Abstract
Superconducting circuits with Josephson junctions are promising candidates for developing future quantum technologies. Of particular interest is to use these circuits to study effects that typically occur in complex condensedmatter systems. Here we employ a superconducting quantum bit—a transmon—to perform an analogue simulation of motional averaging, a phenomenon initially observed in nuclear magnetic resonance spectroscopy. By modulating the flux bias of a transmon with controllable pseudorandom telegraph noise we create a stochastic jump of its energy level separation between two discrete values. When the jumping is faster than a dynamical threshold set by the frequency displacement of the levels, the initially separate spectral lines merge into a single, narrow, motionalaveraged line. With sinusoidal modulation a complex pattern of additional sidebands is observed. We show that the modulated system remains quantum coherent, with modified transition frequencies, Rabi couplings, and dephasing rates. These results represent the first steps towards more advanced quantum simulations using artificial atoms.
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Introduction
The ability to resolve energy variations ΔE occurring in a time interval Δt is fundamentally limited by the energytime uncertainty relation . Consider for example a system of spin1/2 particles filling a porous material (Fig. 1a) in an external magnetic field; then, if the pore wall is paramagnetic, the particles near it will experience a modified local magnetic field. As a result, static particles will produce two spectral peaks in the nuclear magnetic resonance spectrum at frequencies ω_{1} and ω_{2} (Fig. 1b). In contrast, particles moving swiftly back and forth between the two regions on timescales Δt shorter than are not able to discriminate between the two energy values. Then, as the particles move faster, the outcome in the spectroscopy is not simply a continuous broadening and overlapping of the two peaks. Instead, a new peak emerges at the average frequency ω_{0} with a width smaller than the energy separation , denoted as motional averaging and narrowing^{1,2}, respectively. In atomic ensembles and condensedmatter systems, this occurs via fast variations of the electronic state populations, chemical potential, molecular conformation, effective magnetic fields, lattice vibrations, microelectric fields producing acStark shifts and so on^{1,2,3,4,5,6}. Closely related phenomena are the DyakonovPerel effect^{7}, the Dicke line narrowing of Doppler spectra^{8}, and the quantum Zeno effect^{9}.
In this paper, we report the observation of motional averaging and narrowing in a single artificial atom, a quantum bit (qubit), with a simulated fastfluctuating environment under direct experimental control. This should be contrasted with the typical situation in condensedmatter systems, where one has a large number of particles and the experimentalist can only indirectly attempt to change the fluctuation rate, typically by modifying a thermodynamic function of state, such as temperature or pressure.
Results
Frequencymodulated transmon qubit
Our device is a circuit QED system^{10}, consisting of a tunable transmon coupled to a quarterwavelength coplanar waveguide resonator. The circuit schematic and the simplified experimental setup are shown in Fig. 1d and in Supplementary Fig. S1. The applied bias flux Φ through the qubit loop consists of a constant part Φ_{dc} and a timedependent part Φ_{ac}(t) with an amplitude much smaller than Φ_{dc}. The Hamiltonian of the transmon qubit^{11} is
where . Here Φ_{0}=h/2e is the magnetic flux quantum, the Josephson plasma frequency is , denote the Pauli matrices, the singleelectron charging energy is E_{C}=e^{2}/2(C_{G}+C_{B})≈h × 0.35 GHz, E_{J}=E_{J1}+E_{J2}≈24E_{C} is the maximum Josephson energy, and d=(E_{J1}−E_{J2})/E_{J}≈0.11 denotes the junction asymmetry. We choose Φ_{dc} so that ω_{0}/2π=2.62 GHz, which is far detuned from the bare resonator frequency ω_{r}/2π=3.795 GHz, allowing the dispersive measurement of the qubit through the resonator by standard homodyne and heterodyne techniques^{12}.
