Observation of Wave-Packet Branching through an Engineered Conical Intersection

Analog quantum simulators, which efficiently represent model systems, have the potential to provide new insight toward naturally occurring phenomena beyond the capabilities of classical computers. Incorporating dissipation as a resource unlocks a wider range of out-of-equilibrium processes such as chemical reactions. Here, we operate a hybrid qubit-oscillator circuit quantum electrodynamics simulator and model nonadiabatic molecular dynamics through a conical intersection. We identify dephasing of the electronic qubit as the mechanism that drives wave-packet branching when the corresponding oscillator undergoes large amplitude motion. Furthermore, we directly observe enhanced branching when the wave-packet passes through the conical intersection. Thus, the forces that influence a chemical reaction can be viewed from the perspective of measurement backaction in quantum mechanics—there is an effective measurement-induced dephasing rate that depends on the position of the wave packet relative to the conical intersection. Our results set the groundwork for more complex simulations of chemical dynamics using quantum simulators, offering deeper insight into the role of dissipation in determining macroscopic quantities of interest such as the quantum yield of a chemical reaction.

Popular Summary

Most chemical reactions involve the transfer of electrons or protons and are largely driven by thermodynamics. A special class of reactions driven by light sometimes results in ultrafast dynamics where electronic and nuclear motions are strongly coupled. This occurs, for instance, in vision and is responsible for our ability to see. Such dynamics are called nonadiabatic and are generally costly to study numerically on classical computers. Quantum simulators promise to circumvent this cost, but building one that can efficiently represent electronic and nuclear motion with full control has been a challenge. Here, we implement such a quantum simulator and use it to study dynamics through an intriguing feature known as a conical intersection, a hallmark of nonadiabatic dynamics.

Our quantum simulator uses a superconducting circuit consisting of two microwave cavities (representing nuclear modes) and a nonlinear Josephson circuit (representing an electronic degree of freedom). By engineering the system’s energy landscape using microwave drives, we explore a regime where there is competition between coherent motion of a wave packet (much like a ball rolling down a hill) and dissipation. Because of the quantum nature of the conical intersection, however, the dissipation will cause the wave packet to branch from one surface to another. This phenomenon is at the heart of nonadiabatic reactions, which we intimately link to quantum measurement.

Our experiment paves the way for more complex studies of dissipative and strongly interacting out-of-equilibrium processes in nature.

Figure 1

Model molecular Hamiltonian with a conical intersection and experimental setup. (a) Eigenspectrum of ${\widehat{H}}_a$, which consists of two displaced quantum harmonic oscillator Fock ladders (dashed lines) of the tuning mode depending on if the qubit is in $|+⟩$ (blue) or $|-⟩$ (red). (b) In the basis of $|\pm ⟩$ for the qubit, the addition of ${\widehat{g}}_b=g_y{\widehat{\sigma }}_y(\widehat{b}+{\widehat{b}}^{†})$ on top of ${\widehat{H}}_a$ couples the two Fock ladders of the coupling mode (with energy ${\mathrm{\Delta }}_b{\widehat{b}}^{†}\widehat{b}$) via ${\widehat{\sigma }}_y=|+⟩⟨-|+|-⟩⟨+|$ (dashed arrows) for each Fock state of the tuning mode. Hence, dynamics of the coupling mode can cause transitions between $|\pm ⟩$. (c) Schematic of the circuit QED device used in this experiment. The tuning and coupling modes are 3D $\lambda /4$ coaxial resonators which are coupled to an electronic qubit (encoded in a transmon) with a readout resonator and Purcell filter [31]. An additional ancilla module with a transmon mode and readout resonator is coupled to the tuning mode for independent state tomography. The coupling mode is intentionally overcoupled to a $50\mathrm{\Omega }$ transmission line, resulting in a linewidth ${\kappa }_b/2\pi \approx 320\mathrm{kHz}$. (d) Relative mode frequencies and drive configuration used to enact ${\widehat{H}}_{\mathrm{LVC}}$. A full table of system parameters is given in Appendix pp3.

