Ultrafast coherent excitation of a Ca$^+$ ion

TL;DR

This paper demonstrates a high-repetition-rate pulsed laser system for ultrafast coherent excitation of a Ca$^+$ ion, enabling faster quantum gate operations in trapped ion quantum computing.

Contribution

Introduction of a 5 GHz pulsed laser system derived from a stabilized frequency comb for coherent control of Ca$^+$ ions.

Findings

Achieved coherent population transfer with picosecond pulses

Measured ion-laser coupling strength using three techniques

Showed feasibility for implementing fast phase gates

Abstract

Trapped ions are a well-studied and promising system for the realization of a scalable quantum computer. Faster quantum gates would greatly improve the applicability of such a system and allow for greater flexibility in the number of calculation steps. In this paper we present a pulsed laser system, delivering picosecond pulses at a repetition rate of 5 GHz and resonant to the S$_{1/2}$ to P$_{3/2}$ transition in Ca$^+$ for coherent population transfer to implement fast phase gate operations. The optical pulse train is derived from a mode-locked, stabilized optical frequency comb and inherits its frequency stability. Using a single trapped ion, we implement three different techniques for measuring the ion-laser coupling strength and characterizing the pulse train emitted by the laser, and show how all requirements can be met for an implementation of a fast phase gate operation.

Figures

Tables

Table 1. Table 1: Summary of fit parameters for the different experiments

Rep. rate ($\mathrm{GHz}$)	Experiment	$\theta$ ($\pi$)	$\delta {\varphi }_{1,4}$ ($\pi$)
5	interferometer		$0.74$ or 1.26
	many pulses	$0.227\left(12\right)$
	single pulses	$0.195\left(18\right)$	$1.282\left(1\right)$
2.5	interferometer		$0.37$
	many pulses	$0.323\left(5\right)$
	single pulses	$0.312\left(10\right)$	$0.361\left(2\right)$
5/3	interferometer		$0.10$
	many pulses	$0.363\left(5\right)$
	single pulses	$0.339\left(7\right)$	$0.088\left(3\right)$
1.25	interferometer		$0.04$
	many pulses	$0.345\left(5\right)$
	single pulses	$0.358\left(3\right)$	$0.051\left(1\right)$
	Ramsey exp.	$0.377\left(5\right)$

Equations

$$\Delta {\Phi }_i=\Delta {\varphi }_i+k\cdot \Delta x,$$

$${\Gamma }_{\text{PS}}=(1-p_{5/2})\Gamma ,$$

$${\Gamma }_{\text{PD}}=p_{5/2}\Gamma ,$$

$${\rho }^{\prime }={\mathcal{U}}_{\mathcal{R}}{\rho }_0{{\mathcal{U}}_{\mathcal{R}}}^{†},$$

$${\mathcal{U}}_{\mathcal{R}}=\exp \left(\frac{\mathrm{i}}{2}\theta (|2⟩⟨1|+|1⟩⟨2|)\right),$$

$${\rho }^{\prime \prime }={\mathcal{U}}_{\mathcal{Z}}{\rho }^{\prime }{{\mathcal{U}}_{\mathcal{Z}}}^{†},$$

$${\mathcal{U}}_{\mathcal{Z}}=\exp (\mathcal{Z}),$$

$$\mathcal{Z}=\frac{\mathrm{i}}{2}\left[\Delta (|1⟩⟨1|-|2⟩⟨2|)+{\Delta }^{\prime }(|1⟩⟨1|-|3⟩⟨3|)\right]{\tau }_{\text{pulse}}.$$

$$\mathcal{N}=|1⟩⟨1|+\sqrt{1-p-q}|2⟩⟨2|+|3⟩⟨3|,$$

$$\mathcal{D}=\sqrt{p}|1⟩⟨2|+\sqrt{q}|3⟩⟨2|,$$

$${\rho }^{\prime \prime \prime }=\mathcal{N}{\rho }^{\prime \prime }{\mathcal{N}}^{†}+\mathcal{D}{\rho }^{\prime \prime }{\mathcal{D}}^{†},$$

