Engineering bilinear mode coupling in circuit QED: Theory and experiment

Photonic states of high-$Q$ superconducting microwave cavities controlled by superconducting transmon ancillas provide a platform for encoding and manipulating quantum information. A key challenge in scaling up the platform towards practical quantum computation is the requirement to communicate on demand the quantum information stored in the cavities. It has been recently demonstrated that a tunable bilinear interaction between two cavity modes can be realized by coupling the modes to a bichromatically driven superconducting transmon ancilla, which allows swapping and interfering the multiphoton states stored in the cavity modes [Gao et al., Phys. Rev. X 8, 021073 (2018)]. Here we explore both theoretically and experimentally the regime of relatively strong drives on the ancilla needed to achieve fast swap gates but which can also lead to undesired nonperturbative effects that lower the swap fidelity. We develop a theoretical formalism based on linear response theory that allows one to calculate the rate of ancilla-induced interaction, decay, and frequency shift of the cavity modes in terms of a susceptibility matrix. We go beyond the usual perturbative treatment of the drives by using Floquet theory, and find that the interference of the two drives can strongly alter the system dynamics even in the regime where the standard rotating wave approximation applies. The drive-induced ac Stark shift on the ancilla depends nontrivially on the drive and ancilla parameters which in turn modify the strength of the engineered interaction. We identify two major sources of infidelity due to ancilla decoherence. (i) Ancilla dissipation and dephasing lead to incoherent hopping among Floquet states which occurs even when the ancilla is at zero temperature; this hopping results in a sudden change of the swap rate, thereby decohering the swap operation. (ii) The cavity modes inherit finite decay from the relatively lossy ancilla through the inverse Purcell effect; the effect becomes particularly strong when the ac Stark shift pushes certain ancilla transition frequencies to the vicinity of the cavity mode frequencies. The theoretical predictions agree quantitatively with the experimental results, paving the way for using the developed theory for optimizing future experiments and architecture designs.

Figure 1

A schematic showing how a beam-splitter interaction between two cavity modes at different frequencies arises due to their couplings to a driven ancilla. The frequency of cavity mode $a$ at the ancilla input (as a probe) is up- or down-converted by integer multiples $K$ of drive frequency difference ${\omega }_{21}$ at the output (as a response) which then interacts with mode $b$, whereas the frequency of cavity mode $b$ at the ancilla input (as a probe) is down- or up-converted by $K{\omega }_{21}$ at the output (as a response) which then interacts with mode $a$. When the condition ${\omega }_b-{\omega }_a=K{\omega }_{21}$ is satisfied, there arises a resonant beam-splitter interaction between the cavities $a$ and $b$. The energy needed for this process to become resonant is provided by an indirect exchange of $K$ excitations between the two drive reservoirs (indicated by the dashed red arrow), which is a result of their individual interaction with the ancilla (indicated by the blue arrows).

Figure 2

Illustration of quasienergy level anticrossings. Shown on the left is an example of quasienergy spectrum for the case of one blue-detuned drive (drive 1) as a function of the scaled drive power $|{\xi }_1{|}^2$. The detuning of the drive is fixed at ${\delta }_1=\alpha .$ From top to down, the thick solid lines refer to quasienergy levels ${\epsilon }_m$ in Eq. (24) with $m=0$ (black), 1(blue), 2 (red), and 3 (green). Turning on the second drive (drive 2) leads to anticrossings of quasienergy levels when projected into the same Brillouin zone; equivalently, there occurs anticrossing between replicas of quasienergy levels ${\epsilon }_m$ which are shifted from ${\epsilon }_m$ by integer multiples of $\hslash {\omega }_{21}$ as shown by the thin dashed lines. As an illustration, we choose the detuning of drive 2 to be ${\delta }_2=4.5\alpha$. On the right is a schematic for the anticrossing between quasienergy levels at ${\epsilon }_0$ and ${\epsilon }_2+\hslash {\omega }_{21}$. The gap ${\mathrm{\Omega }}_R$ of the anticrossing for a weak drive 2 is given by Eq. (27).

