Gradiometric flux qubits with a tunable gap

The simplest version of the flux qubit (cf figure 1(a)) consists of a small superconducting loop with a diameter of the order of 10 μm intersected by three JJ with lateral dimensions of the order of 100 nm [24]. While two of these JJ have the same area (typically, A_J ≃ 0.03 μm² in our experiments) and, hence, the same critical current (typically, I_c ≃ 600 nA), the third JJ, the so-called α-junction, has a reduced area A_α = αA_J with α ≈ 0.6–0.8, resulting in a reduced critical current I_c,α = αI_c and a reduced junction capacitance C_α = αC_J. Since α = α₀ is fixed in the fabrication process, the qubit gap Δ is also fixed. Consequently, this version of the flux qubit is called the fixed-gap flux qubit. For α ≈ 0.6–0.8, the two-dimensional potential energy landscape of the flux qubit can be simplified. At the symmetry point, where the magnetic flux through the loop is equal to $(n+\frac {1}{2})\Phi _0$, with n being an integer, the potential can be reduced to a one-dimensional double well [26]. The two minima of this potential are associated with two degenerate persistent current states, corresponding to clockwise and counter-clockwise circulating persistent currents ±I_p. Due to the finite tunnel coupling of these states their degeneracy is lifted. The resulting symmetric and anti-symmetric superposition states form the ground and excited states of the flux qubit separated by the minimal energy splitting ℏΔ. Note that our terminology follows the most popular approach based on macroscopic quantum tunneling [21]; however, interpretations based on the resistively and capacitively shunted junction model [34–36] have not been ruled out yet. In the basis of the persistent current states and near the symmetry point, the Hamiltonian describing the flux qubit can be written as [26]

$$\begin{equation} \mathcal{H} = \frac{1}{2} \hbar\varepsilon \sigma_z -\frac{1}{2} \hbar\Delta \sigma_x . \end{equation} \tag{ 1 }$$

Here, σ_z and σ_x are the Pauli spin operators, ℏε = 2I_pδΦ is the magnetic energy bias and $\delta \Phi = \Phi _0 [f-(n+\frac {1}{2})]$ is the deviation of the flux Φ threading the loop from a half-integer multiple of Φ₀. The quantity f = Φ/Φ₀ is the magnetic frustration of the qubit loop and n an integer. The transition frequency between the ground and excited states can be written as

$$\begin{equation} \omega_{\mathrm{q}} = \sqrt{\varepsilon^{2}+\Delta^2} \end{equation} \tag{ 2 }$$

and the qubit gap Δ becomes [26]

$$\begin{equation} \Delta = \sqrt{\frac{4E_{\mathrm{J}} E_{\mathrm{c}} (4\alpha^2 -1)}{\alpha (1+2\alpha)}} \; \exp\left(- a(\alpha) \sqrt{4\alpha(1+2\alpha)\frac{E_{\mathrm{J}}}{E_{\mathrm{c}}}} \right). \end{equation} \tag{ 3 }$$

Here, E_J = ℏI_c/2e is the Josephson coupling energy, E_c = e²/2C is the charging energy and $a(\alpha ) = \sqrt {1-(1/4\alpha ^2)} - [\arccos (1/2\alpha )/2\alpha ]$ with a(α) ≃ 0.15 for α = 0.7. Obviously, Δ is determined by the critical current I_c and the capacitance C_J of the JJ as well as by α = α₀. All these parameters are fixed by the fabrication process. Decreasing α from 1 to 0.5 results in a strong increase of the exponential factor. At the same time, the prefactor (attempt frequency) decreases from the plasma frequency of the JJ to zero, because the double-well potential becomes a single well at α = 0.5. Since the exponential factor dominates within the major part of the interval 0.5 < α < 1, a strong increase of Δ is obtained by reducing α.

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It is evident from (1) and (2) that it is possible to tune ω_q by varying either ε or Δ. Varying the energy bias ε is simply achieved by changing δΦ via the magnetic field generated by an external solenoid or a current fed through an on-chip control line. However, δΦ ≠ 0 causes a shift of the qubit operation point away from the anti-crossing point with a minimal transition frequency Δ. As the energy of the flux qubit is stationary with respect to small variations of the applied magnetic flux (∂ω_q/∂δΦ = 0) only for $f = \left (n+\frac {1}{2} \right )$, any shift away from this symmetry point makes the flux qubit more susceptible to magnetic flux noise and significantly deteriorates the coherence properties [37, 38]. Since the fast tuning of ω_q of flux qubits is a prerequisite for numerous circuit QED experiments [39, 40], it is desirable to realize a tuning of ω_q by a variation of Δ.

