Quantum decoherence: quantum decoherence2

Dirac notation

Using the Dirac notation, let the system initially be in the state $|\psi \rang$ where

$|\psi \rang = \sum_i |i\rang \lang i|\psi \rang$

where the $|i\rang$ s form an einselected basis (environmentally induced selected eigen basis^[4]); and let the environment initially be in the state $|\epsilon\rang$ . The vector basis of the total combined system and environment can be formed by tensor multiplying the basis vectors of the subsystems together. Thus, before any interaction between the two subsystems, the joint state can be written as:

$|\mathit{before}\rang = \sum_i |i\rang |\epsilon \rang \lang i|\psi \rang.$

where $|i\rang |\epsilon \rang$ is shorthand for the tensor product: $|i \rang \otimes |\epsilon\rang$ . There are two extremes in the way the system can interact with its environment: 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 disturbed (e.g. the idealized non-disturbing measurement). In general an interaction is a mixture of these two extremes, which we shall examine:

[edit]System absorbed by environment

If the environment absorbs the system, each element of the total system's basis interacts with the environment such that:

$|i\rang |\epsilon \rang$ evolves into $|\epsilon_i\rang$

and so

$|\mathit{before}\rang$ evolves into $|\mathit{after}\rang = \sum_i |\epsilon_i\rang \lang i|\psi \rang$

where the unitarity of time-evolution demands that the total state basis remains orthonormal and in particular their scalar or inner productswith each other vanish, since $\lang i|j\rang = \delta_{ij}$ :

$\lang \epsilon_i|\epsilon_j\rang = \delta_{ij}$

This orthonormality of the environment states is the defining characteristic required for einselection.^[4]

[edit]System not disturbed by environment

This is the idealised measurement or undisturbed system case in which each element of the basis interacts with the environment such that:

i.e. the system disturbs the environment, but is itself undisturbed by the environment.

and so:

$|\mathit{before}\rang$ evolves into $|\mathit{after}\rang = \sum_i |i,\epsilon_i\rang \lang i|\psi \rang$

where, again, unitarity demands that:

$\lang i,\epsilon_i|j,\epsilon_j\rang = \lang i|j \rang \lang \epsilon_i|\epsilon_j\rang= \delta_{ij} \lang \epsilon_i|\epsilon_j\rang = \delta_{ij}$

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

$\lang \epsilon_i|\epsilon_j\rang \approx \delta_{ij}$

As before, this is the defining characteristic for decoherence to become einselection.^[4] The approximation becomes more exact as the number of environmental degrees of freedom affected increases.

Note that if the system basis $|i\rang$ 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 $|\epsilon_j\rang = |\epsilon'\rang$ . This would correspond to the system basis being degenerate with respect to 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.

[edit]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

ψ

φ

before

ψ

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:

$\mathit{prob}_{\mathit{before}}(\psi \rightarrow \phi) = |\lang \psi |\phi \rang|^2 = |\sum_i\psi^*_i \phi_i |^2 = \sum_{i} |\psi_i^*\phi_i|^2 + \sum_{ij;i \ne j} \psi^*_i \psi_j \phi^*_j\phi_i$

where $\psi_i = \lang i|\psi \rang , \psi_i^* = \lang \psi|i \rang$ and $\phi_i = \lang i|\phi \rang$ etc

Terms appear in the expansion of the transition probability above which involve $i \ne j$ ; these can be thought of as representing interferencebetween 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

ψ

φ

after

ψ

has interacted with its environment, then application of the Born probability rule states we must sum over all the relevant possible states of the environment,

E i

, before squaring the modulus:

$\mathit{prob}_{\mathit{after}}(\psi \rightarrow \phi) = \sum_j|\lang \mathit{after}| \phi, \epsilon_j \rang|^2 = \sum_j|\sum_i \psi_i^* \lang i, \epsilon_i|\phi, \epsilon_j\rang |^2 = \sum_j|\sum_i \psi_i^* \lang i|\phi \rang \lang \epsilon_i|\epsilon_j \rang |^2$

The internal summation vanishes when we apply the decoherence / einselection condition $\lang \epsilon_i|\epsilon_j\rang \approx \delta_{ij}$ and the formula simplifies to:

$\mathit{prob}_{\mathit{after}}(\psi \rightarrow \phi) \approx \sum_j|\psi_j^* \lang j|\phi\rang |^2 = \sum_i|\psi^*_i \phi_i |^2$