The timedependent part of the energy splitting, determined by Φ_{ac}(t), is controlled by an arbitrary waveform generator (AWG) via a fast flux line. The random telegraph noise (RTN) is realized by feeding pseudorandom rectangular pulses to the onchip flux bias coil. Ideally, the dynamics of is a stationary, dichotomous Markovian process, characterized by an average jumping rate χ and symmetrical dwellings at frequency values of ±ξ (see Fig. 1c). The number of jumps is a Poisson process with the probability P_{n}(t)=(χt)^{n}e^{−χt}/n! for exactly n jumps within a time interval t. This Poissonian process simulates the temporal variations causing motional averaging in atomic ensembles and condensedmatter systems (Fig. 1).
Simulation of motional averaging
In order to find the effective transition energies and decoherence rates of the modulated system, we calculate the absorption spectrum^{1,2}. For a qubit subjected to RTN fluctuations as in equation (1), we exploit the quantum regression theorem^{13} to find the spectrum (see details in Supplementary Note S1)
where the expectation value is taken over noise realizations and Γ_{2}=Γ_{ϕ}+Γ_{1}/2 is the decoherence rate in the absence of modulation. For our sample Γ_{2}≈2π × 3 MHz (determined in an independent measurement). For the Poisson process ξ(t), equation (2) gives^{1,2,14,15}
with and . In our measurement, we have recorded the population of qubit's excited state, as shown in Fig. 2 and in Supplementary Fig. S2. The qubit is excited with a transverse drive
The simulation of the occupation probability presented in Fig. 2a (see Methods and Supplementary Note S2) includes also the effect of power broadening due to strong driving amplitudes (g>Γ_{1},Γ_{2}).
Two intuitively appealing limits result from equation (3) (see also Fig. 2) at the opposite sides of a dynamical threshold defined by χ=ξ. In the case of slow jumping, , and resolvable energy variations, , the qubit absorbs energy at ω_{0}±ξ with the total decoherence rate . The correlation time of the displacement process is τ_{ξ}=(2χ)^{−1} and the linewidth (full width at half maximum ) broadens by the amount of the reduced mean lifetime of a qubit excitation (exchange broadening^{1}). In contrast, for fast jumping processes, , and when the variations are not resolvable, , the qubit absorbs energy only at the frequency ω_{0} with . The increase in decoherence rate by ξ^{2}/2χ can be related to the excursions of the accumulated phase (equation (2)) for single noise realizations, especially to the diffusion coefficient of the process^{16}. Surprisingly, the averaged spectral line is welllocalized around the mean value, since . The effect of motional narrowing can be best seen in Fig. 2c as a raise in the height of the centre peak because the additional decoherence ξ^{2}/2χ decreases with increasing jumping rate χ. When χ is comparable with ξ, there is a crossover region where absorption is reduced and the peak broadens due to enhanced decoherence (see Fig. 2). This is important for the improvement of qubit dephasing times as it implies that a longitudinally coupled twolevel system (TLS) fluctuator is the most poisonous when its internal dynamics occurs approximately at the same frequency as the coupling to the qubit.
A counterintuitive aspect of motional averaging is that the system is at any time in either one of the states and spends no time inbetween. Yet spectroscopically it is not seen in the states ; instead it has a clear signature in the middle. Also, the spectrum (3) decays as 1/ω^{4} far in the wings, showing a nonLorentzian character that originates from the nonexponential decay of the correlator ^{14}. This is because the frequent changes have a similar effect to a continuous measurement of the system, slowing its dynamics in analogy with the quantum Zeno effect^{9,17}.
