Adiabatic time evolution in spin systems

Adiabatic processes in the quantum Ising model and the anisotropic Heisenberg model are discussed. The adiabatic processes are assumed to consist in the slow variation of the strength of the magnetic field that environs the spin systems. These processes are of current interest in the treatment of cold atoms in optical lattices and in adiabatic quantum computation. We determine the probability that, during an adiabatic passage starting from the ground state, states with higher energy are excited.

Figure 1

Spectrum of an Ising chain consisting of five particles $\left(N=5\right)$ as a function of the transverse magnetic field $g$. The dotted line marks the critical point at which the quantum phase transition takes place.

Figure 2

Spectrum of the anisotropic Heisenberg model as a function of the external magnetic field $g$. Assumptions: $N=5$, ${\Delta }_x=0.1$, ${\Delta }_y=0.3$, and ${\Delta }_z=1$; the table on the right-hand side gives information about the degeneracy $D$, the group index $n$, and the translational symmetry $k$ of each level.

Figure 3

Plot of the excitation energies of the states $\mid n⟩$ as functions of $g$. The assumed number of particles $N$ is 51.

Figure 4

Excitation probability $p_{0\to n}$ of the levels $\mid n⟩$ in the case of 501 particles $\left(N=501\right)$. The magnetic field is assumed to be varied from $g_0=5$ to $g_1$ with change rate $\dot{g}$ equal to $-0.0001$. The indices $n$ are reshuffled such that states with lower index have lower energy.

Figure 5

Excitation probability $p_E$ as a function of the final field strength $g_1$. Assumptions: 51 particles $\left(N=51\right)$; linear variation of $g$ from $g_0=5$ to $g_1$ with change rate $\dot{g}=-0.01$; solid line, sum of all terms $p_{0\to n}$ given by formula (B2); dotted line, Numerical results (see text); dashed lines, analytical estimations [formulas (4, 5)].

Figure 6

Duration $T$ of the adiabatic process as a function of the particle number $N$. Assumptions: $g$ is linearly decreased from $g_0=5$ to $g_1=0$; the change rate $\dot{g}$ is adjusted such that the excitation probability $p_E$ always amounts to $0.05$; crosses, numerical results (see text); dash-dotted line, analytical result (6). Diamonds: results that were presented in Ref. 6 for the free-end chain.

Figure 7

Expectation value of $H$ relative to the width of the spectrum, i.e., $(⟨H⟩-E_0)∕\Delta E$, as a function of the change rate $\dot{g}$. Assumptions: $N=51$, $g_0=5$, and $g_1=0$; solid line, numerical results (see text); dash-dotted line, indicates variance of $H$; dots, results obtained using the adiabatic approximation.

Figure 8

Excitation probability $p_E$ as a function of the change rate $\dot{g}=(g_1-g_0)∕T$ in the case of five- and nine-particle chains. Assumptions: ${\Delta }_x=0.1$, ${\Delta }_y=0.3$, ${\Delta }_z=1$; variation of $g$ from $g_0=10$ to $g_1=5$; dotted line, excitation probability obtained from a numerical simulation. Solid line: Approximation (7) of the excitation probability in the limit $N⪢1$.

Figure 9

Excitation probability $p_E$ as a function of the number of particles $N$ in the case of change rates $\dot{g}=-6$ and $\dot{g}=-10$. Assumptions: ${\Delta }_x=0.1$, ${\Delta }_y=0.3$, ${\Delta }_z=1$; variation of $g$ from $g_0=10$ to $g_1=5$; dotted line, excitation probability obtained from a numerical simulation. Solid line: Approximation (7) of the excitation probability in the limit $N⪢1$.