Mach-Zehnder Interferometry in a Strongly Driven Superconducting Qubit

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Science  09 Dec 2005:
Vol. 310, Issue 5754, pp. 1653-1657
DOI: 10.1126/science.1119678


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 (29), 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, 59), entanglement of two qubits (11), coherent oscillation (12) and bifurcation (13) in multilevel systems, and the demonstration of basic elements of coherent control (1416). 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 t1), 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 Δθ12, interfere at the second LZ transition (time t2). 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 Math(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 t1 < t < t2. The interference fringes in the occupation probability correspond to integer and half-integer values of Δθ12/2π. Known as Stückelberg oscillations with Rydberg atoms (20, 21), this mechanism can be applied to quantum control (22).

Fig. 1.

MZ interference in a strongly driven qubit. (A) Starting at the dot marker, the qubit state is swept by an rf field. After an LZ transition at the first avoided crossing (time t1), the resulting superposition state of |0 〉 and |1 〉 (dashed lines) accumulates a phase Δθ12 (shaded region) and interferes at the return LZ transition (time t2). The qubit state is subsequently driven away from the avoided crossing and then returns to the starting flux position. This single period of qubit evolution is a single MZ interferometer. Depending on the interference phase Δθ12, amplitude may build in the excited state. a.u., arbitrary units. (B) The corresponding qubit energy variation induced by a periodic rf field, Eq. 2, results in an equivalent optical cascade of MZ interferometers (MZ#1 to #3, top) with resonance condition Eq. 3. (C) The population of the qubit excited state, Eq. 6, as a function of rf amplitude Arf and detuning ϵ0. Note the interference fringes (I to VI) at Arf > ϵ0 and the multiphoton resonances at ϵ0 = nhν.

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 Δθ12 (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) Math(2) Math where Δ is the tunnel splitting, σx and σz are Pauli matrices, ϵ0 is the detuning proportional to dc flux bias, and Arf 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 Arf >|ϵ0| (Fig. 1C). Although the phase Δθ12 equals the shaded area in Fig. 1B and is dependent on Arf, the total phase gained over one period, θ = [1/ħ]∫ϵ(t)dt = 2πϵ0/ħω, equals the difference of the shaded and unshaded areas and is independent of Arf. As consecutive pairs of LZ transitions (consecutive MZ interferometers) interfere constructively when θ = 2πn, the fringes will appear around the resonance detuning values Math(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 fq ≈ Φ0/2, the system exhibits a double-well potential-energy landscape (fig. S1). The classical states of the wells are persistent currents Iq with opposing circulation, described by energy bands ±ϵ0/2 = ±IqΦ0δfq linear in the flux detuning δfqfq – Φ0/2. The double-well barrier allows quantum tunneling of strength Δ, opening the avoided crossing near δfq = 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).

Fig. 2.

Multiple resonances in a strongly driven flux qubit. (A) Circuit schematic of the three-junction flux qubit (inner loop) with circulating current Iq and the DC SQUID readout (outer loop); Josephson junctions are indicated with an ×. A time-dependent flux f(t) ∝ ϵ(t) threading the qubit is a sum of the flux bias due to the dc current Ib and a pulsed ac current at frequency ν irradiating the qubit and driving transitions between its quantum states. The SQUID is shunted by two 1-pF capacitors to lower its resonance frequency. Resistors mark the environmental impedance isolating the SQUID. (B) The time sequence for the rf pulse (duration 1 μs and rf-source voltage Vrf) and SQUID sample current Is. A repetition period of 5 ms allows for equilibration between trials. (C) A cumulative switching-probability distribution of the qubit as a function of Is and the qubit flux detuning δfq under rf excitation at Vrf ≈ 0.12 Vrms and ν = 1.2 GHz. Multiphoton transitions are observed between the qubit states |0 〉 and |1 〉 and are symmetric about the qubit step (δfq = 0 mΦ0). a.u., arbitrary units. (D) The 1D switching probability Psw extracted from (C) (dash-dotted line scan).