$$p=1-\exp \left(-{\Gamma }_{21}{\tau }_{\text{pulse}}\right),$$

$$q=1-\exp \left(-{\Gamma }_{23}{\tau }_{\text{pulse}}\right).$$

$${\rho }_n=\mathcal{N}{\mathcal{U}}_{\mathcal{Z}}{\mathcal{U}}_{\mathcal{R}}{\rho }_{n-1}{{\mathcal{U}}_{\mathcal{R}}}^{†}{{\mathcal{U}}_{\mathcal{Z}}}^{†}{\mathcal{N}}^{†}+\mathcal{D}{\mathcal{U}}_{\mathcal{Z}}{\mathcal{U}}_{\mathcal{R}}{\rho }_{n-1}{{\mathcal{U}}_{\mathcal{R}}}^{†}{{\mathcal{U}}_{\mathcal{Z}}}^{†}{\mathcal{D}}^{†}$$

$$\mathcal{R}\to {\mathcal{R}}^{\text{1st}}=\frac{\mathrm{i}}{2}{\theta }^{\text{1st}}\left(\cos (\delta {\varphi }_{1,4}){\sigma }_x^{12}+\sin (\delta {\varphi }_{1,4}){\sigma }_y^{12}\right).$$

$$P_{\text{P}}=\frac{1}{0.0587}\left(1-{\left(1-P_{\text{D}}\right)}^{\frac{1}{m}}\right).$$

$$P_{\text{P}}=P_{\text{P, max}}\cdot {\sin }^2(\sqrt{P_{\text{light}}}/\omega ),$$

$$|\psi ⟩=\frac{1}{\sqrt{2}}\left(|1⟩+|3⟩\right)\equiv \frac{1}{\sqrt{2}}\left(|\text{4S1/2}⟩+|\text{3D5/2}⟩\right)$$

$$C=\sqrt{{\text{Tr}}^2({\sigma }_x^{\text{SD}}\rho )+{\text{Tr}}^2({\sigma }_y^{\text{SD}}\rho )},$$

$$\Phi =\arg \left[\text{Tr}({\sigma }_x^{\text{SD}}\rho )+i\text{Tr}({\sigma }_y^{\text{SD}}\rho )\right].$$

Full text

1 Introduction

Trapped ions are a promising system for the implementation of a scalable quantum computer [1, 2, 3, 4, 5]. Two-qubit entangling gate operations have been demonstrated [6, 7, 8, 9] and combined with single-qubit gates to build an elementary quantum processor [10, 11, 12]. The entangling gate operations in these experiments rely on spectroscopically-resolved motional sidebands of the ion crystals, a requirement that limits the duration of a gate operation to more than the period of motion of the ions in the trap (typically a few $\mathrm{\SIUnitSymbolMicro s}$ or more). Overcoming this limitation would advance the development of a scalable quantum computer as it would allow one to increase the number of gate operations (computational steps) that can be completed within the coherence time of the ion-qubits.

Two-qubit entangling gate operations in less than one trap period have been proposed by García-Ripoll, Zoller and Cirac in 2003 [13] using counter-propagating laser pulses. Several groups are working on its realization [14, 15] but so far only single-qubit gate operations [16, 17] and single-ion spin-motion entanglement [18] have been reported on time scales shorter than the ion oscillation period. Recently, creation of two-qubit entanglement by a train of ultrafast laser pulses within a few microseconds has been demonstrated in the ground-states of a pair of Yb+ ions [19].

Our goal, beyond the scope of this work, is to implement an ultrafast two-qubit phase gate operation [13, 25] using resonant, counter-propagating laser pulses and to complete it in less than one trap period. The scheme uses pairs of $\pi$-pulses for applying ion-state dependent momentum kicks to a two-ion crystal.

