Measuring a transmon qubit in circuit QED: Dressed squeezed states

Using circuit QED, we consider the measurement of a superconducting transmon qubit via a coupled microwave resonator. For ideally dispersive coupling, ringing up the resonator produces coherent states with frequencies matched to transmon energy states. Realistic coupling is not ideally dispersive, however, so transmon-resonator energy levels hybridize into joint eigenstate ladders of the Jaynes–Cummings type. Previous work has shown that ringing up the resonator approximately respects this ladder structure to produce a coherent state in the eigenbasis (a dressed coherent state). We numerically investigate the validity of this coherent-state approximation to find two primary deviations. First, resonator ring-up leaks small stray populations into eigenstate ladders corresponding to different transmon states. Second, within an eigenstate ladder the transmon nonlinearity shears the coherent state as it evolves. We then show that the next natural approximation for this sheared state in the eigenbasis is a dressed squeezed state and derive simple evolution equations for such states by using a hybrid phase–Fock-space description.

Figure 1

(a) Considered system: a transmon coupled to a pumped resonator. The resonator damping is neglected, since we focus on the resonator ring-up and/or setups with a tunable coupler. (b) Jaynes–Cummings ladder of states. Bare states are shown by solid black lines. Eigenlevels are shown by red dashed lines. When $n\gtrsim n_c$, the eigenlevels are significantly different from bare levels.

Figure 2

Infidelity of coherent-state approximations during resonator ring-up. The infidelity $1-F_{\mathrm{b}}$ of a bare coherent state (dotted red line) is compared with the infidelity $1-F$ of a dressed coherent state (dashed black line). The latter displays two distinct effects: at short time (and small photon number $\overline{n})$ the dominant effect is the leakage of a stray population $P_{\mathrm{stray}}$ (thin solid blue line) out of the correct eigenladder; however, at longer time (and larger $\overline{n})$ the infidelity $1-F_{\mathrm{c}}$ of the renormalized state within the correct eigenladder (thick solid orange line) significantly increases during evolution. Here the system, with parameters ${\omega }_{\mathrm{r}}/2\pi =6\text{GHz},{\omega }_{\mathrm{q}}/2\pi =5\text{GHz},\eta /2\pi =200\text{MHz}$, and $g/2\pi =100\text{MHz}$, is resonantly pumped from its ground state $|0,0\rangle$ with a constant drive envelope $\varepsilon /2\pi =10\text{MHz}$.

Figure 3

Solid blue lines: numerically calculated stray population $P_{\mathrm{stray}}$ as a function of time $t$; dashed black lines: steady-state value $P_{\mathrm{s}.\mathrm{s}.}\left(t\right)$, calculated via Eq. (20). (a) Leaked population in the excited eigenladder $\overline{|n,1\rangle }$ for sudden driving with $\varepsilon /2\pi =60\text{MHz}$ from initial ground state $|0,0\rangle$. The oscillations reach an initial maximum of $P_{\max }\approx 4P_{\mathrm{s}.\mathrm{s}.}\left(0\right)$, then dephase to about $P_{\mathrm{s}.\mathrm{s}.}\left(t\right)+P_{\mathrm{s}.\mathrm{s}.}\left(0\right)$, with decreasing $P_{\mathrm{s}.\mathrm{s}.}\left(t\right)$ because of increasing average photon number $\overline{n}$. (b) The same for adiabatic drive $\varepsilon \left(t\right)$, linearly increasing for the first $10\text{ns}$ to the same constant value of $60\text{MHz}$. The stray population follows the steady state, which increases for 10 ns because of increasing $\varepsilon \left(t\right)$. (c) Sudden driving with $\varepsilon /2\pi =60\text{MHz}$ from an initial excited qubit state $\overline{|0,1\rangle }$, showing population $P_{\mathrm{stray},0}$ leaked to the ground-state eigenladder $\overline{|n,0\rangle }$. This case is fully symmetric with that of panel (a) because it involves the same pair of transmon levels. (d) The same driving as in panel (c), but showing leaked population $P_{\mathrm{stray},2}$ of the second-excited eigenladder $\overline{|n,2\rangle }$. The behavior is similar to that of panel (c), but involves the next pair of transmon levels. For all panels ${\omega }_{\mathrm{r}}/2\pi =6\text{GHz},{\omega }_{\mathrm{q}}/2\pi =5\text{GHz},\eta /2\pi =250\text{MHz},g/2\pi =100\text{MHz}$, and ${\omega }_{\mathrm{d}}$ is on resonance with the resonator frequency, corresponding to each initial state.

