Quantum decoherence

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In quantum mechanics, quantum decoherence is the process by which quantum systems in complex environments exhibit classical behavior. It occurs when a system interacts with its environment in such a way that different portions of its wavefunction can no longer interfere with each other. In the many-worlds interpretation of quantum mechanics, decoherence is responsible for the appearance of wavefunction collapse.

Contents 1 Superposition and entanglement 2 Decoherence can be rapid 3 Decoherence and measurement 4 Mathematics of decoherence 5 Mathematical details 6 Experimental observation of decoherence 7 External links 8 References

Superposition and entanglement

Decoherence occurs when a system loses phase coherence between different portions of its quantum mechanical state. It then no longer exhibits quantum interference between those portions (as might be seen in a double-slit experiment). Decoherence is caused by interactions with a second system which may be thought of as either "the environment" or as "a measuring device". In the latter view, the interactions may be considered to be quantum measurements. As a result of an interaction, the wave functions of the system and the measuring device become entangled with each other. Decoherence happens when different portions of the system's wavefunction become entangled in different ways with the measuring device. For two portions of the entangled system's state to interfere, the original system and the measuring device must both evolve into the same state. If the measuring device has many degrees of freedom, it is very unlikely for this to happen. As a consequence, the system behaves as a classical statistical ensemble of the different portions rather than as a single coherent quantum superposition of them. From the perspective of the measuring device, in each member of the ensemble the system appears to have collapsed onto a state with precise values for the measured attributes.

Decoherence can be rapid

Decoherence represents an extremely fast process for macroscopic objects, since these are interacting with many microscopic objects in their natural environment. The process explains why we tend not to observe quantum behaviour in everyday macroscopic objects despite their existing in a bath of air molecules and photons. It also explains why we do see classical fields from the properties of the interaction between matter and radiation.

Decoherence and measurement

The discontinuous "wave function collapse" postulated in the Copenhagen interpretation to enable the theory to be related to the results of laboratory measurements is now to a large extent describable within the normal dynamics of quantum mechanics via the decoherence process. Consequently, decoherence is an important part of the modern version of the Copenhagen interpretation, based on Consistent Histories. Decoherence shows how a macroscopic system interacting with a lot of microscopic systems (e.g. collisions with air molecules or photons) moves from being in a pure quantum state—which in general will be a coherent superposition (see Schrödinger's cat)—to being in an incoherent mixture of these states. The population of the mixture in case of measurement is exactly that which gives the probabilities of the different results of such a measurement. However, decoherence does not give a complete solution of the measurement problem, since all components of the wave function still exist in a global superposition. Decoherence explains why these coherences are no longer available for local observers.

Mathematics of decoherence

Mathematically, the process results in the off diagonal elements of the density matrix or state operator of the system vanishing very quickly in a basis, which is usually defined by the interaction Hamiltonian between a system and its environment. Technically, the states of the environment are "averaged over".

Decoherence represents a major problem for the practical realization of quantum computers, since these heavily rely on undisturbed evolution of quantum coherences.

Mathematical details

Let's assume for the moment the system in question consists of a subsystem being studied, A and the "environment" E, and the total Hilbert space is the tensor product of a Hilbert space describing A, H_A and a Hilbert space describing E, H_E: that is,

<math>H=H_A\otimes H_E</math>.

This is a reasonably good approximation in the case where A and E are relatively independent (e.g. we don't have things like parts of A mixing with parts of E or vice versa). The point is, the interaction with the environment is for all practical purposes unavoidable (e.g. even a single excited atom in a vacuum would emit a photon which would then go off). Let's say this interaction is described by a unitary transformation U acting upon H. Assume the initial state of the environment is <math>|\mathrm{in}\rangle</math> and the initial state of A is the superposition state

