Dissipative Landau-Zener transitions of a qubit: Bath-specific and universal behavior

We study Landau-Zener transitions in a qubit coupled to a bath at zero temperature. A general formula that is applicable to models with a nondegenerate ground state is derived. We calculate exact transition probabilities for a qubit coupled to either a bosonic or a spin bath. The nature of the baths and the qubit-bath coupling is reflected in the transition probabilities. For diagonal coupling, when the bath causes energy fluctuations of the diabatic qubit states but no transitions between them, the transition probability coincides with the standard Landau-Zener probability of an isolated qubit. This result is universal as it does not depend on the specific type of bath. For pure off-diagonal coupling, by contrast, the tunneling probability is sensitive to the coupling strength. We discuss the relevance of our results for experiments on molecular nanomagnets, in circuit QED, and for the fast-pulse readout of superconducting phase qubits.

Figure 1

(Color online) Adiabatic (solid) and diabatic (dashed) energies for the standard Landau-Zener problem.

Figure 2

(Color online) Sketch of the diabatic energy levels as a function of time for a qubit coupled to a single harmonic oscillator. Energies of the states in the up cluster increase. These states correspond to the qubit state ∣↑⟩. Energies decrease in the down cluster, where the qubit state is ∣↓⟩. According to the no-go-up theorem (A11), the initial state $\mid \uparrow ,0_{+}⟩$ evolves to a superposition in which $\mid \uparrow ,0_{+}⟩$ is the only “up” state. Energies within a band are separated by the oscillator energy $\hslash \Omega$. For a qubit coupled to an oscillator bath, the corresponding crossing clusters would be continuous bands of states.

Figure 3

(Color online) Landau-Zener dynamics for a qubit with $\Delta =0$, in all three cases shown off-diagonally coupled via ${\sigma }_x$ to three oscillators. The various oscillator frequencies ${\Omega }_j$ are given in units of $\sqrt{v∕\hslash }$. All coupling strengths have the same value ${\gamma }_j=\sqrt{\hslash v∕3}$. The dotted line marks the analytical final transition probability corresponding to Eq. (28).

Figure 4

(Color online) Transition probability $P_{\uparrow \to \downarrow }$ as a function of the internal coupling $\Delta$ for $E_0=2\sqrt{\hslash v}$ and $S=0.5\hslash v$. The angles $\vartheta =0$ and $\vartheta =\pi ∕2$ correspond to purely diagonal and off-diagonal couplings, respectively, for which the probability is independent of $\phi$.

Figure 5

(Color online) Time evolution of the bit-flip probability for a qubit with $\Delta =0.5\sqrt{\hslash v}$ diagonally coupled to two harmonic oscillators, two nonlinear oscillators, and seven spins, respectively. The harmonic oscillators are specified by ${\Omega }_1=0.1\sqrt{v∕\hslash }$, ${\Omega }_2=0.5\sqrt{v∕\hslash }$, ${\gamma }_1=2\sqrt{\hslash v}$, and ${\gamma }_2=6\sqrt{\hslash v}$, while the nonlinear oscillators, in addition, have ${\beta }_1={\beta }_2=3\sqrt{\hslash v}$. The values of the $B_j^{\nu }$ and the ${\gamma }_j^z$ are randomly chosen from the range $[-\sqrt{\hslash v}∕10,\sqrt{\hslash v}∕10]$. In all three cases, the transition probability converges to the universal value (41).

Figure 6

(Color online) Sketch of spectral density for an atom near a local defect in a photonic crystal with a band gap. The quadratic free-space spectral density is modified by the crystal that creates a spectral gap around ${\omega }_0$. A narrow defect mode inside a broader band gap allows a controlled atom-defect interaction via LZ sweeps of the atomic transition frequency ${\omega }_A\left(t\right)$.

Figure 7

(Color online) Crossing of two groups of diabatic states: states $\mid a⟩$ whose energy is time independent and states $\mid b⟩$ whose energies are reduced with constant velocities $v_b$.

Figure 8

Relation between the various time variables.