Fast quantum control and light-matter interactions at the 10,000 quanta level

Fast control of quantum systems is essential in order to make use of quantum properties before they are degraded by decoherence. This is important for quantum-enhanced information processing, as well as for pushing quantum systems into macroscopic regimes at the boundary between quantum and classical physics. Bang-bang control attains the ultimate speed limit by making large changes to control fields on timescales much faster than the system can respond, however these methods are often challenging to implement experimentally. Here we demonstrate bang-bang control of a trapped-ion oscillator using nano-second switching of the trapping potentials. We perform controlled displacements which allow us to realize quantum states with up to 10,000 quanta of energy. We use these displaced states to verify the form of the ion-light interaction at high excitations which are far outside the usual regime of operation. These methods provide new possibilities for quantum-state manipulation and generation, alongside the potential for a significant increase in operational clock speed for ion-trap quantum information processing.

The ion is first cooled close to the ground state of motion in a harmonic trapping potential centered at x = 0, and the spin state is pumped to ↓ (see Supplementary Information). At t = 0 the potential center is suddenly displaced to x d and the ion's motional state becomes −α0 . b) Evolution of the motional wave function in the position-momentum phase-space. After a time ∆t in the displaced potential, the trap center is switched back to x = 0, leaving the ion in a coherent state of size given by equation (1). c) Sequence of laser-pulses and potential displacements for monitoring the time evolution of coherent states. d) Sequence for studying the light-matter interaction with highly-excited states. In this sequence, ∆t is an integer number of times j of the oscillator period.
acting directly on the charge (electric monopole) can be used for achieving a strong-interaction regime. This is a key element for transporting trapped-ion qubits in a scalable quantum-computing architecture [9,20], where operations have been performed on timescales comparable to the oscillator frequency [21,22]. In these examples the speed was limited by the bandwidth of the filtered control voltages applied to the electrodes. In this Letter, we demonstrate bang-bang control over trapped-ion harmonic oscillators, which we use to generate and characterize quantum states with up to 10,000 quanta. To achieve this, we wire the trap electrodes according to a new scheme: we prepare constant, filtered voltages at the inputs of switches placed in vacuum close to the trap electrodes, and select the input to be wired to the electrode by means of digital control pulses (for further details about the experimental setup, see Supplementary Information). For this setup we have measured switching times of nano-seconds, which we think were limited by our measurement apparatus [8]. This is much faster than the ion oscillation period, meaning that we can induce quasi-instantaneous changes to the oscillator states (Fig. 1a). Using these techniques, we map out the ion-light coupling strength for high excitations, and track the oscillation trajectory of a single ion throughout multiple oscillation cycles.
In order to examine the timing quality and reproducibility of our control, we prepare a scenario which is highly familiar in classical physics. Starting with a particle at rest at the bottom of a harmonic potential, we suddenly displace the latter by a distance x d . For an instant, the particle finds itself at rest but in an excited state, and subsequently oscillates in the new potential. In our experiments, this situation is reproduced in the quantum regime. In the ideal case, the initial state is the quantum ground state 0 , which has a minimumuncertainty wave-packet of root-mean-squared extension x 0 . The action of the sudden displacement produces a coherent state −α 0 with α 0 = x d /(2x 0 ), which oscillates at a frequency ω m while maintaining the form of the wave-packet. Switching the potential minimum back to x = 0 after a time ∆t leaves the oscillator in a coherent state of size (1) in the original potential (Fig. 1b).
The trapped-ion oscillator we consider also has internal degrees of freedom, from which we can isolate a two-state pseudo-spin system ( ↓ , ↑ ) with transition frequency ω 0 . This transition can be parametrically coupled to the oscillator using electromagnetic radiation at frequency ω L = ω 0 + sω m . For integer s, resonant transitions can be induced on the s th sideband between the states ↓ n ↔ ↑ n + s with a Rabi frequency [23] Ω n,n+s = Ω 0 n + s e ikxx n , where Ω 0 is a constant which contains the internal-state coupling strength and the electric field amplitude, and k x is the size of the radiation wavevector projected on the oscillation direction. For k x x 0 1 and at low excitations, a small parameter expansion of the exponential is sufficient to describe the resulting dynamics. This is the regime in which most quantum control experiments with trapped ions have been operated in the past, and as a result coherent control has only been demonstrated for |s| ≤ 2. In the work presented below, we map out the excitations for 0 ≤ s ≤ 5, confirming the dependence of the light-atom interaction strength for excitations up to n ≈ 10,000.