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

Σ i

from inside of the modulus sign to outside. As a result all the cross- or quantum interference-terms:

$\sum_{ij;i \ne j} \psi^*_i \psi_j \phi^*_j\phi_i$

have vanished from the transition probability calculation. The decoherence has irreversibly converted quantum behaviour (additive probability amplitudes) to classical behaviour (additive probabilities).^[4]^[5]^[6]

In terms of density matrices, the loss of interference effects corresponds to the diagonalization of the "environmentally traced over" density matrix.^[4]

[edit]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 andits environment. The decoherence irreversibly converts the "averaged" or "environmentally traced over"^[4] 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.^[4] The advantage of taking the partial trace is that this procedure is indifferent to the environmental basis chosen.

[edit]Operator-sum representation

Consider a system S and environment (bath) B, which are closed and can be treated quantum mechanically. Let $\mathcal{H_S}$ and $\mathcal{H_B}$ be the system's and bath's Hilbert spaces, respectively. Then the Hamiltonian for the combined system is

$\hat{H} = \hat{H}_{S}\otimes\hat{I}_{B} + \hat{I}_{S}\otimes\hat{H}_{B} + \hat{H}_{I}$

where $\hat{H}_{S},\hat{H}_{B}$ are the system and bath Hamiltonians, respectively, and $\hat{H}_{I}$ is the interaction Hamiltonian between the system and bath, and $\hat{I}_{S}, \hat{I}_{B}$ are the identity operators on the system and bath Hilbert spaces, respectively. The time-evolution of the density operator of this closed system is unitary and, as such, is given by

$\rho_{SB}(t) = \hat{U}(t)\rho_{SB}(0)\hat{U^{\dagger}}(t)$

where the unitary operator is $\hat{U}=e^{\frac{-i\hat{H}t}{\hbar}}$ . If the system and bath are not entangled initially, then we can write $\rho_{SB} = \rho_{S}\otimes\rho_{B}$ . Therefore, the evolution of the system becomes

$\rho_{SB}(t) = \hat{U}(t)[\rho_{S}(0)\otimes\rho_{B}(0)]\hat{U^{\dagger}}(t).$

The system-bath interaction Hamiltonian can be written in a general form as

$\hat{H}_{I} = \sum_{i}\hat{S_{i}}\otimes\hat{B}_{i},$

where $\hat{S_{i}}\otimes\hat{B_{i}}$ is the operator acting on the combined system-bath Hilbert space, and $\hat{S}_{i}, \hat{B}_{i}$ are the operators that act on the system and bath, respectively. This coupling of the system and bath is the cause of decoherence in the system alone. To see this, a partial trace is performed over the bath to give a description of the system alone:

$\rho_{S}(t) = Tr_B[\hat{U}(t)[\rho_{S}(0)\otimes\rho_{B}(0)]\hat{U^{\dagger}}(t)].$

$\mathbf\mathit{{\rho}}_{S}(t)$ is called the reduced density matrix and gives information about the system only. If the bath is written in terms of its set of orthogonal basis kets, that is, if it has been initially diagonalized then $\rho_{B}(0) = \sum_{j}a_{j}|{j}\rangle\langle{j}|.$ Computing the partial trace with respect to this (computational)basis gives:

$\rho_{S}(t) = \sum_{l}\hat{A_{l}}\rho_{S}(0)\hat{A}^{\dagger}_{l}$

where $\hat{A_{l}}, \hat{A}^{\dagger}_{l}$ are defined as the Kraus operators and are represented as

$\hat{A_{l}} = \sqrt{a_{j}}\langle{k}|\hat{U}|{j}\rangle.$

This is known as the operator-sum representation (OSR). A condition on the Kraus operators can be obtained by using the fact that $\mathbf\mathit{{Tr}}(\mathbf{\mathit{\rho}}_{S}(t)) = 1$ ; this then gives

$\sum_{l}\hat{A}^{\dagger}_{l}\hat{A_{l}} = \hat{I}_{S}.$

This restriction determines if decoherence will occur or not in the OSR. In particular, when there is more than one term present in the sum for $\mathbf{\rho_{S}}(t)$ then the dynamics of the system will be non-unitary and hence decoherence will take place.

Quantum decoherence

Tuesday, June 28, 2011

quantum decoherence2