Quantum caustics in resonance-fluorescence trajectories

We employ phase-sensitive amplification to perform homodyne detection of the resonance fluorescence from a driven superconducting artificial atom. Entanglement between the emitter and its fluorescence allows us to track the individual quantum state trajectories of the emitter conditioned on the outcomes of the field measurements. We analyze the ensemble properties of these trajectories by considering trajectories that connect specific initial and final states. By applying the stochastic path-integral formalism, we calculate equations of motion for the most-likely path between two quantum states and compare these predicted paths to experimental data. Drawing on the mathematical similarity between the action formalism of the most-likely quantum paths and ray optics, we study the emergence of caustics in quantum trajectories: places where multiple extrema in the stochastic action occur. We observe such multiple most-likely paths in experimental data and find these paths to be in reasonable quantitative agreement with theoretical calculations.

Figure 1

Resonance fluorescence quantum trajectories. (a) The experiment uses a near-quantum-limited Josephson parametric amplifier to perform homodyne measurement of the fluorescence emitted by an effective two-level system that is realized by resonant interaction of a transmon circuit and a 3D aluminum cavity. (b) The dimensionless homodyne signal (denoted by $d{I}_t$ at time step $t)$ reflects the quantum fluctuations of the measured electromagnetic mode and is normalized so that its variance is $\gamma dt$. Also shown are the (c) $x$ and (d) $z$ components of several trajectories (faded colors) calculated using the SME and homodyne signal. A specific trajectory $(\tilde{x},\tilde{z})$ (dashed lines) is compared to its tomographic reconstruction $(x,z)$ (solid lines). The close agreement between the curves indicates that the SME faithfully tracks the quantum state.

Figure 2

Path probability and most-likely paths. (a) Demonstration of path-probability calculation. The blue (dashed) curve indicates $\sqrt{\eta }\gamma xdt$, which determines the probability distribution (blue background color) for a corresponding homodyne signal (red solid curve). (b) Cross section of the probability density at the time indicated by the long-dashed green line of (a). The Gaussian distribution is shifted from zero by $\sqrt{\eta }\gamma xdt$. The arrow shows the value of the homodyne signal at that point. (c) Pre- and postselected trajectories for initial $\left\{z\right(0),x(0\left)\right\}=\{-0.97,0\}$ and final $\left\{z\right(1.94\mu \mathrm{s}),x(1.94\mu \mathrm{s}\left)\right\}=\{-0.3,-0.6\}$ boundary conditions (only $z$ traces are shown). The top 5% of trajectories having the shortest Euclidean distance from the others are highlighted in black. (d) The solid black line originating from $-0.97$ is the theoretical MLP for $z$ obtained from Eq. (6). The solid blue line originating from 0 is the theoretical MLP for $x$. The dashed lines are the corresponding experimental MLPs obtained by averaging the highlighted trajectories. The shaded area along experimental curves shows the standard deviation of the averaged trajectories (see Appendix pp5 for statistical information).

Figure 3

Comparison of different experimental measures for most-likely paths. We compare the two different measures for experimental MLPs to theory for the initial $\left\{z\right(0),x(0\left)\right\}=\{-0.97,0\}$ and final $\left\{z\right(1.94\mu \mathrm{s}),x(1.94\mu \mathrm{s}\left)\right\}=\{-0.3,-0.6\}$ states; trajectories are shown for (a) $x$ and (b) $z$. The Euclidean distance method (red long-dashed lines) and path probability density maximization (blue short-dashed lines) are both in close agreement with theory (black). The average of the full ensemble of pre- and postselected trajectories (gray) deviates significantly from the MLP.