In our experiments, the oscillator is the axial motion of a single 40 Ca + ion confined in a radio-frequency trap with a frequency between ω m /(2π) ≈ 2.35 MHz and 2.53 MHz, and corresponding values of x 0 ∼ 7 nm. We use the dipole-forbidden transition at wavelength λ ≈ 729 nm to define a pseudo-spin system between levels ↓ ≡ L = 0, J = 1/2, M J = −1/2 and ↑ ≡ L = 2, J = 5/2, M J = −5/2 . The wavevector of the laser addressing this transition makes an angle θ ≈ 45 deg with the motional direction, resulting in a Lamb-Dicke parameter η = k x x 0 ∼ 0.044. Each experimental sequence begins by initializing the trappedion oscillator close to its ground state using a combination of Doppler and electromagnetically-inducedtransparency cooling [25]. We measure a typical mean excitation after cooling to ben th = 0.20 (6). Error bars throughout this paper are given as standard error of the mean.
In a first set of experiments we characterized our control using two successive displacements of the potential well ( Fig. 1c). At t = 0, we switch the center of the potential from 0 to x d , resulting in the ion being excited and subsequently oscillating in the displaced potential well. At t = ∆t we reverse this change, bringing the potential back to its original position. In the ideal case, the ion is then in a coherent state of size |α| given by equation (1). In order to read out the oscillator state, we then perform an analysis using an optical probe pulse of duration t p which is resonant with the s th sideband. This is followed by a detection of the ion's internal state using resonance fluorescence (Supplementary Information). By repeating the experiment a large number of times for each value of t p , we obtain an estimate of the probability of finding the ion spin down as a function of time. This agrees well with the functional form where pn th ,|α| (n) is the occupation of the n th number state for a displaced thermal state with thermal mean quantum numbern th and displacement |α| (this distribution is given in the Supplementary Information). The effects of spin and motional decoherence during the probe pulse are accounted for by an exponential decay at rate Γ [12]. Experiments were performed for a range of values of ∆t, in each case taking data using the first sideband s = 1 for 0 < t p < 100 µs. For a single value of ∆t, the data do not allow the extraction of |α|, Ω 0 and the decay parameter Γ independently. We therefore probe first the non-displaced state directly after cooling. Fitting using equation (3) with |α| = 0 andn th = 0.2, we determine Ω 0 /(2π) = 181(1) kHz and Γ = 2.2(3) ms −1 . These are then fixed in fits to the data for each value of ∆t, from which we extract the corresponding displacement |α|. The resulting values are plotted in Fig. 2a. Also shown is a fitted curve using equation (1), which allows us to determine the oscillation frequency of the ion ω m /(2π) = 2.3505(6) MHz and amplitude α 0 = 5.11(1), corresponding to x d = 75.0(1) nm (more details of this analysis can be found in the Supplementary Information). We observe in the results in Fig. 2a that the amplitude of each oscillation of the ion appears to be different. We attribute this to drifts in the laser intensity at the ion, which result in shifts in the deduced value of |α| (this drift is illustrated by the bounding curves in In order to verify that the switching between potentials does not produce additional effects beyond a simple displacement, we examine our ability to catch the ion in the ground state after a single cycle of oscillation. In an independent measurement we examined data for the region around ∆t = 2π/ω m for ω m /(2π) ≈ 2.53 MHz and an intermediate coherent state of α 0 ≈ 85. To faithfully return these states to the origin after a single trap cycle we need to trigger the experimental sequence from the phase of the radio-frequency trap drive. This can be attributed to a pseudopotential gradient along the axis of our trap [26]. The results are presented in Fig. 2b and show |α| = 0.05(10) for ∆t ≈ 2π/ω m . Figure 2c provides a direct comparison between data for a non-displaced ion withn th = 0.15 (3), and one with the same starting temperature which has been displaced to α 0 ≈ 85 and returned one cycle later. A fit to the data for the latter givesn th = 0.21 (3) and consistent values for Ω 0 and Γ.
