A demonstration of decoherence for beginners Free

Quantum mechanics describes nature in terms of state vectors $|ϕ⟩$ belonging to a Hilbert space, which in the Schrödinger picture evolve in time. This description has the apparent problem that many quantum mechanical states do not correspond to classical macroscopic objects. For example, a particle in state $|ϕX⟩$ localized at position X has a classical counterpart, while a particle in state $|Φ⟩=(|ϕX⟩+|ϕY⟩)/2$ appears to be located at both X and Y simultaneously and thus has no classical counterpart. Furthermore, macroscopic objects initially in classical-like states can evolve into states with no classical analogs. This is demonstrated in the famous Schrödinger-cat paradox, where a macroscopic cat, initially in a live state, evolves into states where it is both dead and alive.¹ 

This problem can be addressed by noting that observable predictions of the quantum theory are expectation values of Hermitian operators A so that physically measurable consequences of the non-classical nature of states, such as $|Φ⟩$⁠, are contained in the interference term $⟨ϕY|A|ϕX⟩$⁠. The decoherence theory demonstrates that an interaction resulting in an entanglement with a many-particle system can produce rapid decay of such interference terms, leading to classical behavior at the macroscopic level. Here, we show how the action of a gas of scatterers on a macroscopic particle via the Schrödinger equation leads to the approximately exponential decay of $⟨ϕY|A|ϕX⟩$ using a simplified scattering model.² In physical terms, scattering effectively “measures” the position of the particle, forcing it into a position eigenstate.

Consider a particle with the center of mass at X, represented by state $|ϕX⟩$ with the corresponding position wavefunction $ϕX(y)=⟨y|ϕX⟩$ and surrounded by a gas of scatterers, each in state $|φ⟩$ and with the corresponding position wavefunction $φ(x)=⟨x|φ⟩$⁠. For simplicity, we assume that scattering is one dimensional and the scatterers are much lighter than the particle.² Scattering is then an elastic reflection from an infinite potential located at x = X, with the momentum of the scatterer changing the sign (see Fig. 1).

Because $p=m dx/dt$⁠, the (scattering) transformation $p→S−p$ is effected by a change in the sign of the position, accompanied by a possible translation, $x→Sa−x$⁠. This transformation is accomplished by the scattered wavefunction $φ(a−x)$⁠, which reverses the momentum distribution associated with $φ(x)$⁠. Because the potential is infinite, the wavefunction must vanish at the wall;¹ thus, the constant a is determined³ by the condition $φ(X)−φ(a−X)=0$ so that $a=2X$⁠. The result is that scattering changes the product of particle and scatterer wavefunctions as

Consider the wavefunction for a superposition of two orthonormal states corresponding to the particle centered at X and Y

Combining Eqs. (1) and (2), scattering changes the superposition of Eq. (2) as

Here, $φ(x)$ cannot be factored from the scattered state, which entangles the scatterer and particle states, and this is a key feature of decoherence.

Non-classical behavior is represented by interference of the two states in expressions for expectation values of Hermitian operators A

where we have defined the interference term $r=⟨ϕY|A|ϕX⟩$ and assumed that the expectation value of A does not depend on the state of the scatterer $|φ⟩$⁠. We now show that the contribution of interference terms to the expectation value tends to zero in the presence of scattering when the expectation value does not depend on $|φ⟩$⁠. The interference amplitude of the scattered state is evaluated as follows

where we have used the properties of the Fourier transform of a complex convolution in Eq. (7) and $φ̃(p)$ is the momentum wavefunction, that is, the Fourier transform of $φ(x)$⁠.

As an aside, Eq. (7) is a special case of the standard positional decoherence expression for environmental scattering^4,5 when the scattering amounts to reflection, that is, the T-matrix satisfies $|T(p,q)|2=Rt δ(p+q)$⁠, where $δ(p)$ is the delta function and R is the scattering rate.

The exponential in Eq. (7) can be expanded in a Taylor series as $e2iαp=1+2iαp−2α2p2+⋯$⁠, giving

where $⟨pn⟩$ denotes the expectation value of pⁿ. Since scattering is assumed to be symmetric with respect to both sides of the particle, we have $⟨p⟩=0$⁠. At a scattering rate R, a total of N = Rt independent scatterers will collide with the particle in time t,² each with initial wavefunction $φ(x)$⁠, so that r varies as

provided that $(X−Y)2⟨p2⟩/ℏ2$ is small. Therefore, if scattering events are independent, which typically occur when the environment consists of a large number of scatterers, interference of particle states decays approximately exponentially, at a rate proportional to the square of the separation for small separations. The effect in Eq. (4) is that after a time inversely proportional to $(X−Y)2$⁠, only the first two terms contribute to the expectation value, and these are the expectation values of A with the particle at X or at Y but not both at the same time. This is what one would expect for a classical mixture of particles in states $|ϕX⟩$ and $|ϕY⟩$⁠, with equal weight 1/2 for both states.

Anticipating a more advanced treatment, we note that the expectation value in Eq. (4) can be written as the trace of a product of matrices

where the second matrix is the density matrix representing the particle state in Eq. (2). It follows from the above paragraph that decoherence diminishes the off-diagonal elements of the density matrix so that with time it tends to a diagonal matrix. Such a density matrix cannot be obtained from a pure quantum state as in Eq. (2) but rather resembles a classical mixture of two quantum states of the particle.

In summary, we have shown that as a consequence of decoherence, the interference of a particle with itself can generally be neglected for macroscopic particles. Ultimately, this leads to a localization of the particle and the fact that classical particles have trajectories, that is, they appear to transition through a set of position eigenstates.⁵