Fundamental loss of quantum coherence related to gravity Jorge Pullin Jorge Pullin Horace Hearne Laboratory for Theoretical Physics Louisiana State University With Rodolfo Gambini (Uruguay) and Rafael Porto (UC Santa Barbara)
It is my guess that everyone who takes a course in quantum mechanics is surprised by the different way in which space and time are treated. ∂ Ψ − = Ψ i H ˆ � ∂ t Ψ < > x x , ˆ ˆ Clearly, Schrödinger’s equation can only be understood as an idealization. In the real world clocks are quantum mechanical and the variable we choose to call t will be associated with a quantum operator.
So how does one do quantum mechanics with real clocks? You do it “relationally”. First of all you choose some physical variable as your “clock”, let us call it T. Such variable will be represented by a quantum operator. Then you choose the variables that will describe the physical system under study. Generically we call them X. One then computes: ρ dt P t P t P t Tr ( ( ) ( ) ( )) ∫ < < >= >= < < >= >= P P X X x x T T t t = = ( ( | | ) ) x x T T T T 0 0 0 0 ρ dt P t Tr ( ( ) ) ∫ T That is, the conditional probability that X takes a value x 0 when T takes a value t 0 . Notice that in the right hand side we have the “ideal” t of Schrödinger’s theory. The density matrices (quantum states) evolve with the traditional Schrödinger equation, we only ask different questions about them than usual.
How does quantum evolution look like when one casts it in terms of T rather than t? Here one needs to make some assumptions. We assume that the density matrix can be written as a direct product of that of the clock and that of the system under study and that one has a unitary independent evolution for the clock and the system, We also define the probability density that the clock variable takes the value T when the ideal time takes the value t, the value T when the ideal time takes the value t, And define an evolution in terms of the variable T,
With these identifications we can rewrite the conditional probability as an ordinary probability in quantum mechanics for the density matrix ρ (T) To get something closer to the usual Schroedinger equation, we assume that the probability for the clock is quite peaked, assume that the probability for the clock is quite peaked, Which gives for the evolution,
So the differential equation that gives the time evolution of the density matrix is given by, ∂ ρ − = ˆ ρ + σ ˆ ˆ ρ + i � H T H H [ , ] ( )[ , [ , ]] ... ∂ T Where σ (T) is the rate of spread of the wavefunction of the clock. Where σ (T) is the rate of spread of the wavefunction of the clock. We have assumed one started with a clock in a quantum state such that the variable T has a distribution that is very peaked around t. In fact, the above expression is approximate, as the spreads increase one gets higher order terms with more commutators. Class.Quant.Grav.21:L51-L57,2004 New J.Phys.6:45,2004
What are the consequences of the extra term? If we assume σ is constant, the equation can be solved exactly and one gets that the density matrix in an energy eigen-basis evolves as, Where the omega’s are the Bohr frequencies associated with the eigenvalues of H. Therefore, the off-diagonal elements of the density matrix decay to zero exponentially, and pure states generically evolve into mixed states. Quantum mechanics with real clocks therefore does not have a unitary evolution.
The effect can be made arbitrarily large simply choosing “lousy clocks” to do physics. This is not usually done, but an interpretation of experiments with Rabi oscillations indicate the effect is there, R . Bonifacio, S . Olivares, P. Tombesi et. al., J. Mod. Optics, 47 2199 ( 2000) Can the effect be eliminated just by choosing better and better clocks? And if not, how much does reality depart from traditional quantum theory? To estimate this we have to ask ourselves the quantum theory? To estimate this we have to ask ourselves the question “what is the best clock we can build”?
This question was considered some time ago by Salecker and Wigner. They consider a clock consisting of two mirrors between which a light ray bounces back and between which a light ray bounces back and forth. Every bounce is a “tick” of the clock. They note that by the time the light bounced off a mirror and returns, the original mirror’s wave-function would have spread. The width of the spread limits the accuracy of the clock. t δ t ≈ = c = � ( 1 ) M
So if you want a better clock, make it more massive. But there is a catch: If you put too much mass your clock becomes a black hole! (Ng and Van Dam, Ann. NY Acad. Sci 755, 579 (1995)) A black hole is, in fact, the best clock you can have in this sense. How is a black hole a clock? Black holes have vibrational modes (quasi-normal modes). Although these modes are heavily damped, at least in principle they allow to think of the black hole as an oscillator. (Think of a bell ) The frequency of oscillation is inversely proportional to the mass of the black hole. Therefore smaller black holes make better clocks. This tension leads to a fundamental limit on how accurate a clock can be if you wish to measure a given time T max : Where t P is Planck’s time: 10 -44 s.
Now that we estimated what the best possible clock can be in nature, we can put a limit on the value of σ (T) in the equation we derived for quantum mechanics with real clocks: ∂ ρ − = ˆ ρ + σ ˆ ˆ ρ + i � H T H H [ , ] ( )[ , [ , ]] ... ∂ T With, So, we have argued that due to the fact that one cannot have a perfectly classical clock in nature, quantum mechanics needs to be modified and we have provided a quantitative estimate of the modification based on the best possible clocks one can construct at least in principle. The above equation conserves energy (good!) but implies that evolution is not unitary (interesting…).
So if we go back to the formula for the decoherence, with our estimate of σ (T), is the decoherence observable experimentally? For a two level system, we get, with the optimal black hole clock, For this effect to be observable, one needs a quantum coherent system with a large separation of energy levels and a long life. system with a large separation of energy levels and a long life. The “Schödinger cat” type experiments are the type of systems we wish to consider. The most promising experiments are Bose-Einstein condensates, (10 N atoms, 10 27-2N s). Simon, Jaksch, quant-ph/0406007 Phys. Rev. A70, 052104 (2004). Interestingly, if one punches in the numbers for LISA, the effect appears observable, but this is fallacious, since classical optical experiments do not measure the off-diagonal elements of the density matrix.
The black hole information paradox: In 1975 Hawking showed that black holes eventually evaporate. The only thing left at the end of the process is an outgoing purely thermal black body radiation. This raises the question of what happened to all the information that went into the creation of the black hole. Information loss is problematic in traditional quantum mechanics because evolution is unitary and this implies information is preserved. In particular a pure state will evolve into a pure state. However, we have argued in this talk that in quantum mechanics with real clocks evolution is not unitary. We also argued that the effect is very small. Could it be large enough to eliminate the black hole information puzzle?
A precise calculation is beyond the current possibilities, since a complete black hole evaporation would require full quantum gravity. To carry out a back-of-the-envelope calculation we make a naïve model of the black hole as a two level system with energy separation given by the temperature. Concretely, One can compute exactly the evolution of the density matrix for One can compute exactly the evolution of the density matrix for such a naïve model. The final result is that the density matrix is given by (approximately, in modulus) as, 2 M 3 ρ ≈ ρ T Planck | ( ) | | ( 0 ) | 12 max 12 M BH So for an astrophysical black hole the loss of coherence is of the order of 10 -28 . One could still argue that the puzzle still exists for smaller black holes…
So we are claiming that in real life one could have never observed the black hole information paradox, since quantum states decohere (or in other ways information is lost) due to our lack of perfect clocks at a rate faster than the one an evaporating black hole makes it disappear. The paradox can still be posed at the level of the “idealized” Schrödinger theory in terms of the perfectly classical time t. Schrödinger theory in terms of the perfectly classical time t. But if one adopts the point of view that the true physical theory is the one formulated in terms of T, then the paradox does not arise. R. Gambini, R. Porto, JP PRL 93, 240401 (2004)
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