According to (2), ω_q can be tuned not only by varying ε but also by varying Δ. This is advantageous, since the operation point of the qubit stays at the symmetry point with optimal coherence properties. Flux qubits with tunable Δ are called tunable-gap flux qubits. As pointed out by Paauw et al [27], an in situ tunability of Δ is achieved by replacing the α-junction by a small α-loop containing two JJ (cf figure 1(b)). That is, the α-junction is replaced by a dc SQUID. Then, the critical current I_c,α of the α-loop and, in turn, the qubit gap Δ can be tuned by a control flux Φ_α threading the α-loop. If we choose the area of the junction in the α-loop to be 0.5α₀A_J, we obtain I_c,α = αI_c with α = α₀|cos(πΦ_α/Φ₀)|. Successful implementations of this design have recently been reported [29, 41–43].

The magnitude of I_p can be calculated as [26]

$$\begin{equation} I_{\mathrm{p}} = \pm I_{\mathrm{c}} \, \sqrt{1- \frac{1}{4\alpha^2}} , \end{equation} \tag{ 4 }$$

yielding the α-dependent transition frequency

$$\begin{equation} \omega_{\mathrm{q}} = \sqrt{4I_{\mathrm{c}}^2 \left[1-(1/4\alpha^2) \right] \delta\Phi^2/\hbar^2 +\Delta^2 (\alpha)} \; \end{equation} \tag{ 5 }$$

with Δ(α) according to (3). We see that for α → 0.5 the persistent current I_p approaches zero.

While the replacement of the α-junction by an α-loop allows for a tunable qubit gap Δ, applying any flux to the α-loop at the same time changes the flux threading the qubit loop and hence the energy bias ε of the flux qubit. This is unintentional and has to be compensated. To keep the energy bias of the flux qubit constant during variations of Φ_α, a gradiometric design can be used. The gradiometric versions of a fixed-gap and a tunable-gap flux qubit are shown in figures 1(c) and (d), respectively.

For the tunable-gap gradiometric flux qubit of figure 1(d), an applied homogeneous magnetic field changes Φ_α, but does not affect the energy bias of the flux qubit, since the screening currents in the two subloops of the eight-shaped gradiometric loop cancel each other on the central line. The immediate consequence is that an inhomogeneous magnetic field is required to adjust the energy bias ε of the flux qubit. This inhomogeneous field can be generated by feeding a small current through the so-called ε-flux line, which couples asymmetrically to the qubit loop (cf figure 1(d)). Furthermore, the outer ring of the gradiometric qubit, denoted as the trapping loop, can be used to trap an integer number of magnetic flux quanta, e.g. by cooling down below T_c in an applied magnetic field. This allows for a pre-biasing of the qubit near the symmetry point. We note, however, that the exact amount of flux threading the qubit loop and the α-loop, respectively, depends on the ratio of the kinetic and geometric inductances. Since an understanding of this point is important for a controlled design of a gradiometric qubit with a tunable gap, it is discussed in more detail in section 2.5. Within this work, we investigate both fixed-gap and tunable-gap gradiometric 3-JJ flux qubits. The former is an ideal model system to study the principle of flux biasing.

In this subsection, we briefly address the flux biasing of gradiometric flux qubits by the trapping of magnetic flux in its outer loop, the so-called trapping loop (cf figures 1(c) and (d)). Flux biasing is based on the phase coherence of the superconducting state. The phase θ of the macroscopic wave function describing the superconducting state is allowed to change only by integer multiples of 2π along a closed integration path

$$\begin{equation} \oint_{\Gamma} {\boldsymbol{\nabla}} \theta \, {\mathrm{d}}\boldsymbol{s} = 2\pi n . \end{equation} \tag{ 6 }$$

In multiply connected superconductors, this leads to the expression for the fluxoid quantization

$$\begin{equation} \oint_{\Gamma} \mu_0\lambda_{\mathrm{L}}^2 \boldsymbol{J}_{\mathrm{s}} \cdot {\mathrm{d}}\boldsymbol{s} + \int_F \boldsymbol{B} \cdot {\mathrm{d}}\boldsymbol{F} = n\Phi_0, \end{equation} \tag{ 7 }$$

where λ_L is the London penetration depth, μ₀ is the vacuum permeability, Γ is a closed integration path encircling the area F, J_s is the supercurrent density along Γ and B is the magnetic flux density. The second term on the left-hand side represents the total magnetic flux Φ threading the area F. For superconductors with large cross-sectional area compared to the London penetration depth λ_L, the first term vanishes, since one can always find an integration path deep inside the superconductor where the supercurrent density J_s = 0. This leads to the expression for flux quantization

$$\begin{equation} \frac{\Phi}{\Phi_0} \equiv f_{\mathrm{tr}} = n , \end{equation} \tag{ 8 }$$

meaning that the total magnetic flux in a closed superconducting loop such as the trapping loop is quantized in units of Φ₀.