Figure 1 shows a level scheme of 40Ca+ with the levels relevant to the gate operation. We encode each qubit in two Zeeman substates of the 4S1/2 ($=\left|\downarrow\right\rangle$) and 3D5/2 ($=\left|\uparrow\right\rangle$) states of a 40Ca+ ion [10]. The laser pulses resonantly excite the 4S1/2 $\leftrightarrow$ 4P3/2 transition [26]. Depending on the qubits’ state, each ion will either not interact with the pulses at all (qubit state $\left|\uparrow\right\rangle$) or absorb one photon from the first pulse of each pair and emit a photon into the second pulse (qubit state $\left|\downarrow\right\rangle$), gaining a momentum of $2\hbar\overrightarrow{k}$ in the direction of the first pulse, where $\overrightarrow{k}$ is the photons’ wave vector. By subjecting the ion crystal to a sequence of momentum kicks and times of free evolution of the crystal in the trap potential, the ions are forced to follow different, state-dependent trajectories through phase-space. The area enclosed by these trajectories corresponds to the phase the state acquires during the pulse sequence [7]. When the relative phase between the state-pairs $\left|00\right\rangle$, $\left|11\right\rangle$ and $\left|01\right\rangle$, $\left|10\right\rangle$ is $\pi/2$ and the pulse sequence returns both the center-of-mass and the breathing mode of motion [27] of the ion crystal to the initial state, the operation will be the desired geometric phase gate [13]. Both conditions can be met by carefully choosing the duration of the times of free evolution in the pulse sequence. In order to complete such a phase gate within one trap period, the rate $f_{\pi}$ at which pairs of $\pi$-pulses are applied to the ions must be much larger than the trap frequency $\nu$ [26]. In general, the larger $f_{\pi}$, the faster the gate operation can be completed.

In order for the laser system to be suitable to implement such a fast gate operation, it should satisfy four requirements: (1) The system needs to provide pulsed laser light with a repetition rate much larger than the trap frequency of $\sim 1\text{\,}\mathrm{MHz}$, in order to provide us with fine-grained control over the timing of pulse sequences. (2) The pulse length $\delta t$ has to be much shorter than the 4P3/2 state’s lifetime of $6.9\text{\,}\mathrm{ns}$ [22] to avoid or at least minimize spontaneous decay. (3) The center frequency has to be resonant with the 4S1/2 $\leftrightarrow$ 4P3/2 transition at $393.366\text{\,}\mathrm{nm}$ and (4) the laser needs to have an intensity such that for every laser pulse $\Omega\,\delta t=\pi$, where $\Omega$ is the corresponding Rabi frequency. Alternatively to (3), non-resonant pulses can be used to apply state-dependent momentum kicks [17]. In order to realize this alternative, we want to be able to tune the laser in between the two fine structure components 4S1/2 $\leftrightarrow$ 4P1/2 and 4S1/2 $\leftrightarrow$ 4P3/2 such that Stark shifts cancel, while limiting overlap of the laser’s optical spectrum with either of the two transition frequencies.

In this paper we describe and characterize a laser system which was designed to generate the light pulses for ultrafast quantum gate operations. Our laser system satisfies requirements (1) to (3), and we will show below, that the intensity of the laser is sufficiently high for generating pulses that flip the state of the ion with $96.4(19)\text{\,}\%$ probability (requirement 4). In order to do that, we present and characterize three methods to extract information on the Rabi frequency and compare them in terms of applicability and prerequisites. The methods allow us to measure the rotation angle per pulse $\theta=\Omega\,\delta t$ and are more accurate than simply deducing the Rabi frequency from a measurement of the laser’s optical intensity at the point of the ion [28, 27].

The paper is structured as follows: Section 2 describes our laser setup in detail. In section 3 we introduce two methods that we developed to gain information on $\theta$ by carrying out measurements on a trapped 40Ca+ ion, and we compare them to measuring $\theta$ by injecting single pulses between the two Ramsey zones of a Ramsey experiment [16].

2 Laser System

Picosecond laser systems with repetition rates on the order of $\mathrm{GHz}$ and a center wavelength of $393.366\text{\,}\mathrm{nm}$ can be constructed by frequency-quadrupling the light generated by commercial lasers operating in the L-band of optical fiber communication ($1565\text{\,}\mathrm{nm}1625\text{\,}\mathrm{nm}$). An overview of the optical setup is provided in figure 2.

The system is seeded by an optical frequency comb, which is described in more detail in subsection 2.1. Filter cavities serve to multiply the laser’s repetition rate. Subsequently, the desired laser wavelength range is selected by spectral filters and the laser output is amplified (subsection 2.2). Fast and slow pulse picking elements enable the selection of arbitrary pulse sequences (subsection 2.3). Finally, the laser frequency is quadrupled by two single-pass second-harmonic generation stages (subsection 2.4). Measurements of the basic pulse characteristics are presented in subsection 2.5.

