Is quantum linear superposition exact on all energy scales? A unique case study with flavour oscillating systems

Transcription

1 Is quantum linear superposition exact on all energy scales? A unique case study with flavour oscillating systems Supervisor: Dr. Beatrix C. Hiesmayr Universität Wien, Fakultät für Physik March 17, 2015

2 Standard quantum mechanics: basic properties linearity of Schrödinger equation allows superpositions: ψ 1, ψ 2 are solutions ψ = c 1 ψ 1 + c 2 ψ 2 is also a solution evolution of quantum system due to Schrödinger equation is deterministic measurement destroys superposition with outcomes distributed due to Born rule: P 1 = c 1 2, P 2 = c 2 2 ( ψ 1 ψ 2 = 0).

3 Transition to the classical world Defining wavefunction as ψ = e is/ we can rewrite the Schrödinger equation: i ψ t = 2 ψ +V (q) S 2m q t = 1 ( S ) 2 i 2 S +V (q) 2m q 2m q 2 and applying 0: S t = 1 ( S ) 2 + V (q) 2m q we get non-linear Hamilton-Jacobi equation: S 1, S 2 are solutions, but S = c 1 S 1 + c 2 S 2 is not a solution.

4 Troubles with standard quantum mechanics Standard quantum mechanics exposes two different regimes: 1 Schrödinger evolution: linear, deterministic and reversible. 2 Measurement: non-linear, stochastic and irreversible. Question: Is there a border between quantum and classical worlds?

5 Solutions for quantum measurement problem Copenhagen interpretation (Bohr, 1928) Bohmian mechanics (Bohm, 1952) many-worlds interpretation (Everett, 1957) decoherence (Zeh, 1970) spontaneous collapse (Ghirardi, Rimini and Weber, 1986) gravity induced collapse (Károlyházy, 1966; Diósi, 1984; Penrose, 1996) A. Bassi et al., Rev. Mod. Phys. 85, 471 (2013).

6 Basic ideas of spontaneous collapse models Universal dynamics should: 1 be non-linear 2 be stochastic 3 include non-unitary evolution 4 not allow for superluminal signaling Proposition: Each particle of a system of n particles experiences a sudden spontaneous localization process with defined rate, and in the time interval between two localizations system evolves due to Schrödinger equation.

7 Classifying the zoo of collapse models Classification by: choice of localization basis: energy, position, momentum,... choice of stochastic noise: white noise: Wiener process + Markovian evolution non-white noise: generic Gaussian noise + non-markovian evolution choice of noise temperature: infinite temperature, finite temperature choice of distinguishability of particles: first quantized models consider distinguishable particles second quantized models consider identical particles

8 GRW: The first model of spontaneous collapse System of N distinguishable particles: state of system: ψ(x 1,..., x N ) L 2 (R 3N ) ignoring spin and other internal degrees of freedom at random times wavefunction undergoes a sudden impulse: ψ t (x 1,..., x N ) L n(x)ψ t (x 1,..., x N ) L n (x)ψ t (x 1,..., x N ) G. C. Ghirardi, A. Rimini, T. Weber, Phys. Rev. D 34, 470 (1986).

9 GRW: The first model of spontaneous collapse collapse operator for n-th particle: L n (x) = 1 (πr 2 C )3/4 e (qn x)2 /2r 2 C coherence length r C 10 7 m is one out of two free parameters probability density for n-th particle undergoing collapse: p n (x) = L n (x)ψ t (x 1,..., x N ) 2 collapses are distributed in time due to a Poissonian process with frequency λ GRW s 1, which is second free parameter G. C. Ghirardi, A. Rimini, T. Weber, Phys. Rev. D 34, 470 (1986).

10 What is particle? GRW model s point of view GRW introduces a density of mass instead of point particles: ρ (n) t (x n ) = m n d 3 x 1...d 3 x n 1 d 3 x n+1...d 3 x N ψ t (x 1,..., x N ) 2 which is associated to n-th particle of the system. A. Bassi et al., Rev. Mod. Phys. 85, 471 (2013). G. C. Ghirardi, R. Grassi, F. Benatti, Found. Phys. 20, 1271 (1995).

11 QMUPL: Stochastic fluctuations in time A simple model with stochastical differential equation: dψ t = Properties: ( i Hdt + λ(q q t )dw t λ ) 2 (q q t) 2 dt ψ t, stochastic fluctuations take place only in time stochastic process acts continuously QMUPL needs only one free constant λ = λ 0 m m 0, λ m 2 s 1 QMUPL is in fact a scaling limit of GRW with r C 0 and λ GRW, but λ GRW r C is a constant L. Diósi, Phys. Rev. A 40, 1165 (1989).

12 CSL: Introducing continuous localization Model with stochastical differential equation for identical particles: dψ t = ( i γ Hdt + dx(m(x) M(x) t )dw t (x) m 0 γ ) dx(m(x) M(x) t ) 2 dt ψ t, 2m 2 0 where M(x) = j m jn j (x), N j (x) = dyg(y x)ψ j (y)ψ j(y), g(x) = 1 ( 2πr C ) 3 e x2 /2r 2 C. This is mass-proportional version of CSL. G. C. Ghirardi, P. Pearle, A. Rimini, Phys. Rev. A 42, 78 (1990). P. Pearle, E. Squires, Phys. Rev. Lett. 73, 1 (1994).

13 Properties of CSL model 1 two parameters: coherence length r C 10 7 m and collapse γ rate λ CSL = s 1 (4πr 2 C )3/2 2 physical interpretation as a random field filling space 3 amplification mechanism:

14 Discrete symmetries: C, P, T C (charge conjugation): {q} { q} P (parity inverse): x x, t t T (time reversal) : x x, t t

15 Neutral kaon system Kaon decays: K 0, K 0 2π, 3π. Mass eigenstates: K S 2π, K L 3π. Flavour eigenstates: K 0, K ) 0 = 1 2 ( K L ± K S. CP eigenstates: CP K 1 = + K 1, CP K 2 = K 2. When CP symmetry is conserved, K 1 = K S, K 2 = K L. Kaon in time: ) K 0 (t) = 1 2 (e Γ L 2 t im L t K L + e Γ S 2 t im S t K S.

16 CP violation Cronin, Fitch (1964): ( sometimes ) K L 2π! 1 K S = K 1 + ε K 2, 2(1+ ε 2 ) ( ) 1 K L = ε K 1 + K 2. 2(1+ ε 2 ) ε 10 3 rate of CP violation

17 Testing collapse models in flavour oscillating system Probabilities of finding K 0 or K 0 in beam: K 0 K 0 (t) 2 p f E K 0, p f U(t) K 0, p i 2, K 0 K 0 (t) 2 E K 0, p f U(t) K 0, p i 2. }{{} p f includes collapse! Up to first order in time with conserved CP symmetry: where Λ = P (1) = 1 ( ) e ΓLt + e ΓSt + 2e Γt e Λt cos[ mt], K 0 K 0 4 P (1) K 0 K = 1 ( ) e ΓLt + e ΓSt 2e Γt e Λt cos[ mt], 0 4 γ m2 16π 3/2 r 3 C m Hz. S. Donadi et al., Found. Phys. 43, 813 (2013).

18 Results: neutrinos, neutral mesons, chiral molecules M. Bahrami et al., Nature Sci. Rep. 3, 1952 (2013).

19 Effect of CP violation (preliminary results) Up to second order in time with violation of CP symmetry: = 1 1 ξ { ( e Γ Lt 1+γf (1) K 0 K 0 L ξ (ξ, m2 )t + γ 2 f (2) L (ξ, m4 )t 2) + ( + e Γ St 1 γf (1) S (ξ, m2 )t + γ 2 f (2) S (ξ, m4 )t 2) P (2) + 2e Γt( 1 γf (1) osc(ξ, m 2 )t + γ 2 f (2) osc(ξ, m 4 )t 2) cos[ mt] where ξ = K L K S = 2 Re ε 1+ ε 2. Results: γf (1) L Hz, γ 2 f (2) L Hz 2, }, γf (1) S Hz, γ 2 f (2) S Hz 2, γf osc (1) Hz, γ 2 f osc (2) Hz 2.

20 Conclusions Collapse models unify macroscopic and microscopic regimes. Collapse models are experimentally falsifiable. Neutral mesons are superpositions of mass eigenstates, thus a unique laboratory for testing the mass-proportional CSL model. CSL collapse affects flavour oscillation by exponential damping of interference term. CP violation is expected to enhance the effect of CSL collapse.

21 Literature Reviews of collapse models and their testing: A. Bassi et al., Rev. Mod. Phys. 85, 471 (2013). A. Bassi, H. Ulbricht, arxiv: (2014). Quantum measurement problem: A. Bassi, G. C. Ghirardi, Phys. Lett. A 275, 373 (2000). GRW model: G. C. Ghirardi, A. Rimini, T. Weber, Phys. Rev. D 34, 470 (1986). QMUPL model: L. Diósi, Phys. Rev. A 40, 1165 (1989). CSL model: P. Pearle, Phys. Rev. A 39, 2277 (1989). G. C. Ghirardi, P. Pearle, A. Rimini, Phys. Rev. A 42, 78 (1990). P. Pearle, E. Squires, Phys. Rev. Lett. 73, 1 (1994). Testing CSL model via oscillating systems (neutrinos, neutral mesons, molecules): M. Bahrami et al., Nature Sci. Rep. 3, 1952 (2013). S. Donadi et al., Found. Phys. 43, 813 (2013).

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