 2 (R d ) C 2 d = 1, 2, 3. Ψ H can be written in the following form Ψ = σ ψ σ χ σ, where the sum runs over σ = ± Ψ, Φ := σ (ψ σ, φ σ ) L 2 Ψ, Φ H. D(S) := C 0 (R d \0) C 2 d = 1, 2, 3 S := I C 2 + I L 2 βˆσ (1) β R +, D(H) := H 2 (R d ) C 2 d = 1, 2, 3 H := I C 2 + I L 2 βˆσ (1) β R +, HΨ = σ ( + β σ ) ψσ χ σ Ψ D(H). R(z)Ψ = σ ( z + β σ ) 1ψσ χ σ Ψ H; z ρ(h),

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction ˆR 0 (z) = R(z) + σ,σ ( (Γ0 (z)) 1) σ,σ Φ z σ, Φ z σ z C\R,

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction Let < α, then { D(Ĥ0) := Ψ H Ψ = Ψ z + q σ Φ z σ ; Ψ z = ψσ z χ σ D(H); z ρ(ĥ0) ; σ σ } q σ = αf σ, d = 1 ; αq σ = f σ, d = 2 ; αq σ = f σ, d = 3 Ĥ 0 Ψ := HΨ z + z σ q σ Φ z σ Ψ D(Ĥ0). ˆR 0 (z) = R(z) + σ,σ ( (Γ0 (z)) 1) σ,σ Φ z σ, Φ z σ z C\R,

Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 249-261 Spin dependent point interaction Let < α, then { D(Ĥ0) := Ψ H Ψ = Ψ z + q σ Φ z σ ; Ψ z = ψσ z χ σ D(H); z ρ(ĥ0) ; σ σ } q σ = αf σ, d = 1 ; αq σ = f σ, d = 2 ; αq σ = f σ, d = 3 Ĥ 0 Ψ := HΨ z + z σ q σ Φ z σ Ψ D(Ĥ0). ˆR 0 (z) = R(z) + σ,σ ( (Γ0 (z)) 1) σ,σ Φ z σ, Φ z σ z C\R, d = 1 Γ 0 (z) = d = 2 Γ 0 (z) = i 2 z β 1 α d = 3 Γ 0 (z) = 0 ln( z β/2) + γ iπ/2 2π z β 4πi 0 0 + α 0 z + β 0 i 2 z + β 1 α 4πi + α 0 + α ln( z + β/2) + γ iπ/2 2π + α

Spin dependent point interaction For d = 1, 2, 3 the essential spectrum is given by σ ess (Ĥ) = [ β, + ).

Spin dependent point interaction For d = 1, 2, 3 the essential spectrum is given by Consider < α < 0 σ ess (Ĥ) = [ β, + ). d = 1. d = 2. d = 3. E 0, = β α2 4 ; E 0,+ = β α2 4. (1) For all < α < 0 the lowest eigenvalue, E 0,, is below the threshold of essential spectrum and 2 2β α < 0 the second eigenvalue is embedded in the continuous spectrum, β E 0,+ < β. E 0, = β 4e 2(2πα+γ) ; E 0,+ = β 4e 2(2πα+γ). (2) The lowest eigenvalue, E 0,, is always below the threshold of essential spectrum if (ln( β/2) + γ)/(2π) α < the second eigenvalue is embedded in the continuous spectrum, β E 0,+ < β. E 0, = β (4πα) 2 ; E 0,+ = β (4πα) 2. (3) The lowest eigenvalue, E 0,, is always below the threshold of essential spectrum and if 2β/(4π) α < 0 the second eigenvalue is embedded in the continuous spectrum, β E 0,+ < β.

Turning to resonances

Turning to resonances Let < α and 0 < ε α, then { D(Ĥε) := Ψ H Ψ = Ψ z + q σ Φ z σ ; Ψ z = σ σ ψ z σ χ σ D(H); z ρ(ĥε) ; q ± = αf ± εf d = 1 ; αq ± + εq = f ± d = 2 ; } αq ± + εq = f ± d = 3 Ĥ ε Ψ := HΨ z + z σ q σ Φ z σ Ψ D(Ĥε).}

Turning to resonances Let < α and 0 < ε α, then { D(Ĥε) := Ψ H Ψ = Ψ z + q σ Φ z σ ; Ψ z = σ σ ψ z σ χ σ D(H); z ρ(ĥε) ; q ± = αf ± εf d = 1 ; αq ± + εq = f ± d = 2 ; } αq ± + εq = f ± d = 3 Ĥ ε Ψ := HΨ z + z σ q σ Φ z σ Ψ D(Ĥε).} ˆR ε (z) = R(z) + σ,σ ( (Γε (z)) 1) σ,σ Φ z σ, Φ z σ z C\R,

Turning to resonances Let < α and 0 < ε α, then { D(Ĥε) := Ψ H Ψ = Ψ z + q σ Φ z σ ; Ψ z = σ σ ψ z σ χ σ D(H); z ρ(ĥε) ; q ± = αf ± εf d = 1 ; αq ± + εq = f ± d = 2 ; } αq ± + εq = f ± d = 3 Ĥ ε Ψ := HΨ z + z σ q σ Φ z σ Ψ D(Ĥε).} ˆR ε (z) = R(z) + σ,σ ( (Γε (z)) 1) σ,σ Φ z σ, Φ z σ z C\R, Γ ε (z) = Γ ε (z) = Γ ε (z) = i 2 z β α ε α 2 ε 2 α 2 ε 2 ε i α 2 ε 2 2 z + β α α 2 ε 2 ln( z β/2) + γ iπ/2 + α ε 2π ln( z + β/2) + γ iπ/2 ε 2π i z β + α ε 4π ε i z + β + α 4π d = 1 d = 2 + α d = 3

