Quantum Damping or Decoherence A Short Review

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Roy Wilkerson
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1 Quantum Damping or Decoherence A Short Review

2 Too Good to Be True Back in the 1970 s we were looking for neutral corrents.. e -proton/nucleus parity violating weak interactions [1]. Differences between L and R chiral isomers? Our guesstimate: ɛ < ev. Or ( 1/1sec) Spectroscopy hopeless But an neat idea: Analogy to K o strangeness oscillations: L-R oscillations H = ( ɛ δ δ ɛ ) δ = tunneling energy ɛ = parity violating energy

3 There exist chiral molecules with δ < 1/sec (In fact most)

4 Tunneling problem L-R, energy splitting δ Both potential wells exactly same if parity conserved. ɛ = 0. Want to see tiny ɛ Easy...

5 Can make ɛ/δ big. Soo easy to measure << ev?????

6 Why not: Decoherence Molecule must hit something in one sec!! How to Treat? Does a collision Stop the Clock or what?? What does oscillation plot become?

7 Introduce density matrix ρ For two-state system [2],[3] ρ = 1/2(1 + P σ), L, R two basis states of a spin, + >,. > Internal evolution (tunneling L,R + parity viol.) H = σ V ρ = i[h, ρ] gives Ṗ = V P P = 1 molecule: V z = ɛ V x = δ

8 Imbed in a big system (x 1...x n ) Ψ = α + > φ + (x 1...x n ) + β > φ (x 1...x n ), If φ + = φ two states of spin perfectly coherent. If < φ + φ >= 0 states of spin cannot interfere. That is, density matrix for spin obtained by not observing (x 1...x n ) is ρ = ( α 2 < φ + φ + > αβ ) < φ + φ > α β< φ φ + > β 2 < φ φ > or ρ = ( how much + (coherence between +, ) coherence between +, how much )

9 The Decherence Rate D [2] Starting with a pure state φ + = φ That is, Spin and surroundings uncorrelated Decreasing < φ + φ > as particles of environment interact with spin. Consider one collision, S-matrix makes initial wavefunction final wavefunction < φ + φ > < φ + S + S φ > Find (slow moving internal state, fast interaction with environment) dρ + dt = Dρ +

10 D = (flux) Im i < i (1 S + S ) i > The Unitarity Deficit - Interesting limits S + = S D = 0, no decoherence Must measure to get decoh. Not all collisions stop the clock LHe example or S = 1 D = 1/2 scattering rate on (+) (Use optical theorem.) D is Im dissipative quanity. Real part also has meaning level shift. Be suspicious of claims where it s not an Im.

11 Final equation for ρ = 1 2 (1 + P σ) Ṗ = P V D P T P T direction chosen by environment, Usually perp. to quality (Assuming environment conserves P z, no barrier hopping )[5] Rotation and shrinking of P, P < 1 implying decoherence. So, knowing V and D can find evolution of quality P z (chirality of molecule, neutrino flavor, value of q-bit... )

12 For V and D comparable: Pz Time Non-zero D, loss of info. For example d(entropy) d( T r[ρlnρ]) = = +DP P T dt dt (small P approx.) Is positive. Arrow of Time is correct[4]. Mis-match of two Unitary Ops is intrinsically positive

13 The Paradoxes Hund s: Why do we see L,R molecules in real life when ground state is 1 2 ( L > ± R >)? Hund said Long tunnel time (not always true, see pg 3) We could say strong parity violation ɛ = V z >> δ = V x Not always Real answer Decoherence: Environment fixes state Solution for Ṗ when D >> V P z = e (V 2 /D)t Big D inhibits relaxation keeps shrinking P T. Collisions interrupt tunneling! Turing s: If measurement causes collapse of the wavefunction, then keep meauring and system will never move?!(aka Watched Pot Effect ) True. Is property of Ṗ eqtn: P z = e (V 2 /D)t means quality never (very slowly) changes for big D.

14 Big D causing initial loss of length followed by Watched Pot

15 Why do outside influences stop tunneling? One nice way of understanding (suggested to me by M. Berry): Tunneling requires near degeneracy of the two states. If these are constantly being moved up-and-down, degeneracy always being lifted. This can be used to answer the worst Paradox : Decay Paradox: Observe if a particle, nucleus, atomic level... has decayed. It is frozen in original, undecayed???? Crazy, but seems to follow from Turing s Pdx. But... Unlike our two-level problem, in decay we have a transition to the continuum, with many (infinite number of) levels involved. If a continuum level is moved away from degeneracy, another one takes it s place.

16 Neutrino Mixing Inhibition of mixing can influence progress of Supernova. Real physical force. Also affects behavior of neutrinos in Early Universe[6][7].

