Quantum-limited measurement of spin qubits via curvature coupling to a cavity

TL;DR

This paper proposes a novel quantum non-demolition readout method for spin qubits using curvature coupling to a microwave resonator, enabling high-fidelity measurement while maintaining qubit stability at the charge noise sweet-spot.

Contribution

It introduces a curvature coupling approach for spin qubits that allows dispersive-like readout without Purcell effect, and enables entanglement via measurement with gate voltage modulation.

Findings

Achieves measurement strength of tens to hundreds of MHz.

Maintains qubit at full charge noise sweet-spot with zero dipole moment.

Enables selective longitudinal readout and multi-qubit entanglement.

Abstract

We investigate coupling an encoded spin qubit to a microwave resonator via qubit energy level curvature versus gate voltage. This approach enables quantum non-demolition readout with strength of tens to hundred MHz all while the qubit stays at its full sweet-spot to charge noise, with zero dipole moment. A "dispersive-like" spin readout approach similar to circuit-QED but avoiding the Purcell effect is proposed. With the addition of gate voltage modulation, selective longitudinal readout and n-qubit entanglement-by-measurement are possible.

Equations

$${\mathcal{H}}_{\mathrm{tot}}={\mathcal{H}}_0+{\mathcal{H}}_{\lambda }+{\mathcal{H}}_{\parallel }+\epsilon \left({\hat{a}}^{+}{e}^{-i{\omega }_{d}t}+h.c.\right)+{\mathcal{H}}_{\mathrm{env}},$$

$$\lambda \equiv \frac{\delta \omega }{{\omega }_r}=\frac{\hslash {\omega }_r}{2}{\left(\frac{\eta }{\hslash }\right)}^2\frac{{\partial }^2{E}_q(V_m^0)}{e^2\partial V_G^2}.$$

$$\frac{\{{\tilde{g}}_{\parallel },{\tilde{g}}_0\}}{{\omega }_r}=\frac{\eta }{e\hslash }\left\{\frac{1}{2}\frac{{\partial }^2{E}_q(V_m^0)}{\partial V_G^2},\frac{{\partial }^2{G}_q(V_m^0)}{\partial V_G^2}\right\}{\tilde{V}}_m$$

$$\frac{d\rho }{dt}=-i\left[{\mathcal{H}}_{\mathrm{tot}},\rho \right]-i\frac{\kappa }{2\hslash }\left[\hat{x}\left\{\hat{p},\rho \right\}\right]-\frac{K_d}{{\hslash }^2}\left[\hat{x}\left[\hat{x},\rho \right]\right]$$

$${\rho }_{+-}^q(t)={\rho }_{+-}^q(0)e^{-i({\omega }_q-2\delta \omega -i{\gamma }_2)t}{e}^{-i2\delta \omega {\int }_0^{t}d{t}^{\prime }{\alpha }_{+}(t^{\prime }){\alpha }_{-}^{\ast }(t^{\prime })}$$

$$\times e^{-i{\tilde{g}}_{\Vert }{\int }_0^{t}d{t}^{\prime }\left[{\alpha }_{+}(t^{\prime })e^{i{\phi }_m}+{\alpha }_{-}^{\ast }(t^{\prime })e^{-i{\phi }_m}\right]},$$

$${\Gamma }_{\lambda }=-2\delta \omega \mathrm{Im}\left[{\alpha }_{+}^{\mathrm{st}}{\alpha }_{-}^{\mathrm{st}\ast }\right]$$

$$=\frac{(2\delta \omega {)}^2(\kappa /2)}{{\left[\delta {\omega }^2+{\kappa }^2/4\right]}^2}\left[{\left(\epsilon +{\tilde{g}}_0/2\right)}^2-{\left({\tilde{g}}_{\Vert }/2\right)}^2\right]$$

$${\Gamma }_{\parallel }=-{\tilde{g}}_{\Vert }\mathrm{Im}\left[{\alpha }_{+}^{\mathrm{st}}+{\alpha }_{-}^{\mathrm{st}\ast }\right]=\frac{{\tilde{g}}_{\Vert }^2(\kappa /2)}{\delta {\omega }^2+{\kappa }^2/4}.$$

$$I_{\pm }=\beta \kappa \left[{\alpha }_{\pm }^{\mathrm{st}}{e}^{-i\varphi }+{\alpha }_{\pm }^{\mathrm{st}\ast }{e}^{i\varphi }\right],$$

$$I_m(t)=\frac{\Delta I}{2}\mathrm{Tr}[{\rho }_m^q(t){\sigma }_z]+\beta \sqrt{\kappa }\xi (t).$$

$${\Gamma }_{\mathrm{m}}^{\max }=\frac{\Delta I_{\max }^2}{4S_0}=|{\alpha }_{+}^{\mathrm{st}}-{\alpha }_{-}^{\mathrm{st}}{|}^2\frac{\kappa }{2},$$

