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

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

2 Metastable states in a model of spin dependent point interactions

3 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, (1990) Resonance theory for Schrödinger operators O.Costin, A.Soffer Comm.Math.Phys. 224, (2001) La regola d'oro di Fermi P.Facchi, S.Pascazio

4 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, (1999) Resonances, metastable states and exponential decay laws in perturbation theory W.Hunzinker Comm.Math.Phys. 132, (1990) Resonance theory for Schrödinger operators O.Costin, A.Soffer Comm.Math.Phys. 224, (2001) La regola d'oro di Fermi P.Facchi, S.Pascazio

5 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, (1999) y x Resonances, metastable states and exponential decay laws in perturbation theory W.Hunzinker Comm.Math.Phys. 132, (1990) Resonance theory for Schrödinger operators O.Costin, A.Soffer Comm.Math.Phys. 224, (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 (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, (2002) 3

6 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

7 Point Interaction: a very short introduction

8 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.

9 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.

10 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)

11 Point Interaction: a very short introduction

12 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

13 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

14 Spin dependent point interaction

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

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

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

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

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

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

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

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

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

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

26 Spin dependent point potentials in one and three dimensions. Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari. J. Phys. A: Math. Theor. 40 (2007) 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 α 0 z + β 0 i 2 z + β 1 α 4πi + α 0 + α ln( z + β/2) + γ iπ/2 2π + α

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

33 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.

34 Turning to resonances

35 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).}

36 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,

37 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

38 Resonances

39 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.

40 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 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) O ( (ε/α) 4) d = 3

41 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 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) O ( (ε/α) 4) d = 3

42 time decay

43 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 +

44 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

45 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

46 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

47 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

48 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

49 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.