17 Quantum Logic Devices, Quantum-Classical Transition For the Quantum Computer decoherence is the most important problem. Need to engineer[8] this and measure relevant D s. φ = flux in SQUID

18 Adiabatic passage for NOT (0 1) Wiggling Wells with noise term makes decoherence Quantum-Classical transition: Fast enough, deco unimportant, have tunneling Too slow, D blocks tunneling, have classical Not too fast, must remain adiabitic Can measure D this way [9]

19 Decoherence Kills the Clock To have a clock Something must move, oscillate. Single state ψ e iet doesn t oscillate. ψ 2 1 Need two states: α e im 1t K 1 > +β e im 2t K 2 > Now ψ cos[(m 1 m 2 )t](the K o clock) Gravity interacts differently with m 1, m 2, decohere s them. (They get different phase factors in a gravtl field) Estimate D for high T near the Planck scale [10] ( ) 2 D T 3 (G M) 2 = T 3 m1 m 2 Mplanck 2 (Can do same for ruler, range of momenta needed to define position)

20 - Some Questions - Decoherence-Fluctuation Relation? D is a dissipative parameter like Resistance in electricty. (Except here not energy but order or info. is being dissipated.) There is the famous Fluctuation-Dissipation theorem, e.g. Johnson noise ( current flucutuations RT ). Exists analog for D? What is conjugate quantity maybe measurement fltns? [11],[12] Noise always equivalent to decoherence? To model our quantum logic devices with decoherence we took the Hamiltonian for the given device (rf Squids) and solved the Shroedinger Eq. many times, each time with a different small random noise signal added H. Then averaged to get the density matrix[5][8]. ρ = ψ ψ

21 Evidently a mixed state. Same as tracing over an external environment? Or under what conditions or approximations is it equivalent? Beyond two levels Would like to extend to multilevels, continuum. Good luck[3].

22 References [1] Quantum Beats in Optical Activity and Weak Interactions, R.A. Harris and L. Stodolsky, Phys Let B78 (1978) 313. [2] Two Level Systems in Media and Turing s Paradox, R.A. Harris and L. Stodolsky, Phys. Let. B 116(1982)464. [3] Quantum Damping and Its Paradoxes, L. Stodolsky, in Quantum Coherence, pg 320 J. S. Anandan ed. World Scientific, Singapore (1990). [4] G. Sigl investigated the D equation more formally (Munich thesis LMU). Some of the results are in Non-Abelian Boltzmann Equation for Mixing and Decoherence G. Raffelt, G. Sigl and L. Stodolsky Phys Rev Let. 70(1993)2363. [5] More general choices for D are give in Studies in a Random Noise Model of Decoherence, with P. Korcyl, and J.Wosiek, arxiv: Quantum Information Processing, Vol. 10, (2011) , DOI: /s

23 [6] On the Treatment of Neutrino Oscillations in a Thermal Environment, L. Stodolsky, Phys. Rev. D 36(1987)2273 and chapter 9 of G.G. Raffelt, Stars as Laboratories for Fundamental Physics (Univ. Chicago Press, 1996). [7] See papers by G.G. Raffelt, and collaborators. Interesting non-linear problems arise when neutrinos create their own index of refraction. [8] Simulations of some quantum gates, with decoherence; with V. Corato, P. Silvestrini, A. Görlich, P. Korcyl, and J.Wosiek arxiv: cond-mat/ Virtual jnl supercondty ( Phys. Rev.B (2007) DOI: /PhysRevB [9] Demonstration of Macroscopic Coherence and Decoherence by Adiabatic Inversion, application to the SQUID, Paolo Silvestrini and L. Stodolsky arxiv:cond mat/ , Physics Letters A (2001) [10] Decoherence Rate of Mass Superpositions, L. Stodo sky, Acta Physica Polonica B27,1915(1996).

24 [11] Decoherence-Fluctuation Relation and Measurement Noise, Physics Reports (1999), (Okun Festschrift ) arxiv:quant-ph/ [12] Measurement Process In a Variable-Barrier System, arxiv:quant-ph/ Phys. Lett. B459 pages , (1999). See also reviews: Quantum Damping and Its Paradoxes, in Quantum Coherence, pg 320 J. S. Anandan ed. World Scientific, Singapore (1990). Coherence and the Clock, Time and Matter, Venice August 2002, I. I. Bigi and M. Faessler, Eds. World Scientific, (2006), pg.117 arxiv:quant-ph/ XXII Solvay Conference on Physics, The Physics of Communication, (XXII Solvay Conference on Physics) I. Anto, V.A. Sadovnichy, and H.Walther, Eds. World Scientific, Singapore, (2003). arxiv: cond-mat/

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