## Abstract

We demonstrate Mach-Zehnder–type interferometry in a superconducting flux qubit. The qubit is a tunable artificial atom, the ground and excited states of which exhibit an avoided crossing. Strongly driving the qubit with harmonic excitation sweeps it through the avoided crossing two times per period. Because the induced Landau-Zener transitions act as coherent beamsplitters, the accumulated phase between transitions, which varies with microwave amplitude, results in quantum interference fringes for *n* = 1 to 20 photon transitions. The generalization of optical Mach-Zehnder interferometry, performed in qubit phase space, provides an alternative means to manipulate and characterize the qubit in the strongly driven regime.

The development of artificial atoms with lithographically defined superconducting circuits presents a new paradigm of quantum solid-state physics (*1*), allowing the realization and exploration of new macroscopic quantum phenomena (*2*–*9*), and also holding promise for applications in quantum computing (*10*). Of the various effects demonstrated with qubits, the most important are time-dependent coherent phenomena. Those include the observation of Rabi oscillations in charge, flux, and phase qubits (*2*, *5*–*9*), entanglement of two qubits (*11*), coherent oscillation (*12*) and bifurcation (*13*) in multilevel systems, and the demonstration of basic elements of coherent control (*14*–*16*). Artificial atoms strongly coupled to photons have opened the arena of “circuit quantum electrodynamics” (c-QED) (*17*, *18*).

Here, we demonstrate an application of superconducting qubits to quantum physics, realized in a strongly driven flux qubit and described in terms of a Mach-Zehnder (MZ) interferometer. The conventional MZ setup uses two beamsplitters: The first divides an optical signal into two coherent waves that travel along paths with different effective lengths, and the second recombines and superposes these waves, leading to quantum interference fringes in the measured output signal. In a driven qubit, according to an idea discussed by Shytov *et al*. (*19*), the beamsplitters can be realized by Landau-Zener (LZ) transitions at a level avoided crossing. Over one oscillation period of the driving field, the qubit is swept through the avoided crossing twice (Fig. 1A). Starting from the marker, at the first LZ transition (time *t*_{1}), the ground state |0 〉 is split into a coherent superposition of the ground and excited states, |1 〉 and |0 〉, which, after evolving independently and accumulating a relative phase Δθ_{12}, interfere at the second LZ transition (time *t*_{2}). The corresponding qubit-state energy evolution (first period, Fig. 1B) between the recurrent LZ transitions (shaded region) provides a phase-space analog to the two arms and the beamsplitters of an optical MZ interferometer (top left, Fig. 1B). The interference phase (1) where *ħ* = *h*/2π, *h* is the Planck constant, and ϵ is the energy difference between states |0 〉 and |1 〉, depends on the magnitude of the qubit energy detuning excursion for times *t*_{1} < *t* < *t*_{2}. The interference fringes in the occupation probability correspond to integer and half-integer values of Δθ_{12}/2π. Known as Stückelberg oscillations with Rydberg atoms (*20*, *21*), this mechanism can be applied to quantum control (*22*).

The qubit MZ interferometer differs in a number of ways from an optical interferometer. First, instead of a photon, the interferometry is performed with the use of the quantum state of a qubit. Second, in the qubit, we have the interference of paths in phase space rather than in coordinate space; the phase Δθ_{12} (Eq. 1) is determined by the qubit level splitting, which plays the role of the optical path length. Finally, because they are more fragile than photons and easier to decohere, qubit states can be manipulated in a coherent fashion only at relatively short time scales.

We used a periodic driving signal, a harmonic variation of the qubit detuning ϵ(*t*) (2) where Δ is the tunnel splitting, σ^{x} and σ^{z} are Pauli matrices, ϵ_{0} is the detuning proportional to dc flux bias, and *A*_{rf} is the radio frequency (rf) field amplitude proportional to the rf flux bias (*23*). In this case (Fig. 1B), we have cascaded LZ transitions which occur when the driving amplitude exceeds detuning, giving rise to the interference fringes at *A*_{rf} >|ϵ_{0}| (Fig. 1C). Although the phase Δθ_{12} equals the shaded area in Fig. 1B and is dependent on *A*_{rf}, the total phase gained over one period, θ = [1/*ħ*]∫ϵ(*t*)*dt* = 2πϵ_{0}/*ħ*ω, equals the difference of the shaded and unshaded areas and is independent of *A*_{rf}. As consecutive pairs of LZ transitions (consecutive MZ interferometers) interfere constructively when θ = 2π*n*, the fringes will appear around the resonance detuning values (3) where *n* = 0, 1, 2,... and ν = ω/2π. Another interpretation of this condition is that the sequential LZ transitions excite multiphoton resonances.