The phase of the superconducting order parameter changes by 2πn around the closed trapping loop. Therefore, in the fully symmetric gradiometric qubit designs of figures 1(c) and (d) the trapping of an odd number (2n + 1) of flux quanta in the trapping loop leads to a phase difference of (2n + 1)π between the points A and B. This corresponds to a flux bias of $(n+\frac {1}{2}) \Phi _0$, i.e. a flux bias at the symmetry point. The biasing with trapped flux has the advantage that it is not affected by the finite noise of current sources required for the biasing with an external magnetic field. On the other hand, once a specific flux state has been frozen in, it can no longer be changed without heating the sample above T_c. Therefore, in practice, flux trapping is often used for pre-biasing at an operation point, while an additional magnetic field is used for making fast changes around this operation point. In order to enable such flux control the width of the superconducting line forming the trapping loop has to be made small enough (of the order of λ_L) to allow for partial penetration of the applied magnetic field. In this case, the first term on the left-hand side of (7) becomes relevant. This term is related to the kinetic energy of the superconducting condensate or, equivalently, the kinetic inductance L_k, whereas the second term is related to the field energy or, equivalently, the geometric inductance L_g of the trapping loop.

We next discuss the influence of the kinetic inductance L_k, which is no longer negligible compared to the geometric inductance L_g of the trapping loop when the width of the superconducting lines is reduced to values of the order of λ_L. In this case the first term on the left-hand side of (7) is no longer negligible. With the supercurrent I_cir = J_sS circulating in the trapping loop, we can rewrite this term as

$$\begin{equation} \oint_\Gamma \mu_0\lambda_{\mathrm{L}}^2 \boldsymbol{J}_{\mathrm{s}} \, {\mathrm{d}} \boldsymbol{s} = \frac{\mu_0\lambda_{\mathrm{L}}^2}{S} \, I_{\mathrm{cir}} \ell = L_{\mathrm{k}}I_{\mathrm{cir}} . \end{equation} \tag{ 9 }$$

Here, we have introduced the kinetic inductance L_k = μ₀λ²_L(ℓ/S) of the trapping loop, with ℓ its circumference and S its cross-sectional area. With Φ_k = L_kI_cir and splitting up the total flux Φ into a part Φ_ex due to an external applied field and a part Φ_g = L_gI_cir caused by the circulating current in the trapping loop with geometric inductance L_g, the fluxoid quantization condition (7) reads as

$$\begin{equation} \frac{\Phi_{\mathrm{k}}}{\Phi_0} + \frac{\Phi_{\mathrm{ex}}}{\Phi_0} + \frac{\Phi_{\mathrm{g}}}{\Phi_0} \equiv f_{\mathrm{k}} + f_{\mathrm{ex}} + f_{\mathrm{g}} = n . \end{equation} \tag{ 10 }$$

Introducing the parameter β = L_g/L_k, we obtain the expression for the net magnetic frustration of the trapping loop as

$$\begin{equation} f_{\mathrm{tr, net}} = f_{\mathrm{ex}} + f_{\mathrm{g}} = \frac{1}{1+\beta}\, f_{\mathrm{ex}}+\frac{\beta}{1+\beta} \, n . \end{equation} \tag{ 11 }$$

The net magnetic frustration of the α-loop in first approximation is obtained by multiplying with the area ratio A_α/A_tr of the α- and the trapping loop:

$$\begin{equation} f_{\alpha, \mathrm{net}} = \frac{A_\alpha}{A_{\mathrm{tr}}} \; f_{\mathrm{tr, net}} . \end{equation} \tag{ 12 }$$

Here, we have neglected effects arising from the fact that the α-loop is not centered in the trapping loop. If the geometric inductance is negligible (β ≪ 1), the contribution of the circulating screening current is negligible and f_tr,net ≃ f_ex. In this case, the superconducting lines cannot screen magnetic fields and we can change the magnetic frustration of the α- and the trapping loop continuously by varying the applied magnetic field. In contrast, if the geometric inductance is dominant (β ≫ 1), the screening is so strong that we can no longer change the flux in the loop by varying the applied field. The frustration of the trapping loop is fixed at the value f_tr,net ≃ n frozen in during cool down. This means that also the frustration of the α-loop can no longer be changed continuously as desired.