Having characterized our control, we perform a second set of experiments to map out the interaction between the ion and the light field as a function of the motional excitation. In contrast to the work described above, we send the probe pulse to the ion while the potential is centered at x d . The potential well is displaced back to x = 0 after the coherent probe pulse has been applied, timing this second displacement to happen once the ion has completed an integer number of oscillation periods. In this way, we prevent the high oscillator excitation from affecting the internal state detection. Experimental measurements of P ↓ are shown in Fig. 3a for sidebands between s = 0 and s = 5 and displacements up to x d ≈ 1.5 µm, which corresponds to α 0 ≈ 100 and n ≈ 10, 000 at the trap frequency of 2.35 MHz. For each value of x d we probe P ↓ using laser-pulse durations between 1.4 and 80 µs. To analyze these measurements, we start by determining Ω 0 /(2π) = 204.3(1) kHz and n = 0.21 (8) from measurements with the non-displaced state using the s = 0 and s = 1 sidebands respectively. We then calibrate the displacement sizes using the data taken for the carrier transition s = 0, fixing Γ = 0 and floating only |α| in equation (3) (for further details, see Supplementary Information). These calibrated values were then fixed, and data for each of the other sidebands was fitted as a full set using a model including motional-excitation-dependent AC Stark shifts due to nearby atomic transitions and a systematic detuning to account for miscalibration (Supplementary Information). We find that the AC Stark shifts become relevant for s = 4 and s = 5 at high motional excitations, because the frequencies of these transitions are close to the seventh and sixth red motional sidebands respectively of an atomic transition which is 25.74 MHz away from the driven transition. Two-dimensional plots showing the fit results are shown in Fig. 3b below the respective dataset.  1200  1200  1200  1000  1000  1000  800  800  800  600  600  600  400  400  400  200  200  200  0  0  0  1400  1400  1400  1200  1200  1200  1000  1000  1000  800  800  800  600  600  600  400  400  400  200  200  200  0  0  0  1400  1400  1400  1200  1200  1200  1000  1000  1000  800  800  800  600  600  600  400  400  400  200  200  200  0  0  0  1400  1400  1400  1200  1200  1200  1000  1000  1000  800  800  800  600  600  600  400  400  400  200  200  200  0  0  0   1400  1400  1400  1200  1200  1200  1000  1000  1000  800  800  800  600  600  600  400  400  400  200  200  200  0  0  0  1400  1400  1400  1200  1200  1200  1000  1000  1000  800  800  800  600  600  600  400  400  400  200  200  200  0  0  0  1400  1400  1400  1200  1200  1200  1000  1000  1000  800  800  800  600  600  600  400  400  400  200  200  200  0  0  1400  1400  1400  1200  1200  1200  1000  1000  1000  800  800  800  600  600  600  400  400  400  200  200  200  0  Our computer-control system imposes tp ≥ 1.4 µs, so for high Rabi frequencies P ↓ can be close to zero even for the first point. b) P ↓ obtained using equation (3) with the parameters determined from fits. c) Mean Rabi frequencies for displaced thermal states as a function of the potential displacement. The data points are given by the center of the Lorentzian curves to which we fit P ↓ (Ω), the Fourier transform of P ↓ (tp). Most error bars are hidden behind the points. The solid lines result from the theory model including AC Stark shifts and a detuning δ off obtained from fitting the data for a given s.
In order to verify the predicted form of the matrix elements in equation (2), we take a Fourier transform of P ↓ (t p ) for each value of x d . This gives us the Rabi frequency distribution, which is peaked due to the limited range of number states which contribute significantly to the coherent state (Supplementary Information). Fig.  3c shows the mean Rabi frequency obtained from each Fourier transform versus the amplitude of the state. We overlay this with theoretical curves using equation (2) which include modifications for the offset detunings and AC Stark shifts, showing good agreement between experiment and theory over the full range of motional occupations. The minima in Rabi frequencies observed for each set of sideband data are separated by between 515 and 525 nm. These arise due to the large amplitude of the ion's motion, which spatially samples multiple wavelengths of the light. The separation between minima is expected to approach the effective wavelength of the light as projected on the trap axis for x d λ. In our case this value is λ/(2 cos θ) ≈ 515 nm (for more details see Supplementary Information). We note that though the s = 5 sideband would take 2 minutes to invert the spin if driven in the ground state, the modulation of the light due to the large ion oscillation allows us to do this in less than 10 µs.
The work presented has a number of possible applications for quantum information science. It may be used to speed-up the transport of trapped-ion qubits in scalable quantum information processing [20][21][22], where the transport would proceed by switching the position of the potential well to two positions x d and 2x d at times which are separated by ∆t = π/ω m [8]. This would result in the ion being "caught" in the ground state of the potential centered at 2x d , with the transport taking half a period of the ion's oscillation. We expect that large squeezed states could be achieved through sudden trap frequency changes, in this case separated by delays of one quarter of the trap period [8]. Bang-bang control routines based on sudden changes to the trap frequency have been proposed for suppressing motional decoherence effects [27]. Using ion chains, the ability to change the trapping potential on timescales which are fast compared to the characteristic frequencies of the interaction between ions has been proposed as a method for generating high levels of continuous variable entanglement by means of the Coulomb interaction [6,7,28].

