Qubit-photon interactions in a cavity: Measurement-induced dephasing and number splitting

We theoretically study measurement-induced dephasing of a superconducting qubit in the circuit QED architecture and compare the results to those obtained experimentally by Schuster et al. [Phys. Rev. Lett. 94, 123602 (2005)]. Strong coupling of the qubit to the resonator leads to a significant ac Stark shift of the qubit transition frequency. As a result, quantum fluctuations in the photon number populating the resonator cause dephasing of the qubit. We find good agreement between the predicted line shape of the qubit spectrum and the experimental results. Furthermore, in the strongly dispersive limit, where the Stark shift per photon is large compared to the cavity decay rate and the qubit linewidth, we predict that the qubit spectrum will be split into multiple peaks, with each peak corresponding to a different number of photons in the cavity.

Figure 1

(Color online) Schematic layout and lumped element version of the circuit QED implementation. A superconducting charge qubit (green) is fabricated inside a superconducting 1D transmission line resonator (blue).

Figure 2

(Color online) Measured spectroscopic lines (blue lines) at (a) intraresonator photon number $\overline{n}\approx 1$ with fit to Lorentzian line shape (solid line) and at (b) $\overline{n}\approx 40$ with fit to Gaussian line shape (solid line). Dashed lines are best fits to (a) Gaussian or (b) Lorentzian line shapes, respectively. The qubit transition frequency ${\tilde{\omega }}_a$ at low measurement power, the half width at half maximum $\delta {\nu }_{\mathrm{HWHM}}$, and the ac Stark shift ${\omega }_{\mathrm{ac}}$ of the lines are indicated.

Figure 3

(Color online) Full red curve: Measurement-broadened qubit linewidth $\delta {\nu }_{\mathrm{HWHM}}$ as a function of the input measurement power or average photon number as predicted by the lowest-order dispersive approximation. Green dotted curve: Same as red but taking into account the nonlinear reduction in the cavity pull and plotted as a function of input power. The symbols are the experimental results. Symbols and color scheme are described in the text. The vertical line indicates the critical photon number $n_{\mathrm{crit}}$. The parameters are those given in Sec. 3. The blue dashed line is the calculated HWHM for ${\Delta }_r∕2\pi =32\mathrm{MHz}$. It clearly shows that measurement-induced dephasing is small at large ${\Delta }_r$ where information about the state of the qubit in the transmitted signal is also small. This can thus be used as the basis of a phase gate.

Figure 4

(Color online) (a) Full red curve: ac Stark shift as a function of the average intracavity photon number using the lowest-order dispersive approximation. Dashed blue curve: ac Stark shift as a function of photon number calculated from the exact eigenvalues of the Jaynes-Cummings Hamiltonian. (b) Blue dots: experimentally measured ac Stark shift as a function of external microwave input power. Full red curve: Predicted ac Stark shift within the lowest-order dispersive approximation. The conversion factor from photon number to external microwave drive power was determined by fitting the red curve to the linear portion of the data at low power. Green dotted curve: Same as red but taking into account the nonlinear reduction in the cavity pull [see (c)] and the nonlinear increase in the average photon number [see (d)] with microwave drive power. For the particular experimental parameters these two effects almost cancel each other out and result in the green dotted line being nearly linear out to much greater input powers than expected. (c) Cavity pull as a function of average photon number $\overline{n}$. The red solid line is the result of the dispersive approximation $(\pm \chi )$ while the dashed blue curve is obtained from the exact eigenvalues of the Jaynes-Cummings model. (d) Average photon number as a function of input power. The full red line is the result of the lowest-order dispersive dispersive approximation Eq. (4.1) fit to the data in (b) at low power. The dotted green line is the nonlinear model with $\chi$ replaced by $\chi \left(\overline{n}\right)$ in Eq. (4.1) The vertical line in all plots indicates the critical photon number $n_{\mathrm{crit}}={\Delta }^2∕4g^2$, which indicates the scale at which the lowest order dispersive approximation breaks down. For the experimental parameters (given in Sec. 3) $n_{\mathrm{crit}}\sim 82$ which corresponds (within the lowest-order dispersive approximation) to $P_{\mathrm{rf}}\sim 110\mu \mathrm{W}$.

