Solvable multistate model of Landau-Zener transitions in cavity QED

We consider the model of a single optical cavity mode interacting with two-level systems (spins) driven by a linearly time-dependent field. When this field passes through values at which spin energy-level splittings become comparable to spin coupling to the optical mode, a cascade of Landau-Zener transitions leads to coflips of spins in exchange for photons of the cavity. We derive exact transition probabilities between different diabatic states induced by such a sweep of the field.

Figure 1

Diabatic levels and couplings between them described by the Hamiltonian (4) at $N_e=1$.

Figure 2

Diabatic levels and couplings between them described by the Hamiltonian (14) at $N_B=0$.

Figure 3

Diabatic levels and couplings between them described by the Hamiltonian (23). For convenience, equal terms are added to all slopes of diabatic levels: $\widehat{H}\to \widehat{H}-1.5t\widehat{1}$. This is equivalent to a change of time-dependent phases of state amplitudes, which does not change transition probabilities. Blue arrows demonstrate the unique semiclassical path connecting the state $|6\rangle$ at $t\to -\infty$ to the state $|5\rangle$ at $t\to +\infty$.

Figure 4

Eigenvalues of the Hamiltonian (23) shown as functions of time (eigenenergy levels). Parameters are $N_B=1,{\epsilon }_1=2.4,{\epsilon }_2=0,{\epsilon }_3=-1,g=0.35$. For convenience, equal terms are added to the main diagonal of the Hamiltonian: $\widehat{H}\to \widehat{H}-2.5t\widehat{1}$.

Figure 5

Comparison of numerical calculations of transition probabilities with semiclassical solution (25). All discrete points correspond to results of the direct numerical solution of the Schrödinger equation with the Hamiltonian (23) starting at $t=-1000$ and changing time up to $t=1000$ with a time step size $dt=0.0001$. Solid lines are theoretical predictions of Eq. (25). In (a) and (b), the independence of transition probabilities of energy-level splittings is tested for ${\epsilon }_1=0.5\epsilon ,{\epsilon }_2=0,{\epsilon }_3=-\epsilon$. Other parameters are given in the panels. In (c)–(f), transition probabilities are shown as functions of the coupling constant $g$ with different initial conditions at parameter values ${\epsilon }_1=0.5,{\epsilon }_2=0,{\epsilon }_3=-1$. For better visibility, not all possible transition probabilities are shown at given initial conditions.

Figure 6

Eigenenergy levels of the Hamiltonian (4) in the sector with $N_s=4$ spins and $N_e=2$ excitations, which corresponds to a total of 11 interacting states. Parameters are ${\epsilon }_1=2.5,{\epsilon }_2=1,{\epsilon }_3=-1,{\epsilon }_4=-2$; $g=0.32$. For convenience, equal terms are added to the main diagonal of the Hamiltonian: $\widehat{H}\to \widehat{H}-3t\widehat{1}$.

Figure 7

Transition probabilities for the degenerate case with total spin $S=3/2$. Different panels correspond to different initial conditions. Discrete points correspond to results of numerical simulations of the evolution with the Hamiltonian (36) from $t=-1000$ to $t=1000$ with a time step size $dt=0.0001$. Solid lines are theoretical predictions of Eq. (39). In all cases, $N_B=1$.

Figure 8

Transition probabilities $P_{1\to n}\equiv P_{n1}^{(2S+1)}$ from the state $n=1$ $\left(S_z=S\right)$ to other states for evolution with the Hamiltonian (33) in the sector with 16 states $\left(S=15/2\right)$ at different coupling strengths. Theoretical predictions (dots on vertical lines) of Eq. (41) are compared to results of direct numerical simulations (empty boxes). Time evolution is from $t=-2000$ to $t=2000$ with time step $dt=0.00005$. In all cases, $N_B=0$.

Figure 9

Comparison of theoretical predictions (dots on vertical lines) of Eq. (42) and numerical simulations (empty boxes) for evolution with the Hamiltonian (33) in the sector with 18 states $\left(S=17/2\right)$ at different coupling strengths. Time evolution is from $t=-2000$ to $t=2000$ with time step $dt=0.00005$. In all cases, $N_B=0$. Evolution starts with the fully polarized spin state $S_z=-S$ with $2S$ bosons, which corresponds to $n=2S+1=18$.