Charge-insensitive qubit design derived from the Cooper pair box

Short dephasing times pose one of the main challenges in realizing a quantum computer. Different approaches have been devised to cure this problem for superconducting qubits, a prime example being the operation of such devices at optimal working points, so-called “sweet spots.” This latter approach led to significant improvement of $T_2$ times in Cooper pair box qubits [D. Vion et al., Science 296, 886 (2002)]. Here, we introduce a new type of superconducting qubit called the “transmon.” Unlike the charge qubit, the transmon is designed to operate in a regime of significantly increased ratio of Josephson energy and charging energy $E_J∕E_C$. The transmon benefits from the fact that its charge dispersion decreases exponentially with $E_J∕E_C$, while its loss in anharmonicity is described by a weak power law. As a result, we predict a drastic reduction in sensitivity to charge noise relative to the Cooper pair box and an increase in the qubit-photon coupling, while maintaining sufficient anharmonicity for selective qubit control. Our detailed analysis of the full system shows that this gain is not compromised by increased noise in other known channels.

Figure 1

(Color online) (a) Effective circuit diagram of the transmon qubit. The two Josephson junctions (with capacitance and Josephson energy $C_J$ and $E_J$) are shunted by an additional large capacitance $C_B$, matched by a comparably large gate capacitance $C_g$. (b) Simplified schematic of the transmon device design (not to scale), which consists of a traditional split Cooper pair box, shunted by a short $(L\sim \lambda ∕20)$ section of twin-lead transmission line, formed by extending the superconducting islands of the qubit. This short section of line can be well approximated as a lumped-element capacitor, leading to the increase in the capacitances $C_{g1}$, $C_{g2}$, and $C_B^{\prime }$ and hence in the effective capacitances $C_B$ and $C_g$ in the circuit.

Figure 2

(Color online) Eigenenergies $E_m$ (first three levels, $m=0,1,2$) of the qubit Hamiltonian (2.1) as a function of the effective offset charge $n_g$ for different ratios $E_J∕E_C$. Energies are given in units of the transition energy $E_{01}$, evaluated at the degeneracy point $n_g=1∕2$. The zero point of energy is chosen as the bottom of the $m=0$ level. The vertical dashed lines in (a) mark the charge sweet spots at half-integer $n_g$.

Figure 3

(Color online) (a) Rotor analogy for the transmon. The transmon Hamiltonian can be understood as a charged quantum rotor in a constant magnetic field $\sim n_g$. For large $E_J∕E_C$, there is a significant “gravitational” pull on the pendulum and the system typically remains in the vicinity of $\phi =0$. Only tunneling events between adjacent cosine wells (i.e., a full $2\pi$ rotor movement) will acquire an Aharonov-Bohm-type phase due to $n_g$. The tunneling probability decreases exponentially with $E_J∕E_C$, explaining the exponential decrease of the charge dispersion. (b) Cosine potential (black solid line) with corresponding eigenenergies and squared moduli of the eigenfunctions.

Figure 4

(Color online) Comparison of numerically exact and asymptotic expressions for the charge dispersion and energy levels. (a) Charge dispersion $\mid {ϵ}_m\mid$ as a function of the ratio $E_J∕E_C$ for the lowest four levels. The solid curves depict the exact results using Mathieu characteristic values, the dashed curves represent the asymptotic expansion, Eq. (2.5). The right vertical scale gives the charge dispersion in MHz for a transition frequency of $7\mathrm{GHz}$. (b) Energy level difference $E_{0m}=E_m-E_0$ at $n_g=1∕2$ as a function of the $E_J∕E_C$ ratio. Solid curves show the exact results; dashed lines are based on the asymptotic expression (2.11). The vertical scale on the right-hand side gives the transition frequencies from the ground state to level $m$ in GHz, assuming a charging energy of $E_C∕h=0.35\mathrm{GHz}$. All numerical data are obtained for $n_g=1∕2$.