$${\rho }_{\mathrm{m}}^q(t,\overline{I})=\frac{{\hat{U}}_{\overline{I},z}{\hat{M}}_{\overline{I},z}{\rho }^q(0){\hat{M}}_{\overline{I},z}^{+}{\hat{U}}_{\overline{I},z}^{+}}{P[\overline{I}(t)]},$$

$${\hat{\Omega }}_{\overline{I},{\hat{c}}_n(\varphi )}=(\frac{t}{\pi S_0}{)}^{\frac{1}{4}}{e}^{-[\overline{I}(t)-{\hat{c}}_n(\varphi ){]}^2\frac{t}{2S_0}}\cdot e^{i\overline{I}(t){\hat{c}}_n(\varphi +\frac{\pi }{2})\frac{t}{S_0}}{e}^{-i{\hat{\phi }}_n(t)}$$

Full text

Introduction.- Quantum measurement of semiconductor spin qubits, e.g. in quantum dots (QD), is usually associated with the spin-to-charge conversion technique, where spin states are mapped to auxiliary charge states, of the system, and sensing the charge is via electron transport in a (nearby) quantum point contact or a single electron transistor Elzerman et al. (2004); Petta et al. (2005); Veldhorst et al. (2015). This method suffers, however, from having to move the qubit away from its operating point, from low sensing efficiency, and/or susceptibility to charge and $1/f$ noise of the QD qubit and detector. Thus, a readout approach using transport of microwave (MW) photons, coupled to the spin via a common superconducting (SC) resonator Petersson et al. (2012); Frey et al. (2012) and utilizing noiseless MW amplification is desirable, as it has been proven suitable to reach a quantum-limited measurement regime for superconducting (SC) qubits Blais et al. (2004); Vijay et al. (2012); Ristè et al. (2013); Roch et al. (2014). The key is then to establish a spin coupling to resonator leading to spin-dependent photon scattering.

In the standard approach, a transverse coupling via the qubit electric dipole moment (e.d.m.), $\bm{d}\neq 0$ is used (as in SC transmonBlais et al. (2004)), leading to a Jaynes-Cummings interaction: ${\cal H}_{\rm tr}\simeq\hbar g_{\perp}(\sigma_{-}\hat{a}^{\dagger}+\sigma_{+}\hat{a})$. In the dispersive limit, the qubit-resonator detuning $\Delta$ is large and the resulting coupling (to leading order in $g_{\perp}/\Delta$), ${\cal H}_{\rm tr}\simeq\frac{\hbar g_{\perp}^{2}}{\Delta}\,\hat{a}^{\dagger}\hat{a}\sigma_{z}$, commutes with the qubit Hamiltonian ${\cal H}_{q}$. Recent studies of the resonant exchange (RX) qubit Taylor et al. (2013), based on a 3-electron triple quantum dot (TQD), have offered strong spin-cavity coupling, at a partial sweet spot to gate detuning fluctuations (see also Ref.Russ and Burkard, 2015).

This approach, however, has several drawbacks for electron spin QD qubits: first, for a finite e.d.m. the spin qubit is more susceptible to charge noise Hu and Das Sarma (2006); Taylor et al. (2013); Russ et al. (2016); Friesen et al. (2016). Secondly, in higher orders of $g_{\perp}/\Delta$, the transverse interaction no longer commutes with ${\cal H}_{q}$, dressing the qubit-resonator states and leading to enhanced qubit relaxation (Purcell effect) and dressed dephasing Boissonneault et al. (2009); Sete et al. (2014) even if the resonator is coherently populated. These effects may increase for a spin qubit (relative to a transmon), since the former usually possesses a small dipole moment, and a trade off between charge noise (less e.d.m.) and a larger resonator photon flux (stronger measurement) may not exist.

In this Letter we propose, alternatively, a spin-to-SC-resonator coupling, utilizing the TQD qubit energy curvature, $\partial^{2}E_{q}(V_{G})/\partial V_{G}^{2}$, with a gate voltage (Fig. 1a). We find that a static, dispersive-like coupling ${\cal H}_{\lambda}=\hbar\lambda\omega_{r}\left(\hat{a}^{+}\hat{a}+\frac{1}{2}\right)\sigma_{z}$, occurs due to the quantized resonator voltage. A dynamical longitudinal coupling: ${\cal H}_{\parallel}=\hbar\left[g_{\parallel}(t)\sigma_{z}+g_{0}(t)\hat{I}\right](\hat{a}+\hat{a}^{+})$ also appears if an additional external voltage modulation $\tilde{V}_{m}(t)$ is applied near the resonator frequency, generating a spin-dependent “force” exerted on the resonator. Similar longitudinal coupling was explored in ion trap quantum gates Mølmer and Sørensen (1999); Milburn et al. (2000); Leibfried et al. (2003); Haljan et al. (2005), and was recently proposed by Kerman Kerman (2013) and others Billangeon et al. (2015); Didier et al. (2015); Richer and DiVincenzo (2016) for SC qubits. The energy curvature can be appreciably large in a regime where e.d.m. is zero (Fig. 1b), and the charge noise to the qubit is minimized (full-sweet-spot), previously referred to as the AEON qubit regime Shim and Tahan (2016). The curvature interactions commute with the qubit Hamiltonian, avoiding Purcell effect. Importantly, we show here that quantum measurements can be performed while each qubit is residing at its full sweet spot, with a measurement time of the order of tens of ns.

