Photon shot noise dephasing in the strong-dispersive limit of circuit QED

We study the photon shot noise dephasing of a superconducting transmon qubit in the strong-dispersive limit, due to the coupling of the qubit to its readout cavity. As each random arrival or departure of a photon is expected to completely dephase the qubit, we can control the rate at which the qubit experiences dephasing events by varying in situ the cavity mode population and decay rate. This allows us to verify a pure dephasing mechanism that matches theoretical predictions, and in fact explains the increased dephasing seen in recent transmon experiments as a function of cryostat temperature. We observe large increases in coherence times as the cavity is decoupled from the environment, and after implementing filtering find that the intrinsic coherence of small Josephson junctions when corrected with a single Hahn echo is greater than several hundred microseconds. Similar filtering and thermalization may be important for other qubit designs in order to prevent photon shot noise from becoming the dominant source of dephasing.

Figure 1

(a) Experimental setup (see also Supplemental Material, Ref. 18). Noise of varying amplitude at the cavity transition frequency is sent into the input port of a 3D resonator. The output port of the resonator has a movable coupler which varies the output side coupling quality factor $Q_c$ from $1.0\times {10}^5$ to $2.5\times {10}^7$. (b) Energy-level diagram. The qubit has a transition frequency that is ac-Stark shifted by $-\chi$ for each photon in the cavity. (c) Photon number splitting of the qubit spectrum. We inject noise and create a mean number $\overline{n}$ of photons in the fundamental mode. The peaks correspond to $N=$ 0,1,2 photons from right to left, with the cavity $Q=1\times {10}^6$, and (*) even without applied noise we measure a photon occupation in the ${\mathrm{TE}}_{101}$ mode of the cavity to be $\overline{n}\sim 0.02$. (d) Cavity population. Rabi experiments (shown in Supplemental Material) performed on each photon peak $N$ for increasing noise power with cavity $Q=2.5\times {10}^5$. The signal amplitude gives the probability of finding $N$ photons in the cavity. Two linear scaling factors, fit globally, provide conversion from homodyne readout voltage (Ref. 25) to probability (vertical axis), and from attowatts within the cavity bandwidth to $\overline{n}$ (horizontal axis). Error bars represent $1\sigma$ fluctuations in the $\left|e\right.〉$ state readout voltage. The solid lines are a thermal distribution using the fit scaling parameters.

Figure 2

Qubit dephasing due to photon noise. (a) Qubit coherence time, determined from Ramsey experiments on the $N=0$ or $N=1$ ($▵$) photon peaks, as a function of both cavity $Q$ and $\overline{n}$. The dashed lines are theory, with an offset due to residual dephasing. Each has a slope proportional to $\kappa$ (or $3\kappa$ for $N=1$ experiments), according to Eq. (2). The ($\circ$) are coherence times vs population in ${\mathrm{TE}}_{103}$ mode, which also dephases the qubit. (b) Ramsey with no noise injected, fundamental mode $Q=1\times {10}^6$, and $T_2^{*}=26\mu$s. The solid line is a fit with an exponentially decaying sine. (c) A Ramsey with moderate noise. Contrast and $T_2^{*}$ are reduced. Fundamental mode $Q=2.5\times {10}^5$, $\overline{n}=0.25$, $T_2^{*}=7.7\mu$s. (d) Ramsey with high noise. Fundamental mode $Q=1\times {10}^6$, $\overline{n}=3.1$, $T_2^{*}=5.2\mu$s. Our selective ($N=0$) pulses produce a loss of contrast and a nonoscillating signal addition (orange) as photon population returns to a thermal distribution. The dashed black line is a numerical simulation (see Supplemental Material, Ref. 18).

Figure 3

(a) Decoherence due to thermal photons. The coherence times extracted from Ramsey ($T_2^{*}$) and Hahn echo ($T_{2\mathrm{E}}$) experiments measured as a function of cryostat temperature. To model dephasing (dashed lines), we predict population in the ${\mathrm{TE}}_{101}$ and ${\mathrm{TE}}_{103}$ modes of the cavity. Then, we sum the total dephasing rate using the measured quality factors for each mode (high $Q$: ${\tau }_{101}=20\mu$s, ${\tau }_{103}=4\mu$s; low $Q$: ${\tau }_{101}=2\mu$s, ${\tau }_{103}=400$ ns). For high $Q$, the use of a Hahn echo pulse leads to a large $T_{2E}$ because either the photon state has much longer correlation time or the remaining dephasing similarly occurs at low frequencies. Although the decline in $T_1$ (not shown) (Ref. 27) contributes to the trend, population in both ${\mathrm{TE}}_{101}$ and ${\mathrm{TE}}_{103}$ are needed for a good fit. (b) Bose-Einstein population of the first two odd-$n$ ${\mathrm{TE}}_{10n}$ modes at 8 and 12.8 GHz (green, A and B, respectively) and the coherence limits they impose individually (blue) and collectively (dashed red) for the low $Q$ values measured above.

Figure 4

Coherence times versus ${\mathrm{TE}}_{101}$ mode decay $\tau$. The ${\mathrm{TE}}_{103}$ cavity, which naturally decays more strongly through the couplers, increases in $Q$ as the entire resonator is decoupled from our coaxial lines. While $T_1$ is nearly constant due to the large qubit detuning from the cavity, its $T_2^{*}$ and $T_{2E}$ increase as the coupling pin is withdrawn from the 3D resonator. This is consistent with diminishing dephasing from cavity modes with $\kappa <\chi$, where a photon transit strongly measures (Ref. 11) the qubit state.