Highly Controllable Qubit-Bath Coupling Based on a Sequence of Resonators

Abstract

Combating the detrimental effects of noise remains a major challenge in realizing a scalable quantum computer. To help to address this challenge, we introduce a model realizing a controllable qubit-bath coupling using a sequence of LC resonators. The model establishes a strong coupling to a low-temperature environment which enables us to lower the effective qubit temperature making ground state initialization more efficient. The operating principle is similar to that of a recently proposed coplanar-waveguide cavity (CPW) system, for which our work introduces a complementary and convenient experimental realization. The lumped-element model utilized here provides an easily accessible theoretical description. We present analytical solutions for some experimentally feasible parameter regimes and study the control mechanism. Finally, we introduce a mapping between our model and the recent CPW system.

Appendices

Appendix 1: Analytical Solution to the Energy Eigenproblem

In the low-energy regime and within the rotating-wave approximation, the time-independent Schrödinger equation $(\hat{H}_{L} + \hat{H}_{R} + \hat{H}_{L-R} + \hat{H}_{q} + \hat{H}_{R-q})|\varphi_{k}\rangle = \epsilon_{k} |\varphi_{k}\rangle$, where k∈{0,1,2,3}, has a ground state given by |φ ₀〉=|0,0,g〉. See Eqs. (2)–(9) for definitions of the Hamiltonians. We select the zero-point energy such that ϵ ₀=0. After rewriting the above Hamiltonian in the basis described in Sect. 3.2, the eigenproblem reduces to solving three simultaneous equations

(30)

(31)

(32)

where $\epsilon'_{k} = \epsilon_{k}/\hbar$. The equations yield the eigenenergies ϵ _k , k=1,2,3, and, similarly, solving γ _k and θ _k produces the excited states as |φ _k 〉=sin(θ _k )|1,0,g〉+cos(θ _k )sin(γ _k )|0,1,g〉+cos(θ _k )cos(γ _k )|0,0,e〉.

Appendix 2: Weak-Coupling Limit

Let us rewrite the total Hamiltonian of the resonator-qubit system [see Eqs. (2)–(9) for definitions] ignoring the resistor as $\hat{H}_{\mathrm{sys}} = \hat{H}_{L} + \hat{H}_{R} + \hat{H}_{q} + \hat{H}'$, where $\hat{H}' = \hat{H}_{L-R} + \hat{H}_{R-q}$ is the small perturbation in the weak coupling limit |α|,|g|≪ω _L ,ω _R ,ω _q and the system is far away from resonance |α|,|g|≪|ω _l −ω _s |, for all l,s∈{L,R,q}. The ground state is unaffected by the perturbation and given by |φ ₀〉=|0,0,g〉 with its eigenenergy selected as the zero-point of energy. After applying the second-order perturbation theory in the small coupling parameters, the eigenenergies of $\hat {H}_{\mathrm{sys}}$ are given by

(33)

(34)

(35)

where we adopted the notation Δ _ls =ω _l −ω _s for the difference between two intrinsic system frequencies. The corresponding eigenstates are

$$\begin{aligned} |\varphi_1\rangle =& A_1 \biggl[ \biggl( \frac{2\Delta_{RL}^{2}-\alpha ^{2}}{2\Delta_{RL}^{2}} \biggr) |1,0,g\rangle + \frac{\alpha}{\Delta_{RL}} |1,0,g\rangle + \frac{\alpha g}{\Delta_{RL}\Delta_{qL}}|0,0,e\rangle \biggr], \end{aligned}$$

(36)

$$\begin{aligned} |\varphi_2\rangle =& A_2 \biggl[ \frac{\alpha}{\Delta_{LR}} |1,0,g\rangle + \biggl(\frac{2\Delta_{qR}^{2}\Delta_{LR}^{2}-|g|^{2}\Delta _{LR}^{2}-\alpha^{2}\Delta_{qR}^{2}}{2\Delta_{qR}^{2}\Delta_{LR}^{2}} \biggr) |0,1,g\rangle \\ &{} +\frac{g}{\Delta_{qR}} |0,0,e\rangle \biggr], \end{aligned}$$

(37)

$$\begin{aligned} |\varphi_3\rangle =&A_3 \biggl[ \frac{g^*\alpha}{\Delta_{Rq}\Delta_{Lq}}|1,0,g\rangle + \frac{g^*}{\Delta_{Rq}}|0,1,g\rangle + \biggl(\frac{2\Delta _{Rq}^{2}-|g|^{2}}{2\Delta_{Rq}^{2}} \biggr) |0,0,e\rangle \biggr], \end{aligned}$$

(38)

and the normalization constants are given by

(39)

(40)

(41)

Appendix 3: Bare Mode Mapping

If we omit the coupling capacitor inserted into the CPW cavity shown in Fig. 3, the cavity regions can be treated as individual resonators. We denote these resonators as bare cavities and write the excitation frequencies for their lowest-energy modes as [28]

(42)

(43)

where c is the constant cavity capacitance per unit length, ℓ _L (ℓ _R ) is the left (right) cavity inductance per unit length, x _c is the position of the dividing capacitor C _c in the central conducting strip of the cavity and x _cav is the length of the cavity. Similarly, the voltage operators for these modes are given by

(44)

(45)

where $(\hat{a}_{l}^{b})^{\dagger}$ and $\hat{a}_{l}^{b}$, l=L,R, are the creation and annihilation operators for the modes, respectively. The prefactors for the operators are $V_{L}^{b0} = \sqrt{\hbar\omega _{L}^{b}/(x_{c} c)}$ and $V_{R}^{b0} = \sqrt{\hbar\omega_{R}^{b}/[(x_{\mathrm{cav}}-x_{c}) c]}$.

We relate the bare cavities to our resonator system in Fig. 1 by demanding that $\omega_{L} = \omega_{L}^{b}$, $\omega_{R} = \omega_{R}^{b}$, $V_{L}^{b0} = V_{L}^{0}$ and $V_{R}^{b0} = V_{R}^{0}$. This causes the resonators in our system and the bare cavities to have equal excitation energies, and the prefactors of the voltage operators at the ends of the bare cavities to match the prefactors of the voltage operators in our system. The resulting mapping is given in Eqs. (26)–(29). Matching the voltage operators in this manner ensures that including the same coupling capacitors into both systems has the same effect on the spectrum. This is because the bare mode voltages at the coupling capacitor effectively determine its charging Hamiltonian.