Resonances

Resonances For d = 1, 2, 3 the point spectrum is given by real roots of equation det Γ ε (z) = 0. There exists ε 0 > 0 such that for all 0 < ε < ε 0 d = 1. If 0 α the point spectrum is empty. If < α < 0 the point spectrum consists of one simple eigenvalue E ε, < β. d = 2. For α = the point spectrum is empty. For all < α < the point spectrum consists of one simple eigenvalue E ε, < β. d = 3. If 0 α the point spectrum is empty. If < α < 0 the point spectrum consists of one simple eigenvalue E ε, < β.

Resonances For d = 1, 2, 3 the point spectrum is given by real roots of equation det Γ ε (z) = 0. There exists ε 0 > 0 such that for all 0 < ε < ε 0 d = 1. If 0 α the point spectrum is empty. If < α < 0 the point spectrum consists of one simple eigenvalue E ε, < β. d = 2. For α = the point spectrum is empty. For all < α < the point spectrum consists of one simple eigenvalue E ε, < β. d = 3. If 0 α the point spectrum is empty. If < α < 0 the point spectrum consists of one simple eigenvalue E ε, < β. Resonances are defined as zeroes in the unphysical sheet of det Γ ε (z) ζ ε = ε2 α 2 [ ( ) 1/2 ] 8β 1 i 16β α 2 1 + O ( (ε/α) 4) d = 1 ( ζ ε = (2πε)2 η 2 + (π/2) 2 η + i π ) + O ( (ε/α) 4) d = 2 2 ζ ε = (4π)4 ε 2 α 2 [ ( ) 1/2 ] 2β 1 i β (4πα) 2 1 + O ( (ε/α) 4) d = 3

time decay

time decay Φ ɛ,λ i = Φ λ i + ( ) 1 φ λ i (0) Γ ɛ (λ) σ σ σ Φ λ (1) (ω) = λ β 4π 3/2 + ( λ β 4πi e iω λ βx χ + + iɛ ) ( λ+β α 4πi G λ σ α + ( ) χ σ ) α ( λ β 4πi λ+β x ei ɛ 2 4π x λ+β 4πi α ) ( λ+β α 4πi ) α λ β x ei ɛ 2 4π x χ λ > β, ω S 2 χ + + Φ λ (2) (ω) = λ + β 4π 3/2 + ( λ β 4πi e iω λ+βx χ + ıɛ ) ( λ+β α 4πi ) α ( λ β 4πi λ β 4πi α ) ( λ+β α λ β x ei ɛ 2 4π x 4πi ) α λ+β x ei ɛ 2 4π x χ + λ > β, ω S 2 χ +

time decay P++(t) = ( β λ 4π e 8πα x 16π 2 x 2 = A e 8πα x x ɛ 2 e ıλt ) ( β+λ α 4πı ( 1 π ( β λ ) α ɛ 2 2 ɛ 2 e ıλt e 2 β λ x 16π 2 x 2 2β (4πα) 2 4π α) ( ı ) e ıλt L(λ; x 0 )dλ 4π dλ ) 2 dλ + α + ɛ 2

time decay P++(t) = ( β λ 4π e 8πα x 16π 2 x 2 = A e 8πα x x ɛ 2 e ıλt ) ( β+λ α 4πı ( 1 π ( β λ ) α ɛ 2 2 ɛ 2 e ıλt e 2 β λ x 16π 2 x 2 2β (4πα) 2 4π α) ( ı ) e ıλt L(λ; x 0 )dλ 4π dλ ) 2 dλ + α + ɛ 2 L(λ; x 0 ) = γ (λ x 0 ) 2 +γ 2 is the Lorentzian distribution in the variable λ and = [ a, a], x 0 = E + + (4πɛ)2 β (4πα) 2, A = 8π 2 α 2β (4πα) 2 γ = (4πα) 3 ɛ 2 α β 2β (4πα) 2

time decay P++(t) = ( β λ 4π e 8πα x 16π 2 x 2 = A e 8πα x x ɛ 2 e ıλt ) ( β+λ α 4πı ( 1 π ( β λ ) α ɛ 2 2 ɛ 2 e ıλt e 2 β λ x 16π 2 x 2 2β (4πα) 2 4π α) ( ı ) e ıλt L(λ; x 0 )dλ 4π dλ ) 2 dλ + α + ɛ 2

time decay P++(t) = ( β λ 4π e 8πα x 16π 2 x 2 = A e 8πα x x ɛ 2 e ıλt ) ( β+λ α 4πı ( 1 π ( β λ ) α ɛ 2 2 ɛ 2 e ıλt e 2 β λ x 16π 2 x 2 2β (4πα) 2 4π α) ( ı ) e ıλt L(λ; x 0 )dλ 4π dλ ) 2 dλ + α + ɛ 2 2πı Res z=(x0 iγ) [ e ızt L(z; x 0 ) ] = π e ı x 0 t e γ t

Conclusions The system we have analyzed is made of a localized q-bit, a two level model atom, and a particle, interacting with the q-bit via zero range forces. We have reproduced in a solvable model the mechanism of formation of a resonance. The generalization to an N-level atom is straightforward. Model with a gas of non interacting particles are under study.