A. Experimental setup
Our experiments are performed using a surfaceelectrode linear radio-frequency (rf) trap in a 5-wire asymmetric configuration [29,30] (Fig. 4). The pseudopotential null line is ≈ 50 µm above the trap chip. The main trapping zone, denoted with a star in the figure, is between the pair of electrodes e5, closer to e5r than e5l due to the asymmetry in the width of the rf electrodes. Electrodes e2, e8 and em are each connected to the output of a Single-Pole-Triple-Throw (SPTT) switch mounted on the Cryo-Electronics Board (CEB, Fig. 5b) and within 3 cm of the trap chip. Voltage offsets of less than a volt at these five electrodes suffice to carry out our experiments. The trap is driven with an rf amplitude of ≈ 100 V and a frequency of ≈ 93 MHz, leading to radial secular motion at ≈ 4 and ≈ 7 MHz at an axial frequency of ω m /(2π) ≈ 2.5 MHz.
The trap is placed in a sealed chamber cooled down to ≈ 4 K. Cryogenic setups yield improved vacuum over room-temperature experiments [31], resulting in longer ion lifetimes. We take advantage of the low outgassing at cryogenic temperatures to use standard Printed-Circuit-Board (PCB) assemblies for in-vacuum electronics. This reduces the technical effort compared to preparing ultrahigh-vacuum-compatible electronics for room temperature operation [32].
The CEB holds the fast switching electronics, 30 lowpass filters for the analog input lines, and a track for guiding the trap rf drive from a quarter-wave helical resonator to the trap chip. For voltage switching we use a commercial CMOS integrated circuit (CD74HC4066M, from Texas Instruments), which implements four bilateral Single-Pole-Single-Throw (SPST) switches. The additional circuitry to implement a SPTT switch is shown in Fig. 5a. The CEB includes five such copies, one per switchable electrode. Measured parameters of this circuit The digital pulses for controlling the switches are produced by room-temperature electronics based on a multichannel delay/pulse generator (P400 from Highland Technology). Sequences of pulses are triggered by a single TTL line locked to the phase of the rf drive and generated by a Field-Programmable-Gate-Array (FPGA), which we also use for the generation of laser-pulse sequences.

B. Cooling and Detection
The experimental sequences are depicted in Fig. 1c-d. Ground-state cooling is achieved in two stages. The first involves Doppler cooling into the Lamb-Dicke regime by applying a laser 10 MHz detuned below the S 1/2 ↔ P 1/2 transition at 397 nm for 500 µs. We then follow this with electromagnetically-induced-transparency cooling, involving the application of two 397 nm laser fields polarized so that they drive simultaneously the σ − and π transitions between the S 1/2 and P 1/2 Zeeman sublevels. With suitable combinations of laser powers and frequency detunings, this allows us to cool the ion close to the ground state of motion [25]. In our setup we reach a steady state atn th ≈ 0.2 after 100 µs, limited by the heating rate of the ion, which is 1 − 2 quanta/ms from the quantum ground state. After cooling, the state is initialized to ↓ by optically pumping with a σ − -polarized resonant 397 nm beam. In all these pulses, 866 nm light is used to repump the population from D 3/2 back to P 1/2 . Coherent operations between the S 1/2 and D 5/2 states are performed with a narrow-linewidth 729 nm laser. These couple the motional state to the spin sys- tem, which is subsequently read out by state-dependent fluorescence with resonant 397 nm and 866 nm light. In a detection time interval of 500 µs, we detect a mean number of 2 photons for an ion initially prepared in ↑ , and 25 photons if the ion is prepared in ↓ .

D. Light-atom interaction
The coupling between the internal electronic state and the ion's motion is described by the Hamiltonian [26] H I c = Ω 0 2σ + e iη(âe −iωm tp +â † e iωm tp ) e −iδtp + H.c. (5) Here,σ + = ↑ ↓ is the spin-flip operator producing transitions from ↓ to ↑ at the Rabi frequency Ω 0 /(2π), the Lamb-Dicke parameter η = k x x 0 relates the motional ground-state wave-packet size x 0 = /(2M ω m ) to the projection of the radiation wave vector k on the trap axis, M is the ion's mass, and δ is the laser detuning from the carrier transition at frequency ω 0 . For integer values of s, laser light detuned by δ ≈ sω m drives near-resonant transitions on the s th motional sideband ↓ n ↔ ↑ n+s . The corresponding matrix element is proportional to n + s e ikxx n and its associated Rabi frequency can be evaluated analytically [23]: Ω n,n+s = Ω 0 n + s e iη(â † +â) n where, n < (n > ) stands for the lesser (greater) of n + s and n, and L α n (x) are generalized Laguerre polynomials.