Figure 3

(a) The scaled ac Stark shift $\delta E_{10}/2\hslash \alpha$ of the transition frequency from the first excited to the ground state of the ancilla as a function of scaled power of the two drives. The solid lines are contours of constant $\delta E_{10}$. For weak drives, the contours are straight lines well described by Eq. (29). For strong drives, the contours become curved and the Stark shift becomes nonlinear in the drive power. For the purpose of comparing with the perturbative result of the Stark shift at weak drives $\delta E_{10}/\hslash \approx -2\alpha {\sum }_{j=1,2}{\left|{\xi }_j\right|}^2{\delta }_j/({\delta }_j+\alpha )$, we have chosen a proper scaling for the $x$ and $y$ axes so that for small values of $x$ and $y$, the scaled Stark shift is simply equal to $-(x+y)$. The detunings of the two drives are ${\delta }_1=\alpha ,{\delta }_2=4.5\alpha$. (b) The ac-Stark-shifted transition frequency ${(E+\delta E)}_{n(n-1)}/\hslash$ of the ancilla in the presence of one drive (drive 1). The transition frequencies are counted from the bare ancilla frequency $E_{10}/\hslash \equiv {\omega }_c$ and scaled by $\alpha$. The black, blue, red, and green lines (from top to down at low drive power) refer to $n=1$, 2, 3, and 4, respectively. The detuning of the drive ${\delta }_1=\alpha$. In (a), due to level anticrossings, the quantity $\delta E_{10}$ is generally discontinuous at particular values of drive amplitudes. We make the contour plot by choosing a discrete set of drive amplitudes and interpolate among them; the discontinuities appear to be smeared out by such interpolation.

Figure 4

Comparison of the scaled ac Stark shift $\delta E_{10}/2\hslash \alpha$ between the theory (solid lines) and the experiment (black dots). (a) ac Stark shift $\delta E_{10}$ as a function of the scaled drive-1 power while drive 2 is turned off. (b) ac Stark shift $\delta E_{10}$ as a function of the scaled drive-2 power for various drive-1 strengths. From top to bottom, the scaled drive-1 power $|{\xi }_1{|}^2{\delta }_1/({\delta }_1+\alpha )=0$, 0.87, 1.35, 1.95. For a relatively large drive-1 power, the ac Stark shift due to drive 2 becomes nonmonotonic in its power. The detunings of the two drives are the same as in Fig. 3: ${\delta }_1=\alpha ,{\delta }_2=4.5\alpha$. The theoretical curve in the top panel and the top curve in the bottom panel are used to calibrate the amplitudes of the two drives, namely, to find out the conversion factor between the drive amplitudes ${\mathrm{\Omega }}_{1,2}$ in the Hamiltonian and the readout of the pulse generator in the experiment.

Figure 5

A schematic showing how the linear susceptibilities arise from virtual transitions between quasienergy states of driven ancillas in the rotating frame of drive 1. (a) A schematic showing the process that gives the first term in $\chi (\omega ,\omega +K{\omega }_{21})$. $\nu$ is the frequency of probe field in the rotating frame of drive 1: $\nu =\omega -{\omega }_1.$ (b) A schematic showing the process that gives the first term in $X(\omega ,2{\omega }_1+K{\omega }_{21}-\omega )$.