Although coherent multiphoton resonances between discrete states of an rf-driven charge qubit have been reported (*5*, *24*) and multiphoton transitions used to drive Rabi oscillations in a flux qubit (*25*, *26*), in these works as well as in the earlier work on quantum dot systems (*27*, *28*), only a few photon transitions could be observed, with coherence quickly weakening as rf amplitude increased (*29*). In contrast, we were able to observe coherent resonances of very high order, up to *n* = 20, which requires driving the system at a high rf amplitude. The fringes for high *n* are as clear as those for *n* ≅ 1, indicating that the qubit preserves a substantial amount of coherence even in the strongly driven regime.

We realized a tunable artificial atom with a niobium persistent-current qubit (Fig. 2A), a superconducting loop interrupted by three Josephson junctions (*30*). When the qubit loop is threaded with a magnetic flux *f*_{q} ≈ Φ_{0}/2, the system exhibits a double-well potential-energy landscape (fig. S1). The classical states of the wells are persistent currents *I*_{q} with opposing circulation, described by energy bands ±ϵ_{0}/2 = ±*I*_{q}Φ_{0}δ*f*_{q} linear in the flux detuning δ*f*_{q} ≡ *f*_{q} – Φ_{0}/2. The double-well barrier allows quantum tunneling of strength Δ, opening the avoided crossing near δ*f*_{q} = 0 (Fig. 1A). Detuning the flux tilts the double well and, thereby, modifies its eigenenergies and eigenstates. The qubit states are read out with a dc superconducting quantum interference device (DC-SQUID), a sensitive magnetometer that distinguishes the flux generated by the circulating currents. The device was fabricated at MIT Lincoln Laboratory (*23*).

We drove transitions between the qubit states by applying a 1-μs rf pulse (Fig. 2B) at frequency ν and rf-source voltage *V*_{rf} (*31*). After a short (≈10-ns) delay, we read out the qubit state by driving the DC-SQUID with a 20-ns “sample” current *I*_{s} followed by a 20-μs “hold” current. The SQUID will switch to its normal state voltage *V*_{s} if *I*_{s} > *I*_{sw,0} (*I*_{s} > *I*_{sw,1}), corresponding to qubit states |0 〉 and |1 〉. By sweeping the sample current and flux detuning while monitoring the presence of a SQUID voltage over many trials, a cumulative switching-distribution function was generated, revealing the “qubit step” (Fig. 2C). At specific values of flux detuning, the rf field at ν = 1.2 GHz becomes resonant with the energy level separation, allowing *n*-photon absorption, Eq. 3; this results in a partial population transfer between the qubits states, manifest as regularly spaced “spikes” in Fig. 2C. We obtained one-dimensional (1D) scans of the “switching probability” *P*_{sw} (the population of state |0 〉) shown in Fig. 2D by following a flux-dependent sample current *I*_{sw,0} < *I*_{s} < *I*_{sw,1} (dash-dotted line in Fig. 2C). Such 1D scans were then accumulated as a function of the rf source parameters *V*_{rf} (Fig. 3) and ν (fig. S2).

The switching probability *P*_{sw} (color scale in Fig. 3) versus qubit flux detuning δ*f*_{q} and voltage *V*_{rf} at frequency ν = 1.2 GHz is shown in Fig. 3 (*23*). The *n*-photon resonances, labeled by *n* = 1 to 20, exhibit MZ interference fringes (I to VI) as a function of *V*_{rf}. The fringes exhibit a Bessel-function dependence, *J*_{n}(λ), so we call the steplike pattern in Fig. 3 a “Bessel staircase.” For each of the *n*-photon resonances, we took a higher resolution scan (e.g., Fig. 3 inset) and fitted the resonance areas and widths in Fig. 4 (*23*).

