The overdamped classical particle performs Brownian motion according to

$$ \dot{x}=K(x)+f(t), $$

(1)

where *K* is the deterministic force experienced by the particle resulting from the metastable potential and *f* is white Gaussian noise with a probability density functional given by [29]

$$\begin{aligned} P\bigl[f(t)\bigr] =&\exp \biggl(-\int_{0}^{t_{\mathrm{f}}} \frac{f^{2}(t)}{2D}\,dt \biggr), \end{aligned}$$

(2)

where we assume [30] the noise intensity *D* to be small compared to the barrier height Δ*U*, see Figure 1. The probability density functional is not normalized *a priori*, subsequent expressions have to be normalized by hand. For a noise driven trajectory it can be obtained by substituting Eq. (1)

$$ P\bigl[x(t)\bigr]=\exp \biggl(-\frac{S[x(t)]}{D} \biggr),\qquad S\bigl[x(t)\bigr]= \frac{1}{2}\int_{0}^{t_{\mathrm{f}}}\,dt\bigl(\dot{x}-K(x) \bigr)^{2}. $$

(3)

For the study of qubit decoherence during a switching event one will need expectation values for a generic observable *O* of the type

$$\begin{aligned}& O(t_{0}) = \bigl\langle \exp \bigl(\lambda\phi\bigl[x(t),s(t,t_{0}) \bigr] \bigr) \bigr\rangle _{\mathrm{sw}}, \end{aligned}$$

(4)

$$\begin{aligned}& \phi\bigl[x(t),s(t,t_{0})\bigr] = \int_{0}^{t_{\mathrm{f}}} x(t) s(t,t_{0})\,dt, \end{aligned}$$

(5)

were \(s(t,t_{0})\) is a time dependent generating field for computing dephasing and bit-flip as will be detailed later. The coupling constant *λ* has been singled out to parameterize the coupling to the qubit and the time \(t_{0}\) indicates when the observable is measured. Effectively, it constitutes a modulation of \(x(t)\) and *λ*. This approach is fairly standard in stochastic path integrals and the choice of \(s(t,t_{0})\) suitable for our problem will be clarified later. This formulation allows for the present formalism to include detectors such as the JBA, where, in a rotating frame, the coupling between the qubit and detector becomes time dependent and periodic. We are interested in the qubit decoherence *during* switching. Thus, the averaging \(\langle\rangle_{\mathrm{sw}}\) is performed only over switching trajectories of the detector, which satisfy the boundary conditions \(x(0)=x_{\mathrm{m}}\) and \(x(t_{\mathrm{f}})=x_{\mathrm{f}}\), with \(x_{\mathrm{m}}\) inside and \(x_{\mathrm{f}}\) outside the metastable well. By choosing \(s(t,t_{0})=0\) at \(t_{\mathrm{f}}>t>t_{0}\), the average becomes post-conditioned by a switching event having taken place at the final time \(t_{\mathrm{f}}\), i.e., experimental verification would need to include recording the time of the switch and post-selecting for different times. This condition makes sure that we only study realizations of the measurement when switching occurs. In order to not over-constrain the trajectories we have verified the dependence of our result on the precise choice of \(x_{\mathrm{f}}\) and found minuscule changes only.

Since the exact switching trajectory between the initial and final point remain random, we average over all possible paths using the weight \(e^{-S[x]/D}\)

$$ \begin{aligned}[b] O(t_{0})={}&\int_{(x_{\mathrm{m}},0)}^{(x_{\mathrm{f}},t_{\mathrm{f}})} \mathcal{D}x(t)\exp \biggl(\lambda\phi\bigl[x(t),s(t,t_{0})\bigr] - \frac{S[x(t)]}{D} \biggr) \\ &{}\times P(x_{\mathrm{m}},0|x_{\mathrm{f}},t_{\mathrm{f}})^{-1}, \end{aligned} $$

(6)

where the total switching probability

$$ P(x_{\mathrm{m}},0|x_{\mathrm{f}},t_{\mathrm{f}})=\int _{(x_{\mathrm{m}},0)}^{(x_{\mathrm{f}},t_{\mathrm{f}})} \mathcal{D}x(t)\exp \biggl(- \frac {S[x(t)]}{D} \biggr) $$

(7)

serves as a normalization.

Switching over a high barrier is a rare event and thus requires very specific noise realizations. Thus, the switching trajectories form a narrow tube in phase space centered around an optimal trajectory [31, 32] which minimizes *S*, and for the present case satisfies

$$ \ddot{x}_{\mathrm{opt}}=K(x_{\mathrm{opt}})K'(x_{\mathrm{opt}}),\qquad x_{\mathrm{opt}}(0)=x_{\mathrm{m}},\qquad x_{\mathrm{opt}}(t_{\mathrm{f}})=x_{\mathrm{f}}. $$

(8)

This optimal trajectory is driven by an optimal realization of the environmental noise. Thus \(S[x(t)]=S[x_{\mathrm{opt}}(t)]+S_{2}[x(t)-x_{\mathrm{opt}}(t)]\) and we perform a saddlepoint approximation around this optimal solution

$$ S_{2}\bigl[x(t)\bigr]\approx\frac{1}{2}\int_{0}^{t_{\mathrm{f}}} \,dt \bigl(\dot{x}(t)^{2}-\Lambda (t)^{2} x(t)^{2} \bigr), $$

(9)

where \(\Lambda(t)^{2}=-(K'(x)^{2}+K(x)K''(x))|_{x=x_{\mathrm{opt}}(t)}\). Divergences due to the emergence of a slow mode on the barrier top [30, 33] are avoided by the appropriate choice of the initial (kinetic) energy, \(0<\dot{x}(0)^{2}/2 \ll\Delta U\) in order to satisfy the boundary conditions (8). Thus, the switching event takes place, with non-vanishing probability, within a *finite* time \(t_{\mathrm{f}}\). Within the saddle point approximation (9), the path integral (6) becomes Gaussian and can be evaluated analytically [34]

$$ O(t_{0})=\exp \biggl(\lambda\phi \biggl[x_{\mathrm{opt}}(t)+ \frac {x_{0}(t)}{2},s(t,t_{0}) \biggr] \biggr), $$

(10)

where \(x_{0}\) is the solution of

$$ \ddot{x}_{0}+\Lambda^{2} x_{0}+D\lambda s(t,t_{0})=0,\qquad x_{0}(0)=x_{0}(t_{\mathrm{f}})=0. $$

(11)

The two linearly independent solutions of the homogeneous part of Eq. (11) are

$$ x_{1}(t)=\dot{x}_{\mathrm{opt}}(t),\qquad x_{2}(t)=\dot{x}_{\mathrm{opt}}(t)\int_{0}^{t} \frac {dt'}{x^{2}_{\mathrm{opt}}(t')}, $$

(12)

and the full \(x_{0}(t)=x_{1}(t)c_{1}(t)+x_{2}(t)c_{2}(t)\) can be determined by variation of parameters.

We consider the case of a potential given by \(U(x)/\Omega=x/2-x^{3}/6\), which can, for example, approximate a Josephson junction biased at half the critical current. Note that this approximation does not account for retrapping of the Josephson junction in the supercurrent state after *n* full rotations of the phase, hence it applies when the bias is large enough to precvent such retracing or phase diffusion. \(\Omega=K'(x_{\mathrm{m}})\) plays the role of a characteristic frequency in our effective system but physically corresponds to the damping coefficient.