With the net magnetic frustration (12), the critical current of the α-loop is obtained as I_c,α(f_α,net) = α(f_α,net)I_c with

$$\begin{eqnarray} \alpha (f_{\alpha, \mathrm{net}}) & = & \alpha_{0} \left|\cos\left(\pi f_{\alpha\mathrm{,net}}\right)\right| \nonumber \\ & = & \alpha_{0}\, \left|\cos\left(\pi\frac{A_{\mathrm{\alpha}}}{A_{\mathrm{tr}}} \; \left[ \frac{1}{1+\beta} \, f_{\mathrm{ex}} + \frac{\beta}{1+\beta} \, n \right] \right) \right| . \end{eqnarray} \tag{ 13 }$$

For a suitable width of the superconducting lines and, hence, a suitable value of β, we can vary α both by changing f_ex via an external magnetic field and by changing the number n of flux quanta frozen into the trapping loop during cool down. For example, n could be used for pre-biasing at a specific α value and the external magnetic field provided by a current sent through an on-chip control line for small variations around this value. The pre-biasing with trapped flux has the advantage that it is not affected by the noise added by the current source, while the variations with the on-chip control line can be very fast.

For zero applied magnetic field, (13) reduces to

$$\begin{equation} \alpha (f_{\alpha, \mathrm{net}})|_{f_{\mathrm{ex}} = 0} = \alpha_{0} \, \left|\cos\left(\pi\frac{A_{\mathrm{\alpha}}}{A_{\mathrm{tr}}} \; \frac{\beta}{1+\beta}\, n \right) \right| . \end{equation} \tag{ 14 }$$

This expression applies to the experimental situation, where an odd number (2n + 1) of flux quanta is frozen into the trapping loop to bias the gradiometric flux qubit at its symmetry point and no additional external magnetic field is applied. Fixing α₀ ≃ 1 by the fabrication process, we can change the number of trapped flux quanta to choose α in the desired regime 0.5 < α < 1. Of course, flux trapping only allows for a step-wise variation of α. For continuous and fast variations of α, magnetic fields generated by external solenoids or on-chip control lines have to be used. For a typical value of β ≃ 0.8, we obtain f_α,net ≃ 0.08 n for f_ex = 0. This shows that we need only a small number of trapped flux quanta to significantly modify α. Furthermore, we obtain f_tr,net = 0.55f_ex for n = 0, meaning that about half of the applied magnetic flux is shielded by the trapping loop.

A perfect gradiometer should be completely insensitive to a homogeneous magnetic field. However, in reality there are always imperfections such as slight differences of the areas A₁ and A₂ of the two subloops of the eight-shaped gradiometer and/or of the geometric inductances L_g1 and L_g2 and kinetic inductances L_k1 and L_k2 of the superconducting lines forming the subloops. Due to these imperfections there will be a finite imbalance δf_imb of the magnetic frustration of the two subloops. According to (10), δf_imb can be expressed as

$$\begin{eqnarray} \delta f_{\mathrm{imb}} & = & \delta f_{\mathrm{ex}} + \delta f_{\mathrm{g}} + \delta f_{\mathrm{k}} = \frac{\delta \Phi_{\mathrm{ex}}}{\Phi_0} + \frac{\delta\Phi_{\mathrm{g}}}{\Phi_0} + \frac{\delta\Phi_{\mathrm{k}}}{\Phi_0} \nonumber\\ & = & \frac{\Phi_{\mathrm{ex}}}{\Phi_0} \frac{\delta A}{A} + \frac{I_{\mathrm{cir}}\delta L_{\mathrm{g}}}{\Phi_0} + \frac{I_{\mathrm{cir}}\delta L_{\mathrm{k}}}{\Phi_0} . \end{eqnarray} \tag{ 15 }$$

With I_cir = (n − f_ex)Φ₀/(L_g + L_k) we can rewrite this expression to

$$\begin{equation} \delta f_{\mathrm{imb}} = f_{\mathrm{ex}} \underbrace{\left( \frac{\delta A}{A} - \frac{\delta L_{\mathrm{g}} +\delta L_{\mathrm{k}}}{L_{\mathrm{g}} + L_{\mathrm{k}}} \right)}_{\equiv 1/Q_{\rm grad,ex}} +\, n \underbrace{\left( \frac{\delta L_{\mathrm{g}} +\delta L_{\mathrm{k}}}{L_{\mathrm{g}} + L_{\mathrm{k}}} \right)}_{\equiv 1/Q_{\mathrm{igrad,n}}} . \end{equation} \tag{ 16 }$$

The total gradiometer quality Q is given by Q⁻¹ = Q⁻¹_grad,ex + Q⁻¹_igrad,n. The first term describes imbalances of the frustration when a homogeneous external field is applied, and the second those when an integer number of flux quanta is frozen in. Obviously, the higher the Q the lower is δf_imb. As shown below, Q values of the order of 500 are feasible.