Figure 5

(Color online) Relative (a) and absolute (b) anharmonicity at the degeneracy point as a function of the ratio $E_J∕E_C$. The solid curves show the exact results from Eq. (2.2), the dashed curves depict the perturbative result from Eq. (2.12). (c) Minimum pulse duration ${\tau }_p$ [blue (gray) line] and dephasing time due to charge fluctuations $T_2$ [black line, Eq. (5.9)] as a function of $E_J∕E_C$. (Arrows indicate the corresponding axis.) Starting in the charge regime, an increase of the $E_J∕E_C$ ratio initially leads to a strong increase in the minimum pulse duration, which diverges when the anharmonicity $\alpha$ crosses zero (“anharmonicity barrier”). Above the barrier, the operation time scales with a weak power law $\sim (E_J∕E_C{)}^{1∕2}$. At the same time, the inverse charge dispersion determining the dephasing time due to charge noise increases exponentially in $(E_J∕E_C{)}^{1∕2}$.

Figure 6

(Color online) Solutions to the qubit Hamiltonian (2.1) in the charge basis. Panel (a) shows plots of the probabilities $\mid ⟨n\mid {\psi }_m⟩{\mid }^2$ for the presence of $n$ Cooper pairs when residing in the transmon eigenstates $m=0$ and $1$ for three different $E_J∕E_C$ ratios $(n_g=1∕2)$. In the limit $E_J∕E_C\gg 1$, the solutions converge to (a discretized version of the) harmonic oscillator wave functions with increasing width. (b) Fluctuations of the number of Cooper pairs $n$ as a function of $E_J∕E_C$ for the first three transmon levels. Solid lines show numerically exact results, dashed lines correspond to the asymptotic solution (2.14). The Cooper pair number in the ground state (first excited state) fluctuates by approximately 1 (2) for an $E_J∕E_C$ ratio of 100.

Figure 7

(Color online) (a) Off-diagonal matrix elements of the Cooper pair number operator as a function of the energy ratio $E_J∕E_C$ $(n_g=1∕2)$. Solid curves represent the exact result, dashed curves depict the asymptotic behavior, Eq. (3.4). The results illustrate that coupling between neighboring transmon states is the only relevant coupling in the limit of large $E_J∕E_C$. (b) Level scheme for the coupled transmon system. Transmon states are denoted by $\mid i⟩$, $i=1,2,3$ for the ground, first, and second excited state. Photon numbers $n_{\mathrm{ph}}$ in the cavity are plotted vertically. The two arrows on the $\mid i=1,n=2⟩$ level illustrate the perturbative level repulsions leading to the dispersive shift.

Figure 8

(Color online) Mechanical analogy illustrating the fundamental difference between dc and ac response. (a) For a slow (adiabatic) change in the suspension point of the oscillator, the oscillator mass is displaced but the oscillator frequency does not change. (b) For an ac drive at resonance, energy is forced into or extracted from the system. Quantum mechanically, this leads to the coupling between the transmon and the cavity field.

Figure 9

(Color online) (a) Dispersive frequency shift $\chi$ as a function of detuning ${\Delta }_0$ between cavity and transmon and energy ratio $E_J∕E_C$ (top panel), for fixed transition frequency ${\omega }_{01}$, as well as $\beta =1∕2$ and $n_g=1∕2$. (b) Cross sections of the three-dimensional plot for $E_J∕E_C=20$ (a), $40$ (b), $60$ (c), $80$ (d). The right-hand side $y$ axis shows the dispersive shift in MHz for the representative parameters chosen in the text. In the general case, the universal axis for $\hslash \chi$ (left-hand side) can be translated into MHz with the conversion factor $\frac{(g_{01}∕2\pi {)}^2}{({\omega }_{01}∕2\pi )\mid ⟨0\mid \widehat{n}\mid 1⟩{\mid }^2}$. Inset, level configuration and regimes for the dispersive frequency shift.

Figure 10

(Color online) Number of quasiparticles and contributions to the relaxation time $T_1$ due to inelastic quasiparticle tunneling as a function of temperature at $E_J∕E_C=60$. From this estimate, tunneling of quasiparticles is not expected to limit the performance of the transmon at cryogenic temperatures. In typical dilution refrigerator experiments, (phonon) temperatures are of the order of $20\mathrm{mK}$, marked in the plot by a vertical bar.

Figure 11

(Color online) Model for the estimate of relaxation times due to flux coupling, describing (i) flux coupling between the transmon’s SQUID loop and the external flux bias with mutual inductance $M$, and (ii) flux coupling between the transmon circuit and an external flux bias circuit via the mutual inductance $M^{\prime }$.

Figure 12

(Color online) (a) Full capacitance network for the transmon device. (b) Simplified schematic of the transmon device design (not to scale). (c) Reduced network.