Figure 6

Comparison between perturbation theory and full Floquet theory on the susceptibility ${\chi }_0(\omega ,\omega +{\omega }_{21})$ which is responsible for the ancilla-mediated beam-splitter interaction between two cavity modes with frequencies ${\omega }_b-{\omega }_a={\omega }_{21}$. (a) Susceptibility ${\chi }_0(\omega ,\omega +{\omega }_{21})$ as a function of probe frequency $\omega$ at fixed drive strengths: $|{\xi }_1|=1.5,|{\xi }_2|=0.14$. The detunings of the drives are ${\delta }_1/\alpha =1,{\delta }_2/\alpha =16.1$ as indicated by the vertical dashed lines. The red and black dots refer to results of the full Floquet theory and perturbation theory [Eq. (36)], respectively. We have chosen the relative phase of the two drives to be zero, so that ${\chi }_0(\omega ,\omega +{\omega }_{21})$ is real in the absence of ancilla decoherence. (b) The scaled beam-splitter interaction strength $|{\overline{g}}_{\text{BS}}|$ between the two cavity modes as a function of driving amplitudes $|{\xi }_1|$ and $|{\xi }_2|$ calculated using Floquet theory. ${\overline{g}}_{\text{BS}}$ is related to the beam-splitter strength $g_{\mathrm{BS}}$ in Eq. (12) by a constant factor: $g_{\mathrm{BS}}=\zeta {\overline{g}}_{\mathrm{BS}},\zeta =-\frac{2\alpha g_a^{*}{g}_b}{{\delta }_a{\delta }_b}\frac{{\delta }_a+{\delta }_2}{{\delta }_a+{\delta }_2+\alpha }$. The scaling factor is chosen so that at weak drives, ${\overline{g}}_{\mathrm{BS}}\approx {\xi }_1{\xi }_2^{*}.$ The solid lines are contours of constant ${\overline{g}}_{\mathrm{BS}}$. As a comparison, the dashed lines are contours of constant $|{\xi }_1{\xi }_2|$. The frequencies of cavity modes are detuned from the ancilla frequency by ${\delta }_a/\alpha =-6.9,{\delta }_b={\delta }_a+{\omega }_{21}$ as also indicated as a vertical dashed line in the top panel.

Figure 7

Comparison between the theory and experiment on the ancilla-induced beam-splitter rate $g_{\mathrm{BS}}$ between the two cavity modes whose frequencies satisfy ${\omega }_b-{\omega }_a={\omega }_{21}$. The red solid and black dashed lines refer to the results of Floquet theory and perturbation theory in Eq. (36), respectively. The black dots refer to the experimental results. The detunings of the drives and cavity modes from the ancilla frequency are the same as in Fig. 6. (a) $|{\overline{g}}_{\mathrm{BS}}({\xi }_1,{\xi }_2)|$ as a function of $|{\xi }_1|$ at fixed $|{\xi }_2|=0.14.$ (b) $|{\overline{g}}_{\mathrm{BS}}({\xi }_1,{\xi }_2)|$ as a function of $|{\xi }_2|$ at fixed $|{\xi }_1|=2.14.$ The dots are experimental results. In the experiment, parameter $\left|\zeta \right|/2\pi =0.33$ MHz corresponding to cavity-ancilla coupling strengths $|g_a/{\delta }_a|=0.047,|g_b/{\delta }_b|=0.054$.

Figure 8

Comparison between the theory (solid lines) and experiment (dots) on the heating from the Floquet ground state ${\mathrm{\Psi }}_0$ in the presence of one drive (drive 1). The detuning of the drive ${\delta }_1=\alpha =2\pi *71.68\mathrm{MHz}$. (a) The steady-state population of not in the ground Floquet state as a function of the scaled drive power. The experimental data are taken for a pump duration $\sim 100\mu \mathrm{s}$. The blue, black, and green dots refer to 48, 100, and 1000 ns ramping time, respectively. The spike around $|{\xi }_1{|}^2=0.5$ is likely due to leakage from the mixer that excites the ancilla when the ancilla frequency is Stark shifted to the frequency of this tone. The theoretical curve is a result of Eq. (44): $P_0\equiv {\rho }_{00}$. We use a constant $\gamma$ and ${\gamma }_{\mathrm{ph}}^{\left(\mathrm{hf}\right)}$ whose ratio is taken to be ${\gamma }_{\mathrm{ph}}^{\left(\mathrm{hf}\right)}/\gamma =0$ (black), 1/60 (red), and 1/30 (blue); see the text for details. The dashed lines are results of semiclassical calculation (see Appendix pp6-s2a). (b) The transmon population not in the ground Floquet states as a function of the dimensionless time $\gamma t$ for various scaled drive powers. From bottom to top, the scaled drive powers are $|{\xi }_1{|}^2=0$ (black), 0.55 (blue), 1.52 (red), 4.85 (green), and 8.66 (purple). At $t=0$, when the drive is turned on, the ancilla is in an effective thermal equilibrium with its environment with a thermal population $n_{\mathrm{th}}\approx 0.006$. The independently measured decay rate of the ancilla is $\gamma =90$ kHz. The solid lines refer to the results of simulation using Eq. (44) where we have neglected the finite time of ramping up and down the drive which is much shorter than the timescale set by $1/\gamma$ and assumed that the ancilla adiabatically evolves from Fock states $|n\rangle$ to the adiabatically connected Floquet states $|{\mathrm{\Psi }}_n\rangle$ during these times.

