Quantum decoherence

In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. Decoherence occurs when a system interacts with its environment, or any complex external system, in such a thermodynamically irreversible way that ensures different elements in the quantum superposition of the system+environment's wavefunction can no longer interfere with each other.

Decoherence does not provide a mechanism for the actual wave function collapse; the quantum nature of the system is simply "leaked" into the environment so that a total superposition of the wavefunction still exists, but exists beyond the realm of measurement; rather decoherence provides a mechanism for the appearance of wavefunction collapse.

Before an understanding of decoherence was developed the Copenhagen interpretation of quantum mechanics treated wavefunction collapse as a fundamental, a priori process. Decoherence provides an explanatory mechanism for the appearance of wavefunction collapse and was first developed by David Bohm in 1952 who applied it to Louis DeBroglie's pilot wave theory, producing Bohmian mechanics, the first "successful" hidden variables interpretation of quantum mechanics. Decoherence was then used by Hugh Everett in 1957 to form the core of his many-worlds interpretation . However decoherence was largely ignored for many years, and not until the 1980s /90s did decoherent-based explanations of the appearance of wavefunction collapse become popular, with the greater acceptance of the use of reduced density matrices. The range of decoherent interpretations have subsequently been extended around the idea, such as consistent histories. Some versions of the Copenhagen Interpretation have been rebranded to include decoherence.

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

Mechanisms

Decoherence isn't a new theoretical framework, but instead a set of new theoretical perspectives in which the environment is no longer ignored in modeling systems. To examine how decoherence operates we will present an "intuitive" model (which, alas, does require some familiarity with the basics of quantum theory) making analogies between visualisable vectors spaces and Hilbert spaces before presenting the conclusion (that decoherence destroys interference effects and the "quantum nature" of systems) in Dirac notation. Then the density matrix approach will be presented for perspective (there are many different ways of understanding decoherence).

"Lost in Hilbert-space" picture

A physical system is represented in quantum theory by a wavefunction, also known as a state vector, in a Hilbert space. Different previously isolated, non-interacting systems occupy different Hilbert spaces, or alternatively we can say they occupy different, lower-dimensional subspaces in the Hilbert space of the joint system. (The dimension of a system's Hilbert space is simply the number of degrees of freedom present (in non-relativistic models = 3 x the number of a system's "free" particles.)) When two systems (and the environment would be a system) start to interact, though, their associated state vectors are no longer constrained to the subspaces, but instead the combined state vector time-evolves a path through the "larger volume", whose dimensionality is the sum of the dimensions of the two subspaces. (Think, by analogy, of a square (2-d surface) extended by just one dimension (a line) to form a cube. The cube has a greater volume in, some sense, than its component square and line axes.) The relevance of this is that the extent that two vectors interfere with each other is a measure of how "close" they are to each other (formally, their overlap or scalar product together) in the Hilbert space. When a system couples to an external environment the dimensionality of, and hence "volume" available, to the joint state vector increases enormously -- each environmental degree of freedom contributes an extra dimension.

The original system's wavefunction can be expanded as a sum of elements in a quantum superposition, in a quite arbitrary way. Each expansion corresponds to a projection of the wave vector onto a basis, and the bases can chosen at will. Let us choose any expansion where the resulting elements interact with the environment in an element-specific way; such elements will -- with overwhelming probability -- be rapidly separated from eachother by their natural unitary time evolution along their own independent paths -- so much in fact that after a very short interaction there is almost no chance of any further interference and the process is effectively irreversible; the different elements effectively become "lost" from each other in the expanded Hilbert space created by the coupling with the environment. The elements of the original system are said to have decohered. The environment has effectively selected out those expansions or decompositions of the original state vector that decohere (or lose phase coherence) with each other. This is called "environmentally-induced-superselection", or einselection. The decohered elements of the system no longer exhibit quantum interference between each other, as might be seen in a double-slit experiment. Any elements that decohere from each other via environmental interactions are said to be quantum entangled with the environment. (Note the converse is not true: not all entangled states are decohered from each other.)

Any measuring device, in this model, acts as an environment since any measuring device or apparatus, at some stage along the measuring chain, has to be large enough to be read by humans; it must possess a very large number of hidden degrees of freedom. In effect 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 einselected elements of the entangled system's state to interfere, both the original system and the measuring in both elements device must significantly overlap, in the scaler product sense. As we have seen 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 elements rather than as a single coherent quantum superposition of them. From the perspective of each ensemble member's measuring device, the system appears to have irreversibly collapsed onto a state with a precise value for the measured attributes, relative to that element.