We note that the ε-flux line shown in figure 1(c) generates different flux densities B in the two subloops of area A_tr,1 ≃ A_tr,2, leading to different amounts of total flux $\Phi _1 = \int _{A_{\mathrm { tr,1}}} B \,{\mathrm {d}} A$ and $\Phi _2=\int _{A_{\mathrm { tr,2}}} B \,{\mathrm {d}} A$. This results in the magnetic frustration

$$\begin{eqnarray} f_{12} & = & \frac{\Phi_1 - \Phi_2}{\Phi_0} \; \end{eqnarray} \tag{ 17 }$$

by the ε-flux line, which is used to change the energy bias ε of the gradiometric flux qubit. Correspondingly, the deviation of f₁₂ from the value $(n+\frac {1}{2})$ at the symmetry point is

$$\begin{eqnarray} \delta f_{12} & = & f_{12} - \left( n + \frac{1}{2} \right) . \end{eqnarray} \tag{ 18 }$$

The critical current density of the JJ is determined to be J_c(30 mK) = 1.5–3.5 kA cm⁻² by measuring the current–voltage characteristics (IVCs) of the readout SQUIDs fabricated on the same chip and by determining the junction area A_J by SEM. The J_c values can be varied by changing the oxidation process. For J_c = 2 kA cm⁻² and the typical junction area of A_J ≃ 0.03 μm², we have I_c ≃ 600 nA. The specific capacitance of the junctions is derived from the analysis of resonances in the IVCs of the readout SQUIDs [45]. For junctions with J_c = 2 kA cm⁻², we find that C/A = (195 ± 10) fFμm⁻², resulting in C_J ≃ 6 fF for A_J ≃ 0.03 μm². The geometric inductance L_g of the superconducting loops is estimated according to [46]. In order to estimate the kinetic inductance of the superconducting lines, we use the dirty limit expression L_k = ℏρ_nℓ/πΔ₀S [47, 48], where Δ₀ = 0.18 meV is the zero-temperature energy gap of Al. The use of this expression is justified, since the mean free path in our 90 nm thick Al films is limited by the film thickness and therefore is much smaller than the coherence length ξ ≃ 1.5 μm of Al. The normal resistivity ρ_n is determined by suitable test structures fabricated on the same chip. For the cross-sectional area S = 500 × 90 nm² of the superconducting line forming the trapping loop, we obtain a kinetic inductance per unit length of L_k/ℓ ≃ 1 pH μm⁻¹.

We first discuss the properties of fixed-gap, non-gradiometric flux qubits serving as reference samples. The qubit gap Δ and the persistent current I_p are determined by qubit spectroscopy [37, 44]. Figure 3 shows typical spectra obtained for two 3-JJ flux qubits with fixed α-junction by sweeping the qubit frustration $\delta f = f-(n+\frac {1}{2})=\delta \Phi /\Phi _0$ at a fixed microwave frequency. The qubit state is read out repeatedly by the readout dc SQUID. Only at those δf values where the microwave driving is resonant with the qubit transition frequency ω_q, a 50% population of the excited state is detected. This manifests itself in characteristic peak and dip structures in the switching current I_sw of the readout SQUID at frequency-dependent δf values. Plotting these values versus the microwave frequency as shown in figure 3 yields ω_q(δf). Assuming that J_c has the same value for all three junctions, the value of α = α₀ = A_α/A_J can be determined from the measured area ratio. Then a two-parameter fit of (5) to the spectroscopy data yields Δ and I_p = ℏε/2δΦ. The spectra in figure 3 are obtained for two flux qubits differing only in their α₀ values. For α₀ = 0.75 and 0.6, we obtain Δ/2π = 1.39 and 10.76 GHz and I_p = 583 and 283 nA, respectively. Obviously, for α₀ values closer to 0.5 (1.0), large (small) Δ and small (large) I_p values are obtained in agreement with (3) and (4). A consistency check can be made by calculating the I_p values from (4). Here, the unknown critical current I_c = J_cA is estimated from the measured junction area and using the J_c value of the junctions of the readout SQUID. We obtain I_p = 619 and 310 nA in good agreement with the values derived from the spectroscopy data.

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We have also performed numerical simulations based on the diagonalization of the full qubit Hamiltonian using E_J, E_c and α = A_α/A_J as the input parameters. They are based on the J_c values derived from the IVCs of the readout SQUID and the measured junction areas. As shown in figure 3, there is very good agreement between the simulation result and the two-parameter fit for α = 0.6. However, significant deviations appear for α = 0.75. The reason is that there are not enough data points around δΦ = 0, where the readout of the qubit state by the dc SQUID fails. This leads to large uncertainties in Δ for the two-parameter fit. Therefore, small Δ values tend to have larger error bars. Nevertheless, figure 3 clearly demonstrates that the numerical simulation describes the experimental data very well.

