Probe spectroscopy of quasienergy states

The present qubit technology, in particular, in Josephson qubits, allows an unprecedented control of discrete energy levels. This motivates a new study of the old pump-probe problem, where a discrete quantum system is driven by a strong drive and simultaneously probed by a weaker one. The strong drive is included by the Floquet method and the resulting quasienergy states are then studied with the probe. We study a qubit where the harmonic drive has a significant longitudinal component relative to the static equilibrium state of the qubit. Both analytical and numerical methods are used to solve the problem. We present calculations with realistic parameters and compare the results with recent experimental results. A short introduction to the Floquet method and the probe absorption is given.

Figure 1

Illustration of the absorption spectrum of a strongly driven atom. The atom with an energy separation ${\varepsilon }_0$ is driven with an angular frequency $\omega =2\pi /\tau$. The drive changes the atomic energy (quasienergy ${\Delta }_{\mathrm{q}}$) leading to a shift of the resonance peak in the absorption spectrum (dynamic Stark shift).

Figure 2

Schematics of the longitudinally driven and weakly probed two-level system both in the temporal space and the quasienergy space. The temporal space: a two-level system, whose energy splitting $\varepsilon \left(t\right)$ oscillates sinusoidally around the mean ${\varepsilon }_0$ with the period $\tau$. The quasienergy space: transformation to the Floquet formalism results in the temporally static quasienergy levels that repeat in energy with the period $\hslash \omega$. The weak probe field (blue arrows) acts between the atomic states or between the quasienergy states. In the case of ${\omega }_{\mathrm{P}}<\omega$, the probe resonance is met when ${\Delta }_{\mathrm{q}}=\hslash {\omega }_{\mathrm{P}}$ or $\hslash \omega -{\Delta }_{\mathrm{q}}=\hslash {\omega }_{\mathrm{P}}$.

Figure 3

Landscapes of the quasienergy ${\Delta }_{\mathrm{q}}$ in the ${\varepsilon }_0$-$A$ plane with different values for the tunnel amplitude: (a) $\Delta /\hslash \omega =0.10$, (b) $\Delta /\hslash \omega =0.37$ corresponding to an experimental realization,[8] (c) $\Delta /\hslash \omega =0.84$ corresponding to another experimental realization,[9] and (d) $\Delta /\hslash \omega =1.50$. The energy scale is the same in all panels. For the contour lines, we use the following color coding: numerical (solid gray), analytic in the diabatic basis Eq. (20) (dashed red), and analytic in adiabatic basis, that is, the adiabatic version of Eq. (20) (dashed black). The numerical contour lines are spaced by $0.10\hslash \omega$, except for two contours in the panel (b), where the comparison between the numerical and analytical contour lines is done with ${\Delta }_{\mathrm{q}}/\hslash \omega =0.092$ and ${\Delta }_{\mathrm{q}}/\hslash \omega =0.918$ since they correspond to the values of the experimentally measured weak probe resonances.[8]

Figure 4

Comparison between the numerical (solid line) and RWA analytical (dotted line) transition amplitudes $|F_{+-}{|}^2$ (24), calculated with the parameters of Fig. 3b. The transition amplitudes are picked by following the corresponding resonance conditions (23), i.e., from the parametrized line ${\varepsilon }_0\left(A\right)$ where ${\Delta }_{\mathrm{q}}=0.092\hslash \omega$ or ${\Delta }_{\mathrm{q}}=0.918\hslash \omega$. The curves are labeled by the index $n$ in Eq. (17).

Figure 5

The transition rate (13) of the strongly driven and weakly probed qubit presented as a gray-scale plot in the ${\varepsilon }_0\text{−}A$ plane. The parameters are the same as in the corresponding panels in Fig. 3. In addition to those, the parameter $\kappa$ describing the Ohmic spectral density is chosen so that $\gamma /\omega =0.016$ in the absence of driving and detuning in Eq. (25) and $\beta =\hslash \omega /kT=2.24$. The discontinuities of the lines are a consequence of the roots of $J_n(A/\hslash \omega )$ related to the coherent destruction of the tunneling. The $\mathcal{P}$ vs $A$ plot along the vertical dashed line in (b) is shown in Fig. 6.

Figure 6

The transition rate $\mathcal{P}$ of the strongly driven and weakly probed qubit and the corresponding spectrum $S\left({\omega }_{\mathrm{P}}\right)$, both scaled with their maximum value. The parameters are ${\varepsilon }_0/\hslash \omega =1.05$, $\Delta /\hslash \omega =0.37$, and ${\omega }_{\mathrm{P}}/\omega =0.092$. The parameter $\kappa$ describing the Ohmic spectral density is chosen so that $\gamma /\omega =0.016$ in the absence of driving and detuning in Eq. (25) and $\beta =\hslash \omega /kT=2.24$. The transition rate $\mathcal{P}$ is calculated with the numerical Floquet method [solid line, vertical projection from Fig. 5b], which is contrasted with the spectrum $S\left({\omega }_{\mathrm{P}}\right)$, see Eq. (27), (circles) of the driven qubit ($1/T_2=\gamma$ and $1/T_1=\gamma /2$) at the weak probe frequency.

Figure 7

The probe absorption $\mathcal{P}$ calculated as a function of level spacing ${\varepsilon }_0$ and driving amplitude $A$. The parameters are $\omega /2\pi =7.0$ GHz, $\Delta /\hslash \omega =0.37$, ${\omega }_{\mathrm{P}}/\omega =0.092$, and $T=150$ mK. The parameter $\kappa$ describing the Ohmic spectral density is chosen so that $\gamma /\omega =0.045$ in the absence of driving and detuning in Eq. (25), corresponding to the experimental estimate for qubit dephasing. This plot should be compared with the experimental plot in Ref. 8. For the comparison we have given the axis scales also using the units of this reference.

Figure 8

The probe absorption $\mathcal{P}$ calculated as a function of level spacing ${\varepsilon }_0$ and driving amplitude $A$. The parameters are $\omega /2\pi =4.15$ GHz, $\Delta /\hslash \omega =0.84$, ${\omega }_{\mathrm{P}}/\omega =0.005$, and $T=70$ mK. The parameter $\kappa$ describing the Ohmic spectral density is chosen so that $\gamma /\omega =0.17$ in the absence of driving and detuning in Eq. (25), corresponding to the experimental estimate for qubit dephasing. This plot should be compared with the experimental plot in Ref. 9. For the comparison, we have given the axis scales also using the units of this reference.

Figure 9

Quasienergy in the generalized Floquet method. (a) Quasienergy landscape in the ${\varepsilon }_0\text{−}A$ plane with the parameters $\Delta /\hslash \omega =0.37$, ${\omega }_{\mathrm{P}}/\omega =0.10$, and $A_{\mathrm{P}}/\hslash \omega =0.20$; the solid black line shows the weak probe resonance condition (23) deduced from the corresponding single-mode quasienergy landscape [see Fig. 3b]. (b) Projection of (a) at $A/\hslash \omega =5.6$, denoted with arrows and dashed line. The quasienergy ${\epsilon }_{r,n,m}$ (black) is periodic so that ${\epsilon }_r$ (blue) is shifted by $n\hslash \omega +m\hslash {\omega }_{\mathrm{P}}$, where $n,m=0,\pm 1,\pm 2,....$ The red dash-dotted line shows the corresponding quasienergy calculated with the single-mode Floquet-method. (c) Comparison of the two-mode quasienergies with the probe amplitude $A_{\mathrm{P}}/\hslash \omega =0.20$ (solid) and $0.40$ (dashed). Magnified view of the box in (b).