Figure 2

Calibrating coherent and dissipative conditional displacement interactions on the tuning mode (a),(b) and the coupling mode (c). (a) Depiction of tuning mode dynamics under ${\widehat{H}}_a$. The oscillator’s equilibrium position shifts by $+{\alpha }_g$ when the qubit is in $|-⟩$ (red) and $-{\alpha }_g$ when the qubit is in $|+⟩$ (blue). Top: an initial wave packet $|-⟩\otimes |\alpha =0{⟩}_a$ oscillates along the red potential surface associated with $|-⟩$. Bottom: in phase space, the state performs a circular trajectory around the equilibrium location, indicated by the red triangle for the top schematic. Smaller and larger equilibrium positions are indicated by the circle and square, respectively. (b) Coherent state revivals during evolution under ${\widehat{H}}_a$. We prepare an initial state $|-⟩\otimes |\alpha =0{⟩}_a$ and simultaneously measure the vacuum projector of the tuning mode ${\widehat{P}}_0^a=|0{⟩}_a⟨0{|}_a$ (top) and ${\widehat{\sigma }}_x$ (bottom) for different calibrated values of ${\mathrm{\Delta }}_a$. Solid lines are fits to a simple analytic model, extracting values $g_x/2\pi =450\mathrm{kHz}$ and ${\mathrm{\Delta }}_a/2\pi =457\mathrm{kHz}$ (circles), 355 kHz (triangles), and 246 kHz (squares) which determine the values of ${\alpha }_g$. Approximate locations of the coherent state for each calibration at a delay time marked by the magenta arrow are shown in the bottom panel of (a). At the same time, the qubit largely remains in $|-⟩$ up to intrinsic decoherence. (c) Top: coupling mode state trajectories starting in vacuum $|0{⟩}_b$ for various values of ${\mathrm{\Delta }}_b$ (rates from the bottom panel) with fixed values of ${\widehat{g}}_b$ and ${\kappa }_b$, splitting in phase space depending on if the qubit is in $|\pm y⟩$. Full coherent state distributions are not depicted because we are in the weak measurement regime where there is significant overlap between the coherent states. Bottom: measurement-induced dephasing of $|-⟩$ with $g_y/2\pi \approx 117\mathrm{kHz}$, ${\mathrm{\Delta }}_b/2\pi \approx \{0(\text{circles}),400(\text{triangles}),800(\text{squares})\}\mathrm{kHz}$ and ${\kappa }_b/2\pi \approx 320\mathrm{kHz}$ obtained from independent calibrations. Solid lines represent equivalent time-domain master equation simulation results (see Appendix pp6). All error bars indicate standard deviations and may not be visible compared to marker size.

Figure 3

Wave-packet branching under white noise dephasing. (a) A ground state reactive wave packet $|-⟩\otimes |{\alpha }_g⟩$ branches from the red potential surface to the blue potential surface when a dephasing event flips the electronic qubit from $|-⟩$ to $|+⟩$ (and vice versa). When the wave packet branches at random times, it will subsequently evolve along the blue potential surface which diffuses the oscillator state toward a mixed state. (b) Measured (left) and simulated (right) Wigner functions of the tuning mode conditioned on the ${\widehat{\sigma }}_x$ measurement outcome being either $|-⟩$ (top, $P_{-}=91\%$) or $|+⟩$ (bottom, $P_{+}=9\%$) after evolving $|{\psi }_0⟩=|-⟩\otimes |+{\alpha }_g{⟩}_a\otimes |0{⟩}_b$ for $10\mathrm{\mu }\mathrm{s}$ under ${\widehat{H}}_x$ with the parameters listed in the main text. Experimentally, the overall phase of the measured Wigner functions is determined by the phase acquired by the cavity relative to the initial displacement due to Stark shifts and is free to be adjusted [40]. The phase in the simulation is aligned to that of the coherent projected data ($|-⟩$). Additional deviations of the experimental data from the simulation are likely due to higher order Hamiltonian terms, whose precise calibration and control remain an open task left for future experiments.

