Thermalization Induced by Quantum Scattering

We use quantum scattering theory to study a fixed quantum system $Y$ subject to collisions with massive particles $X$ described by wave packets. We derive the scattering map for system $Y$ and show that the induced evolution crucially depends on the width of the incident wave packets compared to the level spacing in $Y$. If $Y$ is nondegenerate, sequential collisions with narrow wave packets cause $Y$ to decohere. Moreover, an ensemble of narrow packets produced by thermal effusion causes $Y$ to thermalize. On the other hand, broad wave packets can act as a source of coherences for $Y$, even in the case of an ensemble of incident wave packets given by the effusion distribution, preventing thermalization. We illustrate our findings on several simple examples and discuss the consequences of our results in realistic experimental situations.

Popular Summary

Decoherence and thermalization are well understood within open quantum-system theory when the system of interest is in constant and weak contact with its surrounding. However, in many cases a better description of a quantum system interacting with its surrounding is to consider that the system undergoes repeated collisions with moving particles. In such situations, one expects that the state of the moving particles, described by wave packets (i.e., superpositions of waves traveling through space at different speeds), plays an important role in determining the system dynamics as well as the conditions for decoherence and thermalization to arise. Surprisingly, this important problem has been little studied.

We thus consider collisions between a fixed quantum system and moving particles described by narrow or broad wave packets, depending on whether their energy width is smaller or larger than the typical energies of the system. We find that narrow wave packets always lead to eigenstate decoherence in the system, while broad wave packets can be a source of coherence. Thermalization instead, is induced by collisions with a statistical ensemble of narrow wave packets, which are thermally effusing from a thermal source. This collision-based approach to open quantum-system theory should be of relevance to many areas of quantum physics ranging from atomic and mesoscopic physics to cosmology.

Figure 1

Narrow (upper plate) and broad (lower plate) wave packets. The blurred gray zones represent the scatterer, which undergoes the transitions $|j⟩\to |j^{\prime }⟩$ and $|k⟩\to |k^{\prime }⟩$. The figure shows the two transmitted wave packets corresponding to these specific transitions (notice that the collision will generate, in general, much more outgoing packets than the two depicted). The crucial difference is that the overlap of the outgoing wave packets is zero in the narrow case, unless both have exactly the same energy: ${\mathrm{\Delta }}_{j^{\prime }j}={\mathrm{\Delta }}_{k^{\prime }k}$. For broad packets, coherences between states do not vanish and can even be created due to the overlap (dark area) of the two outgoing packets depicted in the lower plate.

Figure 2

Thermalization via collisions is achieved if the system (scatterer) is bombarded by narrow wave packets coming from equilibrium reservoirs at the two sides. If we remove one of the reservoirs, the setup is out of equilibrium. Notice however that, if the scatterer is symmetric, that is, if the interaction potential $V(x)$ is even, $V(x)=V(-x)$, then the second reservoir is not necessary.

Figure 3

Bloch sphere representation of the state ${\rho }_Y^{(n)}$ under repeated collisions ($n=200$) with narrow and broad Gaussian wave packets (blue circles and red triangles, respectively). The colored arrows show the direction of the evolution of each state in the sphere starting from the initial states, which are pure (i.e., they lie on the surface of the sphere) and given by $({\rho }_Y{)}_{11}^{(0)}=1$ and $({\rho }_Y{)}_{12}^{(0)}=0$ (upper panel) and $({\rho }_Y{)}_{11}^{(0)}=({\rho }_Y{)}_{12}^{(0)}=0.5$ (lower panel). All parameters are identical in the two panels, ${\mathrm{\Delta }}_Y=m=1$, $p_0=10$, except the widths, which are $\sigma /(m{\mathrm{\Delta }}_Y/2p_0)=0.2$ for narrow wave packets and $\sigma /(m{\mathrm{\Delta }}_Y/2p_0)=20$ for broad wave packets.

Figure 4

Effect of free evolution between collisions on the populations (left panel), modulus and phase of coherences (middle and right panel, respectively) for a two-level system colliding with narrow ($N$) and broad ($B$) Gaussian wave packets. The system evolves freely for $0$ or $3/4$ cycles between each collision. The initial state is $({\rho }_Y{)}_{22}^{(0)}=0.9$ and $({\rho }_Y{)}_{11}^{(0)}=0.1,({\rho }_Y{)}_{12}^{(0)}=0.3$. The remaining parameters are ${\mathrm{\Delta }}_Y=m=1$, $p_0=6.3$, and $\sigma /(m{\mathrm{\Delta }}_Y/2p_0)=0.12$ for narrow wave packets and $\sigma /(m{\mathrm{\Delta }}_Y/2p_0)=12$ for broad wave packets.

Figure 5

Plot of ${\mathsf{B}}_{jk}^{(n)}$ defined in Eq. (58) computed for an ensemble of narrow Gaussian wave packets weighted by the effusion distribution in Eq. (45) and the Maxwell-Boltzmann (MB) distribution (upper and lower panel, respectively) with $\beta =3$. In both cases, the initial state is the thermal state $e^{-{\beta }^{\prime }{H}_Y}/Z_Y$ with ${\beta }^{\prime }=1$ and the mass of the wave packet is $m=0.5$. The orange line corresponds to $\beta =3$. For clarity, not all pairs of indices $(j,k)$ are plotted.

Figure 6

Evolution of ${\rho }_Y^{(n)}$ when the state of $X$ is a mixture of broad Gaussian wave packets weighted with the effusion distribution. Upper panel: The values $({\rho }_Y{)}_{ij}^{(n)}$ are obtained by composing the map ${\mathbb{S}}_Y$. In black we depict $({\rho }_Y{)}_{11}^{(n)}$ and $({\rho }_Y{)}_{22}^{(n)}$ in blue, while the real and imaginary parts of $({\rho }_Y{)}_{12}^{(n)}$ are depicted in green and red, respectively. Lower panel: The iterated map is given in Eq. (5) with random Poissonian times ${\tau }_i$. In both panels: The orange lines indicates the thermal values $({\rho }_Y{)}_{11}$ and $({\rho }_Y{)}_{22}$ with inverse temperature $\beta$. The light blue lines indicate the thermal values $({\rho }_Y{)}_{11}$ and $({\rho }_Y{)}_{22}$ with inverse temperature ${\beta }_C$. The gray lines indicate the values $({\rho }_Y{)}_{11}$ and $({\rho }_Y{)}_{22}$ computed by considering the master equation for the populations ruled by ${\mathbb{W}}_{ik}={\mathbb{S}}_{ii}^{kk}$ as given in Eq. (59). The initial state is $({\rho }_Y{)}_{11}^{(0)}=0.3$ and $({\rho }_Y{)}_{12}^{(0)}=0$. The remaining parameters are ${\mathrm{\Delta }}_Y=m=0.5$ with $e_2=2.5,e_1=2$, $\sigma =0.31$ and $\beta =3$.