2.1 Seed laser

The seed laser is a fiber-based optical frequency comb (Menlo Systems FC1500-250-WG) with a repetition rate of $250\text{\,}\mathrm{MHz}$. The mode locked laser creates short pulses of $70\text{\,}\mathrm{fs}$ pulse width with a center wavelength of $1550\text{\,}\mathrm{nm}$ and a spectral bandwidth of $50\text{\,}\mathrm{nm}$. We lock both the carrier envelope offset and the repetition rate to a frequency reference provided by a GPS-disciplined oven-controlled crystal oscillator (Menlo Systems GPS 6-12) with a fractional frequency instability of $\sim 2\text{\times}{10}^{-13}$.

2.2 Pulse manipulation

The pulses produced by the frequency comb need to be amplified and frequency-upconverted in order to create $\pi$-pulses on the ions. Furthermore, the spectral bandwidth should be limited to about $1\text{\,}\mathrm{THz}$ ($\binrel@{=}\binrel@@{\mathop{\kern 0.0pt=}\limits^{\scriptscriptstyle\wedge}}$0.5\text{,}\mathrm{nm}$$ at $393\text{\,}\mathrm{nm}$) to avoid residual off-resonant excitation of the 4S1/2 $\leftrightarrow$ 4P1/2 transition, and the center wavelength should be resonant to the 4S1/2 $\leftrightarrow$ 4P3/2 transition. In order to have a higher resolution of the pulse timings, we additionally multiply the repetition rate by a factor of 20.

Erbium doped fiber (pre-)amplifiers (EDFAs) and semiconductor optical amplifiers (SOAs) are used to compensate insertion loss at various stages in the set-up (panels B in figure 2). After the first preamplifier, the pulse train travels through a stretcher fiber which adds dispersion and stretches the pulses from $70\text{\,}\mathrm{fs}$ to $\sim 50\text{\,}\mathrm{ps}$ for chirped pulse amplification [29]. The pulse train then travels through a spectral filter which selects the new center wavelength of $4\cdot$393\text{,}\mathrm{nm}$=$1572\text{,}\mathrm{nm}$$ and reduces the spectral bandwidth to $8\text{\,}\mathrm{nm}$. Two filter cavities with a free spectral range of $5\text{\,}\mathrm{GHz}$ each then increase the repetition rate from $250\text{\,}\mathrm{MHz}$ to $5\text{\,}\mathrm{GHz}$ by transmitting only the light’s spectral modes that are $5\text{\,}\mathrm{GHz}$ apart and suppressing all others. The second cavity’s purpose is to increase the extinction ratio and therefore equalize optical intensity of the output pulses. The subsequent fast pulse picker (bandwidth of $7\text{\,}\mathrm{GHz}$) will be described in more detail in the next subsection. Furthermore, we use a second spectral filter to compensate amplifier-induced frequency shifts [30] of up to $4\text{\,}\mathrm{nm}$ and further limit the bandwidth to $6.4\text{\,}\mathrm{nm}$. Next, a high power EDFA amplifies the pulse train from $15\text{\,}\mathrm{mW}$ to a maximum average power of $2.8\text{\,}\mathrm{W}$. A free-space third order dispersion compressor reduces the pulse width to $680\text{\,}\mathrm{fs}$ (time bandwidth product 0.53) which is close to the transform limited pulse width of $560\text{\,}\mathrm{fs}$ for a Gaussian-shaped pulse of the given bandwidth.

2.3 Pulse picking

In order to select the pulse sequences described in section 1 out of the $5\text{\,}\mathrm{GHz}$ pulse train, we need an optical element that is able to select pulses at this rate and to withstand up to $2.8\text{\,}\mathrm{W}$ of laser power after the high power EDFA. To satisfy both requirements, we chose a twofold approach using a fast switching element before the high power EDFA (where the average laser power can be limited to $10\text{\,}\mathrm{mW}$) and a slow element after the amplifier to create the desired pulse sequences (see panel C in figure 2).