Sinusoidal modulation of frequency
To further explore these effects, we have used sinusoidal waves to modulate the qubit energy splitting: . In Fig. 3a, the experimental data and the corresponding numerical simulation are shown. Resolved sidebands appear at ω=ω_{0}±kΩ (k=0,1,2,…), and are amplitudemodulated with Bessel functions J_{k}(δ/Ω), where δ and Ω denote the modulation amplitude and frequency, respectively. The Hamiltonian (equations (1) and (4)) can be transformed to a nonuniformly rotating frame^{18,19} (see Methods). The theoretical prediction for the steadystate occupation probability on the resolved sidebands (Ω>g>Γ_{2}) is
which is compared with experimental data in Fig. 3c and in Supplementary Fig. S3. These population oscillations can be alternatively understood as a photonassisted version of the standard Landau–Zener–Stückelberg interference^{20,21} (see Methods and Supplementary Fig. S4). Also, when the qubit drive is tuned close to the resonator frequency ω≈ω_{r} and when the driving amplitude g is large and equal to the frequency of the modulation g=Ω, the system can be seen as a realization of a quantum simulator of the ultrastrong coupling regime^{22,23}. In our setup, the simulated coupling rate is estimated to reach over 10% of the effective resonator frequency (see Supplementary Note S3), comparable to earlier reports^{24,25} where the ultrastrong coupling was obtained by sample design. The sinusoidal modulation allows a different perspective on motional averaging. The RTN noise comprises many different modulation frequencies Ω (Fig. 1c). Each frequency creates different sidebands (k=1,2,3,…), which overlap and average away. At large Ω's only the central band k=0 survives because the only nonvanishing Bessel function of zero argument is J_{0}(0)=1.
Coherence of the modulated system
Finally, we show that it is possible to use the modulated system as a new, photondressed qubit. Indeed, we can drive Rabi oscillations on the central band and on the resolved sidebands (see Fig. 4 and Supplementary Fig. S5). For fast RTN modulation, , and under the strong drive in equation (4), we construct (see Supplementary Note S2) a master equation^{15} describing the Rabioscillation with the detuned Rabi frequency and with the total decoherence rate . In the case of sinusoidal modulation, the Rabi frequencies on the sidebands are obtained as .
Discussion
Our work could open up several research directions. For example, a key limitation in solidstate based quantum processing of information is the decoherence due to fluctuating TLSs in the dielectric layers fabricated onchip^{26,27,28}. We anticipate, resting on the motional narrowing phenomenon, that the dephasing times of the existing superconducting qubits may be dramatically improved if one is able to accelerate the dynamics of the longitudinally coupled TLSs. The quantum coherence of the resolved sidebands and of the central band suggest that lowfrequency modulation offers an additional tool for implementing quantum gates^{29}, in analogy with the use of vibrational sidebands in the ion trap computers^{30}. Also, if the two tones used to drive and modulate the qubit satisfy certain conditions, the system can be mapped into an effective qubitharmonic oscillator system with ultrastrong coupling, opening this regime for experimental investigations. Finally, the recent progress in the field of nanomechanical oscillators makes possible the study of frequency jumps in nanomechanical resonators^{16}, which is predicted to be accompanied by squeezing^{31}. Our work paves the way for further simulations of quantum coherence phenomena using superconducting quantum circuits^{32,33}.
Methods
Measurement setup
The electronic measurement setup at room temperature is illustrated in Supplementary Fig. S1. For qubit spectroscopy with RTN modulations, the dc flux bias and RTN modulations are generated by an Agilent 81150A AWG. The qubit driving signal from an Agilent E8257D analogue signal generator and the coplanar waveguide cavity probe signal from an Agilent N5230C PNAL network analyser are combined together by a MiniCircuits ZFSC210G power splitter/combiner and sent to the cavity input line of the dilution refrigerator. The signal from the cavity output line of the dilution refrigerator is amplified and detected by a PNAL network analyser. For sinusoidal modulations, the tones are generated by a R&S SMR27 microwave signal generator and they are added to the dc bias generated by the Agilent 81150A via a bias tee (not shown). Between the radio frequency instruments (analogue signal generator and network analyser) and their corresponding lines, dc blocks are used for breaking possible ground loops. All instruments are synchronized with a SRS FS725 Rubidium frequency standard (not shown in the figure).
For Rabioscillation measurements, high ON/OFF ratio Rabi pulses are generated by mixing a continuous microwave signal from the Agilent E8257D with rectangular pulses from a SRS DG645 digital delay generator via two identical MiniCircuits frequency mixers. The Agilent N5230C PNAL is used as a signal generator. Its output signal is split into two parts. One part is used for generating measurement pulses in a similar fashion to the Rabi pulses; the other part acts as a local oscillator signal to mix the output signal from the coplanar waveguide cavity down via a Marki IQ mixer. The IQ data is filtered, amplified by a SRS SR445A preamplifier, and digitized by an Agilent U1082A001 digitiser.