Dispersive-like and longitudinal curvature couplings.- We consider a TQD 3e-qubit coupled to a resonator via a voltage variation on the middle dot, $V_{G}=V_{m}^{0}+\tilde{V}_{m}(t)+\hat{V}_{r}$, with $V_{m}^{0}$ at the full-sweet-spot; $\hat{V}_{r}\propto\sqrt{\omega_{r}}\left(\hat{a}^{+}+\hat{a}\right)$ is the resonator quantized voltage, and $\tilde{V}_{m}(t)$ is an external modulation, Fig. 1. The Hamiltonian (including resonator driving and environment) is:

[TABLE]

where ${\cal H}_{0}=\hbar\omega_{r}\hat{a}^{+}\hat{a}+{\cal H}_{q}$, $\hat{a}$ is a mode annihilation operator, and the couplings, ${\cal H}_{\lambda}$, ${\cal H}_{\parallel}$, derive from the qubit energy: ${\cal H}_{q}=E_{g}|g\rangle\langle g|+E_{e}|e\rangle\langle e|\equiv G_{q}(V_{G})\hat{I}+\frac{E_{q}(V_{G})}{2}\tilde{\sigma}_{z}$, expanded to second order. With no gate modulation, a static dispersive-like interaction ${\cal H}_{\lambda}\propto\hat{V}^{2}_{r}\sigma_{z}$ (neglecting contra-rotating terms $\sim\hat{a}^{2},\hat{a}^{+2}$) leads to a spin-dependent detuning, $\delta\omega_{s}=\pm\lambda\omega_{r}$:

[TABLE]

The ratio $\frac{\eta}{\hbar}\equiv\alpha_{c}\,\sqrt{\frac{Z_{r}}{\hbar/e^{2}}}$ includes the middle dot coupling capacitances to the resonator and the ground (Fig. 1a) via $\alpha_{c}\equiv\frac{C_{c}}{C_{c}+C_{d}}$, and a ratio of the resonator impedance, $Z_{r}\simeq\omega_{r}L_{r}\gg R_{r}$, to the resistance quantum. Here, the zero-point fluctuation (for a resonator circuit, $mass\leftrightarrow L_{r}$) is $\Delta x_{0}=e\sqrt{\frac{\hbar/e^{2}}{2\omega_{r}L_{r}}}$, and a ratio of $\frac{\eta}{\hbar}\gtrsim 1$ is possible for high kinetic inductance ($L_{r}\gtrsim 100\,{\rm nH}$) resonators, reached in SC wires with disorderSamkharadze et al. (2016).

By switching on a voltage modulation $\tilde{V}_{m}(t)$, a term $\propto\tilde{V}_{m}\,\hat{V}_{r}$ gives the longitudinal Hamiltonian: ${\cal H}_{\parallel}=\hbar\left[g_{\parallel}\sigma_{z}+g_{0}\hat{I}\right](\hat{a}+\hat{a}^{+})$ with couplings modulated in time: $g_{\parallel,0}=g_{\parallel,0}^{\rm st}+\tilde{g}_{\parallel,0}(V^{0}_{m})\,\cos(\omega_{m}t+\varphi_{m})$ at a frequency $\omega_{m}\sim\omega_{r}$ (a static coupling, $g_{\parallel,0}^{\rm st}\propto\frac{\partial E_{q}(V^{0}_{m})}{\partial V_{G}}$, is zeroed at sweet spot). In a frame rotating with $\omega_{m}$, the longitudinal couplings read:

[TABLE]