The fast element is a pulse picker (custom-made ModBox by Photline) which contains a Mach-Zehnder interferometer with an electro-optic modulator of $7\text{\,}\mathrm{GHz}$ bandwidth. Since its maximum optical input power is on the order of a few $\mathrm{mW}$, we install it before the high power EDFA where the light intensity is sufficiently low. Considering the amplifier’s need to be seeded continuously with a maximally allowed dark time on the order of $20\text{\,}\mathrm{ns}$, we need an additional switching element after the amplifier with a high damage threshold and a switching time of less than $20\text{\,}\mathrm{ns}$. For this we use a Pockels cell (Leysop BBO-3-25-AR790) with a driver (custom-made by Bergmann Meßgeräte Entwicklung) that enables switching of the cell with a rise/fall time of $7\text{\,}\mathrm{ns}$ at a maximum repetition rate of $10\text{\,}\mathrm{MHz}$ and a measured optical extinction ratio of $30\text{\,}\mathrm{dB}$. Both the pulse picker and the Pockels cell are controlled by an arbitrary waveform generator (Tektronix AWG 70002A) with a sample rate of $25\text{\,}\mathrm{G}\mathrm{S}\mathrm{/}\mathrm{s}$ which is synchronized with the seed laser by a $250\text{\,}\mathrm{MHz}$ RF signal derived directly from the laser’s pulse train.

Figure 3 shows three oscilloscope traces recording an arbitrary pulse pattern (including a dark time) which was detected by a photodiode. To generate the three traces, either only the pulse picker, only the Pockels cell or both were used. It shows that the pulse picker can switch individual pulses in arbitrary sequences.

2.4 Frequency up conversion

The fundamental $1572\text{\,}\mathrm{nm}$ light is converted to $393\text{\,}\mathrm{nm}$ by frequency doubling the light twice in two separate nonlinear crystals (panel D in figure 2). The first one is an MgO-doped PPLN (periodically-poled lithium niobate) crystal (Covesion MSHG1550-0.5-5), the second crystal is a PPKTP (periodically-poled potassium titanyl phosphate) crystal (custom-made by Raicol Crystals Ltd.). Since the doubling efficiency of the crystals scales with the square of the peak power, the measured extinction ratio of the Pockels cell/PPKTP crystal system is $56\text{\,}\mathrm{dB}$, i.e. almost the square of the Pockels cell extinction ratio. At the current maximum average $1572\text{\,}\mathrm{nm}$ light power of about $2.8\text{\,}\mathrm{W}$ and at a repetition rate of $5\text{\,}\mathrm{GHz}$ ($1.25\text{\,}\mathrm{GHz}$, and therefore four times higher peak power) we can produce $0.77\text{\,}\mathrm{W}$ ($1.18\text{\,}\mathrm{W}$) of $786\text{\,}\mathrm{nm}$ light, which corresponds to a conversion efficiency of $28\text{\,}\%$ ($42\text{\,}\%$), and $25\text{\,}\mathrm{mW}$ ($110\text{\,}\mathrm{mW}$) of $393\text{\,}\mathrm{nm}$ light, which corresponds to a conversion efficiency of the second frequency doubling step of $3.2\text{\,}\%$ ($9.3\text{\,}\%$). The maximum total conversion efficiency (from $1572\text{\,}\mathrm{nm}$ to $393\text{\,}\mathrm{nm}$) is $0.9\text{\,}\%$ ($3.9\text{\,}\%$). Measurements of the $393\text{\,}\mathrm{nm}$ light power and conversion efficiencies for four different repetition rates are presented in figure 4. The inset in panel (a) of that figure is a log-log graph of the same data. The graph, together with the line $y\propto x^{4}$ visualizes the expected fourth-power dependency of the $393\text{\,}\mathrm{nm}$ light power on the $1572\text{\,}\mathrm{nm}$ light power.

After each up-conversion of the laser pulses, the remaining fundamental light is split off by a dichroic mirror and in case of the 2nd SHG stage further suppressed by a shortpass filter (Thorlabs FESH0700). Next, the light is coupled into a polarization-maintaining single-mode fiber and sent to the ion trap. After the $5\text{\,}\mathrm{m}$ long fiber, we measure a pulse width of $3\text{\,}\mathrm{ps}$. Finally, a collimator equipped with a $250\text{\,}\mathrm{mm}$ lens focuses the laser beam to a waist measured to be $w_{0}=$11.8(3)\text{,}\mathrm{\SIUnitSymbolMicro m}$$ and directs the beam such that the ion is located in the beam waist.