We drove transitions between the qubit states by applying a 1-μs rf pulse (Fig. 2B) at frequency ν and rf-source voltage Vrf (31). After a short (≈10-ns) delay, we read out the qubit state by driving the DC-SQUID with a 20-ns “sample” current Is followed by a 20-μs “hold” current. The SQUID will switch to its normal state voltage Vs if Is > Isw,0 (Is > Isw,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” Psw (the population of state |0 〉) shown in Fig. 2D by following a flux-dependent sample current Isw,0 < Is < Isw,1 (dash-dotted line in Fig. 2C). Such 1D scans were then accumulated as a function of the rf source parameters Vrf (Fig. 3) and ν (fig. S2).

Fig. 3.

Multiphoton interference fringes show a Bessel staircase. Switching probability Psw is plotted as a function of qubit flux detuning δfq and voltage Vrf at frequency ν = 1.2 GHz. n-photon resonances are labeled 1 to 20. Each n-photon resonance exhibits oscillations in Psw resulting from a MZ-type quantum interference that results in a Bessel dependence Jn(λ), where λ is the rf amplitude scaled by hν (Eq. 4). Roman numerals mark the interference fringes of Jn(λ) (solid white lines). The n-photon resonances are symmetric about the qubit step (0 mΦ0). (Inset) Close-up of the n = 4 photon resonance.

The switching probability Psw (color scale in Fig. 3) versus qubit flux detuning δfq and voltage Vrf 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 Vrf. The fringes exhibit a Bessel-function dependence, Jn(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).

Fig. 4.

Analysis of the resonance area and width. (A) Resonance area An versus voltage Vrf for the n = 1 and n = 5 photon transitions. The Bessel dependence Jn(λ) is observed over several lobes. The data are best fit by functions that include decoherence (blue line) rather than omit it (red line). (Insets) Decoherence becomes more pronounced as photon number increases. (B) The resonance width wn versus voltage Vrf for n = 1 and n = 5 also exhibits a Bessel dependence. (C and D) The area (C) and the width (D) plotted for resonances n = 1 to 9 and n = 19.

Multiphoton transitions at the resonances (Eq. 3) in the strong driving regime, |Arf|,hν ≫ Δ, occur by means of fast LZ transitions. The notion of quasi-stationary qubit levels Math is inadequate in this regime and, instead, we use a different approach, transforming the Hamiltonian (Eq. 2) to a nonuniformly rotating frame, Math, where ϕ(t) = λsinωt with dimensionless rf-field amplitude Math(4) The rf field disappears from the detuning term, reappearing as a phase factor of the off-diagonal term: Math, where h.c. is hermitian conjugate. Given that Δ ≪ hν, near the nth resonance nhν ≅ ϵ0 we can replace the phase factor eiϕ(t) by its nth Fourier harmonic, Jn(λ)einωt, where Jn is the Bessel function. The resulting effective Hamiltonian Math is of a “rotating-field” form Math(5) where Δn = ΔJn(λ). The resonance approximation (Eq. 5) describes transitions at an arbitrary ratio Arf /hν. Standard Rabi dynamics analysis of the Hamiltonian (Eq. 5) with the initial state |0 〉 gives the time-averaged occupation probability of the excited state Math. This expression predicts Lorentz-shaped resonances of width δϵ = |Δn|. The result, a sum of independent contributions with different n, Math(6) is displayed in Fig. 1C. The agreement with the observed resonances is notable: The oscillations in rf power, described by Jn(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 Psw for voltages Vrf = 71 mVrms and Vrf = 7.1 mVrms (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 An, which, in contrast with the widths of the resonances wn, are not affected by inhomogeneous broadening. The standard Bloch approach yields Math(7) Math where T1,2 represents the longitudinal and transverse relaxation times, and T2* describes the inhomogeneous broadening. These are aggregate relaxation times averaged over the periodic qubit detuning, which, in the operating limit ϵ0 ≫ Δ, tends to overestimate T1 and underestimate T2 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 T1,2. 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: T1 ≈ 20 μs, T2 ≈ 15 to 25 ns, T2* ≈ 5 to 10 ns, and Δ/ħ ≈ (2π)4 MHz. The T1, T2, and Δ estimates are similar to those reported by Yu et al. (26). The nearly linear behavior at the nodes of Jn (Fig. 4A) indicates that the decoherence is small compared with the splitting: T1T2(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 Mq = 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

Materials and Methods

SOM Text

Figs. S1 and S2

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