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Low-frequency characterization of quantum tunneling in flux qubits

We propose to investigate flux qubits by the impedance measurement technique (IMT), currently used to determine the current--phase relation in Josephson junctions. We analyze in detail the case of a high-quality tank circuit coupled to a persistent-current qubit, to which IMT was successfully applied in the classical regime. It is shown that low-frequency IMT can give considerable information about the level anticrossing, in particular the value of the tunneling amplitude. An interesting difference exists between applying the ac bias directly to the tank and indirectly via the qubit. In the latter case, a convenient way to find the degeneracy point in situ is described. Our design only involves existing technology, and its noise tolerance is quantitatively estimated to be realistic.


I. INTRODUCTION
Josephson-junction flux qubits are known to be candidates for solid-state quantum computing circuits. 1 This qubit variety has good tolerance to external noise, especially to dangerous background-charge fluctuations. 2 A flux qubit is a superconducting loop, the two lowestenergy states of which differ in the direction of circulating persistent current. For many flux qubits, these two states become degenerate when the external flux Φ x threading the loop equals Φ 0 /2 (Φ 0 = h/2e is the flux quantum), and quantum tunneling between them becomes possible.
Moving Φ x away from Φ 0 /2 lifts the degeneracy and applies a bias between the two states. When the biasing energy exceeds the tunneling amplitude ∆ the tunneling stops, but the relative phase between the two states will still evolve in time. This, together with coherent tunneling, provides single-bit quantum gate operations. To have a universal set of gates, necessary for quantum computing, one needs to be able to couple two qubits. The methods of coupling two flux qubits and performing gate operations are beyond the present scope. Instead, we propose a method to characterize the quantum behavior of a flux qubit by coupling it to a tank circuit. The discussion will be quite general and can be applied to different types of flux qubit such as rf-SQUID, 3 three-Josephsonjunction (3JJ), 4,5 multi-terminal, 6 etc. We will use the example of the 3JJ qubit, where quantum superposition of the macroscopic current states has been observed. 5 Due to the loop self-inductance, the total qubit flux Φ may differ from Φ x , depending on the direction of the persistent current. Figure 1 shows the Φ-Φ x curve for a typical flux qubit. The solid lines correspond to classical behavior. Near the degeneracy point, the diagram is hysteretic, a signature of the qubit's bistability. This has been observed for the 3JJ in Refs. 7,8. In the quantum regime, tunneling between the states at degeneracy may eliminate the hysteresis (dashed line in Fig. 1). This phenomenon will be discussed in detail below.
In general, one can plot the classical (local) minimum energies of a flux qubit as in Fig. 2a. The left (right) branch then corresponds to (counter-)clockwise flow of the spontaneous current. The hysteresis is also evident from this diagram. In the quantum regime, there will be discrete local states in each of the qubit's bistable potential wells. From now on we denote the lowest-lying such states as Ψ l and Ψ r , corresponding to "left" and "right" directions of the persistent current respectively. At Φ x = Φ 0 /2, resonant tunneling will render the lowest eigenstates of the full Hamiltonian as superpositions (Ψ l ±Ψ r )/ √ 2. A small splitting equal to 2∆ will appear between their energies (Fig. 2b). Starting with the qubit in its ground state (lower band in Fig. 2b), adiabatically changing Φ x will keep it in the ground state. This means that by passing through the degeneracy point, the qubit will continuously transform from Ψ l to Ψ r . This pure quantum behavior is shown by the dashed line in Fig. 1.
On the other hand, if Φ x changes rapidly, there is a considerable probability to excite the qubit and therefore continue on the same classical branch (left or right). This so-called Landau-Zener effect can be used to distinguish the classical from the quantum energy curves.
The curvature of the energy profile is related to the qubit's effective inductance and is therefore important for measurement. Figure 3 displays the second derivative of the curves in Fig. 2. In the classical regime (Fig. 3a), the hysteretic behavior is the same as for the energy. On the other hand, in the quantum regime the hysteresis is replaced by a sharp spike, due to the level anticrossing. The appearance of this spike can be ascribed to enhanced susceptibility of the system due to tunneling. Its size and width can provide information about ∆. A simple experimental implementation is to inductively couple the qubit to an LC tank circuit with known inductance L T , capacitance C T , and quality Q through a mutual inductance M (Fig. 4). The resonant characteristics of the tank circuit (frequency, phase shift, etc.) will then be sensitive to the qubit inductance and therefore to its energy curvature. In particular, the spike in Fig. 3b appears as sharp dips in both phase shift and tank voltage as a function of Φ x (see Section III).
This method, known as impedance measurement technique (IMT), has been used for current-phase measurements of Josephson junctions. It originates from the pioneering work of Rifkin and Deaver, 9 and is analyzed in detail in Ref. 10. IMT has also successfully been applied to a 3JJ qubit in the classical regime, 8 and the hysteretic dependence of the ground-state energy on Φ x (cf. Fig. 2a) was observed as predicted in Ref. 4. The method has also been used for the investigation of quantum transitions in an rf-SQUID (Ref. 11 and references therein). First of all, in Section II we calculate the two qubit energies in more detail than in Ref. 4. In Section III, we study the qubit's interaction with a high-quality resonant tank, showing that low-frequency IMT yields useful information about the qubit's quantum behavior. Finally, in Section IV, the effect of noise is considered.