Unless stated otherwise, all following measurements and experiments presented in this paper were conducted at a high power EDFA output power of $\approx 600\text{\,}\mathrm{mW}$. During times when the Pockels cell was off and the pulses directed into a beam dump, the high power EDFA was seeded with pulses at $1.25\text{\,}\mathrm{GHz}$ repetition rate.

2.5 Pulse switching characteristics

Our pulse picking scheme described in subsection 2.3 necessitates blocking the pulses with the fast pulse picker during the rise and fall time of the Pockels cell for typically $12\text{\,}\mathrm{ns}$. This results in an equally long dark time of the amplifiers after the pulse picker. We examined the characteristics of the pulses during the first nanoseconds after such dark time by measuring pulse areas and relative phase of the pulses.

We characterize the phase shifts of the pulses by interfering them in a Michelson interferometer (see figure 5(a)). At the beam splitter the input pulse train is split into two copies, one of which is temporally delayed by one pulse period $\tau_{\text{pulse}}$ with respect to the other before being recombined. In this way, every pulse $i$ interferes with its successive pulse $i+1$ and the total phase difference $\Delta\Phi_{i}$ of the two pulses at the output of the interferometer is a function of the phase difference $\Delta\phi_{i}$ of pulses $i$ and $i+1$, and the path length difference $\Delta x=2x_{2}-2x_{1}\approx c\cdot\tau_{\text{pulse}}$:

[TABLE]

with $x_{1}$ ($x_{2}$) the length of interferometer arm 1 (2), $c$ the speed of light and $k$ the length of the wave vector. The interference pulses are detected on a fast photodiode and their pulse areas, which are functions of $\Delta\Phi_{i}$, extracted by integrating the photodiode’s signal over the pulse length.

We can tune $\Delta x$ by manually moving one of the retro-reflecting mirrors which is fixed on a manual translation stage, but we can not deterministically change $\Delta x$ on a sub-wavelength scale which would have allowed to keep track of the changes in $\Delta\Phi_{i}$ due to changes in $\Delta x$. Nevertheless, we found its fluctuations to be much smaller than the light’s wavelength and consider $\Delta x$ to be constant on time scales of our measurements of about $100\text{\,}\mathrm{ms}$. For each measurement we therefore randomly choose $\Delta x$ and repeat the measurement for different $\Delta x$. Accordingly, the interference pulse areas of each measurement are both random but also correlated in size since $\Delta x$ is random but equal for all pulses in a given measurement, and if the pulse areas of two interference pulses $i$ and $j$ are different, there is also a change $\delta\phi_{i,j}=\Delta\phi_{i}-\Delta\phi_{j}$ in their phase differences.

In order to determine $\delta\phi_{i,j}$ we repeatedly send a five-pulse pulse train into the interferometer at a rate of $1\text{\,}\mathrm{kHz}$ and average over 100 consecutive trains. We measure the interference pulse areas for different and random $\Delta x$ by moving the translation stage back and forth on the order of $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ between two 100 pulse trains-long measurements. Figure 5(b) shows two example interference signals, taken at $2.5\text{\,}\mathrm{GHz}$. For every measurement at different $\Delta x$ we plot the pulse areas of two interference pulses against each other and present the data in panels (c) through (f) in figure 5. Every abscissa is the area of interference pulse 4; the ordinates are the areas of interference pulses 1 through 3, respectively.

The data points fall on a straight line at a [math] angle, if $\delta\phi_{i,j}$ between the two interference pulses is zero or an integer multiple of $2\pi$, and at a [math] angle if $\delta\phi_{i,j}=(2n+1)\pi,n\in\mathbb{Z}$. In general, for any other value of $\delta\phi_{i,j}$ the data points are located on an ellipse. This allows us to extract $\delta\phi_{i,j}$ by fitting an ellipse to the data.