TQD couplings estimations.- For a triple QD (TQD) 3-electron qubit, we are seeking a configuration where, ideally, the electric dipole moment is zero, avoiding both spurious transverse coupling, static longitudinal coupling, and also minimizing susceptibility to charge noise. In a recently established full-sweet-spot parameter regime Shim and Tahan (2016) the relevant qubit states are made of the two bare $(111)$-states with spin projection $S_{z}=1/2$: $|1\rangle=|S\rangle_{13}|\uparrow\rangle_{2}=\frac{1}{2}\left[|\uparrow_{1}\uparrow_{2}\downarrow_{3}\rangle-|\downarrow_{1}\uparrow_{2}\uparrow_{3}\rangle\right]$, and $|2\rangle=-\frac{1}{\sqrt{6}}\left[|\uparrow_{1}\uparrow_{2}\downarrow_{3}\rangle+|\downarrow_{1}\uparrow_{2}\uparrow_{3}\rangle-2|\uparrow_{1}\downarrow_{2}\uparrow_{3}\rangle\right]$ with small admixture of the other charge configurations, like $(201)$, $(102)$, etc., Fig. 1b. Unlike the RX-regime Taylor et al. (2013); Medford et al. (2013), the Coulomb energy cost, $\tilde{U}_{i}$, for a double occupation of the i-th dot is large compared to the interdot tunnelings: $\tilde{U}_{i}\gg t_{l},t_{r}$. Coupling the resonator through the middle dot, Fig. 1a, allows the sweet-spot to remain largely intact (compare with Refs. Reed et al. (2016); Martins et al. (2016)).

Introducing the independent energy detunings, $\epsilon_{v}\equiv(V_{3}-V_{1})/2$, and $V_{m}\equiv(V_{3}+V_{1})/2-V_{2}$ ($V_{i}$ is the gate voltage applied to the i-th dot) the effective qubit Hamiltonian is recast to the form Taylor et al. (2013); Shim and Tahan (2016): ${\cal H}_{\rm q,eff}=-J(\epsilon_{v},V_{m})\,\hat{I}-\frac{E_{q}(\epsilon_{v},V_{m})}{2}\tilde{\sigma}_{z}$, where the exchange energies $J_{l,r}$, $J\equiv(J_{l}+J_{r})/2$, determine the qubit splitting $E_{q}=\sqrt{J_{l}^{2}+J_{r}^{2}-J_{l}J_{r}}$. At the sweet spot ($\frac{\partial E_{q}}{\partial\epsilon_{v}}=\frac{\partial E_{q}}{\partial V_{m}}=0$): $J_{l,r}^{0}\equiv J_{l,r}(\epsilon_{v}^{0},V_{m}^{0})=\frac{8t_{l,r}^{2}}{a_{l,r}}$, with $a_{l}=\tilde{U}_{1}+\tilde{U}_{2}^{{}^{\prime}}$, $a_{r}=\tilde{U}_{2}+\tilde{U}_{3}$. For coupling estimations, we consider: $a_{l,r}=a$, $t_{l,r}=t$, resulting in $\frac{\partial^{2}E_{q}^{0}}{\partial V_{m}^{2}}=64t^{2}/a^{3}$. For typical Si QD charging energies (see Refs. Maune et al. (2012); Medford et al. (2013); Reed et al. (2016)) $U_{1}=U_{3}=U=0.5\,{\rm meV}$, and $V_{12}=V_{23}=0.1U$, $V_{13}=0.05U$, tunneling $t=20(40)\,\mu{\rm eV}$, and modulation amplitude $\tilde{V}_{m}=0.1\,{\rm meV}$ one obtains a resonator frequency shift $\delta\omega\simeq 10.3\,{\rm MHz}$ ($\delta\omega=41.4\,{\rm MHz}$), a longitudinal coupling $\tilde{g}_{\parallel}\simeq 25\,{\rm MHz}$ ($\tilde{g}_{\parallel}=100\,{\rm MHz}$), and a qubit splitting $E_{q}=J\simeq 1\,{\rm GHz}$ ($3.9\,{\rm GHz}$), well off resonance with $\omega_{r}/2\pi=10\,{\rm GHz}$. Since the scalings, $\delta\omega_{s}\propto\eta^{2}\omega_{r}^{2}\,t_{l,r}^{2}/U^{3}$ and $\tilde{g}_{\|}\propto\eta\omega_{r}\,t_{l,r}^{2}/U^{3}$ ($\eta\propto\sqrt{\omega_{r}}$), a large range of parameters can be explored e.g., for slightly larger dots, $U\approx 0.4\,{\rm meV}$, the couplings increase twice. Higher curvature corrections to $\delta\omega$, $\tilde{g}_{\parallel}$ are due to $\partial^{4}E_{q}^{0}/\partial V_{m}^{4}$, and reach 5-15%. There appear also small non-linearities of the form, $\zeta_{0}(\hat{a}^{+}\hat{a})^{2}$, $\zeta_{\|}(\hat{a}^{+}\hat{a})^{2}\sigma_{z}$, $\zeta_{\hat{n}}(\hat{a}^{+}\hat{n}+\hat{n}\hat{a})^{2}\sigma_{z}$, that are Purcell free as well.