We discuss next the properties of fixed-gap gradiometric flux qubits to demonstrate the operation of the gradiometric qubit design shown in figures 4(a) and (b). The flux qubit is biased close to its symmetry point by freezing in an odd number (2n + 1) of flux quanta in the trapping loop during cool down. This results in a phase difference of (2n + 1)π between points A and B, equivalent to a flux bias of $(n+\frac {1}{2}) \Phi _0$ of the gradiometric flux qubit at its symmetry point. To change the energy bias ε after cool down, a spatially inhomogeneous magnetic field is required, which is generated by the current I_ε sent through the ε-flux line. The qubit state is read out via the readout dc SQUID inductively coupled to the trapping loop of the qubit. The operation point of the readout dc SQUID can be optimized by applying a homogeneous magnetic field (e.g. by a solenoid) that does not affect the energy bias of the qubit due to its gradiometric design.

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Figure 4(c) shows the typical spectroscopy data of a gradiometric flux qubit with α₀ = 0.65. Since we are measuring ω_q(δI_ε) and not ω_q(δf), the only problem in evaluating these data is to determine the calibration factor

$$\begin{equation} \kappa \equiv \frac{\partial \delta f}{\partial \delta I_\varepsilon} , \end{equation} \tag{ 19 }$$

where δI_ε = I_ε − I^sym_ε is the deviation of the current I_ε sent through the ε-flux line from the value I^sym_ε needed for biasing the qubit at the symmetry point. This is done by calculating ω_q(δf) by numerical simulations using E_J, E_c and α = A_α/A_J as the input parameters. The scaling factor κ is then obtained by re-scaling the measured ω_q(δI_ε) dependence to obtain optimum agreement with the simulation result. For the sample in figure 4(b), we obtain κ = 0.7 mA⁻¹, meaning that a current of about 1 mA results in δf = 1. In general, the agreement between the experimental data and the simulation was found to be very good. The simulated values for the sample in figure 4(c) are Δ/2π = 5.1 GHz, I_p = 420 nA and α₀ = 0.65. Again, we can make a consistency check by calculating the I_p value according to (4) as discussed above. We obtain I_p = 485 nA in good agreement with the value derived from the simulation.

We note that we can also trap an even number 2n of flux quanta in the trapping loop. In this case the phase difference between points A and B is 2πn. This corresponds to a flux bias of the gradiometric qubit by 2nΦ₀/2 = nΦ₀ instead of $(n+\frac {1}{2}) \Phi _0$ for an odd number of trapped flux quanta. That is, the qubit is biased far away from its symmetry point and no qubit transitions should be observable. This is in full agreement with the experimental observation. We finally note that I_cir = (n − f_ex)Φ₀/(L_g + L_k) is not allowed to exceed the critical current of the trapping loop. For our samples, this fact limits the number of trapped flux quanta to a maximum value of 10–15.

We also use the simple fixed-gap gradiometric qubit to check the quality of the gradiometer discussed in section 2.6. Figure 5(a) shows the switching current of the readout SQUID as a function of $\delta f_{12} = f_{12} - \left (n+\frac {1}{2} \right )$ (cf (18)) recorded for a fixed microwave frequency of 19.33 GHz. The peaks and dips in the I_sw(δf₁₂) curves mark the δf₁₂ positions where the qubit transition frequency ω_q/2π = 19.33 GHz. On varying the number n of trapped flux quanta, these positions shift due to the imperfect balance of the gradiometer. From the measured shift we derive Q_grad,n = 943 ± 19. In figure 5(b), δf₁₂ is plotted against f_ex generated by a homogeneous applied magnetic field. From the measured slope the quality factor Q_grad,ex = 1076 ± 16 is determined. The total quality of the gradiometer is then Q ≃ 500, corresponding to a gradiometer imbalance of only 0.2%. This means that the qubit operation point is shifted by about 2mΦ₀ when we apply a homogeneous field generating one Φ₀ in the trapping loop. The measured quality factors are plausible. For example, the limited precision of the electron beam lithography process causes a finite precision δA/A_tr of the trapping loop area as well as δS/S of the cross-sectional area and δℓ/ℓ of the length of the superconducting lines. The measured quality factor corresponds to δA ≃ 0.2 μm², δS ≃ 50 nm² or δℓ ≃ 60 nm. These values agree well with the values expected for the precision of the fabrication process.

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In figure 5(b), we also plot the distance between the peak and dip positions in the I_sw(δf₁₂) curves. This distance is almost independent of f_ex. This demonstrates that the qubit potential is not affected by the homogeneous background field. In total, our results show that the gradiometric flux qubits can be fabricated in a controlled way and work as expected. The fact that the qubit operation point is not affected by a homogeneous background field allows us to integrate these qubits into large-scale circuits where several qubits have to be operated and read out simultaneously without affecting each other.