Figure 4

Lagrange manifold. (a) Schematic representation of the Lagrange manifold folding leading to multiple most-likely paths. For an initial state $\{z_i,x_i\}$ we consider all possible initial momenta, represented as a vertical red line. Evolution under Eqs. (8) results in different final states for different initial momenta specifying the most-likely path that connects initial and final states, as shown in the left panels. When the red curve fails the vertical line test as shown in the right panels; multiple initial momenta result in the same final state. (b) and (c) The Lagrange manifold used to find MMLPs in this system is a two-dimensional object in a four-dimensional Hamiltonian phase space; we show its projection into the $x–z$ plane of the Bloch sphere. A sampled range of momenta $\{p_{zi},p_{xi}\}$ are initialized for the state $\left(\{z_i,x_i\}=\{-1,0\}\right)$ at $t=0$ and allowed to evolve up to (b) $T=1.4\mu \text{s}$ and (c) $T=2.25\mu \text{s}$; each point in the scatter plot corresponds to a path generated by a different initial momentum. Compare with (a): Just as a line of all $p$ may intersect the manifold one or more times at a given $q_f$, we may now imagine sticking a pin into the figure at a particular point $\{z_f,x_f\}$, which might go through one or more layers of the manifold. Each intersection or layer corresponds to a distinct path reaching the chosen final state. Caustic regions emanating from a catastrophe in the manifold are clearly visible at both of the times shown above. The counterclockwise spiraling of the manifold is the result of the Rabi drive applied to the qubit; we see that overlap in the manifold due to this spiraling matures into a cusp catastrophe over time. Color denotes stochastic energy $E$ for each path. The two theory MLPs that are compared with data in Fig. 5 are shown superposed in black in (b); these are the paths that two distinct points on the manifold trace out as each evolves and folds over itself.

Figure 5

Experimental multiple most-likely paths. (a) and (b) By applying a clustering algorithm to experimental trajectories that attain the chosen initial and final boundary conditions, we obtain two groups that minimize the mean Euclidean distance between members of each group. The trajectories in each group with lowest Euclidean distance from other members in the group are highlighted in black $(z$ trajectories are shown). (c) and (d) Comparison to theory; the solid curves are theory predictions and the dashed lines are obtained from averaging the highlighted experimental trajectories. The curves starting from 0 are the $x$ trajectories and the curves starting from $-0.97$ are the $z$ trajectories. The shaded area along experimental curves shows the standard deviation of the averaged trajectories (see Appendix pp5 for statistical information).

Figure 6

Relative probability of multiple most-likely paths. (a) The peak of the probability density distribution of the Euclidean distance from the MLP (green, left distribution) is related to the extremum of the stochastic action $\left({\mathcal{S}}_{\max }\right)$. The observed distribution (blue, center distribution) has a maximum that is shifted from 0 due to the increasing multiplicity of trajectories (right distribution) with a larger Euclidean distance from the MLP. (b) To compare the experimental distributions to the theoretical stochastic entropy maxima, we use the multiplicity $n\left(\mathcal{E}\right)$ and observed distribution $H\left(\mathcal{E}\right)$ to determine $P\left(0\right)$. We compare the relative probability $P{\left(0\right)}_{\mathrm{cluster}1}/P{\left(0\right)}_{\mathrm{cluster}2}$ to the predicted relative probability $S_1-S_2$. Error bars indicate the uncertainty in the fit to the relative distribution (see Appendix pp5 for statistical information). We apply this analysis to four different MMLP pairs with initial state $\left\{z\right(0),x(0\left)\right\}=\{-0.97,0\}$ and final states $\left\{z\right(T),x(T),T(\mu \mathrm{s}\left)\right\}=\{0.65,-0.08,1.2\},\{0.19,-0.93,1.4\},\{0.36,0.47,1\}$, and $\{0.21,-0.62,1.4\}$ (from left to right on the graph).