Figure 4

Branching through a conical intersection. (a) Top: measured expectation value of ${\widehat{\sigma }}_x$ with standard error bars over time for the three different initial states $|-⟩\otimes |{\alpha }_0{⟩}_a\approx \{|0{⟩}_a(\text{magenta}),|{\alpha }_g{⟩}_a(\text{blue green}),|2{\alpha }_g{⟩}_a(\text{orange})\}\otimes |0{⟩}_b$, where ${\alpha }_g=g_x/{\mathrm{\Delta }}_a\approx 1.26$. The magenta wave packet prepared at the CI immediately dephases, whereas the other two dephase more slowly, as they are farther away. After half of an oscillation period $\tau \sim 1/(2{\mathrm{\Delta }}_a)$, the orange wave packet arrives at the CI and dephases. Solid lines are predictions from a master equation simulation using independently fitted parameters. Dashed lines represent the negligible background decoherence due to $T_{2\rho }^x$ on the timescale of the interaction and dissipation. Bottom: unconditional Wigner tomography on the tuning mode at $\tau =2\mathrm{\mu }\mathrm{s}$ (left) and $6\mathrm{\mu }\mathrm{s}$ (right) for preparing $|{\alpha }_0{⟩}_a\approx |2{\alpha }_g{⟩}_a$, revealing a coherent wave packet before and dephased state after passage through the CI. The distortion of the Wigner function from a Gaussian at $\tau =2\mathrm{\mu }\mathrm{s}$ suggests the presence of a residual self-Kerr nonlinearity in the oscillator. (b) A semiclassical interpretation of the dephasing versus location of the three initial wave packets prepared in (a). The potential surfaces are shown in dashed lines to emphasize that they are no longer exact in the diagonalized basis of ${\widehat{H}}_{\mathrm{LVC}}$. The dissipative dynamics arises from a competition between the ${\widehat{\sigma }}_x$ energy gap $E$ with the finite strength of the dephasing ${\mathrm{\Gamma }}_{\mathrm{meas}}$.

Figure 5

Optimizing the static detuning. By numerically diagonalizing ${\widehat{H}}_q/\hslash$ for various values of ${\mathrm{\Delta }}_R$, we can plot the dimensionless factor that contributes to the residual cross-Kerr for ${\mathrm{\Omega }}_R/2\pi =80\mathrm{MHz}$ and ${\alpha }_q/2\pi =244\mathrm{MHz}$. We find an optimal value of ${\mathrm{\Delta }}_R/2\pi \approx 7\mathrm{MHz}$.

Figure 6

Experimental wiring diagram.

Figure 7

Tracking system stability over time. Top panel: simultaneous measurements of the Rabi frequency and the ambient temperature (inferred from the measured thermistor resistance) where the active microwave components are held. The data shows that the two are correlated. Middle panel: extracting driven coherences $T_{2\rho }$ suggests a stable amplitude noise spectrum within an acquisition time ${\tau }_{\mathrm{acq}}=3\min$ (bottom left-hand panel: a typical time-domain Ramsey trace), with a single instance where the amplitude drift was large (bottom right-hand panel), as confirmed via looking at the raw data binned 10 shots at a time over ${\tau }_{\mathrm{acq}}$.

Figure 8

Free evolution of the driven Rabi qubit. (a) A Ramsey-style pulse sequence is used to calibrate the decode rate. The final $\pi /2$ rotation has a phase that depends on both the delay time $\tau$ and the programmed decode rate ${\mathrm{\Omega }}_d$. (b) The 2D plot reveals an optimal decode rate which is inferred to be equal to the true Rabi frequency offset by the static detuning ${\mathrm{\Omega }}_R-{\mathrm{\Delta }}_R$. (c) The decode rate can then be fixed to the optimal value to extract a driven transverse relaxation time $T_{2\rho }\approx 27\mathrm{\mu }\mathrm{s}$ via fitting to an exponential decay function (solid line).

