Gravity-related spontaneous collapse in bulk matter Lajos Di osi - PowerPoint PPT Presentation

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

Schr¨ odinger Cats, Catness 1 Different catness in CSL and DP 2 Master equation of spontaneous decoherence 3 Decoherence of acoustic d.o.f. 4 Decoherence of acoustic modes 5 Center of mass decoherence Universal dominance of spontaneous decoherence Strong spontaneous decoherence at low heating Concluding remarks 6 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 f 1 , f 2 | Cat � = | f 1 � + | f 2 � √ 2 Catness: squared-distance ℓ 2 ( f 1 , f 2 ) [dim: energy] Standard QM: Cat collapses immediately if we measure f In ”new” QM: we postulate spontaneous collapse ⇒ either | f 1 � or | f 2 � with collapse rate ℓ 2 / � | Cat � = Testable consequence: spontaneous decoherence (of ˆ ρ ) ⇒ 1 2 | f 1 �� f 1 | + 1 2 | f 2 �� f 2 | with decoherence rate ℓ 2 / � | Cat �� Cat | = 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 ℓ 2 ( f 1 , f 2 ) = C 11 + C 22 − 2 C 12 � � � f i ( r 1 ) f j ( r 2 ) d r 1 d r 2 CSL : C ij = Λ f i ( r ) f j ( r ) d r DP : C ij = G r 12 Spatial cut-off σ is needed (by Gaussian g σ of width σ ): � f ( r ) = m g σ ( r − x a ) a CSL : σ = 10 − 5 cm , 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 ˆ dt = − i ρ � [ˆ H , ˆ ρ ] + D ˆ ρ Key quantity: ˆ f ( r ) = m � a g σ ( r − ˆ x a ) ρ ’s diagonalization in f at rate ℓ 2 / � : Dynamics of ˆ ρ = − G �� ρ ]] d r 1 d r 2 [ˆ f ( r 1 ) , [ˆ D ˆ f ( r 2 ) , ˆ 2 � r 12 � � ρ = − Λ � [ˆ f ( r ) , [ˆ CSL : D ˆ f ( r ) , ˆ ρ ]] d r 2 � Useful detailed Fourier form: � 4 π e − k 2 σ 2 ρ = − Gm 2 ρ ]] d k [ e i kˆ x a , [ e − i kˆ x b , ˆ � D ˆ 2 � k 2 (2 π ) 3 a , b 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 ˆ u ( r ), canonically conj. momentum field ˆ π ( r ): � � 1 π 2 + f 0 � ˆ 2 c 2 u ) 2 H = 2 f 0 ˆ ℓ ( ∇ ˆ d r , f 0 = M / V is mass density; c ℓ is (longitudinal) sound velocity. x a = x a + ˆ u ( x a ); x a are fiducial positions. Recall D , insert ˆ Assume ˆ u ( r ) ≪ σ , exp[ i kˆ u ( x a )] ≈ 1+ i kˆ u ( x a ); etc. ρ = − 1 � 2 � f 0 ( ω nucl ) 2 D ˆ [ ˆ u ( r ) , [ ˆ u ( r ) , ˆ ρ ]] d r . G √ Gf nucl ∼ 1kHz ω nucl = G i.e.: frequency of Newton oscillator in density f nucl = m / (4 πσ 2 ) 3 / 2 ∼ 10 12 g/cm 3 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 1 � � u k e i kr , π k e i kr u ( r ) = √ π ( r ) = √ ˆ ˆ ˆ ˆ V V k k Hamiltonian and decoherence: � 1 � H =1 ρ = − 1 ˆ � � π † π k + f 0 c 2 ℓ k 2 ˆ u † f 0 ( ω nucl ) 2 [ ˆ u † f 0 ˆ k ˆ k ˆ u k , D ˆ k , [ ˆ u k , ˆ ρ ]] G 2 2 � k k Master equation of acoustic modes spontaneous decoherence: � − i � d ˆ d t = 1 ρ � π † u † u † ρ ] − i f 0 c 2 ℓ k 2 [ ˆ ρ ] − f 0 ( ω nucl ) 2 [ ˆ f 0 [ ˆ k ˆ π k , ˆ k ˆ u k , ˆ k , [ ˆ u k , ˆ ρ ]] G 2 � k Recall: summation over acoustic wave numbers k . u † k k 2 [ ˆ ρ ∼ � Note: CSL would have D ˆ k , [ ˆ u k , ˆ ρ ]]. 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 √ 1 ˆ ˆ √ X = ˆ u 0 , P = V ˆ π 0 V (we set the fiducial c.o.m. to the origin) Identify c.o.m. part in master equation: d ˆ d t = 1 ρ � − i � � π † ρ ] − i f 0 c 2 ℓ k 2 [ ˆ u † ρ ] − f 0 ( ω nucl ) 2 [ ˆ u † f 0 [ ˆ π k , ˆ u k , ˆ k , [ ˆ u k , ˆ ρ ]] k ˆ k ˆ G 2 � k Get closed master equation for c.o.m.: � ˆ � P 2 d ˆ ρ c . o . m . = − i − 1 ) 2 [ ˆ X , [ ˆ 2 � M ( ω nucl 2 M , ˆ ρ c . o . m . X , ˆ ρ c . o . m . ]] , G d t � 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: � − i � d ˆ d t = 1 ρ � π † ρ ] − i f 0 c 2 ℓ k 2 [ ˆ u † ρ ] − f 0 ( ω nucl ) 2 [ ˆ u † f 0 [ ˆ k ˆ π k , ˆ k ˆ u k , ˆ k , [ ˆ u k , ˆ ρ ]] G 2 � k Harmonic potential and decoherence terms: quadratic in ˆ u k . Although structures are different, they compete, decoherence wins if: c ℓ k ≪ ω nucl ∼ 1kHz = ⇒ 1 / k ≫ 1m (e.g. in solids) G The master equation for these modes: d ˆ d t = 1 ρ � − i � � π † ρ ] − f 0 ( ω nucl ) 2 [ ˆ u † f 0 [ ˆ π k , ˆ k , [ ˆ u k , ˆ ρ ]] . k ˆ G 2 � 1 / k ≫ 1 m 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: d ˆ H dt = D ˆ H = N × ˙ ǫ ( N : number of d.o.f.) For a single acoustic mode ˆ u j k ≡ ˆ u , ˆ π j k ≡ ˆ π , heating rate: π † ˆ 2 f 0 = − f 0 π † ˆ ǫ = D ˆ π � � u , ˆ π �� = 1 ) 2 ∼ 10 − 21 erg ) 2 u † , 2 � ( ω nucl 2 � ( ω nucl ˙ ˆ ˆ / s G G 2 f 0 In M =1g, the # of d.o.f. N ∼ 10 23 = ⇒ N ˙ ǫ ∼ 100erg / s: far too much! Refine DP-model: Spontaneous collapse for modes 1 / k ≫ λ only: − f 0 d ˆ d t = − i ρ � 1 � � ) 2 � π † ρ ]+ f 0 c 2 ℓ k 2 [ ˆ u † u † 2 � ( ω nucl f 0 [ ˆ π k , ˆ u k , ˆ ρ ] [ ˆ k , [ ˆ u k , ˆ ρ ]] k ˆ k ˆ G 2 � k 1 / k ≫ λ E.g.: λ =10 − 5 cm, # of d.o.f. N ∼ 10 14 = ǫ ∼ 10 − 7 erg ⇒ N ˙ / 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 ˆ ρ showed spontaneous DP-decoherence dominates at large scales reduced spontaneous heating in DP, kept macrosopic predictions spared spontaneous collapse stoch. eqs. for | ψ � 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