In this subsection, we discuss the results obtained with tunable-gap gradiometric flux qubits as sketched in figure 1(d). Besides a step-wise variation of α by freezing in an odd number of flux quanta in the trapping loop, we can make a continuous variation of α by an applied magnetic field generated by either the current I_coil fed through an external solenoid or the current I_α fed through the on-chip α-flux line. We first discuss the experiments using a homogeneous magnetic field of a solenoid placed underneath the sample. The homogeneous magnetic field generates the frustrations f_tr,net and f_α,net of the trapping and α-loop, respectively, which are given by (11) and (12).

Spectroscopy data of a tunable-gap gradiometric flux qubit are shown in figure 6(c). The different α values were generated by the homogeneous magnetic field of the solenoid, whereas the flux trapped during cool down was constant at a single flux quantum, i.e. n = 1. We can fit the data by a two-parameter fit yielding Δ and the slope ∂ω_q/∂δI_ε at large ω_q values. Here, δI_ε = I_ε − I^sym_ε is the deviation of the current I_ε sent through the ε-flux line from the value I^sym_ε needed for biasing the qubit at the symmetry point. To derive the persistent current I_p = (ℏ/2Φ₀)(∂ω_q/∂δf) from this slope, we have to calibrate the horizontal axis. For this we need the calibration factor κ ≡ ∂δf/∂δI_ε (cf (19)), which has already been discussed above.

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For the analysis of the Δ(α) dependence we need a second transfer function, relating the coil current I_coil sent through the solenoid to the frustration f_α,net of the α-loop. With (11) and (12) we obtain

$$\begin{equation} \zeta \equiv \frac{\partial f_{\alpha,{\rm net}}}{\partial I_{\mathrm{coil}}} = \frac{A_{\mathrm{\alpha}}}{A_{\mathrm{tr}}} \; \frac{1}{1+\beta} \, \frac{\partial f_{\mathrm{ex}}}{\partial I_{\mathrm{coil}}} . \end{equation} \tag{ 20 }$$

With this transfer function and the expressions (4) and (13) for I_p and α, respectively, we obtain

$$\begin{equation} \frac{\partial \omega_{\mathrm{q}}}{\partial \delta f} = \frac{2\Phi_0 I_{\mathrm{p}}}{\hbar} = \frac{2\Phi_0 I_{\mathrm{c}}}{\hbar} \sqrt{1-\left[ 2\alpha_0 \left| \cos \left( \pi \zeta I_{\mathrm{coil}} + \pi \frac{A_\alpha}{A_{\mathrm{tr}}} \frac{\beta}{1+\beta} n \right) \right| \right]^{-2} } . \end{equation} \tag{ 21 }$$

Using the abbreviations η = 2Φ₀I_cκ/ℏ and I_n = (A_α/A_tr)(β/1 + β)(n/ζ) this simplifies to

$$\begin{equation} \frac{\partial \omega_{\mathrm{q}}}{\partial \delta I_\varepsilon} = \eta \sqrt{1-\left[ 2\alpha_0 \left| \cos \left( \pi \zeta [I_{\mathrm{coil}} + I_n] \right) \right| \right]^{-2} } . \end{equation} \tag{ 22 }$$

We can use this expression to fit the measured ∂ω_q(I_coil)/∂δI_ε dependence using η, I_n and ζ as fitting parameters.

In figure 7(a), the measured ∂ω_q/∂δI_ε values are plotted against I_coil together with a fit by (22). Evidently, the data points are clustered near specific I_coil values. The reason is that the homogeneous magnetic field produced by I_coil also changes the frustration of the readout SQUID and that the sensitivity of this SQUID is sufficient only in a limited range of frustration. Figure 7(a) shows that the expression (22) fits the experimental data well, yielding values for ζ and I_n. With these fitting parameters we can calculate $\alpha = \alpha _0 \left | \cos \left ( \pi \zeta [I_{\mathrm { coil}} + I_n] \right ) \right |$. The resulting curve is also shown in figure 7(a). We note, however, that in this case the fit parameters I_n and ζ cannot be used to directly determine β from the expression I_n = (A_α/A_tr)(β/1 + β)(n/ζ), because the value of I_n can be distorted by an additional background magnetic field. Therefore, we use only differences ΔI_n to determine β (cf (23)). Knowing the α(I_coil) dependence, we can adjust α to any desired value by adjusting I_coil and then do spectroscopy at these values. Fitting the spectroscopy data (cf figure 6(c)), we can derive the qubit gap Δ and plot it against α. The result is shown in figure 7(b) together with the dependence obtained from numerical simulations based on the full Hamiltonian. The agreement between the experimental data and the numerical simulation is best for E_J/h = 200 GHz and E_c/h = 1.6 GHz, i.e. E_J/E_c = 125. We note that the E_J value agrees well with the one estimated independently of the measured junction areas and the J_c value measured for the junctions of the readout SQUID. This clearly shows the consistency of the data analysis and demonstrates the good control on the junction parameters fabricated on the same chip. Knowing the Δ(α) and α(I_coil) dependences, we can adjust the qubit gap in situ by I_coil, while operating the qubit at the symmetry point with optimal coherence properties. This is a key prerequisite for many applications of flux qubits.