Figure 9

The decoherence rate of a superposition of Floquet states ${\mathrm{\Psi }}_0$ and ${\mathrm{\Psi }}_n$ as a function of the scaled drive power for the case of one blue-detuned drive with detuning ${\delta }_1/\alpha =1$. The solid and dashed lines refer to the full Floquet results and semiclassical results in Appendix pp6-s2b, respectively. (a) The dissipation-induced decoherence rate $V_{0n}^{\gamma }$ in Eq. (51). (b) The dephasing-induced decoherence rate $V_{0n}^{{\gamma }_{\mathrm{ph}}}$ in Eq. (52). The ratio ${\gamma }_{\mathrm{ph}}^{\left(\mathrm{hf}\right)}/{\gamma }_{\mathrm{ph}}\left(0\right)$ is chosen to be 1/60 as we found in Sec. 4b3.

Figure 10

Illustration of the inverse Purcell effect in the presence of one drive (drive 1) on the ancilla. (a) An example of partial susceptibility spectrum $\mathrm{Im}{\chi }_0^{\mathrm{st}}(\omega ,\omega )$ of the driven ancilla calculated using Eq. (54). The scaled drive power is $|{\xi }_1{|}^2=2$ and drive detuning is ${\delta }_1/\alpha =1.$ (b) Comparison between the experiment (red dots) and theory (curves) on the rate of drive-induced decay of cavity $a$ as a function of the scaled drive power. As the drive power increases, the cavity becomes close to the resonance at frequency ${\nu }_{030}$ that excites the ancilla from the state ${\mathrm{\Psi }}_0$ to ${\mathrm{\Psi }}_3$ by absorbing two drive-1 photons and the cavity photon [the inset shows the susceptibility spectrum ${10}^3\alpha \mathrm{Im}{\chi }_0^{\mathrm{st}}(\omega ,\omega )$ as a function of the drive power]. The vertical axis is the total decay rate of cavity $a$ minus the same quantity in the absence of drive on the ancilla. The points with different colors (red and magenta) represent two sets of measurements with different cavity decay rates: ${\kappa }_a({\xi }_1=0)=6.13\pm 0.08$ (kHz) (red) and $5.71\pm 0.06$ (kHz) (magenta). In the theoretical calculation using Eq. (57) (black line), we used the previously found dephasing rate ${\gamma }_{\mathrm{ph}}^{\left(\mathrm{hf}\right)}=\gamma /60$ to calculate the rate of dephasing-induced hopping among ancilla Floquet states, and the Ramsey dephasing rate ${\gamma }_{\mathrm{ph}}^{\left(R\right)}\approx \gamma$ to calculate the rate of incoherent hopping between the cavity and ancilla. As explained in the text, this choice overestimates the latter hopping rate when the cavity is far detuned from the ancilla resonance, but matches the experiment well near the resonance which occurs at strong drive. As a comparison, the green line shows the result where an overall constant ${\gamma }_{\mathrm{ph}}\left(\omega \right)={\gamma }_{\mathrm{ph}}^{\left(\mathrm{hf}\right)}$ was used. The dashed black line represents the partial contribution from the resonant peak at frequency ${\nu }_{030}$. The coupling strength $g_a$ and the detuning ${\delta }_a$ between cavity $a$ and the ancilla frequency ${\omega }_c$ is the same as in the beam-splitter experiment: $|g_a/{\delta }_a|=0.047,{\delta }_a/\alpha =-6.9.$