II. QUANTUM DYNAMICS OF THE 3JJ QUBIT
The 3JJ qubit 4 consists of three Josephson junctions in a loop with very small inductance L, typically in the pH range. This insures effective decoupling from the environment. Two junctions have equal critical current I c and (effective) capacitance C, while those of the third junction are slightly smaller: αI c and αC, with 0.5 < α < 1. If the Josephson energy E J = I c Φ 0 /2π is much larger than the Coulomb energy E C = e 2 /2C, the Josephson phase is well defined. Near Φ x = Φ 0 /2, this system has two low-lying quantum states. 4,12 The energy splitting between them in the presence of a small flux bias has been given in Ref. 5, but only for a particular choice of, e.g., α and g ≡ E J /E C . In this section we derive the splitting with its explicit dependence on the qubit parameters. The energy levels are derived from the Hamiltonian [see Eq. (12) where ϕ = (ϕ 1 +ϕ 2 )/2, θ = (ϕ 1 −ϕ 2 )/2 with ϕ 1,2 the phase differences across the two identical junctions, In contrast to Ref. 4, we define the flux bias f x = Φ x /Φ 0 − 1 2 as a small deviation from degeneracy. Since the qubit is assumed to have small L and I c (typically L ≈ 10 pH, I c ≈ 100 nA), the shielding factor LI c /Φ 0 ≈ 0.001. Hence, in (1) we have neglected the shielding current, considering Φ as an external flux.
At f x = 0, the potential (2) has two minima at ϕ = 0, θ = ±θ * , with cos θ * = 1/2α (θ * > 0). Tunneling lifts their degeneracy, leading to energy levels E ± = ε 0 ± ∆. To find the levels for |f x | ≪ 1 we expand Eq. (2) near its minima, retaining linear terms in f x and quadratic terms in φ, θ. Define θ r/l * as the minima, shifted due to f x : that is, the upper (lower) sign refers to the right (left) well. The potential energy then reads Near degeneracy, the eigensolutions of H 0 Ψ ± = E ± Ψ ± can be written as superpositions Ψ ± = a ± Ψ l + b ± Ψ r , yielding the well-known eigenenergies E ± = (ε l +ε r )/2 ± (ε l −ε r ) 2 /4 + ∆ 2 , with ε r/l = Ψ r/l |H 0 |Ψ r/l . The matrix element ∆ cannot accurately be found in terms of Ψ r/l . In what follows it is assumed constant, neglecting its dependence on f x . To find the dependence of E ± on f x , we take Ψ r/l to be oscillator ground states in their respective wells: corresponding to where Combining the above, one finds the eigenenergies where The splitting given by Eq. (10) differs from that of Eq. (1) in Ref. 5 by a factor λ(α) which explicitly accounts for the dependence of E ± on α and g. For stationary states, the current in the qubit loop can be calculated either as the average of the current operatorÎ q = I c sin(ϕ + θ) over the eigenfunctions, or as the derivative of the energy over the external flux: where ω 0 = E + − E − . In equilibrium at finite temperature T , Eq. (13) readily generalizes to with the density matrix elements ρ eq ++ = e −E+/kBT /Z and ρ eq −− = e −E−/kBT /Z, where Z = e −E+/kBT + e −E−/kBT .

III. QUBIT-TANK INTERACTION
We propose here to extract information about the quantum dynamics of a flux qubit with the aid of a classical linear high-quality tank circuit, coupled to the qubit via a mutual inductance M . The tank consists of a capacitor C T , inductor L T , and a resistor R T which are connected in parallel and driven by a current source I b (t) (Fig. 4). The problem of coupling a quantum object to a dissipative classical one has no unique theoretical solution. However, if we assume that the classical object is much slower than the quantum one we may solve for the latter's motion, accounting for the coupling coordinates of the former as mere external parameters. 11 Here, the characteristic frequency ∆/h of the qubit is in the GHz range, while the resonances ω T of our tank circuit lie below 100 MHz. There exist two different schemes of coupling a tank circuit to the qubit. First we consider direct biasing, where a current I b (t) = I 0 cos ωt is fed directly into L T (Fig. 4).