Qubit dephasing via the resonator relaxation- The qubit plus a SC resonator system is described via a Caldeira-Leggett master equation Caldeira et al. (1989) (plus qubit relaxation and dephasing):

[TABLE]

where $\hat{x}\equiv\Delta x_{0}(\hat{a}+\hat{a}^{+})$, $\hat{p}\equiv-i\Delta p_{0}(\hat{a}-\hat{a}^{+})$ are the “position” and “momentum” operators, and $K_{d}=\frac{L_{r}\hbar\omega_{r}\kappa}{2}\,\coth{\frac{\hbar\omega_{r}}{2k_{B}T_{r}}}$ is the temperature dependent diffusion. For the qubit readout one considers a zero resonator temperature $T_{r}=0$, where under the evolution of Eq. (4), a coherent state remains coherent (pure) state. This is also preserved by continuous measurements of the resonator, (see Ref. Ruskov et al. (2005) and references therein).

The qubit-resonator density matrix can be expanded in the complete set of qubit operators $|s\rangle\langle s^{\prime}|$ ($s=\pm$ for a single qubit): $\rho=\sum_{s,s^{\prime}}\hat{\rho}_{s,s^{\prime}}|s\rangle\langle s^{\prime}|$ where the partial matrices $\hat{\rho}_{s,s^{\prime}}$ act only on the resonator Wiseman and Milburn (2010). In this case it is sufficient to solve for the partial density matrices, $\hat{\rho}_{s,s^{\prime}}$, using positive $P^{(+)}$-representation Gardiner and Zoller (2000). By making a Gaussian (coherent state) anzatz $\hat{\rho}_{s,s^{\prime}}(t)=\rho^{q}_{ss^{\prime}}(t)\,|\alpha_{s}(t)\rangle\langle\alpha_{s^{\prime}}(t)|$ for the partial density matrices Gambetta et al. (2008) one obtains the spin-dependent resonator oscillations under driving and modulation (in rotating wave approximation) $\dot{\alpha}_{\pm}(t)=-i\{\left[(\tilde{\omega}_{r}\pm\delta\omega)-i\frac{\kappa}{2}\right]\alpha_{\pm}+\varepsilon+\frac{e^{-i\varphi_{m}}}{2}\left[\pm\tilde{g}_{\|}+\tilde{g}_{0}\right]\}$, where $\tilde{\omega}_{r}=\omega_{r}-\omega_{m}$, $\delta\omega\equiv\lambda\omega_{r}$. For the non-diagonal qubit density matrix one gets the solution:

[TABLE]

where $\gamma_{2}\equiv 1/T_{2}$ is the qubit internal dephasing, and the last two terms can be written in the long-time limit ($t\gg 1/\kappa$) as a resonator-induced dephasing, $e^{-\Gamma_{\rm qb,res}t}$, with $\Gamma_{\rm qb,res}\equiv\Gamma_{\lambda}+\Gamma_{\parallel}$. With the stationary solutions, $\alpha_{\pm}^{\rm st}=-\frac{\varepsilon+\frac{e^{-i\varphi_{m}}}{2}(\pm\tilde{g}_{\|}+\tilde{\eta}_{0})}{(\tilde{\omega}_{r}\pm\delta\omega)-i\frac{\kappa}{2}}$, one obtains (at resonance $\tilde{\omega}_{r}=0$ and $\varphi_{m}=0$) the dispersive-like and longitudinal contributions to $\Gamma_{\rm qb,res}$:

[TABLE]

One also obtains qubit frequency (ac Stark) shifts: $\delta\omega_{\lambda}=-2\delta\omega\,{\rm Re}\left[\alpha_{+}^{\rm st}\alpha_{-}^{\rm st*}\right]=\frac{2\delta\omega\left[\delta\omega^{2}-\kappa^{2}/4\right]}{\left[\delta\omega^{2}+\kappa^{2}/4\right]^{2}}\left[\left(\varepsilon+\tilde{g}_{0}/2\right)^{2}-\left(\tilde{g}_{\|}/2\right)^{2}\right]$, $\delta\omega_{\|}=-\tilde{g}_{\|}{\rm Re}\left[\alpha_{+}^{\rm st}+\alpha_{-}^{\rm st*}\right]=\frac{\tilde{g}_{\|}^{2}\,\delta\omega}{\delta\omega^{2}+\kappa^{2}/4}$.

