Centre of mass decoherence due to time dilation: paradoxical - PowerPoint PPT Presentation

Centre of mass decoherence due to time dilation: paradoxical frame-dependence Lajos Diósi Wigner Research Centre for Physics H-1525 Budapest 114, POB 49, Hungary 16 Sept 2016, Castiglioncello Acknowledgements go to: EU COST Action MP1209 ‘Thermodynamics in the quantum regime’

Two stories for one model Newtonian Equivalence Principle Relativistically: c.o.m. couples to internal d.o.f. C.o.m. positional decoherence due to g Frame-dependence of positional decoherence? Frame-dependence of positional decoherence! Summary: Pikovski et al. theory for pedestrians

Two stories for one model Effect: Positional decoherence of composite objects, ∝ g / c 2 . Pikovski-Zych-Costa-Brukner, Nature Phys. 11 , 668 (2015). ◮ Method: 1 / c 2 GR correction to composite object QM. ◮ Arguments: relativistic, semiclassical ◮ Claim: universal decoherence due to gravitational time dilation Same Hamiltonian, pedestrian story [L.D. arXiv:1507.05828]: ◮ Method: 1 / c 2 SR correction to composite object QM. ◮ Arguments: non-relativistic, exact dynamics ◮ Claim: frame-dependent decoherence due to 1 / c 2 coupling between c.o.m. and i.d.o.f. SR/GR arguments for frame-dependence: Bonder-Okun-Sudarski PRD92, 124050, (2015) Pang-Chen-Khalili PRL117, 090401 (2016)

Newtonian Equivalence Principle http://wigner.mta.hu/ ∼ diosi/tutorial/freefalltutor.pdf Free-Falling observer: g = 0. Laboratory observer: g = 9 . 81cm/s 2 . Example: center-of-mass (c.o.m.) motion of free mass m . p 2 � � Free-Falling: � x , � p ; H 0 = 2 m � P 2 Laboratory: � X , � � 2 m + mg � P ; H g = X ( X : vertical ) Canonical transformation: � � � � � � � igt 2 � img 2 t 3 / 6 U = exp − p / 2 exp imgt � x exp U † = � X = � � x � x − gt 2 / 2 U � U † = � P = � � p � U � p − mgt U † − i ˙ H g = � � U � H 0 � U � � U †

Relativistically: c.o.m. couples to internal d.o.f. Internal Hamiltonian � H i is additive: � 0 / g = � H 0 / g + � H tot H i . Special relativistic correction, try m → m + � H i / c 2 . p 2 � � + � H tot Free-Falling: � x , � p , � o i ; = H i 0 2 ( m + � H i / c 2 ) � P 2 Laboratory: � X , � P , � � +( m + � H i / c 2 ) g � X + � H tot O i ; = H i g 2 ( m + � H i / c 2 ) Canonical transformation � U (as before, just m → m + � c 2 ): H i / U † = � X = � � x � x − gt 2 / 2 pure kinematics, as before U � U † = � P = � � U � P � p − ( m + � H i / c 2 ) gt mixing i.d.o.f. to � p U † = exp ( ic − 2 gt � O i = � � o i � ic − 2 gt � U � H i � x ) � o i exp ( − H i � x ) mixing � x to i.d.o.f. U † = � Note: � U � H i � H i .

C.o.m. positional decoherence due to g � 2 m + g P 2 � X � � H i + � H tot = H i g c 2 A wonderful coupling betwen Laboratory c.o.m. � X and � H i . ρ tot = � ρ i = Z − 1 exp ( − β � If initial state � ρ cm ⊗ � ρ i where � H i ) , that’s typical system-bath situation, yields c.o.m. positional decoherence: ρ cm ( t ) | x 2 � ≈ e − 1 2 t 2 /τ 2 dec × � x 1 − 1 2 gt 2 | � 2 gt 2 � � x 1 | � ρ cm ( 0 ) | x 2 − 1 � 1 g decoherence rate: = k B CT | x 1 − x 2 | . τ dec � c 2 m = 1 µ g, C = 10 − 5 cal/K, T = 300K, x 1 − x 2 = 1 µ m: ⇒ τ dec ∼ 1ms. ◮ Positional decoherence ∝ g in Laboratory frame ◮ No positional decoherence in Free-Fall frame

Frame-dependence of positional decoherence? Hm ..., that’s counterintuitive. 2 gt 2 � If | x 1 � + | x 2 � decays in the Laboratory and | X � = | x − 1 then in the Free-Fall frame | X 1 � + | X 2 � should, too, decay. 2 gt 2 � . This argument is just false: | X � � = | x − 1 No closed map exists between Laboratory eigenstates | x � and Free-Fall eigenstates | X � ! Why: U † = � X = � � x � x − gt 2 / 2 pure kinematics U � U † = � P = � � U � P � p − ( m + � H i / c 2 ) gt mixing i.d.o.f. to � p C.o.m. generic observables are frame-dependent. Split H cm ⊗ H i is frame-dependent. Hilbert space H cm is frame-dependent. You don’t expect this. It is just so if you start with p 2 � � + � H tot FF = H i 2 ( m + � H i / c 2 ) and change for Laboratory frame, or vice versa.

Frame-dependence of positional decoherence! Yes! In Earth gravity g : ◮ Free-Falling screen detects no decoherence ◮ Laboratory (fixed) screen detects positional decoherence In gravity-free ( g = 0 ) frame: ◮ Static screen detects no decoherence ◮ Accelerated screen detects positional decoherence Lucid proof: Pang-Chen-Khalili [PRL 117, 090401 (2016)]: Fringes shifted ∝ arrival time: � p ( x 1 − x 2 ) / L � �� x Lm cos x screen − v screen p � v x p c 2 . m is random since m → m + H i / 2 x Visibility supressed ∝ v screen . 1 Choice v screen = gt recovers τ dec screen just like in Earth’s Laboratory L frame.

Summary: Pikovski et al. theory for pedestrians Pedestrian = non-relativistic thinker, sees different depths. i) SR (not GR) correction to standard Hamiltonian: p 2 � � + � H = H i 2 ( m + � H i / c 2 ) A piece of SR, but no Lorentz inv., no general cov. ii) Exact Galilean inv. and Newtonian Equivalence Principle. iii) We can interpret everything in non-relativistic terms - plus the fact that m contains the correction � H i / c 2 . iv) Positional decoherence is missing in inertial frames. It emerges in accelerating frames only. v) Moving ( v ≪ c ) detector sees different interference fringes, accelerating detector sees same fringe as static one in gravity. With these pedestrian lessons can we put the theory back to SR/GR context (and re-attribute positional decoherence to time dilation).