Measurement controlling and data processing are done by MATLAB running on a measurement computer. The communication between the measurement computer and the instruments is realized through IEEE488 GPIB buses. To generate RTN pulses, we use MATLAB’s internal Poisson random number generator poissrnd(λ) to obtain binary RTN sequences, and load the sequences into the Agilent 81150A AWG. Each binary RTN sequence consists of 50,000 data points and around 5,000 random jumps in average. The mean jumping rate χ (of AWG's output) is modulated by changing the clock frequency ν of the AWG: χ≈5,000 × ν. To verify this relation between χ and ν, we observe the RTN sequences at different ν's by a fast oscilloscope (10 GS per s), count the number n of edges (jumping events) for certain period of time t, and calculate the real mean jumping rate by its original definition, χ=n/t. As an example, a 0.5μs long RTN sequence with estimated mean jumping rate χ≈5,000 × 10 KHz=50 MHz is shown in Supplementary Fig. S1. Twentyfive jumps are counted during 0.5 μs, which gives χ=50 MHz. As long as ν<100 KHz, the formula χ≈5,000 × ν gives a good estimation of the mean jumping rate. For ν>100 KHz, the real mean jumping rate is smaller than the estimated one, due to the intrinsic ∼2 ns rising/falling time of the AWG.
Numerical simulations
The occupation probability of Figs 2a, c, 3b, and Supplementary Figs S2 and S4 is a result of solving numerically the master equation
in the frame rotating at ω and applying the rotating wave approximation (the pure dephasing rate Γ_{ϕ}=Γ_{2}−Γ_{1}/2). For the random modulation (Figs 2a, c, and Supplementary Fig. S2), we apply the method of quantum trajectories^{13} and the imperfections of the experimentally realized wave forms (the raising/falling time ∼2.0 ns and the sampling time 0.5 ns) are included in the simulation.
Multiphoton transition process in sinusoidal modulation
The effects of sinusoidally modulating the transition frequency of an atom is a generic problem in theoretical physics, and has attracted a lot of interest in the past, most notably in the quantumoptics community^{34,35}. We address now this problem in the context of superconducting qubits. We consider a transmon qubit whose energy splitting is modulated sinusoidally: . Even though , the qubit can be excited with the lowfrequency signal in the presence of an additional highfrequency drive ω. This can be seen as a multiphoton transition process involving quanta from both fields (in equation (5)), or, as a photonassisted Landau–Zener–Stückelberg (LZS) interference^{18,20,21,36} between the dressed states ↓, n〉 and ↑, n−1〉.
Let us first consider the multiphoton transition processes associated with the sinusoidalmodulated Hamiltonian, defined in equations (1) and (4),
The qubit is driven both in the longitudinal () and in the transverse () direction. We show below how to eliminate the timedependence from the longitudinal drive. After moving to a nonuniformly rotating frame with the unitary transformation^{18,19}
the effective Hamiltonian is , and by using the JacobiAnger relations, we get
By assuming that the transverse drive is close to a resonance, that is, ω≈ω_{0}±kΩ (k=0,1,2…) and that the resonances are resolvable Ω>g>Γ_{2}, we transform back with and ignore all but the resonant terms, that is, all the fast rotating terms (rotating wave approximation, RWA). The resulting RWA Hamiltonian reads
describing, in the Bloch spin representation, precessions around the vector Ω=(gJ_{k}(δ/Ω),0,ω_{0}+kΩ−ω) with the effective Rabi frequency
Exactly at the multiphoton resonance ω=ω_{0}±kΩ (k=0,1,2…), the effective Rabi frequency is gJ_{k}(δ/Ω), and it is plotted in Fig. 4c.