A. Direct biasing scheme
The voltage across the tank circuit evolves as Here, Q = ω T R T C T ≫ 1 and ω T = 1/ √ L T C T ; I q is given by (13) or (14), and depends on the qubit flux V dt is the current in L T and Φ x is time independent. Below we study the simplest case k B T ≪ ∆, so that the qubit is definitely in its ground state E − . Then, Eq. (15) takes the form and f = [Φ x +M I L (t)]/Φ 0 − 1 2 (|f | ≪ 1). Thus, (16) is nonlinear in V . Since the coupling to the qubit is small one may apply the method of harmonic balance, which is well known in rf-SQUID theory. 13 Accordingly, if ω ≈ ω T , then V oscillates with frequency ω, while its amplitude v and phase χ are slow functions of time: with the detuning ξ 0 = (ω T −ω)/ω T , and where with η = E J λ(α)/∆ and γ = M v/ω T L T Φ 0 . Settingv =χ = 0 in (18) and (19) one obtains the stationary tank voltage and phase, where we introduced a flux-dependent detuning We have used Eqs. (21)-(23) to find voltage-flux v(f x ), phase-current χ(I 0 ), and phase-flux χ(f x ) characteristics at resonance ω = ω T . We take the qubit parameters I c = 400 nA, α = 0.8, L = 15 pH, g = 100, a tank with L T = 50 nH, Q = 1000, ω T /2π = 30 MHz, and k = 10 −2 . The χ(I 0 ) curves for several f x are shown in Fig. 5. The χ(f x ) and v(f x ) curves are shown in Figs. 6 and 7 for various ∆. The sharp dips in Figs. 6 and 7 correspond to the spike in the second derivative of the energy profile in Fig. 3b. Clearly, the width of the dips is correlated with ∆: with the increase of ∆ the width of the dips also increases. The χ(f x ) curves for different I 0 are shown in Fig. 8. The shape and the value of χ are seen to be very sensitive to I 0 . The dependence of the voltage modulation δv ≡ v(f x =0) − v(f x =10 −3 ) on I 0 is shown in Fig. 9. In this scheme, a bias flux Φ b (t) = Φ ac sin ωt is applied to the qubit loop from a separate coil (Fig. 10). The tank response is similar to (16): where Φ ac is the flux which the external coil couples directly into the tank and d 2 Φ E − is given by (17) Rewriting the first term on the rhs of (24) as makes its time dependence manifest: The advantage of a separate driving coil is that one can effectively decouple the tank from the fundamental harmonic of the bias, since the qubit signal G(t) contains not only ω but also 2ω, 3ω, etc. At f x = 0, G(t) contains only odd harmonics. This can be used to find the degeneracy point in practice, e.g., by tuning f x so that the tank response vanishes (reaches its maximum) at frequency 2ω (3ω). We have studied the higher harmonics by solving Eq. (24) numerically with Φ ac = 0 for ω = ω T /2 and ω = ω T /3, see Fig. 11. Since the full amplitudes contain contributions from all harmonics, at ω = ω T /2, f x = 0 one observes a finite dip rather than a zero.

IV. REQUIREMENTS ON NOISE SOURCES
Figures 6 and 7 clearly reveal the quantum nature of the flux qubit within a range |δf x | ≤ 5 · 10 −4 from the degeneracy point f x = 0. Therefore, the unavoidable external flux noise coupled to the qubit must be much smaller than this value. The most important sources are the Nyquist noise I n = 4k B T/R T and the current noise I a of the preamplifier. The former generates the qubitflux noise Φ n = M I n Q √ B, where B = ω T /2πQ is the tank bandwidth. With T = 20 mK and the tank parameters of Section III, one gets Φ n ≈ 8 · 10 −6 Φ 0 . For I a = 10 −14 A/ √ Hz, we estimate the corresponding flux noise as Φ a = M I a Q √ B ≈ 7 · 10 −6 Φ 0 . Thus, the noise these sources couple to the qubit is at least two orders smaller than the peak widths in Figs. 6 and 7. On the other hand, these sources give rise to directly detected voltage noise across the tank circuit. The thermal tank noise is V n = I n ω T L T Q √ B ≈ 17.6 nV. The noise due to I a is V a1 = I a ω T L T Q √ B ≈ 16 nV. And finally, if we take V a2 = 40 pV/ √ Hz for the preamplifier's own voltage noise, we get V a2 √ B ≈ 7 nV for its contribution in the tank bandwidth. Comparing these values with the voltage modulation in Figs. 7, 9, and 11, we see they are at least several times smaller than the qubit signal.

V. CONCLUSION
We have shown that IMT can be used for low-frequency characterization of the ground (in general: equilibrium) state of a flux qubit. The method allows determining the tunnel splitting between qubit states for a broad class of devices; with the term "flux (as opposed to phase) qubit" we stress that the two states must differ not only in Josephson phase, but in actual magnetic flux visible to the outside. 1 The design exclusively employs presentday technology, and the expected noise levels have been shown not to disrupt the measurement. On the qubit time scale, the method is a quasi-equilibrium one; as such, it can determine ∆ but not, e.g., the "T 2 " dephasing time. 14 Efforts to adapt IMT to this and related problems such as qubit readout and control are underway.