Dirac Notation

Let the system initially be in the state $|\psi \rangle$ where

$|\psi \rangle =\sum _{i}|i\rangle \langle i|\psi \rangle$ where the $|i\rangle$s form an einselected basis (environmentally induced selected eigen basis); and let the environment initially be in the state $|E\rangle$. The state of the total combined system & environment, before any interaction between the two subsystems, is therefore:

$|before\rangle =\sum _{i}|i\rangle |E\rangle \langle i|\psi \rangle$

The system can interact with its environment in three ways: either (1) the system loses its distinct identity and merges with the environment (e.g. photons in a cold, dark cavity get converted into molecular excitations within the cavity walls), or (2) the system is not disturbed at all, even though the environment is effected (e.g. the idealised non-disturbing measurement), or (3) the system is disturbed by the environment but maintains its identity to a degree. (3) can modelled as a mixture of (1) and (2), so we shall analyse the two extremal cases, (1) and (2). The latter case, (2), is of most interest.

In (1), where the environment absorbs the system, each element of the total system's basis interacts with the environment such that:

$|i\rangle |E\rangle$ evolves into $|E_{i}\rangle$

and so $|before\rangle$ evolves into

$|after\rangle =\sum _{i}|E_{i}\rangle \langle i|\psi \rangle$

where the unitarity of time-evolution demands that the total state basis remains orthonormal and in particular their scalar or inner products with each other vanish:

$\langle E_{i}|E_{j}\rangle =\delta _{ij}$ since $\langle i|j\rangle =\delta _{ij}$

Additionally decoherence requires that for each decomposition of

$|E_{i}\rangle =\sum _{k(i)}|k(i)\rangle \langle k(i)|E_{i}\rangle$ that:

$\langle k(i)|k'(i')\rangle \approx \delta _{kk'}\delta _{ii'}$

In (2), the idealised measurement/undisturbed system case, each element of the basis interacts with the environment such that:

$|i\rangle |E\rangle$ evolves into the product $|i,E_{i}\rangle =|i\rangle |E_{i}\rangle$

and so $|before\rangle$ evolves into

$|after\rangle =\sum _{i}|i,E_{i}\rangle \langle i|\psi \rangle$

where, again, unitarity demands that:

$\langle i,E_{i}|j,E_{j}\rangle =\delta _{ij}$

and additionally decoherence requires, by virtue of the large number of hidden degrees of freedom in the environment, that

$\langle E_{i}|E_{j}\rangle \approx \delta _{ij}$

Note that if the system basis $|i\rangle$ were not an einselected basis then the last condition is trivial since the disturbed environment is not a function of i and we have the trivial disturbed environment basis $|E_{j}\rangle =|E'\rangle$. This would correspond to the system basis being degenerate with respect the environmentally-defined-measurement-observable. For a complex environmental interaction (which would be expected for a typical macroscale interaction) a non-einselected basis would be hard to define.

Loss of Interference and the Transition from Quantum to Classical

The utility of decoherence lies in its application to the analysis of probabilities, before and after environmental interaction, and in particular to the vanishing of quantum interference terms after decoherence has occurred. If we ask what is the probability of observing the system making a transition or quantum leap from $\psi$ to $\phi$ before $\psi$ has interacted with its environment, then application of the Born probability rule states that the transition probability is the modulus squared of the scalar product of the two states:

$prob_{before}(\psi \rightarrow \phi )=|\langle \psi |\phi \rangle |^{2}=|\sum _{i}\psi _{i}^{*}\phi _{i}|^{2}=\sum _{i}|\psi _{i}^{*}\phi _{i}|^{2}+\sum _{ij;i\neq j}\psi _{i}^{*}\psi _{j}\phi _{j}^{*}\phi _{i}$

where $\psi _{i}=\langle i|\psi \rangle ,\psi _{i}^{*}=\langle \psi |i\rangle$ and $\phi _{i}=\langle i|\phi \rangle$ etc

Terms appear in the expansion of the transition probability above which involve $i\neq j$; these can be thought of as representing interference between the different basis elements or quantum alternatives. This is a purely quantum effect and represents the non-additivity of the probabilities of quantum alternatives.