Figure 11

Comparison of the susceptibility spectra ${\chi }_m(\omega ,\omega +{\omega }_{21})$ in Eq. (30) for different $m$. The red, blue, black, and green dots correspond to $m=0$, 1, 2, and 3, respectively. (a) The scaled anharmonicity $\alpha /{\delta }_1=0.14$. The ratio of the drive detunings ${\delta }_2/{\delta }_1=3.1$. The scaled drive amplitudes ${\xi }_1=1.5,{\xi }_2=0.15.$ (b) $\alpha /{\delta }_1$ is reduced by a factor of 7 compared to (a), the drive amplitudes ${\xi }_1$ and ${\xi }_2$ are both increased by a factor of $\sqrt{7}$ so that the quantity $\alpha {\xi }_1{\xi }_2/{\delta }_1$ (which sets the beam-splitter rate) remains unchanged. The ratio ${\delta }_2/{\delta }_1$ remains the same. The inset shows the ratio $({\chi }_m-{\chi }_0)/{\chi }_0$ for $m=1$ (blue), 2 (black), and 3 (green) for the frequency region inside the dashed box. At particular probe frequencies, the difference between ${\chi }_{m\ne 0}$ and ${\chi }_0$ vanishes. The comparison between (a) and (b) clearly shows that the difference between the spectra ${\chi }_m(\omega ,\omega +{\omega }_{21})$ for different $m$ goes down as the scaled anharmonicity $\alpha /{\delta }_1$ goes down.

Figure 12

Comparison of the susceptibility spectra ${\chi }_m(\omega ,\omega )$ in Eq. (30) for different $m$. The red, blue, black, and green dots correspond to $m=0$, 1, 2, and 3, respectively. The parameters used in (a) and (b) are the same as those in Figs. 11 and 11, respectively. The inset shows the difference ${\chi }_m-{\chi }_0$ for $m=1$ (blue), 2 (black), and 3 (green) for the part of the spectra inside the dashed box. It shows that ${\chi }_{m\ne 0}$ can become equal to ${\chi }_0$ at certain probe frequency $\omega$. The horizontal dashed red line is to guide the eye. The comparison between (a) and (b) clearly shows that the difference between the spectra ${\chi }_m(\omega ,\omega )$ for different $m$ goes down as the scaled anharmonicity $\alpha /{\delta }_1$ goes down. We note that the seemingly isolated dots in the spectra including the two near zero frequency in (b) are a result of the finite sampling along the frequency axis which does not suffice to resolve the sharp change in the spectra in a narrow frequency range due to weak resonances.

Figure 13

A schematic of the four-wave mixing process between the two cavity modes and two drives that leads to the beam-splitter interaction between the dressed cavity modes $A$ and $B$. The process becomes resonant when ${\omega }_A+{\omega }_2={\omega }_B+{\omega }_1$. The green box represents the transmon ancilla as a frequency mixer.

Figure 14

A 2D lattice that represents the tight-binding Hamiltonian $H_{\mathrm{tb}}$ in Eq. (E1). Quantum number $N$ indicates the level of the ancilla in the Fock space. $K$ indicates the change in the number of excitations in drive-2 reservoir. Direct transfer of excitations between the drives and the ancilla corresponds to hopping on the 2D lattice: hopping along the horizontal direction (indicated by ${\mathrm{\Omega }}_1$) means exchange of excitations between drive 1 and the ancilla, and along one of diagonal directions (indicated by ${\mathrm{\Omega }}_2$) means exchange of excitations between drive 2 and the ancilla. Indirect hopping along vertical direction can happen via two direct hopping along the diagonal and horizontal directions meaning exchange of excitations between drive 1 and drive 2. The solid lines are contours of constant on-site energy obtained by treating $N$ and $K$ as continuous variables for the parameters: ${\delta }_1/\alpha =1.5,{\omega }_{21}/\alpha =4$. For these parameters, sites (0,0) and (2,1) have the same on-site energy indicating multiphoton resonance as discussed in Sec. 3b1.