In the time domain, the measured Rabi oscillations are presented in Supplementary Fig. S5 together with numerical simulations of Bloch equations exploiting the effective Rabi frequency in equation (11). To find the steadystate occupation probability in the presence of relaxation with rate Γ_{1} and decoherence with rate Γ_{2}, we solve the Bloch equations analytically. In the RWA, that is when Ω>g>Γ_{2} is satisfied, one is allowed to add up independent contributions from all the resonances (the resolved sidebands). The result for the steadystate excited state occupation probability is equation (5). In the nonmodulated case, the corresponding expression for the occupations is simply
The occupation probabilities in equations (5) and (12) are compared with the experimental steadystate occupation probabilities in Supplementary Fig. S3. An interesting effect is that the linewidth of the modulated qubit on the central band is always smaller than the linewidth on the qubit in the absence of the modulation due to reduced power broadening.
Photonassisted Landau–Zener–Stückelberg interference
The spectra seen in the experiment under sinusoidal modulation can be also interpreted as a photonassisted LZS effect. Note that in the absence of the driving field standard LZS processes are not possible: indeed the qubit energy separation is one order of magnitude higher than the modulation frequency, therefore the standard LZS probability is negligibly small. However, the system can still perform LZS transitions by absorbing a photon from the driving field. This photonassisted LZS interference can be seen by transforming the Hamiltonian from equation (7) into the frame rotating at ω around the z axis (unitary transformation ), where it has exactly the same form (in the RWA) as that of LZS interference^{18,21,36}, namely
As illustrated in Supplementary Fig. S4, LZS transition events may occur when the modulation amplitude δ is of the order of, or larger than, the detuning between the driving frequency and the qubit splitting ω–ω_{0}. The LZS transitions occur between the transversely dressed states ↓, n〉 and ↑, n−1〉, where n refers to the photon number of the transverse driving field. The phase difference of the two states gathered between the consecutive tunnelling events leads to either constructive or destructive interference observed as maxima or minima in the occupation probability of the excited state, shown in Supplementary Fig. S4. The LZS interference seen in our system corresponds to the socalled fast passage limit^{21}. In this limit, the expression for the occupation probability can be calculated analytically^{21} and the result agrees exactly with our equation (5).
To prove experimentally that this picture is valid, we have scanned the modulation amplitude δ and the driving frequency ω at fixed Ω. We observe an interference pattern of the steadystate occupation probability, which is in good agreement with the theoretical prediction, see Supplementary Fig. S4.
Additional information
How to cite this article: Li. J. et al. Motional averaging in a superconducting qubit. Nat. Commun. 4:1420 doi: 10.1038/ncomms2383 (2013).
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Acknowledgements
We thank S. Girvin and J. Viljas for discussions. This work was done under the Center of Excellence `Low Temperature Quantum Phenomena and Devices' (project 250280) of the Academy of Finland. G.S.P. acknowledges support from the Academy of Finland, projects 141559, 253094, and 135135. M.P.S. was supported by the Magnus Ehrnrooth Foundation and together with J.M.P. by the Finnish Academy of Science and Letters (Vilho, Yrjö ja Kalle Väisälä Foundation). J.M.P. acknowledges also the financial support from Emil Aaltonen Foundation and KAUTE Foundation. The contribution of M.A.S. was done under an ERC Starting Grant. J.L. and M.P.S. acknowledge partial support from NGSMP.
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J.L., K.S.K., and G.S.P. designed and performed the experiment. M.P.S. carried out the theoretical work. A.V., W.C.C., and M.P.S. contributed to experiments. J.M.P. fabricated the sample. J.T. and E.V.T. provided theoretical support. J.L. and M.P.S. cowrote the manuscript in cooperation with all the authors. M.A.S., P.J.H., E.V.T. and G.S.P. provided support and supervised the project.
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Li, J., Silveri, M., Kumar, K. et al. Motional averaging in a superconducting qubit. Nat Commun 4, 1420 (2013). https://doi.org/10.1038/ncomms2383
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DOI: https://doi.org/10.1038/ncomms2383
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