<math> c_1 | \psi_1 \rangle + c_2|\psi_2\rangle </math>

where |ψ₁> and |ψ₂> are orthogonal and there is no entanglement initially. Also, choose an orthonormal basis for H_A, <math> \{ |e_i\rangle \}_i</math>. (This could be a "continuously indexed basis" or a mixture of continuous and discrete indexes, in which case we would have to use a rigged Hilbert space and be more careful about what we mean by orthonormal but that's an inessential detail for expository purposes.) Then, we can expand

<math>U(|\psi_1\rangle\otimes|\mathrm{in}\rangle)</math>

and

<math>U(|\psi_2\rangle\otimes|\mathrm{in}\rangle)</math>

uniquely as

<math>\sum_i |e_i\rangle\otimes|f_{1i}\rangle</math>

and

<math>\sum_i |e_i\rangle\otimes|f_{2i}\rangle</math>

respectively uniquely. One thing to realize is that the environment contains a huge number of degrees of freedom, a good number of them interacting with each other all the time. This makes the following assumption reasonable in a handwaving way, which can be shown to be true in some simple toy models. Assume that there exists a basis for H_A such that <math>|f_{1i}\rangle</math> and <math>|f_{1j}\rangle</math> are all approximally orthogonal to a good degree if i is not j and the same thing for <math>|f_{2i}\rangle</math> and <math>|f_{2j}\rangle</math> and also <math>|f_{1i}\rangle</math> and <math>|f_{2j}\rangle</math> for any i and j (the decoherence property).

This often turns out to be true (as a reasonable conjecture) in the position basis because how A interacts with the environment would often depend critically upon the position of the objects in A. Then, if we take the partial trace over the environment, we'd find the density state is approximately described by

<math>\sum_i (\langle f_{1i}|f_{1i}\rangle+\langle f_{2i}|f_{2i}\rangle)|e_i\rangle\langle e_i|</math>

(i.e. we have a diagonal mixed state and there is no constructive or destructive interference and the "probabilities" add up classically). The time it takes for U(t) (the unitary operator as a function of time) to display the decoherence property is called the decoherence time.

Experimental observation of decoherence

The collapse of a quantum superposition into a single definite state was quantitatively measured for the first time by Haroche and his co-workers at the Ecole Normale Superieure in Paris in 1996 [1]. Their approach involved sending individual rubidium atoms, each in a superposition of two states, through a microwave-filled cavity. The two quantum states both cause shifts in the phase of the microwave field, but by different amounts, so that the field itself is also put into a superposition of two states. As the cavity field exchanges energy with its surroundings, however, its superposition collapses into a single definite state.

Haroche and his colleagues measured the resulting decoherence via correlations between the energy levels of pairs of atoms sent through the cavity with various time delays between the atoms.

External links

• http://www.decoherence.info
• http://plato.stanford.edu/entries/qm-decoherence/
• Decoherence, the measurement problem, and interpretations of quantum mechanics
• Measurements and Decoherence

References

• {{cite book

| first = R. | last = Omnes
| year = 1999
| title = Understanding Quantum Mechanics
| location = Princeton
| publisher = Princeton University Press
}}

• {{cite book

| first = E. | last = Joos
| coauthors = et al.
| year = 2003
| title = Decoherence and the Appearance of a Classical World in Quantum Theory
| edition = 2nd edition
| location = Berlin
| publisher = Springer
}}

• Zurek, Wojciech H. (2003). "Decoherence and the transition from quantum to classical — REVISITED", Template:Arxiv (An updated version of PHYSICS TODAY, 44:36-44 (1991) article)
• {{cite journal

| authorlink = Wojciech H. Zurek
| first = Wojciech H. | last = Zurek
| year = 2003
| title = "Decoherence, einselection, and the quantum origins of the classical"
| journal = Reviews of Modern Physics
| volume = 75
| issue = 715
| id = Template:Arxiv
}}

• {{cite journal

| first = Maximilian | last = Schlosshauer
| year = 23 February 2005
| title = "Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics"
| journal = Reviews of Modern Physics
| volume=76(2004)
| pages = 1267–1305 
| id= Template:Arxiv, Template:Doi
}}de:Dekohärenz

pl:Dekoherencja kwantowa zh:量子脫散