Figure 5

(Color online) Measurement-induced dephasing rate ${\Gamma }_{\mathrm{m}}$ as a function of the detuning ${\Delta }_r$ between the bare resonator frequency ${\omega }_r$ and the measurement drive frequency ${\omega }_d$. The dephasing rate is divided by the measurement power ${\epsilon }_{\mathrm{rf}}^2$ since it only changes the overall scale and not the structure of ${\Gamma }_{\mathrm{m}}$. The blue (dashed) line corresponds to the experimental parameters given in Sec. 3. The red (full) line has the same parameters but a larger cavity pull $\chi ∕2\pi =5\mathrm{MHz}$.

Figure 6

(Color online) Dephasing rate ${\Gamma }_{\mathrm{m}}$ as a function of the detuning ${\Delta }_r$ between the bare resonator frequency ${\omega }_r$ and the measurement drive frequency ${\omega }_d$ for fixed ${\overline{n}}_{-}=2$. (a) Experimental parameters given in Sec. 3. (b) Same as (a) but with a larger cavity pull $\chi ∕2\pi =5\mathrm{MHz}$.

Figure 7

(Color online) Spectrum $S\left(\omega \right)$ as a function of detuning ${\Delta }_r∕\kappa$ at fixed cavity pull $\chi ∕\kappa =100$. The dephasing rate was set to the conservative value of ${\gamma }_2=7.6\kappa$. The average photon number in the cavity was fixed to ${\overline{n}}_{-}=2$ in (a) $({\Delta }_r⩾0)$ and we choose ${\overline{n}}_{+}=2$ in (b) $({\Delta }_r⩽0)$. This implies that the measurement beam power changes with detuning. Inset: Spectrum at ${\Delta }_r=\chi =100\kappa$. At this detuning, the peaks are split by $2\chi$. More generally, they are split by $\chi +\Delta$. Large detuning yields large splitting but, as shown in Fig. 5, this can be at the expense of a small measurable phase shift in the transmitted field.

Figure 8

(Color online) Spectrum $S\left(\omega \right)$ as a function of the cavity pull $\chi ∕\kappa$. The detuning is ${\Delta }_r=\chi$ such that the cavity is always driven at the dressed frequency ${\omega }_r-\chi$. Other parameter values are the same as in Fig. 7. Inset: Spectrum at $\chi ∕\kappa =20$. At this experimentally realistic value of the cavity pull, the number splitting should be resolvable.

Figure 9

(Color online) Distance $D\left(t\right)={\mid {\alpha }_{+}\left(t\right)-{\alpha }_{-}\left(t\right)\mid }^2$ as a function of time for ${\Delta }_r=\chi$ and ${\epsilon }_{\mathrm{rf}}=2\kappa$. The red (full) line corresponds to $\chi ∕\kappa =5$ and the blue (dashed) line to $\chi ∕\kappa =0.1$. Other parameter values are the same as in Fig. 7. Inset: Real and imaginary parts of the cavity field ${\alpha }_{+}\left(t\right)$ as a function of time. The arrows indicate direction of time.

Figure 10

(Color online) Pictorial representation of the dispersive Hamiltonian in a frame defined by Eq. (5.33) at ${\Delta }_r=0$. When calculating the correlator $⟨{\widehat{\sigma }}_{-}\left(t\right){\widehat{\sigma }}_{+}\left(0\right)⟩$, the qubit is flipped at time $t=0$. In the displaced frame, this corresponds to both flipping the qubit and displacing the oscillator from the vacuum state to a coherent state with mean photon number $4\overline{n}$. This distribution is shown on top of the harmonic oscillator corresponding to the qubit in its excited state. This simple picture explains the observed peaks in the qubit power spectrum and the presence of negative shifts.