4. LOS RETOS DE LA INDUSTRIA 0
4.2. La importancia de la innovación
distributions1[27].
2.1.2
The Collapse Time
The fact that it is impossible to define a Killing vector field for a superposition of two different spacetimes, makes it plausible that the superposition state is in fact unstable. In that case an estimate for its mean lifetime is given by Heisenberg’s uncertainty relation to be
τcoll=
h
E. (2.4)
This time can then be interpreted as an effective collapse time. At times greater then the mean lifetime the superposition state typically will have decayed into one of the two gravitationally stable component states.
None of the above considerations give us any clue as to how the collapse process would work in practice, nor do they give us any indication of which direction to take to even begin looking for a theory that might unite quantum mechanical and gravitational time evolution. However, if we assume for the moment that such a collapse process exists, and that it is caused by the incompatibility of superpositions and general covariance giving rise to an uncertainty in energy, then it is clear that the timescale at which the collapse should take place is of the order ofτcoll.
The main point in Penrose’s observation is that the size of this timescale is rather special. For microscopic particles it is enormously large. A simple estimate shows that a single proton can be expected to stay in a superposition state for at least a few million years, and thus its decay will never be observed. On the other hand the timescale becomes short for macroscopic bodies. Even a drop of water with a diameter of only a hundredth of a millimeter will not sustain a superposition for more than a millionth of a second, and it will thus for all practical purposes always seem to be in a classical state. The surprising observation is that the regime in between, where the mass is such that the collapse time becomes measurable, has not been experimentally explored at all yet and in fact seems to be just beyond the reach of what can currently be experimentally tested [28, 131]. That region though is the place to look if one wants to establish whether gravity has anything to do with wavefunction collapse.
2.2
The Schrödinger-Newton Equation
To theoretically study the possibility of gravitationally induced quantum collapse any further, one will need to introduce a specific model, which describes exactly
1Notice that this expression also includes an infinite self energy term that comes from the interaction
92 PART IV, CHAPTER 2. PENROSE’S OBSERVATION
how gravity alters the usual quantum mechanics. In particular, the model will have to provide a time evolution for the actual collapse process, and it will have to single out one specific basis into which classical (or macroscopic) bodies will collapse. In order to provide just such a basis, Penrose introduced the so called Schrödinger- Newton equation [26, 27, 29]
−h2
2m∇
2ψ+Uψ=Eψ
∇2U=4πGm2|ψ|2, (2.5)
whereψis the quantum mechanical wavefunction,Uis the gravitational self energy which acts as a potential energy, andEis the total energy eigenvalue.
The Schrödinger-Newton equation is a non-linear eigenvalue equation in which the expectation value of the wavefunction itself serves to generate the potential en- ergy which helps determine the eigenfunctions of the equation, and thus the stable wavefunctions of the system. This extended form of the Schrödinger equation has been studied in the spherically symmetric limit and a set of "bound state" solutions has been found [29]. What has not been considered yet is what this form of the total energy operator would imply for the actual time evolution of a microscopic system. Different forms of a time evolution operator involving gravity have been intro- duced by others [58–60]. Most of these approaches have the disadvantage that they need to introduce either a large number of randomly fluctuating fields or, equiv- alently, a random localization process. Since none of these randomly fluctuating fields has actually been observed, it is difficult to experimentally test or distinguish these different theories.
Before we turn to a discussion of the time evolution implied by the Schrödinger- Newton equation, we will first show in the following chapter that it is possible to make an experimental prediction based on Penrose’s ideas without invoking any specific scheme for the description of the exact time evolution.
93
Chapter 3
An Experimental Test
Assuming that gravity has something to do with the quantum collapse process, and assuming that this leads to a timescaleτcoll, as proposed by Penrose, it is possible
to make at least some experimental predictions without referring to any underlying description of the collapse process. In particular, we will consider in this chapter what the idea of having a gravitationally induced collapse would imply for the su- perposition state of counter rotating supercurrents in a flux-qubit.
3.1
The Flux Qubit
In recent years there has been an enormous experimental effort to create and con- trol quantum superposition states [36, 79, 80]. The pursuit is fueled by the hope that one day we’ll be able to control the quantum world with such accuracy that we could build a quantum computer. The concept of the quantum computer was con- ceived by various people already in the 1970’s, and popularized in 1981 by Richard Feynman [132]. After Peter Shor showed in 1994 that a quantum computer could be used to factorize large integers and thus potentially break encryption codes used by conventional computers [133], the race for creating a quantum computer in the lab broke loose. Many different proposals and realizations of single qubit systems have been studied since. One of the proposed setups for creating a qubit (i.e. a con- trollable quantum superposition) is to take a superconducting loop which contains a Josephson junction, and use a magnetic flux to put it into a superposition of a clockwise and a counterclockwise rotating supercurrent [78, 80, 81].
Because a supercurrent is a collective, coherent current of a macroscopic num- ber of electrons, the superposition state that is achieved in the flux qubit is a sort of macroscopic Schrödinger cat like state [78]. The combined weight of all the elec- trons in the supercurrent can in principle be made so large that gravitational effects
94 PART IV, CHAPTER 3. AN EXPERIMENTAL TEST
Figure 3.1: An STM image of the superconducting flux qubit used in Delft [80]. Supercurrent circulates both clockwise and anti-clockwise through the central ring. The ’obstructions’ in the ring are the Josephson junctions.
of the type envisioned by Penrose might become observable. If we have an indica- tion of the timescale on which to expect the onset of gravitational collapse, and a way to distinguish gravitational collapse from the usual processes of decoherence, then the flux qubit could prove to be a window through which we can study the quantum collapse process.
3.1.1
Trains and Wagons
In the proposal by Penrose the uncertainty in energy due to gravity equals the self energy of the difference of the mass distributions of the superposed states. If we were to apply that principle blindly to our scenario, then we would immediately be faced by an infinite lifetime of the superposition state in the flux qubit. After all, there may be a macroscopic current which is moving both clockwise and coun- terclockwise, but the mass distribution associated with that current is the same for both directions, and constant in time1. Thus, a uniform current would be allowed to flow in superposition forever. This certainly is not a property that we want to have in a quantum collapse theory. After all, the exact same reasoning could be used to argue that a sufficiently smooth soccer ball could be shot at the goal in a su- perposition of top and bottom spin, or that a sufficiently densely packed passenger train could ride a circular track in opposite directions at the same time.
The way out of this dilemma is in fact a very simple one: the supercurrents as a whole may have a vanishing difference of mass distributions, but if we consider
1There may be a relativistic correction to the uniform mass distributions associated with the different
framedraggings induced by the current but since this effect has a prefactor of orderv/c, it is neglected in the present discussion.
3.2. SELF ENERGY 95