Longitudinal readout- The rate $\Gamma_{\rm qb,res}$ can be interpreted as the maximal measurement rate of a qubit. Indeed, performing a homodyne measurement of the resonator , the measurement signal is Wiseman and Milburn (2010): $I_{\rm m}(t)=\beta\left[\kappa\langle\hat{X}_{\phi}\rangle+\sqrt{\kappa}\xi(t)\right]$, where $\hat{X}_{\phi}=\hat{a}\,e^{-i\phi}+\hat{a}^{+}\,e^{i\phi}$ is the measured resonator quadrature, $\langle\ldots\rangle\equiv\mbox{\rm Tr}[\rho_{m}\ldots]$ is the quantum average ($\rho_{m}$ is the conditioned system density matrix), $\beta$ and $\phi$ are respectively the strength and phase of the local oscillator (Fig. 1a), and $\xi(t)$ is the detector noise. In the “bad cavity limit” (when $\kappa\gg\Gamma_{\rm qb,res}$), the photon leakage out of the resonator is much faster than the qubit internal evolution, implying that homodyne measurement of the resonator field is a qubit measurement. In the stationary regime, for fixed qubit states $|\pm\rangle$, the average currents read

[TABLE]

and the current signal can be expressed via the qubit (conditioned) density matrix $\rho^{q}_{m}(t)$, with $\Delta I\equiv I_{+}-I_{-}$:

[TABLE]

The (single sided) current spectral density, $S_{0}$, is related to the photon shot noise $I_{0}(t)\simeq\beta\sqrt{\kappa}\xi(t)$ via: $\langle I_{0}(t)\,I_{0}(t^{\prime})\rangle_{\xi}=\beta^{2}\kappa\delta(t-t^{\prime})\equiv\frac{S_{0}}{2}\,\delta(t-t^{\prime})$. For each of the qubit states $|+\rangle$, $|-\rangle$, the random finite time average $\overline{I}(t)=\frac{1}{t}\int_{0}^{t}I_{\rm m}(t^{\prime})dt^{\prime}$ is Gaussian distributed with the averages $I_{\pm}$ and variance $\frac{S_{0}}{2t}$ [assuming weak response, $\Delta I\ll\beta\kappa$]. Then, a measurement time $\tau_{\rm meas}\equiv 1/\Gamma_{\rm m}$ can be introduced, as the time needed to distinguish between the two state-dependent distributions Korotkov (2001): $\tau_{\rm meas}=\frac{4S_{0}}{\left(I_{+}-I_{-}\right)^{2}}$. The detector response takes the form $\Delta I=2\beta\kappa|\alpha_{+}^{\rm st}-\alpha_{-}^{\rm st}|\,\cos{\Delta\phi}$, and is maximized by choosing the measured quadrature $\hat{X}_{\phi}$, so that $\Delta\phi\equiv\phi-\phi_{\rm dif}=0$, where $\phi_{\rm dif}={\rm arg}[\alpha_{+}^{\rm st}-\alpha_{-}^{\rm st}]$ is the phase of the difference for the two resonator fields. Thus, the maximal measurement rate is:

[TABLE]

which is equal to the dephasing rate, $\Gamma_{\rm qb,res}=\Gamma_{\lambda}+\Gamma_{\parallel}$, of Eqs. (6),(7). For quadrature measurement with $\phi\neq\phi_{\rm dif}$ one gets a rate $\Gamma_{\rm m}=\Gamma_{\rm m}^{\rm max}\cos^{2}{(\phi-\phi_{\rm dif})}$ [see also Ref. Gambetta et al. (2008)].

The qubit density matrix at time $t$, given the measurement record $\overline{I}(t)$, will be updated via a quantum Bayesian rule Korotkov (2001):

[TABLE]

where $P[\overline{I}(t)]\equiv{\rm Tr}[\rho^{q}(0)\,\hat{M}_{\overline{I},z}^{+}\,\hat{M}_{\overline{I},z}]$ is the total probability of the “event” $\overline{I}(t)$, the measurement operators are $\hat{M}_{\overline{I},z}=(\frac{t}{\pi S_{0}})^{\frac{1}{4}}\,e^{-[\overline{I}(t)-\frac{\Delta I_{\rm max}}{2}\cos\Delta\phi\,\hat{\sigma}_{z}]^{2}\,\frac{t}{2S_{0}}}$, and an additional unitary backaction is induced by the homodyne measurement Wiseman and Milburn (2010); Gambetta et al. (2008); Korotkov (2016): $\hat{U}_{\overline{I},z}=e^{i\overline{I}(t)\,\frac{\Delta I_{\rm max}}{2}\sin\Delta\phi\,\hat{\sigma}_{z}\,\frac{t}{S_{0}}}e^{\!-i\hat{\varphi}_{1}(t)}$. Measuring the quadrature with $\phi=\phi_{\rm dif}$ eliminates this backaction (except the deterministic phases $\hat{\varphi}_{1}(t)$, see below) and leads to maximum information inferred from the qubit. The measurement operators, $\hat{M}_{\overline{I},z}$ are derived from two requirements: (i) the qubit diagonal density matrix elements $\rho^{\rm m}_{++}(t)$, $\rho^{\rm m}_{--}(t)$, are updated according to a classical Bayesian rule with likelihood probabilities $P_{\pm}(\overline{I})=(\frac{t}{\pi S_{0}})^{\frac{1}{2}}e^{-(\overline{I}(t)\mp\frac{\Delta I_{\rm max}}{2}\cos\Delta\phi)^{2}\,\frac{t}{S_{0}}}$, obeying a quantum-classical correspondence Korotkov (2001); (ii) the evolution of the non-diagonal element obeys the rule: $\rho^{\rm m}_{+-}(t)=\rho^{\rm m}_{+-}(0)\sqrt{\rho^{\rm m}_{++}(t)\rho^{\rm m}_{--}(t)/\rho^{\rm m}_{++}(0)\rho^{\rm m}_{--}(0)}$, which follows from the saturation of the inequality $|\rho_{ij}|\leq\sqrt{\rho_{ii}\rho_{jj}}$ averaged over all possible records $\overline{I}(t)$ Korotkov (2001). By differentiating Eq. (11) a stochastic evolution equation can be obtained.

