Relational time and intrinsic decoherence

Transcription

1 Relational time and intrinsic decoherence G. J. Milburn David Poulin Department of Physics, The University of Queensland, QLD 4072 Australia. 1

2 Quantum state of the universe. Page & Wooters,Phys. Rev Unruh &Wald, Phys. Rev. D The state of the universe should be stationary. It cannot depend on changes in coordinate time. The value of ǫ is unknown, Ĥ ǫ = ǫ ǫ ρ U = ǫ p ǫ ǫ ǫ p ǫ arbitrary. 2

3 Quantum state of the universe. Introduce an un-physical coordinate time through a group average 1 ρ U = lim T T 0 e iĥt Ψ Ψ e iĥt Ψ = ǫ c ǫ ǫ p ǫ = c ǫ 2 c ǫ arbitrary. 3

4 The two spring universe. Ĥ = ω 1 a 1a 1 + ω 2 a 2a 2 a ia i n i = n n i, n = 0, 1, 2,... ǫ = n 1 m 2 ǫ = ω 1 n + ω 2 m 4

5 The two spring universe. Scale energy in units of ω 1 : ǫ = n + ω 2 m where ǫ = ǫ/ω 1, ω 2 = ω 2 /ω 1. assume ω 1,2 and ǫ are commensurate. ω 2 = 1 N, ǫ = M 5

6 The two spring universe. N,M are integers and n,m are related by m = N(M n) ǫ M : N = M n=0 c n n 1 N(M n) 2 Bayesian reasoning suggests state of universe: ρ N = M p M M : N M : N 6

7 Phase difference. The state M : N = 1 M n 1 N(M n) 2 M + 1 n=0 is a near eigenstate of the phase difference operator: ˆΦ = T 1 (ˆΦ 1 N ˆΦ 2 ) which commutes with the total Hamiltonian. What is the phase operator? 7

8 Phase estimation problem. Susskind &Glowgower, 1964: Holevo,1982 Optimal phase estimation, p(φ θ) = tr(e iθa a ρe iθa a Ê(φ)) Projection Operator Valued Measure: Ê(φ)dφ Ê(φ) = φ φ φ = n=0 e inφ n 8

9 Canonical phase. Define, ˆP = 2π 0 e iφ Ê(φ)dφ Canonical displacement operator for number ˆP n = n + 1 Define a phase operator, ˆΦ = 2π 0 φê(φ)dφ 9

10 Joint phase distribution. ) P(φ 1,φ 2 ) = tr (Ê1 (φ 1 ) Ê2(φ 2 ) M : N M : N P(φ 1,φ 2 ) = 1 M e i(m+1)(φ 1 Nφ 2 ) 1 e i(φ Nφ 2 ) When M >> 1 this is sharply peaked at 2 φ 1 Nφ 2 = 2πk k integer M : N is a near eigenstate of the phase difference operator. 10

11 Poincaré section. Select data sets for which φ 1 = ω 2 = 15 ω oscillator 2-1 oscillator 1 11

12 Coordinate time? Use group average representation: ρ U 1 = lim Ω Ω = lim Ω Ω + 1 Ω α=0 Ω α=0 e iαĥ ( Ψ Ψ ) e iαĥ e iαĥ ( ψ 1 ψ ξ 2 ξ ) e iαĥ Ψ is arbitrary, take as an initial state of the two. Ψ = ψ 1 ξ 2 12

13 Good clocks. Take ψ 1 as the clock state. We would like it to evolve rapidly compared to everything else. ψ (a 1a 1 ) 2 ψ ( ψ a 1a 1 ψ ) 2 >> 1 need c n 1 n ψ 1 almost independent of n over some large range K n K + L with L >> 1. 13

14 Conditional states. select clock state for measurements of ˆΦ 1 with result φ. ρ (φ) 2 = tr 1,2 ( φ 1 φ I 2 ρ U ) ρ (φ) 2 = Ω α=0 P(φ α)e i α N a 2 a 2 ξ 2 ξ e i α N a 2 a 2 P(φ α) = [p(φ)] 1 2 c n e i(α φ)n n=0 G.J.Milburn, Phys. Rev. A44, (1991); R. Gambini, R. A. Porto and J. Pullin, New J. Physics, 6, 45 (2004). 14

15 Conditional states. ρ (φ) 2 = Ω α=0 P(φ α)e i α N a 2 a 2 ξ 2 ξ e i α N a 2 a 2 for a good clock, P(φ α) is sharply peaked at α = φ. ρ (φ) 2 e i φ N a 2 a 2 ξ 2 ξ e i φ N a 2 a 2 change variable: φ = ω 1 t and use ω 2 /ω 1 = N ρ (φ) 2 ρ 2 (t) e iω 2ta 2 a 2 ξ 2 ξ e iω 2ta 2 a 2 corrections intrinsic decoherence. 15

