Gravity-related spontaneous collapse in bulk matter Lajos Di´osi Wigner Center, Budapest 29 Apr 2014, Frascati Acknowledgements go to: Hungarian Scientific Research Fund under Grant No. 75129 EU COST Action MP1006 ‘Fundamental Problems in Quantum Physics’ Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 1 / 11 1 Schr¨odinger Cats, Catness 2 Different catness in CSL and DP 3 Master equation of spontaneous decoherence 4 Decoherence of acoustic d.o.f. 5 Decoherence of acoustic modes Center of mass decoherence Universal dominance of spontaneous decoherence Strong spontaneous decoherence at low heating 6 Concluding remarks Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 2 / 11 Schr¨ odinger Cats, Catness Schr¨odinger Cats, Catness Well-defined spatial mass distributions f1 , f2 |Cati = |f1 i + |f2 i √ 2 Catness: squared-distance `2 (f1 , f2 ) [dim: energy] Standard QM: Cat collapses immediately if we measure f In ”new” QM: we postulate spontaneous collapse |Cati =⇒ either |f1 i or |f2 i with collapse rate `2 /~ Testable consequence: spontaneous decoherence (of ρˆ) 1 1 |CatihCat| =⇒ |f1 ihf1 | + |f2 ihf2 | with decoherence rate `2 /~ 2 2 I discuss spontaneous decoherence (collapse would come easily). Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 3 / 11 Different catness in CSL and DP Different catness in CSL and DP `2 (f1 , f2 ) = C11 + C22 − 2C12 Z ZZ dr1 dr2 CSL : Cij = Λ fi (r)fj (r)dr DP : Cij = G fi (r1 )fj (r2 ) r12 Spatial cut-off σ is needed (by Gaussian gσ of width σ): X f (r) = m gσ (r − xa ) CSL : σ = 10−5 cm, a D(P) : σ = 10−12 cm DP: ’nuclear’ σ, weak G; CSL: ’macroscopic’ σ, strong Λ DP and CSL: same (similar) collapse for c.o.m. of a bulk DP: too much spontaneous heating, CSL: tolerable heating DP: significance for acoustic modes, CSL: no significance DP: large scale dominance; CSL: Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 4 / 11 Master equation of spontaneous decoherence Master equation of spontaneous decoherence d ρˆ i ˆ = − [H, ρˆ] + Dρˆ dt ~ Key quantity: fˆ(r) = m a gσ (r − ˆ xa ) Dynamics of ρˆ’s diagonalization in f at rate `2 /~: ZZ G dr1 dr2 Dρˆ = − [fˆ(r1 ), [fˆ(r2 ), ρˆ]] 2~ r12 Z Λ ˆ ˆ CSL : Dρˆ = − [f (r), [f (r), ρˆ]]dr 2~ Useful detailed Fourier form: Z 2 2 4πe−k σ X ikˆ dk Gm2 xa −ikˆ xb ρ ˆ ]] [e , [e , Dρˆ = − 2 2~ k (2π)3 a,b P Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 5 / 11 Decoherence of acoustic d.o.f. Decoherence of acoustic d.o.f. Elasto-hydrodynamics (acoustics) in homogeneous bulk Displacement field u ˆ(r), canonically conj. momentum field π ˆ (r): Z 1 2 f0 2 ˆ = H u)2 dr, π ˆ + c` (∇ˆ 2f 0 2 f 0 = M/V is mass density; c` is (longitudinal) sound velocity. Recall D, insert ˆ xa = xa + u ˆ(xa ); xa are fiducial positions. Assume u ˆ(r)σ, exp[ikˆ u(xa )]≈1+ikˆ u(xa ); etc. Z 1 u(r), [ˆ u(r), ρˆ]]dr. Dρˆ = − f 0 (ωGnucl )2 [ˆ 2~ ωGnucl √ = Gf nucl ∼ 1kHz i.e.: frequency of Newton oscillator in density f nucl = m/(4πσ 2 )3/2 ∼ 1012 g/cm3 Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 6 / 11 Decoherence of acoustic modes Decoherence of acoustic modes Fourier modes in rectangular bulk: 1 X 1 X u ˆ(r) = √ u ˆk eikr , π ˆ (r) = √ π ˆ k eikr V k V k Hamiltonian and decoherence: 1X 1 † −1 X 0 nucl 2 † 0 2 2 † ˆ H= π ˆ π ˆ + f c k u ˆ u ˆ , D ρ ˆ = f (ωG ) [ˆ uk , [ˆ uk , ρˆ]] k k ` k 2 k f0 k 2~ k Master equation of acoustic modes spontaneous decoherence: dˆ ρ 1 X −i † 0 nucl 2 † 0 2 2 † uk , [ˆ = [ˆ π π ˆ k , ρˆ] − if c` k [ˆ uk u ˆk , ρˆ] − f (ωG ) [ˆ uk , ρˆ]] dt 2~ k f0 k Recall: summation over acoustic numbers k. P wave 2 † Note: CSL would have Dρˆ ∼ k k [ˆ uk , [ˆ uk , ρˆ]]. Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 7 / 11 Decoherence of acoustic modes Center of mass decoherence Center of mass decoherence C.o.m.dynamics: k = 0 acoustic mode √ ˆ = √1 u ˆ = Vπ X ˆ0 , P ˆ0 V (we set the fiducial c.o.m. to the origin) Identify c.o.m. part in master equation: dˆ ρ 1 X −i † 0 nucl 2 † 0 2 2 † uk , [ˆ uk , ρˆ]] = [ˆ π π ˆ k , ρˆ] − if c` k [ˆ uk u ˆk , ρˆ] − f (ωG ) [ˆ dt 2~ k f0 k Get closed master equation for c.o.m.: " # 2 ˆ dˆ ρc.o.m. −i P 1 ˆ [X, ˆ ρˆc.o.m. ]], = , ρˆc.o.m. − M(ωGnucl )2 [X, dt ~ 2M 2~ Full accordance with old derivations in DP-model. Compare it to richness of acoustic mode spontaneous decoherence! Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 8 / 11 Decoherence of acoustic modes Universal dominance of spontaneous decoherence Universal dominance of spontaneous decoherence Inspect long wavelength feature of master equation: dˆ ρ 1 X −i † 0 nucl 2 † 0 2 2 † uk , ρˆ]] uk , [ˆ ˆk , ρˆ] − f (ωG ) [ˆ uk u [ˆ π π ˆ k , ρˆ] − if c` k [ˆ = dt 2~ k f0 k Harmonic potential and decoherence terms: quadratic in u ˆk . Although structures are different, they compete, decoherence wins if: c` k ωGnucl∼1kHz =⇒ 1/k1m (e.g. in solids) The master equation for these modes: 1 X −i † dˆ ρ 0 nucl 2 † uk , [ˆ = [ˆ π π ˆ k , ρˆ] − f (ωG ) [ˆ uk , ρˆ]] . dt 2~ f0 k 1/k1m Wavelength 1m: ’free motion’ plus spontaneous decoherence. Example: Bulk of rock as big as 100m, sub-volume about a few m’s =⇒ C.o.m. moves and decoheres like free-body. Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 9 / 11 Decoherence of acoustic modes Strong spontaneous decoherence at low heating Strong spontaneous decoherence at low heating Side-effect of spontaneous decoherence: spontaneous warming up: ˆ dH ˆ = N × ˙ = DH (N : number of d.o.f.) dt For a single acoustic mode uˆjk ≡ uˆ, π ˆjk ≡ π ˆ , heating rate: π ˆ†π ˆ 1 π ˆ†π ˆ −f 0 nucl 2 † (ωG ) uˆ , uˆ, 0 = ~(ωGnucl )2 ∼ 10−21 erg/s ˙ = D 0 = 2f 2~ 2f 2 In M=1g, the # of d.o.f. N∼1023 =⇒ N ˙ ∼ 100erg/s: far too much! Refine DP-model: Spontaneous collapse for modes 1/k λ only: dˆ ρ −i X 1 † f 0 nucl 2 X † 0 2 2 † u ˆ ,ˆ ρ ] − k [ˆ u = [ˆ π π ˆ ,ˆ ρ ]+f c (ω ) [ˆ uk ,[ˆ uk ,ˆ ρ]] k ` k k dt 2~ k f 0 k 2~ G 1/kλ E.g.: λ=10−5 cm, # of d.o.f. N∼1014 =⇒ N ˙ ∼ 10−7 erg/s: fairly low! DP-collapse of macroscopic acoustic modes (c.o.m., too) remains. Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 10 / 11 Concluding remarks Concluding remarks We killed Cats by collapse or just by decoherence compared spontaneous decoherence in DP and CSL derived G-related spontaneous decoherence of acoustic modes derived spontaneous decoherence master eq. for ρˆ showed spontaneous DP-decoherence dominates at large scales reduced spontaneous heating in DP, kept macrosopic predictions spared spontaneous collapse stoch. eqs. for |ψi claimed spontaneous decoherence is the only testable local effect claime spontaneous collapse is untestable global effect - for DP, CSL, GRW,... mention spontaneous collapse becomes testable in extended DP-model E-print: arXiv1404.6644 Lajos Di´ osi (Wigner Center, Budapest) Gravity-related spontaneous collapse in bulk matter 29 Apr 2014, Frascati 11 / 11
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