$n$-qubit readout- Consider simultaneous measurement of $n$ qubits coupled to a resonator. The basis spin states are of the form $|s\rangle=|\uparrow\uparrow\dots\downarrow\rangle$. The $n$-qubit couplings $G_{\parallel,s}$ and detunings $\delta\omega_{s}$ appear to be: $G_{\parallel,s}=\langle s|\,\frac{e^{-i\varphi_{m}}}{2}\sum_{j}\left[\tilde{g}^{(j)}_{0}+\tilde{g}^{(j)}_{\parallel}\sigma_{z}^{(j)}\right]\,|s\rangle$, and $\delta\omega_{s}=\langle s|\sum_{j}\delta\omega_{j}\sigma_{z}^{(j)}|s\rangle$. With the solution of $\dot{\alpha}_{s}(t)=-i\{\left[(\tilde{\omega}_{r}+\delta\omega_{s})-i\frac{\kappa}{2}\right]\alpha_{s}+\varepsilon+G_{\parallel,s}\}$ one gets the average currents $I_{\rm s}(\phi)=\beta\kappa\left[\alpha_{s}^{\rm st}\,e^{-i\phi}+\alpha_{s}^{\rm st*}\,e^{i\phi}\right]$ and the accumulated phases $\varphi_{s}(t)$, $\dot{\varphi}_{s}={\rm Re}[(\varepsilon+G_{\parallel,s})^{*}\alpha_{s}(t)]$ for each state $|s\rangle$. The $n$-qubit measurement operator reads ($\hat{\Omega}_{\overline{I},\hat{c}_{n}}\equiv\hat{M}_{\overline{I},\hat{c}_{n}}\cdot\hat{U}_{\overline{I},\hat{c}_{n}}$):

[TABLE]

with $\hat{c}_{n}(\phi)={\rm diag}\{I_{1}{\scriptstyle(\phi)},\ldots\!,I_{2^{n}}{\scriptstyle(\phi)}\}$, $\hat{\varphi}_{n}(t)={\rm diag}\{\varphi_{1},\ldots\!,\varphi_{2^{n}}\}$ being diagonal operators. In Eq. (12) the “pure” measurement operator $\hat{M}_{\hat{c}_{n}}$ is in general form and is equivalent to a quantum Bayesian update of the $n$-qubit density matrix, while $\hat{U}_{\hat{c}_{n}}$ is a unitary backaction derived using a “history tails” approach Korotkov (2016). By averaging over all possible realizations of $\overline{I}(t)$ one obtains the ensemble evolution: $\dot{\rho}^{q}_{ss^{\prime}}=(-i\omega_{ss^{\prime}}-\Gamma_{ss^{\prime}})\rho^{q}_{ss^{\prime}}$, where $\Gamma_{ss^{\prime}}=\frac{\Delta I_{{\rm max};ss^{\prime}}}{4S_{0}}=|\alpha_{s}^{\rm st}-\alpha_{s^{\prime}}^{\rm st}|^{2}\frac{\kappa}{2}$ are the partial dephasing rates, analogous to Eq. (10), and $\omega_{ss^{\prime}}=(\delta\omega_{s}-\delta\omega_{s^{\prime}})\,{\rm Re}\left[\alpha_{s}^{\rm st}\alpha_{s^{\prime}}^{\rm st*}\right]+{\rm Re}\left[(G_{\parallel,s}-G_{\parallel,s^{\prime}})^{*}(\alpha_{s}^{\rm st}+\alpha_{s^{\prime}}^{\rm st*})\right]$ are the ac-Stark shifts, similar to the single-qubit case.