16 Intrinsic decoherence Parameterise the distribution P(φ 1 α) to implicitly define the clock state. Make the Poisson choice: P(φ 1 α) = ( γφ 1) α α! e γφ 1 change of variables: ω 1 t = φ 1, γ = γ/ω 1 P(t α) = (γt)α α! e γt 16

17 Intrinsic decoherence ( dρ 2 (t) = γ e ia 2 a2/γ ρ 2 (t)e ia 2 a2/γ ) ρ 2 (t) dt Stationary states remain stationary. Constants of motion remain constant. Expect γ >> 1 First order correction to Schrödinger dynamics: dρ 2 (t) dt = i[a 2a 2,ρ 2 (t)] 1 2γ [a 2a 2, [a 2a 2,ρ 2 (t)]]

18 Decoherence. See GJM. Phys. Rev. 44, 5401, (1991) γ >> 1 dρ(t) dt dρ(t) dt = γ [ e iĥ/γ ρ(t)e iĥ/γ ] ρ(t) i[ĥ,ρ(t)] 1 [Ĥ, [Ĥ,ρ(t)]] 2γ Decay of coherence in energy basis: ρ ǫ,ǫ = ǫ ρ ǫ ρ ǫ,ǫ t = i(ǫ ǫ)ρ ǫ,ǫ 1 2γ (ǫ ǫ) 2 ρ ǫ,ǫ 18

19 Estimation problem for time: clocks. See GJM. Phys. Rev. 44, 5401, (1991) dρ(t) dt = γ [ e iĥ/γ ρ(t)e iĥ/γ ] ρ(t) Uncertainty in parameter estimation: ψ t=0 = 1 2 ( E 1 + E 2 ) 2 E variance in energy. δt E e 2 E t 2γ 19

20 Temporal translations δt t Clocks age! 20

21 Frequency cutoffs. Simple harmonic oscillator: H = ω 0 a a Average amplitude: a(t) = a(0) exp [ γt(1 e iω0/γ ) ] When ω 0 /γ << 1: a(t) a(0) e iω 0t ω0t/2γ 2 If ω 0 = 2nπγ, a(t) = a(0) 21

22 Lorentz invariant uncertainty principle. Braunstein et al. Ann. Phys. 247, 135 (1996) Objective: estimate a spacetime translation using a quantum field. Generator: energy-momentum four vector ˆP = ˆP α e α = ˆP 0 e 0 + ˆ P = ˆP 0 e 0 + ˆP j e j. The spacetime translation: X = Sn = Sn α e α with n = n 0 e 0 + n 22

23 ψ S = e isn ˆP/ h ψ 0 n ˆP = η αβ n α ˆP β = n α ˆPα = n 0 ˆP0 + n ˆ P (δs) 2 S (n ˆP = (δs) 2 n α n β ˆP α ˆP β h2 4 When n is time like this is a time-energy uncertainty relation for the observer whose 4-velocity is n, when n is space-like, this is a position-momentum uncertainty relation for an observer whose 4-velocity is orthogonal to n. 23

24 Electromagnetic field. ˆP = k,σ hkâ k,σ â k,σ where k = ωe 0 + k = ωe 0 + k j e j is a null wave 4-vector with ω = k = k and the sum is over all wave 3-vectors k and polarisation σ. a a is generator of phase shifts for EM field spatial and temporal translations reduce to phase estimation. 24

25 Lorentz invariant spacetime translations. ( dρ(s) ds = γ e in ˆP/γ ρ(s)e in ˆP/γ ) ρ(s) 25

26 Modified dispersion relations. Need a field with a non-zero average amplitude on a space-like hypersurface. Ê( x) = i k ω k (u k ( x)a k u k ( x) a k) Choose a coherent state: tr[a k ρ] = α k This is a semiclassical state with field amplitude on t = 0 E( x) = k iω k (u k ( x)α k u k ( x) α k) 26

27 Modified dispersion relations. Calculate the new amplitudes displaced proper time τ, E( x,τ) = tr[ê( x)ρ(τ)] Need α k (τ) = tr[a k ρ(τ)] = α k exp [ τγ(e ik/γ 1) ] 27

28 Write as α k (τ) = α k e iω(k)τ e Γ(k)τ where the observed frequency of this mode amplitude is ω(k) = γ sin(k/γ) and the amplitude decays due to intrinsic decoherence at the rate As γ Γ(k) = γ(cos(k/γ) 1) ω(k) = k(1 k2 6γ ) 28

29 Conclusion. Relational time can emerge in a timeless state. leads to an intrinsic decoherence mechanism while conserving energy and momentum modified uncertainty constraints generalise to a Lorentz invariant field formulation which leads to modified dispersion relations 29