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For the sample of figure 7, the qubit gap could be varied between values close to zero and about 5 GHz. For comparison, the data of a sample with a larger ratio E_J/E_c = 800 are shown in figure 8. The overall behavior of this sample is very similar but the qubit gap can be tuned to values above 10 GHz. Moreover, we investigate the tunability of this sample for different amounts of trapped flux quanta n. In figure 8(c) we plot ∂ω_q/∂δI_ε against the coil current I_coil through the solenoid for three different values of the trapped flux ranging from n = −3 to +3. Evidently, the general shape of the three curves is very similar as well as the obtained fitting parameters ζ and η. The shift along the horizontal axis is expected from (22) and can now be used to calculate β. Starting with the expression I_n = (A_α/A_tr)(β/1 + β)(n/ζ), we only use the differences ΔI_n = I_n,i − I_n,j. They correspond to the differences Δn = n_i − n_j and result in ΔI_n = (A_α/A_tr)(β/1 + β)(Δn/ζ). For our sample, we find a mean value of ΔI_n/Δn = 0.43 mA, finally yielding

$$\begin{equation} \beta = \left( \frac{\Delta n}{\Delta I_n\zeta} \frac{A_{\mathrm{\alpha}}}{A_{\mathrm{tr}}} -1 \right)^{-1} = 0.52. \end{equation} \tag{ 23 }$$

This value is in reasonable agreement with that derived from the L_g and L_k values which can be estimated from the qubit geometry, the cross-sectional area of the superconducting lines and the dirty limit expression of L_k. We note that the result from (23) is more precise because it is computed directly from the sample. All in all, our results show that the measured data agree well with the behavior expected from theory. Moreover, the values of E_J and E_c obtained from fitting the data agree well with those obtained for junctions fabricated on the same chip. This demonstrates that the gap of gradiometric flux qubits can be reliably tuned over a wide range, making them attractive for a large number of applications.

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We finally address the tuning of Δ by the on-chip α-flux line. Since the maximum current through this line is limited by its critical current and by heating effects in contacts, only small variations of the frustration of the α-loop are possible. Therefore, a constant applied magnetic field or a proper number of trapped flux quanta are used to pre-bias the qubit at a value α_b, where the slope of the Δ(α) dependence is steep. Then, I_α is used to vary α around this value. In our experiments, a constant applied field is used to set α_b. Since the variation of the frustration of the α-loop is generated by I_α instead of I_coil, we have to use the modified calibration factor

$$\begin{equation} \widetilde{\zeta} \equiv \frac{\partial f_{\alpha,{\rm net}}}{\partial I_\alpha} . \end{equation} \tag{ 24 }$$

With this factor, we obtain that

$$\begin{equation} \frac{\partial \omega_{\mathrm{q}}}{\partial \delta I_\varepsilon} = \eta \sqrt{1-\left[ 2\alpha_0 \left| \cos \left( \arccos (\alpha_{\rm b}) + \pi \widetilde{\zeta}I_\alpha \right) \right| \right]^{-2} } . \end{equation} \tag{ 25 }$$

We can use this expression to fit the measured ∂ω_q/∂δI_ε versus I_α dependence using $\widetilde {\zeta }$ as the fitting parameter. Based on these results, we can calculate $\alpha = \alpha _0 \left | \cos \left ( \arccos (\alpha _{\mathrm { b}}) + \pi \widetilde {\zeta }I_\alpha \right ) \right |$. Knowing α, we can use the Δ values obtained from two-parameter fits of the spectroscopy data to obtain the Δ(α) dependence. Experimental data for the two samples of figures 7 and 8 are shown in figure 9. In figures 9(b) and (d), we compare the experimental Δ(α) curves with numerical simulations based on the full qubit Hamiltonian with the same E_J and E_c values as those obtained by tuning α with the coil current (cf figures 7(b) and 8(d)). The very good agreement between measurement data and calculation demonstrates again the consistency of our data analysis. All in all, our data clearly show that the qubit gap can be varied in a controlled way over a wide range by varying the frustration of the α-loop of the gradiometric flux qubits either by an external solenoid or an on-chip control line.

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