Figure 9

Calibrating individual exchange interactions. (a) Pulse sequence. The transmon is initialized in either $|g⟩$ or flipped to $|e⟩$ with a $\pi$ rotation depending on if the red or blue sideband is activated. The sideband is turned on with a ramp time ${\tau }_{\mathrm{ramp}}^{\mathrm{SB}}\gg 1/{\mathrm{\Delta }}_{\mathrm{SB}}$, where ${\mathrm{\Delta }}_{\mathrm{SB}}={\omega }_{\mathrm{pump}}-{\omega }_c$ is the detuning from the sideband to the Stark-shifted cavity frequency. A frequency selective $\pi$ rotation of the ancilla qubit maps the vacuum projector of the tuning mode onto the state of the ancilla, which is subsequently measured with the qubit simultaneously. (b) Interaction strengths $g$ for individual sideband interactions between the driven qubit and the tuning mode (triangles) or coupling mode (squares). $g$ is extracted via a fit using Eq. (d4) for the red sidebands (red markers) and blue sidebands (blue markers). Error bars are smaller than the markers. $\xi$ is calibrated via independent measurements of ${\chi }_{a/b}$ and the qubit’s Stark shift as a function of pump amplitude. Dashed and dotted lines represent the expected interaction strength in the ideal case $g={\chi }_{a/b}\xi /2$ for the tuning and coupling modes, respectively. Deviations of the data from a linear relation for the tuning mode coincides with regions where the Stark shift becomes nonlinear in pump power. Slight disagreement between the data and theory expectation for the coupling mode may stem from an inaccurate estimate of ${\chi }_b$ and from a sensitivity of the extracted fit values to a true value of ${\kappa }_b$. (c) Raw calibration data of the qubit population for a fixed pump amplitude [dashed circles in (b)] for the tuning mode (top) and coupling mode (bottom). The sideband detunings are referenced to the cavity frequencies, showing that we are operating at Rabi frequencies ${\mathrm{\Omega }}_R/2\pi \approx 80\mathrm{MHz}$.

Figure 10

Calibrating conditional displacements. (a) Pulse sequence. The displacement pulse is played after the Rabi drive is turned on and the cross-Kerr is nulled. An echo sequence eliminates the residual entanglement between the qubit and cavity during the ramp off of the Rabi drive. (b) Top: coherent revivals for various initial displacements along the interaction axis, probed through measurement of the vacuum projector. The contrast is maximized when the wave packet passes through the origin, and oscillations vanish when the ground state ${\alpha }_g\approx 1.3$ is prepared. The data in Fig. 3 are taken at the location of the yellow star. Bottom: simultaneous measurement of $T_{2\rho }^x$ reveals driven coherence times with a slight dependence on $⟨{\widehat{a}}^{†}\widehat{a}⟩$.

Figure 11

Echoing away residual entanglement. Simultaneous measurement of the vacuum projector on the tuning mode via a transmon ancilla (left) and the driven qubit (right) while ${\widehat{H}}_a$ is active. The pulse sequence used for this calibration is shown in Fig. 10 without (top) and with (bottom) the $\pi$ rotation and echo delay ${\tau }_{\mathrm{ED}}$ and only sidebands on the tuning mode. There is no initial displacement amplitude (${\alpha }_0=0$). Time dynamics of the tuning mode’s vacuum projector reveals coherent revivals as expected [as in Fig. 2 of the main text], independent of the decode rate on the driven qubit. Scanning the decode rate without the echo sequence results in distortions to the driven Ramsey data that are correlated with the cavity photon distribution. Implementing the echo sequence eliminates this effect, suggesting that the systems remain unentangled at all delay times. Here, we use an optimized value of the delay time ${\tau }_{\mathrm{ED}}=144\mathrm{ns}$.

Figure 12

Extended data and leakage statistics. (a)–(c) Probability of leaking out of the qubit subspace to $|f⟩$ for the data presented in Figs. 2, 2, and 10, respectively. (d) The same dataset as appears in Fig. 4 (left), but including two additional initial states with $|{\alpha }_0{⟩}_a=|{\alpha }_g/2{⟩}_a$ (blue) and $|3{\alpha }_g/2{⟩}_a$ (red), with the associated leakage probabilities (right). All of the leakage probabilities are equivalent to the percentage of data that is postselected away for the rest of the data in the paper, which is always less than 5%.