We see a dependence of $\delta\phi_{i,j}$ on the length of the pulse period $\tau_{\text{pulse}}=1/f_{\text{rep}}$ if there was a long (here: $12\text{\,}\mathrm{ns}$) dark time before the first pulse: For the maximum repetition rate of $f_{\text{rep}}=$5\text{,}\mathrm{GHz} we find that between the first (second) and fourth interference peak (pulses 1&2 (2&3) and 4&5, respectively) $\delta\phi_{1,4}=$0.74\text{\,}\pi ($\delta\phi_{2,4}=$0.12\text{,}\pi). For later interference peaks ($i\geq 3$) $\delta\phi_{i,4}$ is vanishingly small. As a trend we observe that for larger $\tau_{\text{pulse}}$, $\delta\phi_{i,j}$ becomes smaller and is vanishing for $\tau_{\text{pulse}}\geq$800\text{\,}\mathrm{ps}$=1/$1.25\text{\,}\mathrm{GHz}.

In addition to the phase shift, we observe repetition rate-dependent deviations of the pulse intensity of the first pulses after the dark time. Figure 6 shows the pulse areas of the pulses during the first $2\text{\,}\mathrm{ns}$ after the dark time. In the case of $5\text{\,}\mathrm{GHz}$ repetition rate we observe that all pulses from the second onward are weaker by a factor of about 3 with respect to the first pulse. At $2.5\text{\,}\mathrm{GHz}$ repetition rate, the intensity of the second pulse decreases by about $10\text{\,}\%$ with respect to the first. At the other repetition rates we do not observe this effect. For all repetition rates we observe an increase in pulse intensity of about $10\text{\,}\%$ for the shown time of $2\text{\,}\mathrm{ns}$ of pulses after the possible initial decrease.

Both the phase shift and the change in pulse area will be problematic for experiments using coherent manipulations of the qubit. Since for our planned two-ion phase gate every pulse needs to act as a $\pi$-pulse, the pulse area needs to be constant, but due to the way we plan to create the counter-propagating pulse pairs, the phase shift does not: We intent to generate the two pulses of every required $\pi$-pulse pair by splitting a larger-area pulse. Therefore, every pulse pair will consist of identical copies and the two pulses will add up to a $2\pi$-rotation of the Bloch vector around the same axis regardless of phase difference with respect to other pulse pairs. For this reason only the intensity changes should pose an issue for our phase gate, the phase shifts should not.

We believe that both effects are caused by a semiconductor optical amplifier (SOA2 in figure 2) which serves as a preamplifier to the high power EDFA and are due to the finite carrier lifetime of the SOA of $500\text{\,}\mathrm{ps}$. This causes the dynamical behavior of the SOA to depend on the input signal of the past $\sim 500\text{\,}\mathrm{ps}$ [31] and is known as the “pattern effect”[32]. We therefore plan to replace the SOA with another fiber-based preamplifier.

3 Methods to measure the Rabi frequency of a pulsed laser-ion system

We have employed three different techniques to measure the rotation angle per pulse $\theta=\Omega\,\delta t$. The model for the ion-laser interaction used for simulating our experiments is presented in subsection 3.1. Subsection 3.2 presents measurements using the three techniques. The first approach uses many pulses – $\mathcal{O}(${10}^{3}$)$ – to pump the ion into the metastable 3D5/2 state. Measuring the population in that state and fitting the model to the data allows $\theta$ to be extracted. Next, we use the same principle but in the regime of single pulses, allowing pulse dynamics and characteristics to be extracted. Finally, the third method is based on preparing the ion in a coherent superposition of the two qubit states. We then use single laser pulses to reduce and restore coherence between the qubit states by transferring population of one of the two states back and forth to a third state.

3.1 Pulse model for simulation and fits

We calculate the interaction of a 40Ca+ ion with a pulsed laser field resonant to the ion’s 4S1/2 $\leftrightarrow$ 4P3/2 transition (see figure 1). We consider spontaneous decay of the 4P3/2 state into the 4S1/2 state with decay rate $\Gamma_{\text{PS}}$ and into the 3D5/2 state with decay rate $\Gamma_{\text{PD}}$. For the experiments that we describe, the 3D3/2 state does not need to be considered as any population in this state is pumped back to 4S1/2 via 4P1/2 on a timescale of $10\text{\,}\mathrm{ns}$ with an $866\text{\,}\mathrm{nm}$ laser. Therefore, any decay to 3D3/2 effectively becomes a decay to 4S1/2 and the modified decay rates are

[TABLE]

[TABLE]

where \Gamma=\frac{1}{\tau}=\frac{1}{6.9240(19)\text{,}\mathrm{ns}} [22] and $p_{5/2}=$0.0587(2)$$ the probability of 4P3/2 to decay to 3D5/2 [20].

