[Other parts in this series: 1,2,3,4,5,6,7,8.]
A common mistake made by folks newly exposed to the concept of decoherence is to conflate the Schmidt basis with the pointer basis induced by decoherence.
[Show refresher on Schmidt decompsition]
Given any two quantum systems
and a pure joint state
, there always exists a Schmidt decomposition of the form
where and are local orthonormal Schmidt bases on and , respectively.
Now, any state in such a joint Hilbert space can be expressed as for arbitrary fixed orthonormal bases and . What makes the Schmidt decomposition non-trivial is that it has only a single index rather than two indices and . (In particular, this means that the Schmidt decomposition constains at most non-vanishing terms, even if .) The price paid is that the Schmidt bases, and , depend on the state .
When the values in the Schmidt decomposition are non-degenerate, the local bases are unique up to a phase. As evolves in time, this decomposition is defined for each time . The bases and evolve along with it, and can be considered to be a property of the state . In fact, they correspond to the eigenvectors of the respective reduced density matrices of and .
In the ideal case of so-called pure decoherence, the environment begins in an initial state and is coupled to the system through a unitary of the form
with as , where is a conditional unitary on and . The elements of the density matrix of the system evolve as , i.e.… [continue reading]
[Other parts in this series: 1,2,3,4,5,6,7,8.]
Although it is possible to use the term “vacuum fluctuations” in a consistent manner, referring to well-defined phenomena, people are usually way too sloppy. Most physicists never think clearly about quantum measurements, so the term is widely misunderstood and should be avoided if possible. Maybe the most dangerous result of this is the confident, unexplained use of this term by experienced physicists talking to students; it has the awful effect of giving these student the impression that their inevitable confusion is normal and not indicative of deep misunderstandinga .
Here is everything you need to know:
- A measurement is specified by a basis, not by an observable. (If you demand to think in terms of observables, just replace “measurement basis” with “eigenbasis of the measured observable” in everything that follows.)
- Real-life processes amplify microscopic phenomena to macroscopic scales all the time, thereby effectively performing a quantum measurement. (This includes inducing the implied wave-function collapse). These do not need to involve a physicist in a lab, but the basis being measured must be an orthogonal one.b
- “Quantum fluctuations” are when any measurement (whether involving a human or not) is made in a basis which doesn’t commute with the initial state of the system.
- A “vacuum fluctuation” is when the ground state of a system is measured in a basis that does not include the ground state; it’s merely a special case of a quantum fluctuation.
… [continue reading]
The Master equation in Lindblad form (aka the Lindblad equation) is the most general possible evolution of an open quantum system that is Markovian and time-homogeneous. Markovian means that the way in which the density matrix evolves is determined completely by the current density matrix. This is the assumption that there are no memory effects, i.e. that the environment does not store information about earlier state of the system that can influence the system in the future.a Time-homogeneous just means that the rule for stochastically evolving the system from one time to the next is the same for all times.
Given an arbitrary orthonormal basis of the space of operators on the -dimensional Hilbert space of the system (according to the Hilbert-Schmidt inner product ), the Lindblad equation takes the following form:
with .… [continue reading]
[This post was originally “Part 1” of my HTTAQM series. However, it’s old, haphazardly written, and not a good starting point. Therefore, I’ve removed it from that series, which now begins with “Measurements are about bases”. Other parts are here: 1,2,3,4,5,6,7,8. I hope to re-write this post in the future.]
It’s often remarked that the Aharonov–Bohm (AB) effect says something profound about the “reality” of potentials in quantum mechanics. In one version of the relevant experiment, charged particles are made to travel coherently along two alternate paths, such as in a Mach-Zehnder interferometer. At the experimenter’s discretion, an external electromagnetic potential (either vector or scalar) can be applied so that the two paths are at different potentials yet still experience zero magnetic and electric field. The paths are recombined, and the size of the potential difference determines the phase of the interference pattern. The effect is often interpreted as a demonstration that the electromagnetic potential is physically “real”, rather than just a useful mathematical concept.
The magnetic Aharanov-Bohm effect. The wavepacket of an electron approaches from the left and is split coherently over two paths, L and R. The red solenoid in between contains magnetic flux
. The region outside the solenoid has zero field, but there is a non-zero curl to the vector potential as measured along the two paths. The relative phase between the L and R wavepackets is given by
However, Vaidman recently pointed out that this is a mistaken interpretation which is an artifact of the semi-classical approximation used to describe the AB effect.… [continue reading]
[Added 2015-1-30: The paper is now in print and has appeared in the popular press.]
One criticism I’ve had to address when proselytizing the indisputable charms of using decoherence detection methods to look at low-mass dark matter (DM) is this: I’ve never produced a concrete model that would be tested. My analysis (arXiv:1212.3061) addressed the possibility of using matter interferometry to rule out a large class of dark matter models characterized by a certain range for the DM mass and the nucleon-scattering cross section. However, I never constructed an explicit model as a representative of this class to demonstrate in detail that it was compatible with all existing observational evidence. This is a large and complicated task, and not something I could accomplish on my own.
I tried hard to find an existing model in the literature that met my requirements, but without luck. So I had to argue (with referees and with others) that this was properly beyond the scope of my work, and that the idea was interesting enough to warrant publication without a model. This ultimately was successful, but it was an uphill battle. Among other things, I pointed out that new experimental concepts can inspire theoretical work, so it is important that they be disseminated.
I’m thrilled to say this paid off in spades. Bateman, McHardy, Merle, Morris, and Ulbricht have posted their new pre-print “On the Existence of Low-Mass Dark Matter and its Direct Detection” (arXiv:1405.5536). Here is the abstract:
Dark Matter (DM) is an elusive form of matter which has been postulated to explain astronomical observations through its gravitational effects on stars and galaxies, gravitational lensing of light around these, and through its imprint on the Cosmic Microwave Background (CMB).
… [continue reading]
I asked a question back in November on Physics.StackExchange, but that didn’t attract any interest from anyone. I started thinking about it again recently and figured out a good solution. The question and answer are explained below.a
Q: Is there a good notion of a “diagonal” operator with respect the overcomplete basis of coherent states?
A: Yes. The operators that are “coherent-state diagonal” are those that have a smooth Glauber–Sudarshan P transform.
The primary motivation for this question is to get a clean mathematical condition for diagonality (presumably with a notion of “approximately diagonal”) for the density matrix of a system of a continuous degree of freedom being decohered. More generally, one might like to know the intuitive sense in which , , and are all approximately diagonal in the basis of wavepackets, but is not, where is the unitary operator which maps
(This operator creates a Schrodinger’s cat state by reflecting about .)
For two different coherent states and , we want to require an approximately diagonal operator to satisfy , but we only want to do this if . For , we sensibly expect to be within the eigenspectrum of .
One might consider the negativity of the Wigner-Weyl transformb of the density matrix (i.e. the Wigner phase-space quasi-probability distribution aka the Wigner function) as a sign of quantum coherence, since it is known that coherent superpositions (which are clearly not diagonal in the coherent state basis) have negative oscillations that mark the superposition, and also that these oscillations are destroyed by decoherence.… [continue reading]