To calculate the probability of observing the system making a quantum leap from $\psi$ to $\phi$ after $\psi$ has interacted with its environment, then application of the Born probability rule states we must sum over the all the relevant possible states of the environment, $E_{i}$, before squaring the modulus:

$prob_{after}(\psi \rightarrow \phi )=\sum _{i}|\langle after|\phi ,E_{i}\rangle |^{2}=\sum _{i}|\sum _{j}\psi _{j}^{*}\langle j,E_{j}|\phi ,E_{i}\rangle |^{2}$

The internal summation vanishes when we apply the decoherence condition $\langle E_{i}|E_{j}\rangle \approx \delta _{ij}$ and the formula simplifies to:

$prob_{after}(\psi \rightarrow \phi )\approx \sum _{i}|\psi _{i}^{*}\langle i|\phi \rangle |^{2}=\sum _{i}|\psi _{i}^{*}\phi _{i}|^{2}$

If we compare this with the formula we derived before the environment introduced decoherence we can see that the effect of decoherence has been to move the summation sign $\sum _{i}$ from inside of the modulus sign to outside. As a result all the cross- or quantum interference-terms:

$\sum _{ij;i\neq j}\psi _{i}^{*}\psi _{j}\phi _{j}^{*}\phi _{i}$

have vanished from the transition probability calculation. The decoherence has irreversibly converted quantum behaviour (additive probability amplitudes) to classical behaviour (additive probabilities).

In terms of density matrices, the loss of interference effects corresponds to the diagonalization of the "environmentally traced over" density matrix.

Density Matrix approach

The effect of decoherence on density matrices is essentially the decay or rapid vanishing of the off-diagonal elements of the partial trace of the joint system's density matrix, i.e. the trace, with respect to any environmental basis, of the density matrix of the combined system and its environment. The decoherence irreversibly converts the "averaged" or "environmentally traced over" density matrix from a pure state to a reduced mixture; it is this that gives the appearance of wavefunction collapse. Again this is called "environmentally-induced-superselection", or einselection. The advantage of taking the partial trace is that this procedure is indifferent to the environmental basis chosen.

Timescales

Decoherence represents an extremely fast process for macroscopic objects, since these are interacting with many microscopic objects, with an enormous number of degrees of freedom, 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 emerge from the properties of the interaction between matter and radiation. The time taken for off-diagonal components of the density matrix to effectively vanish is called the decoherence time, and is typically extremely short for everyday, macroscale process.

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 weighting of each outcome in 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, which is explicitly acknowledged in the many-worlds interpretation. All decoherence explains is why these coherences are no longer available for inspection by local observers.

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,

$H=H_{A}\otimes H_{E}$.

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 $|\mathrm {in} \rangle$ and the initial state of A is the superposition state

$c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle$

where $|\psi _{1}\rangle$ and $|\psi _{2}\rangle$ are orthogonal and there is no entanglement initially. Also, choose an orthonormal basis for H_A, $\{|e_{i}\rangle \}_{i}$. (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

$U(|\psi _{1}\rangle \otimes |\mathrm {in} \rangle )$

and

$U(|\psi _{2}\rangle \otimes |\mathrm {in} \rangle )$

uniquely as

$\sum _{i}|e_{i}\rangle \otimes |f_{1i}\rangle$

and

$\sum _{i}|e_{i}\rangle \otimes |f_{2i}\rangle$

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 $|f_{1i}\rangle$ and $|f_{1j}\rangle$ are all approximally orthogonal to a good degree if i is not j and the same thing for $|f_{2i}\rangle$ and $|f_{2j}\rangle$ and also $|f_{1i}\rangle$ and $|f_{2j}\rangle$ 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

$\sum _{i}(\langle f_{1i}|f_{1i}\rangle +\langle f_{2i}|f_{2i}\rangle )|e_{i}\rangle \langle e_{i}|$

(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

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 [2]. 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 appears to collapse 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
• A Detailed introduction (graduate student project)
• Quantum Bug : Qubits might spontaneously decay in seconds Scientific American Magazine (October 2005)

References

.

Further reading

• Omnes, R. (1999). Understanding Quantum Mechanics. Princeton: Princeton University Press.
• Joos, E. (2003). Decoherence and the Appearance of a Classical World in Quantum Theory (2nd edition ed.). Berlin: Springer. {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Zurek, Wojciech H. (2003). "Decoherence and the transition from quantum to classical — REVISITED", arXiv:quant-ph/0306072 (An updated version of PHYSICS TODAY, 44:36-44 (1991) article)
• Schlosshauer, Maximilian (23 February 2005). ""Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics"". Reviews of Modern Physics. 76(2004): 1267–1305. arXiv:quant-ph/0312059, doi:10.1103/RevModPhys.76.1267.
• J.J. Halliwell, J. Perez-Mercader, Wojciech H. Zurek, eds, The Physical Origins of Time Asymmetry, Part 3: Decoherence, ISBN 0521568374