Environment-induced decoherence

Superposition principle is one of the most fundamental features of quantum mechanics. It means as long as some specific states exist; then any possible linear combinations of those states should also exist. Therefore, there should be always a large number of superposition states in our ordinary life. How-ever in fact we usually don’t see a lot of those superposition states, but only a few classical states instead. For example, an object is always located on definite position, but not on the superposition of different positions. Why do those superposition states lose their coherences to be classical states?

The answer to this question, which lies in the interaction and entanglement of an open quantum system with its environment, was first pointed out by Zeh [4] and developed in detail by Zurek [5]. In literatures, it is referred as

“environment-induced decoherence”.

A concrete description for environment-induced decoherence is as follows:

Because of the system-environment interaction, the superposition state in the pointer states basis of the system loses its coherence and is entangled with the environment. The pointer states are the most robust states un-der the system-environment interaction and lose minimum information into the environment, which means, after time evolution, the pointer states still

maintain almost the same form. The pointer states become classical states at the classical limit~ → 0[6].

Moreover, it can be said that the coherence, which originally only exists among the pointer states of the system, has been delocalized and spread out into the total system. Therefore from the view point of the total system, it is still a superposition state.

Such decoherence is also encoded in the off-diagonal terms of the reduced density matrix with the pointer basis, which describe the quantum correla-tion of different point states. Therefore, the study of the evolucorrela-tion of the off-diagonal terms of the reduced density matrix provides a direct way to understand how a system loses its coherence as time goes on. As the off-diagonal terms disappear, the system decoheres completely. Of course, the precondition to achieve this is that the pointer basis must to be known.

In [5], Zurek proposed a criterion to determine if a set of states is pointer basis or not for a particular limit, in which the energy scales of the interaction Hamiltonian dominate. The criterion is as follows: In such a limit, referred as quantum-measurement limit, if any observable which commutes with the interaction Hamiltonian exists, then the eigenstates of such an observable is a pointer basis. This criterion matches with our intuition that the pointer states are the eigenstates of the interaction Hamiltonian such that they are the robustest states under the system-environment interaction.

For general cases, the methods to find out the pointer basis, which are developed by [7, 8], are to prepare a lot of different initial states, and then find the time evolution of “purity” (defined as Trρ²) or “von Neumann en-tropy” (defined as −Trρ log ρ) of the reduced density matrix corresponding to those different initial states. Finally, by comparing the results, the pointer states are the states corresponding to the minimal decrease in purity or the minimal increase in von Neumann entropy. The minimal decrease in purity and the minimal increase in von Neumann entropy both reflect the minimal information loss of the pointer states under the system-environment interac-tion.

Purity and von Neumann entropy are not only helpful to find the pointer states, but also very useful to characterize the decoherence by themselves.

the basis choice. It means even if we don’t know what the pointer states are, we still can characterize decoherence by these two scalar quantities[9].

Besides the off-diagonal term, purity and von Neumann entropy of the reduced density matrix, the Wigner function is another useful tool to char-acterize the decoherence, which is defined as the Fourier transformation of the reduced density as follows

One of the advantage of the Wigner function is that it is defined in the phase-space such that its classical counterpart is the distribution function.

Therefore, it is easier to derive the equation of motion of the distribution function (i.e. Liouville Equation) from the equation of motion of the Wigner function, by taking classical limit~ → 0.

Unlike its classical counterpart, the Wigner function is not positive-definite, but it will be as long as complete decohrence occurs. It means the negative part of the Wigner function can be used to characterize de-coherence. In addition, this quantity is also one of the basis-independent quantities which characterize decoherence. In [9], these basis-independent quantities are calculated for a system with scalar fields.

Of course, the total time for a system to decohere completely must be very short, such that we wouldn’t have any chance to see the intermediate process of the decoherence usually. But how short is it? What is its time scale? In [5], Zurek considered a particle with mass m at position x coupled with a scalar field ϕ (a heat bath of harmonic oscillators with temperature T ) through the interaction Hamiltonian Hint = ϵxdϕ/dt and estimated the decoherence rate. In this case, the initial wave function of the particle is prepared as the superposition of two Gaussians at positions x+^∆x₂ and x−^∆x₂ respectively. The off-diagonal part of the master equation is approximately

dρ^off

dt ∼ −τ_D⁻¹ρ^off⇒ ρ^off∝ e^−τDt , (1.2) where τ_D = τ_R(^λ_∆x^dB)²is the decoherence time. Here τ_R= ^4m_ϵ2 is the relaxation time and λ_dB is the thermal de Broglie wavelength defined as~(2mkBT )^−1/2. Such estimation matches with our intuition: The stronger the coupling is, the more quickly the system decoheres. The higher the temperature is, the

more quickly the system decoheres. The larger the separation ∆x is, the more quickly the system decoheres (the reason is that the larger the separation is, the smaller the quantum correlation between these two Gaussians is). We can say more about these factors affecting decoherence time. τDis dominated more by the coupling ϵ and the separation ∆x than by the temperature T . Unlike τ_R, τ_D doesn’t depend on the mass of the particle.

To get more physical sense about how quickly the system decoheres, we put practical number into the dechorence time and relaxation time and calculate their ratio. For example, if a system is at temperature T = 300 kelvins with mass m = 1 gram and separation ∆x = 1 centimeter, then the ratio is τ_D/τ_R∼ 10⁻⁴⁰. It is an extremely small value.

To get purity, von Neumann entropy, Wigner function or the off-diagonal terms of reduced density matrix, the fundamental thing we need to know is the evolution of the reduced density matrix. To get it, we need to integrate out the degree of freedom of the environment. It can be carried out according to the Feynman-Vernon formalism, which deals with the integration-out by path-integral method and encodes all information of environment into the influence functional. We will introduce the Feynman-Vernon formalism in the following subsection.