Figure 7

We show lines of constant stochastic energy (top) and lines of constant $\dot{S}$ (bottom). Here $E$ and $\dot{S}$ are in the same units of MHz as $\mathrm{\Omega }$ and $\gamma$. Green (top) and red (bottom) lines are those for the energy $E_c=-2.13$ MHz containing the lone fixed point at $(\overline{\theta }=4.16,\overline{p}=0.967)$. Paths in region $A$ all travel right to left (with the Rabi drive) and remain relatively probable over long periods of time provided they have a modest momentum or stochastic energy. Paths in region $B$ contain both branches $p_{+}$ and $p_{-}$ (A6), which move in opposite directions; these paths may also have reasonably high relative probabilities to occur for initial conditions just to the left of the fixed point, over modest time evolutions (approximately less than the time required to almost a period or approach the next fixed point), but asymptote into regions of very negative $\dot{S}$ over long time intervals. Paths in region $C$ only go from left to right (against the Rabi drive) and inhabit regions of more negative $\dot{S}$, meaning that they are relatively improbable over any appreciable time interval.

Figure 8

We plot the locations of fixed points $\overline{\theta }$ against the dimensionless parameter $\omega =\mathrm{\Omega }/\gamma$. Color and size denote the momentum $\overline{p}$. Experimentally we operate at $\omega =0.63\times 2\pi$, where there is only one fixed point. This fixed point is a location in the phase space where the Rabi drive and fluorescence cancel. Another pair forms for $\omega <0.145={\omega }_c$, splitting around $\theta =5.18$. As the drive is eliminated entirely, these show that the system is able to stall at either the ground $(\theta =0$ or $2\pi )$ or excited $\left(\theta =\pi \right)$ states. All fixed points exist on the half of the Bloch sphere where the fluorescence and drive compete (the drive pushes the state from ground to excited, whereas the fluorescence pushes the state from excited to ground); none exist on the half where the fluorescence and drive work together (both push towards the ground state), indicating that no optimal readout can fight both the drive and spontaneous emission at the same time to hold the system still at some particular state.

Figure 9

We show MMLPs with different winding numbers, from ${\theta }_i=\pi$ to either ${\theta }_f^{\left(B\right)}=-1.24$ (black thick-dashed line) or ${\theta }_f^{\left(A\right)}=-1.24-2\pi$ (blue solid line). These are shown as (a) $\theta \left(t\right)$ and (b) and (c) $x\left(t\right)$ and $z\left(t\right)$ and (d) in phase space. In the phase-space plot we show ${\theta }_i$ as a yellow dash-dotted line (the Lagrangian manifold at $t=0)$, the two ${\theta }_f$ as cyan dash-dotted lines, the separatrix as a red dotted line, and the Lagrangian manifold at $T=1.4\mu \text{s}$ as a green dash-dotted green line. Finally, in (e) we show the pure state paths (still blue solid and black thick-dashed lines) in the $x–z$ plane of the Bloch sphere. The red dotted line marks the edge of the sphere; the pure-state curves are given an artificial radius outside the sphere for added visibility. The cyan solid and green thin-dashed curves in (e) are the theory calculations for the example with $\eta =0.45$ from Figs. 4 and 5, shown for comparison.

Figure 10

We include the phase portrait for $\omega =0.13$ and $\gamma =1$ (in the regime with three fixed points), with a complete picture (top) and reduced range to zoom in on the features of greatest interest (bottom). Two of the fixed points are hyperbolic (pink $\times )$, at $(\overline{\theta }=6.04,\overline{p}=-0.201)$ and $(\overline{\theta }=4.63,\overline{p}=-1.25)$, and one is elliptical (pink $+)$, at $(\overline{\theta }=5.55,\overline{p}=-5.63)$. The latter two points are responsible for partitioning off the new regions $D$ and $E$ inside region $B$. Periodic paths are possible within the island $D$, which is bounded by a separatrix of energy $E_s=-0.938$ MHz. We note that for smaller $\omega$ compared with Fig. 7, we have a larger region $B$, which now encroaches asymmetrically into $A$ (making winding-number MMLPs less likely), and region $C$ has grown (indicating that it is easier for trajectories to travel against the weaker Rabi drive).