E. Dependence of deduced oscillation amplitude on carrier Rabi frequencies
For the first set of experiments we carried out the sequence in Fig. 1c and obtained the results in Fig. 2. Here we focus on one of the consequences of the slow drifts in laser power, which are not included in the model used for theoretical calculations.

F. Motional-state dependent AC Stark shifts
The model used for calculating the expected P ↓ in our experiment (Fig. 3) includes a detuning from the driven motional sideband as well as an AC Stark shift due to off-resonant coupling to transitions other than the one probed. The latter is given by and plotted in Fig. 6 for our experimental parameters. The first term arises from coupling to sidebands of the spin transition, while the second is due to the fact that the 729 nm laser drives off-resonantly the secondary transition ↓ ↔ L = 2, J = 5/2, M J = 3/2 . The coupling strength to the secondary transition is reduced relative to that of the resonantly driven transition by a factor W − 1 2 → 3 2 = √ 5. At our magnetic field of ≈ 3.83 G, the frequency gap between both carrier transitions is ∆ − 1 2 → 3 2 /(2π) ≈ 25.74 MHz. This is close to an integer times the motional frequency, causing the s = 4 (5) sideband of the main transition to be almost resonant with the s = −7 (−6) sideband of the secondary transition, and therefore strongly AC Stark shifting the ↓ state. For a total detuning δ tot (n, s) = δ off (s)+δ AC (n, s), equation (3) can be re-written as P ↓ (n th , |α| , Ω 0 , s, t p ) = 1 − n≥0 pn th ,|α| (n) Ω 2 n,n+s Ω 2 n,n+s + δ 2 tot (n, s) where we have ignored the exponential-decay term due to decoherence, since our probe times go up to 80 µs and we found previously Γ ≈ 2 ms −1 .

G. Light-matter-interaction data analysis
For this experiment we carried out the sequence in Fig.  1d. The results are shown in Fig. 3a. To analyze the data and compare it to the theoretical model, we first calibrated the displacements with the data obtained for the carrier (s = 0). We fitted the measurement results for each value of x d to equation (8) while fixing Ω 0 andn to values determined previously by driving Rabi oscillations on the s = 0 and s = 1 transitions, respectively. This yielded a coherent-state size which we converted into displacements according to α 0 = x d /(2x 0 ). For the rest of the sidebands we used the calibrated displacements and II: Fit results for the light-matter-interaction experiment. The probe-pulse frequencies for sidebands s = 0 to 2 were pre-calibrated with non-displaced states. Higher-order sidebands are too weak for a direct measurement with nondisplaced states, so probe-pulse frequencies for s > 2 where estimated from the lower sideband frequencies. This explains the large systematic shifts in δ off resulting from the fits. fitted the measurement data to find Ω 0 and δ off for each individual sideband. The best fit parameters are given in Tab. II and yield the results in Fig. 3b-c.
To analyze the evolution of the mean Rabi frequency for a given driven sideband s and as a function of the state size (Fig. 3c), we calculate the discrete Fourier transform of the measured P ↓ (t p ) and propagate the shot-noise uncertainties to the frequency domain according to [34]. The number-state distribution pn th ,|α| (n) for low values ofn th is narrow compared to the features which result from equation (6), so the frequency spectrum shows a single peak at the mean Rabi frequency (Fig. 7). We fit these to symmetric Lorentzian functions, from which we determine the center frequencies given in the plots. We compare this to the theoretical mean Rabi frequency for a displaced thermal state, given bȳ Ω th = n pn th ,|α| (n) Ω 2 n,n+s + δ 2 tot .
The separation between minima in mean Rabi frequency depends on the radiation wavelength λ. For large values of n, the Laguerre polynomials in equation (6) can be approximated by Bessel functions J α as [35] L |s| n (η 2 ) ≈ √ n η |s| e η 2 2 J |s| (2η √ n).
The zeros in the right-hand side correspond to zeros of the Bessel functions, whose separation tends to π for large arguments. This implies that the separation between minima in mean Rabi frequencies tends to λ/(2 cos θ) for large values of x d .