Quantum nondemolition measurements of a qubit coupled to a harmonic oscillator

We theoretically describe the weak measurement of a two-level system (qubit) and quantify the degree to which such a qubit measurement has a quantum nondemolition (QND) character. The qubit is coupled to a harmonic oscillator, which undergoes a projective measurement. Information on the qubit state is extracted from the oscillator measurement outcomes, and the QND character of the measurement is inferred from the result of subsequent measurements of the oscillator. We use the positive operator valued measure (POVM) formalism to describe the qubit measurement. Two mechanisms lead to deviations from a perfect QND measurement: (i) the quantum fluctuations of the oscillator, and (ii) quantum tunneling between the qubit states $|0⟩$ and $|1⟩$ during measurements. Our theory can be applied to QND measurements performed on superconducting qubits coupled to a circuit oscillator.

Figure 1

(Color online) (a) Schematics of the 4-Josephson junction superconducting flux qubit surrounded by a SQUID. (b) Measurement scheme: (b1) two short pulses at frequency $\sqrt{{ϵ}^2+{\Delta }^2}$, before and between two measurements prepare the qubit in a generic state. Here, $ϵ$ and $\Delta$ represent the energy difference and the tunneling amplitude between the two qubit states. (b2) Two pulses of amplitude $f$ and duration ${\tau }_1={\tau }_2=0.1\text{ns}$ drive the harmonic oscillator to a qubit-dependent state. (c) Perfect QND: conditional probability $P\left(0\mid 0\right)$ for $\Delta =0$ to detect the qubit in the state 0 vs driving time ${\tau }_1$ and ${\tau }_2$, at Rabi frequency of 1 GHZ. The oscillator driving amplitude is chosen to be $f/2\pi =50\text{GHz}$ and the damping rate $\kappa /2\pi =1\text{GHz}$. (d) Conditional probability $P\left(0\mid 0\right)$ for $\Delta /ϵ=0.1$, $f/2\pi =20\text{GHz}$, $\kappa /2\pi =1.5\text{GHz}$. A phenomenological qubit relaxation time $T_1=10\text{ns}$ is assumed.

Figure 2

(Color online) Schematic description of the single-measurement procedure. In the bottom panel the coherent states $|{\alpha }_0⟩$ and $|{\alpha }_1⟩$, associated with the qubit states $|0⟩$ and $|1⟩$, are represented for illustrative purposes by a contour line in the phase space at half-width at half maximum (HWHM) of their Wigner distributions, defined (Ref. 37) as $W\left(\alpha ,{\alpha }^{\ast }\right)=\left(2/{\pi }^2\right)\text{exp}\left(2{\left|\alpha \right|}^2\right)\int d\beta ⟨-\beta \left|\rho \right|\beta ⟩\text{exp}\left(\beta {\alpha }^{\ast }-{\beta }^{\ast }\alpha \right)$. The corresponding Gaussian probability distributions of width $\sigma$ centered about the qubit-dependent “position” $x_s$ are shown in the top panel.

Figure 3

(Color online) $\text{Prob}\left(0,t=0.1\text{ns}\right)$ for the initial state $|0⟩⟨0|$, as given by Eq. (49), plotted as a function of the detuning $\Delta \omega /2\pi$. The values of the parameters used are listed in Table .

Figure 4

(Color online) Probability to detect the outcome $s=1$, corrected by (a) the real part of $F^{\left(1\right)}$, for the initial state ${|+⟩}_X⟨+|$, and (b) the imaginary part of $F^{\left(1\right)}$, for the initial state ${|+⟩}_Y⟨+|$, plotted vs the detuning $\Delta \omega /2\pi$ for several values of the amplitude $f$. The values of the parameters used are listed in Table .

Figure 5

(Color online) Plot of the second order correction ${\text{Prob}}^{\left(2\right)}\left(s=1\right)$ to detect 1 for the initial state $|0⟩⟨0|$, for $\Delta t=\Delta /ϵ$, as a function of the detuning $\Delta \omega /2\pi$, for several values of the driving amplitude $f$. The values of the parameters for the evaluation used are listed in Table .

Figure 6

(Color online) Conditional probability to obtain (a) $s^{\prime }=s=1$, (b) $s^{\prime }=-s=1$, (c) $s^{\prime }=-s=-1$, and (d) $s^{\prime }=s=-1$ for the case $\Delta t=\Delta /ϵ=0.1$ and $T_1=10\text{ns}$, when rotating the qubit around the $y$ axis before the first measurement for a time ${\tau }_1$ and between the first and the second measurement for a time ${\tau }_2$, starting with the qubit in the state $|0⟩⟨0|$. Correction in $\Delta t$ are up to second order. The harmonic oscillator is driven at resonance with the bare harmonic frequency and a strong driving together with a strong damping of the oscillator are assumed, with $f/2\pi =20\text{GHz}$ and $\kappa /2\pi =1.5\text{GHz}$.

Figure 7

(Color online) Comparison of the deviations from QND behavior originating from different mechanisms. Conditional probability $P\left(0\mid 0\right)$ vs qubit driving time ${\tau }_1$ and ${\tau }_2$ starting with the qubit in the state $|0⟩⟨0|$, for (a) $\Delta =0$ and $T_1=\infty$, (b) $\Delta =0$ and $T_1=2\text{ns}$, and (c) $\Delta =0.1ϵ$ and $T_1=\infty$. The oscillator driving amplitude is $f/2\pi =20\text{GHz}$ and a damping rate $\kappa /2\pi =1.5\text{GHz}$ is assumed.