Figure 15

Spectroscopic Poincaré section of the Hamiltonian $g(Q,P,t)$. The scaled driving amplitudes ${\overline{\mathrm{\Omega }}}_1=0.89,{\overline{\mathrm{\Omega }}}_2=4.5$. The scaled frequency ${\omega }_{21}/\left|{\delta }_1\right|=15.$

Figure 16

The squeezing parameter $sinh\varphi$ and the squared classical response $Q_0^2$ as a function of the scaled drive power ${\overline{\mathrm{\Omega }}}_1^2$. In terms of ${\xi }_1$ defined in the main text, ${\overline{\mathrm{\Omega }}}_1^2=\alpha {\left|{\xi }_1\right|}^2/{\delta }_1.$

Figure 17

Quasienergy level anticrossing due to multiphoton resonance. Top: Spectroscopy of the two-tone driven ancilla as a function of the scaled drive-2 power. The power of drive 1 is fixed at $|{\xi }_1{|}^2{\delta }_1/({\delta }_1+\alpha )=0.22$. The scaled drive detunings ${\delta }_1/\alpha =1,{\delta }_2/\alpha =4.5.$ The vertical axis is the frequency of the spectroscopy tone (a $\pi$ pulse) counted from the ancilla transition frequency ${\omega }_c$. The color indicates the population of the ancilla not in the ground state. Bottom: The quasienergy spectrum of the driven ancilla for the same range of drive strengths as in the top panel. From top to down at ${\xi }_2=0$, the blue, green, red, and black lines refer to the quasienergy levels ${\epsilon }_1, {\epsilon }_3, {\epsilon }_2$, and ${\epsilon }_0$, respectively, projected into the same Brillouin zone. The anticrossing between ${\epsilon }_0$ and ${\epsilon }_2$ indicates a multiphoton resonance where the ancilla is excited from the ground to the second excited state by absorbing three drive-1 photons and emitting one drive-2 photon. The gap of the anticrossing is too weak to be seen on the scale of the plot. In the top panel, the dip at around $-30$ MHz at the drive strength where the anticrossing occurs is likely due to that the ancilla decays from the second to the first excited state and then be de-excited to the ground state by the spectroscopy tone.

Figure 18

The scaling of the rates $W_{01}^{\gamma }$ and $W_{01}^{{\gamma }_{\mathrm{ph}}}$ with respect to $\alpha /{\delta }_1$ for fixed $\alpha |{\xi }_1{|}^2/{\delta }_1=0.315$ and $\alpha |{\xi }_2{|}^2/{\delta }_2=0.001$. The ratio ${\delta }_2/{\delta }_1=3.1$ and $n_{\mathrm{th}}=0$. The dots show the Floquet calculation using Eq. (46). The dashed lines show the semiclassical result in Eq. (F13). For the chosen parameters, the transition rates are primarily due to the much stronger drive 1. We clarify that the rate $W_{01}^{{\gamma }_{\mathrm{ph}}}$ is nonzero even when the ancilla is linear ($\alpha =0$) and is proportional to the scaled drive power $|{\xi }_1{|}^2+{\left|{\xi }_2\right|}^2$ for weak drives; see Eq. (48). The dependence on $\alpha /{\delta }_1$ shown in the figure is a result of decreasing $\alpha /{\delta }_1$ while keeping $\alpha |{\xi }_1{|}^2/{\delta }_1$ and $\alpha |{\xi }_2{|}^2/{\delta }_2$ fixed, which requires increasing ${\xi }_1,{\xi }_2$ proportionally.