By performing a joint measurement of qubits by a single resonator one can entangle them without a direct qubit interaction Ruskov and Korotkov (2003). In the simplest case of two qubits this is achieved provided the measurement cannot distinguish certain two-qubit subspaces Ruskov and Korotkov (2003); Mao et al. (2004); Ruskov et al. (2006); Ristè et al. (2013); Roch et al. (2014). Assuming equal couplings, $\lambda_{1}=\lambda_{2}$, $\tilde{g}_{\|,1}=\tilde{g}_{\|,2}$ (i.e. symmetric measurement), one gets $I_{\uparrow\downarrow}=I_{\downarrow\uparrow}$, a necessary condition for two-qubit entanglement by a joint measurement Ruskov and Korotkov (2003). In the limit, $\kappa\gg\delta\omega,\tilde{g}_{\|}$ the current differences are equal: $I_{\downarrow\downarrow}-I_{\uparrow\downarrow}=-(I_{\uparrow\uparrow}-I_{\uparrow\downarrow})=4\beta\tilde{g}_{\|}\sin\phi$, corresponding to a linear detector response which leads to an effective measurement of the total spin $(\overrightarrow{\sigma}_{1}+\overrightarrow{\sigma}_{2})^{2}$ and probabilistic entanglement, e.g., to the spin-zero subspace, starting from any separable initial state Ruskov and Korotkov (2003). By choosing the local oscillator phase $\phi=\pi/2$ one can eliminate the additional unitary backaction, while maximizing the qubit response.

For measurement rate estimation one considers the “bad cavity limit”, when $\omega_{r}\gg\kappa\gg\Gamma_{\rm m}$, also requiring $\kappa\gg\delta\omega,\tilde{g}_{\parallel}$. With an external driving $\epsilon\approx 20(15)\,{\rm MHz}$ (at $\omega_{d}=\omega_{r}$), no modulation, and for the parameters of $\omega_{r}/2\pi=5(10)\,{\rm GHz}$, $\eta/\hbar=0.35(1.)$, tunneling $t_{l,r}=40(20)\,\mu{\rm eV}$, $U_{\rm charge}=0.32(0.5)\,{\rm meV}$, and loaded resonator $Q$-factor $1.25(2.5)10^{2}$, one gets: $E_{q}\approx 6(1)\,{\rm GHz}$, $\delta\omega=5.2(10.3)\,{\rm MHz}$, and $\kappa/2\pi=40\,{\rm MHz}$, thus obtaining $\Gamma_{\rm m}/2\pi\simeq 4.7(7.4)\,{\rm MHz}$, average photon number $\bar{n}\approx 0.9(0.4)$, and a measurement time of $\tau_{m}\simeq 34(21.3)\,{\rm ns}$. Similarly, with modulation $\tilde{V}_{m}=0.1\,{\rm meV}$ (at $\omega_{m}=\omega_{r}$) and no driving, for the parameters $\omega_{r}/2\pi=5(10)\,{\rm GHz}$, $\eta/\hbar=0.35(1.)$, tunneling $t_{l,r}=20\,\mu{\rm eV}$, $U_{\rm charge}=0.4(0.5)\,{\rm meV}$, and $Q=10^{2}$, one gets: $E_{q}\approx 1\,{\rm GHz}$, $\delta\omega=0.6(10.3)\,{\rm MHz}$, $\tilde{g}_{\parallel}=8.8(25)\,{\rm MHz}$, and $\kappa/2\pi=100\,{\rm MHz}$, thus obtaining $\Gamma_{\rm m}/2\pi\simeq 3.1(13.4)\,{\rm MHz}$, photon number of $\bar{n}\approx 0.3(0.5)$, and a measurement time of $\tau_{m}\simeq 50.8(11.8)\,{\rm ns}$, much smaller than typical $T_{2}$ of tens of $\mu{\rm s}$ Medford et al. (2013); Reed et al. (2016).

Summary.- The proposed dispersive-like and longitudinal curvature couplings of a TQD spin-qubit to a SC resonator of tens to hundred MHz can be much larger than the transverse dispersive coupling $g_{\perp}^{2}/\Delta$ for a similar TQD system Taylor et al. (2013); Russ and Burkard (2015), which needs a large e.d.m. These couplings can be comparable to superconducting qubits Walter et al. (2017), allowing fast spin readout at tens of ns. As opposed to Jaynes-Cummings interaction, curvature couplings are Purcell free, admitting higher photon numbers and even shorter readout times. The curvature couplings allow for spin measurements at a sweet-spot 111These can be used in a recently proposed quadrupole qubit Friesen et al. (2016)., with zero QDs’ e.d.m.​​ and minimized qubit dephasing, allowing for high readout efficiency. With the dispersive-like coupling $\lambda\omega_{r}$, a quantum-limited readout of individual qubits can be performed, as in CQED. On the other hand, in a regime where $\lambda\omega_{r}\ll\tilde{g}_{\|}$, and using the $n$-qubit measurement result Eq. (12), it is possible to utilize designated resonator(s) that selectively couple to a number of spin-qubits, which could be a viable route to generate spin entanglement within a cluster of qubits, and to create medium range spin entanglement across chip, which can be a resource in quantum computations Raussendorf and Briegel (2001).