Multiphoton transitions at the resonances (Eq. 3) in the strong driving regime, |*A*_{rf}|,*h*ν ≫ Δ, occur by means of fast LZ transitions. The notion of quasi-stationary qubit levels is inadequate in this regime and, instead, we use a different approach, transforming the Hamiltonian (Eq. 2) to a nonuniformly rotating frame, , where ϕ(*t*) = λsinω*t* with dimensionless rf-field amplitude (4) The rf field disappears from the detuning term, reappearing as a phase factor of the off-diagonal term: , where h.c. is hermitian conjugate. Given that Δ ≪ *h*ν, near the *n*th resonance *nh*ν ≅ ϵ_{0} we can replace the phase factor *e*^{–iϕ(t)} by its *n*th Fourier harmonic, *J*_{n}(λ)*e*^{–inωt}, where *J*_{n} is the Bessel function. The resulting effective Hamiltonian is of a “rotating-field” form (5) where Δ_{n} = Δ*J*_{n}(λ). The resonance approximation (Eq. 5) describes transitions at an arbitrary ratio *A*_{rf} /*h*ν. Standard Rabi dynamics analysis of the Hamiltonian (Eq. 5) with the initial state |0 〉 gives the time-averaged occupation probability of the excited state . This expression predicts Lorentz-shaped resonances of width δϵ = |Δ_{n}|. The result, a sum of independent contributions with different *n*, (6) is displayed in Fig. 1C. The agreement with the observed resonances is notable: The oscillations in rf power, described by *J*_{n}(λ), accurately predict both the overall profile of the fringes (Fig. 3) and the fine details, such as positions of the nodes (Fig. 4).

In the frequency dependence of *P*_{sw} for voltages *V*_{rf} = 71 mV_{rms} and *V*_{rf} = 7.1 mV_{rms} (fig. S2), the resonances approach the qubit step as frequency decreases, in accordance with the linear energy versus flux-detuning dependence. MZ interference fringes are again visible. The number of resonances increases at low frequencies, due primarily to the frequency dependence of λ and in lesser part, a frequency-dependent mutual coupling.

Our analysis of peak profile accounts for the relaxation and dephasing, as well as for the inhomogeneous broadening due to low-frequency noise. These effects can be separated from one another by considering the peak areas *A*_{n}, which, in contrast with the widths of the resonances *w*_{n}, are not affected by inhomogeneous broadening. The standard Bloch approach yields (7) where *T*_{1,2} represents the longitudinal and transverse relaxation times, and *T*_{2}* describes the inhomogeneous broadening. These are aggregate relaxation times averaged over the periodic qubit detuning, which, in the operating limit ϵ_{0} ≫ Δ, tends to overestimate *T*_{1} and underestimate *T*_{2} compared with their values at the avoided crossing.

Figure 4A shows the Bessel dependence of the *n* = 1 and *n* = 5 resonance areas fit by Eq. 7 including (blue) and omitting (red) times *T*_{1,2}. The corresponding resonance widths and their fittings are shown in Fig. 4B. Figure 4, C and D, show the resonance area and width, respectively, for 10 resonances, including *n* = 19. Fitting the areas and widths yields self-consistent estimates: *T*_{1} ≈ 20 μs, *T*_{2} ≈ 15 to 25 ns, *T*_{2}* ≈ 5 to 10 ns, and Δ/*ħ* ≈ (2π)4 MHz. The *T*_{1}, *T*_{2}, and Δ estimates are similar to those reported by Yu *et al*. (*26*). The nearly linear behavior at the nodes of *J*_{n} (Fig. 4A) indicates that the decoherence is small compared with the splitting: *T*_{1}*T*_{2}(Δ/*ħ*)^{2} ≈ 250 for *n* = 1 and decreases slightly for *n* = 5. The fit/data discrepancy for the first fringe for *n* = 1, which disappears as *n* increases, is traced to ∼20% thermal population of the excited state because of its proximity to the qubit step (supporting online material text).

This MZ interferometry technique can be applied to qubit characterization and model validation, two increasingly important research areas in quantum information science. In addition to coherence times, which can be obtained by multiple means, MZ interferometry allows the direct calibration of the microwave amplitude driving the qubit through the Bessel argument λ; we found the rf mutual coupling (±SD) to be *M*_{q} = 100 ± 2 fH over all 20 resonances. The agreement between the two-level Hamiltonian in Eq. 2 and the observed resonances *n* = 1 to 10 in Fig. 3 is notable. The MZ technique also reveals shortcomings of the two-level model at strong driving. For example, the influence of a second MZ interferometer at the avoided crossing between the first and second excited states results in the moiré-like pattern observed for resonances *n* > 12. Also notable is an observed (∼0.1 GHz) shift in the resonance positions at strong driving [Fig. 3 inset]. Both effects require the presence of higher excited states modeled by the full qubit Hamiltonian (*26*, *30*). The high stability and coherence of the strongly driven qubit, even at *n* = 20 photon transitions, illustrates not only the potential for nonadiabatic control methods (*22*), but also indicates the high potential of niobium devices fabricated in a fully planarized, scalable process for superconductive quantum computation.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/1119678/DC1

Materials and Methods

SOM Text

Figs. S1 and S2

References and Notes