Local sensing with the multilevel ac Stark effect

Analyzing weak microwave signals in the GHz regime is a challenging task if the signal level is very low and the photon energy widely undefined. A superconducting qubit can detect signals in the low photon regime, but due to its discrete level structure, it is only sensitive to photons of certain energies. With a multilevel quantum system (qudit) in contrast, the unknown signal frequency and amplitude can be deduced from the higher level ac Stark shift. The measurement accuracy is given by the signal amplitude, its detuning from the discrete qudit energy level structure, and the anharmonicity. We demonstrate an energy sensitivity in the order of ${10}^{-3}$ with a measurement range of more than $1\mathrm{GHz}$. Here, using a transmon qubit, we experimentally observe shifts in the transition frequencies involving up to three excited levels. These shifts are in good agreement with an analytic circuit model and master equation simulations. For large detunings, we find the shifts to scale linearly with the power of the applied microwave drive. Exploiting the effect, we demonstrated a power meter which makes it possible to characterize the microwave transmission from source to sample.

Figure 1

Analytically calculated ac Stark shift for (a) a weakly anharmonic circuit (${\omega }_{\mathrm{q}}/2\pi =5\mathrm{GHz}, \gamma /2\pi =0.4\mathrm{GHz}$, solid lines) using perturbation theory up to fourth order. For strong driving amplitudes, a crossing of the multiphoton transitions can be observed. The dashed lines represent the level structure at a five times larger detuning $\mathrm{\Delta }$. The solid and dashed lines in (b) depict the same level structure but for a three times larger anharmonicity $\gamma$, showing that our theory is not only applicable to transmons but also to other qudits with larger anharmonicity. (c) Using the qudit as a sensor for an unknown drive: the frequency of the first transition ${\omega }_{01}$ compared to the bare frequency ${\omega }_{01}^0={\omega }_{\mathrm{q}}-\gamma /2$ and the resulting anharmonicity provide the detuning $\mathrm{\Delta }$ and the drive amplitude $A_{\mathrm{D}}$.

Figure 2

(a) Micrograph of the transmon sample. The microwave tones are coupled via the transmission line (left), bandpass filtered by the resonator (center), and finally reach the transmon (right). (b) Level scheme for the strongly driven circuit. Gray arrows indicate the ac Stark shift of the qudit by the strong drive. Also shown are multiphoton transitions by the drive (red) and by drive and probe combined (blue). (c) Schematic diagram of the employed microwave setup for reflection measurements. The resonator probe tone $P_{\mathrm{VNA}}$ and transmon probe tone $P_{\mathrm{P}}$ are merged in a power combiner and the drive tone $P_{\mathrm{D}}$ is added via a directional coupler. The combined signals are sequentially attenuated at various temperature stages of the cryostat before reaching the the sample. Circulators are placed after the sample to screen thermal noise.

Figure 3

Qudit spectroscopy for the concentric transmon at varying probe power and frequency without an additional drive tone. The emergence of multiphoton transitions at higher probe powers is clearly visible. The shadowlike transition next to the ${\omega }_{04}/4$ transition and the transition around $4.36\mathrm{GHz}$ can be assigned to transitions between higher states, showing that the system was initially not in the ground state at these high powers. The vertical dashed line at $P_{\mathrm{P}}=5\mathrm{dBm}$ indicates the probe power chosen for the following measurements, as an even higher probe power would strongly increase the linewidth for the first transition and hence reduce the visibility.

Figure 4

Left: ac Stark shift of the transmon levels under a power-varying drive at ${\omega }_{\mathrm{D}}/2\pi =4.95\mathrm{GHz}$. The analytic solution from Eq. (10) (colored lines) fits well to the measured transitions. Additional multiphoton transitions consisting of one or two drive photons and one or two probe photons are shown as dotted lines. Deviations from the analytic solutions are attributed to using a simplified transmon Hamiltonian, Eq. (4), which does not represent higher transmon levels correctly. The data has been columnwise normalized, as the drive also influences the resonator's resonance frequency. Right: QuTiP simulations with ${\omega }_{\mathrm{D}}/2\pi =4.95\mathrm{GHz}$. Line colors are the same as in the left plot. Due to a slightly higher $A_{\mathrm{P}}$, more transitions including the $|4\rangle$ state are visible that can be explained as well with the analytic approximation.

Figure 5

Verification of the sensing technique: scanning the drive frequency at a constant drive power. From the fitted ${\omega }_{01}$ and ${\omega }_{02}/2$ transitions, we calculate the drive amplitude (a) and frequency (b), where both $A_{\mathrm{D}}$ and ${\omega }_{\mathrm{D}}$ are free (blue crosses) or ${\omega }_{\mathrm{D}}$ is fixed to the known value of the drive frequency (green circles). The gray and green shaded areas show the uncertainty if we take into account an accuracy of $\pm 1\mathrm{MHz}$ for both ${\omega }_{01}/2\pi$ and ${\omega }_{02}/4\pi$. As the frequency determination fails for more than $0.5\mathrm{GHz}$ detuning between drive and the first qudit transition at ${\omega }_{\mathrm{D}}/2\pi =4.755\mathrm{GHz}$, no error bars are shown for this area. To verify the method of fixing ${\omega }_{\mathrm{D}}$ to the known value, we calculate the ${\omega }_{01}$ and ${\omega }_{02}/2$ transitions from the values in (a) and (b), plot them on top of the measurement data (c), and find a good agreement.

Figure 6

ac Stark shift of transmon transition frequencies as obtained through resonator population. The axes above the image show the average resonator photon population and the effective drive power on the feedline of the chip, both calculated from the analytic drive amplitude $A_{\mathrm{D}}$ on the bottom of the image. The axis on top of the graph shows the room temperature drive power of the VNA probing the resonator. A good agreement between our measured data and the analytic solution (colored lines) can be seen. The data is normalized columnwise as the probe frequency of the VNA had to be adjusted for each drive power. The reduced visibility of the qudit for higher drive powers clearly demonstrates the advantages of the off-resonant ac Stark shift.

Figure 7

Simulation of the first transitions in the spectrum at a drive amplitude of $A_{\mathrm{D}}/2\pi =0.3\mathrm{GHz}$ for different $(T_1/T_2/T_{\mathrm{sim}})$ times. It can be clearly seen that the transition frequencies do not depend on the choice of times. However, the absolute level population varies, as the system is in a driven steady state at the end of the simulation time and therefore the absolute level population depends on the strength of the decay channels. The combination of $T_1=0.25\mu \mathrm{s},T_2=0.25\mu \mathrm{s},\text{and}{T}_{\mathrm{sim}}=0.5\mu \mathrm{s}$ was chosen as a good compromise between visibility of all levels and computation time. The lines are guides for the eye.