Figure 4

Model validation for stray population $P_{\mathrm{stray}}$ in the neighboring eigenladder, using a sudden resonant [off-resonant in panel (d)] drive and starting with $|0,0\rangle$ [panels (a)–(g)] or $\overline{|0,1\rangle }$ [panels (h) and (i)]. (a)–(d) Testing of Eq. (21) for the maximum stray population $P_{\max }$ against numerical results, by varying (a) the drive amplitude $\varepsilon$, (b) coupling $g$, (c) resonator-qubit detuning $\mathrm{\Delta }$, and (d) drive frequency ${\omega }_{\mathrm{d}}$. (e) Testing that the time-dependent oscillation frequency evolves as ${\mathrm{\Omega }}_{\overline{n}}={\mathrm{\Delta }}_{\overline{n}}$ given by Eq. (16). (f), (g) Testing of Eq. (23) for the decay time $t_{\mathrm{decay}}$ of the eigenladder oscillations [as in Fig. 3], using a prefactor of 1.23 for decay to $1/3$ amplitude. (h), (i) Similar to panels (a) and (g), but for the leakage to the second-excited eigenladder $\overline{|n,2\rangle }$ starting from the excited state $\overline{|0,1\rangle }$; in this case Eqs. (21) and (23) need the following replacements: $g\mapsto \sqrt{2}g,\mathrm{\Delta }\mapsto \mathrm{\Delta }+\eta ,{\mathrm{\Omega }}_0\mapsto {\mathrm{\Omega }}_0+\eta ,\chi \mapsto {\chi }^{\prime }={\omega }_{\mathrm{r}}^{\left(2\right)}-{\omega }_{\mathrm{r}}^{\left(1\right)}$. In all panels, blue dots show numerical results, while red lines are calculated analytically. We use the following parameters: ${\omega }_{\mathrm{r}}/2\pi =6\text{GHz},{\omega }_{\mathrm{q}}/2\pi =5\text{GHz},\eta /2\pi =200\text{MHz},g/2\pi =100\text{MHz},\varepsilon /2\pi =10\text{MHz}$, except for parameters, which are varied, and in panel (g) $\varepsilon /2\pi =50\text{MHz}$ and in panels (h) and (i) $\eta /2\pi =300\text{MHz}$.

Figure 5

(a) Numerically simulated evolution of the dressed Husimi $Q$ function for the state remaining in the correct eigenladder, given an initial state of $|0,0\rangle$ and a resonant drive. Snapshots taken at $50\text{ns}$ intervals show the progressive shearing of the state caused by resonator nonlinearity. Inset shows $n$ dependence of the difference $\mathrm{\Delta }{\omega }_{\mathrm{r}}^{\left(k\right)}={\omega }_{\mathrm{r}}^{\left(k\right)}-{\omega }_{\mathrm{r}}$ between the effective and bare resonator frequencies. The solid blue (upper) line shows $\mathrm{\Delta }{\omega }_{\mathrm{r}}^{\left(0\right)}\left(n\right)$ for the ground-state eigenladder, the solid orange (lower) line shows $\mathrm{\Delta }{\omega }_{\mathrm{r}}^{\left(1\right)}\left(n\right)$ for the excited-state eigenladder, and the red dashed line indicates the applied drive frequency. (b) Detail of the $Q$ function at $200\text{ns}$. The analytical result for a dressed squeezed state (dashed red) shows good agreement with the numerically simulated state (solid black). The agreement is significantly better for earlier times (not shown). Parameters are ${\omega }_{\mathrm{r}}/2\pi =6\mathrm{GHz},{\omega }_{\mathrm{q}}/2\pi =5\mathrm{GHz},\eta /2\pi =200\mathrm{MHz},g/2\pi =100\mathrm{MHz}$, and $\varepsilon /2\pi =10\mathrm{MHz}$. The contours of the $Q$ function are drawn at the levels of $0.1/\pi ,0.2/\pi ,\cdots ,0.8/\pi$.

Figure 6

Infidelity $1-F_{\mathrm{c}}$ of the dressed-coherent-state approximation within the initial eigenladder, starting with $|0,0\rangle$ (upper lines, blue) or $\overline{|0,1\rangle }$ (lower lines, red). Parameters are the same as for Fig. 2. The solid lines show numerical results, upper (blue) dashed line is calculated via Eq. (32) with the frequency derivative taken at $n=0$, and the lower (red) dashed line is calculated by integrating Eq. (31).

Figure 7

Comparison between the dressed-squeezed-state and dressed-coherent-state models within the “correct” eigenladder. Parameters are the same as in Figs. 2 and 5, evolution starts with $|0,0\rangle$. Blue (lower) lines show time dependence of the infidelity $1-F=1-|{}_0{\langle \beta ,\xi |\psi \rangle }_0{|}^2$ for the dressed squeezed states, orange (upper) lines show infidelity $1-|{}_0{\langle \alpha |\psi \rangle }_0{|}^2$ for the dressed coherent states. For solid lines the state centers $\beta \left(t\right)$ and $\alpha \left(t\right)$ are calculated as average values of the operator $\overline{a}$. For the dashed lines, $\beta \left(t\right)$ and $\alpha \left(t\right)$ are obtained from Eq. (50). For the dotted lines, in Eqs. (48, 49, 50) we use correction (51) for the drive amplitude. The dressed squeezed-state model is about two orders of magnitude more accurate than the dressed coherent-state model.

Figure 8

(a) Solid lines: infidelity $1-|{\langle {\psi }_{\mathrm{dp}}|\alpha \rangle }_0{|}^2$ of approximating the dressed coherent state ${|\alpha \rangle }_0$ with a direct-product state $|{\psi }_{\mathrm{dp}}\rangle$ given by Eq. (A8), as a function of ${\left|\alpha \right|}^2$. For comparison, the dots show similar infidelity for the eigenstates $\overline{|n,0\rangle }$, i.e., dressed Fock states, as a function of $n$ (axes of $n$ and ${\left|\alpha \right|}^2$ coincide). We assume $({\omega }_{\mathrm{r}}-{\omega }_{\mathrm{q}})/2\pi =1$ GHz, $\eta /2\pi =200$ MHz, and $g/2\pi =100$ MHz (lower blue line and dots, $n_c=25)$ or $g/2\pi =141.4$ MHz (upper orange line and dots, $n_c=12.5)$. (b) Entanglement of formation $E_F$ (coinciding with entropy of entanglement) for the dressed coherent states ${|\alpha \rangle }_0$ (lines) and dressed Fock states (dots) with the same parameters as in (a).