Consequently, we assume a three-level system and identify the 4S1/2 state with $\left|1\right\rangle$, the 4P3/2 state with $\left|2\right\rangle$ and the 3D5/2 state with $\left|3\right\rangle$, as indicated in figure 1. Furthermore, we assume infinitesimally short pulses and neglect any decay of $\left|3\right\rangle$, since the 3D5/2 state’s lifetime of $1.2\text{\,}\mathrm{s}$ [24] is much longer than the cycle time of our experiments of about $10\text{\,}\mathrm{ms}$. The system’s excited state $\left|2\right\rangle$ decays spontaneously with rates $\Gamma_{21}\equiv\Gamma_{\text{PS}}$ and $\Gamma_{23}\equiv\Gamma_{\text{PD}}$ into state $\left|1\right\rangle$ and $\left|3\right\rangle$, respectively.

To model the time evolution of the quantum state over one pulse period of duration $\tau_{\text{pulse}}=1/f_{\text{rep}}$ we start with a rotation of the Bloch vector of the $\left|1\right\rangle$ - $\left|2\right\rangle$ subsystem around the x-axis to account for the effect of a single pulse, followed by a rotation around the z-axis to account for the detuning during the dark time between two pulses. Next, a rotation of the Bloch vector of the $\left|1\right\rangle$ - $\left|3\right\rangle$ subsystem around its z-axis takes into consideration an effect of the pulses on this transition’s frequency. Finally, we account for spontaneous decay of state $\left|2\right\rangle$ during the pulse period by applying the appropriate Kraus operators to the quantum state.

Hence, we start by applying an x-rotation operator $\mathcal{U_{R}}$ to a given density operator $\rho_{0}$:

[TABLE]

with

[TABLE]

where $\theta$ is the rotation angle.

Next, we allow for a possible detuning $\Delta$ of the laser light with respect to the 1 $\leftrightarrow$ 2 transition and a shift $\Delta^{\prime}$ of the state $\left|3\right\rangle$ by applying a z-rotation $\mathcal{U_{Z}}$ to the result. Here, the angle of rotation is proportional to $\tau_{\text{pulse}}$, $\Delta$ and $\Delta^{\prime}$, respectively:

[TABLE]

with

[TABLE]

[TABLE]

Finally, we calculate the Kraus operators $\mathcal{N}$, $\mathcal{D}$ to allow for spontaneous decay of the excited state $\left|2\right\rangle$:

[TABLE]

[TABLE]

[TABLE]

with

[TABLE]

[TABLE]

Calculating the decay of the excited state only after applying the x- and z-rotation operators is acceptable, since the pulse length of $3\text{\,}\mathrm{ps}$ is very short compared to the state’s lifetime of $6.9\text{\,}\mathrm{ns}$, and the z-rotation and decay do not influence each other.

To find the density operator $\rho_{n}$ after a train of $n$ pulses we iteratively apply these operators $n$ times

[TABLE]

and finish the calculation by letting state $\left|2\right\rangle$ decay completely. From the resulting density operator we can easily calculate experimentally accessible observables such as populations $\Tr(\left|1\right\rangle\left\langle 1\right|\rho_{n})$, $\Tr(\left|3\right\rangle\left\langle 3\right|\rho_{n})$ and coherences $\Tr(\left|1\right\rangle\left\langle 3\right|\rho_{n})$, $\Tr(\left|3\right\rangle\left\langle 1\right|\rho_{n})$.

To account for our observations described in section 2.5 we allow the first pulse in a pulse train to have a different (usually higher) peak power and therefore allow for its rotation angle $\theta^{\text{1st}}$ to be different. Additionally, we allow the first pulse to turn the Bloch vector of the subsystem around an axis in the equatorial plane rotated by $\delta\phi_{1,4}$ from the x-axis to account for its phase offset relative to the other pulses. We therefore replace $\mathcal{R}$ with $\mathcal{R}^{\text{1st}}$ in equation 4:

[TABLE]

3.2 Experimental results

Experiments are conducted in the same linear Paul trap as described in [33]. Its trap axis is aligned with the quantization axis defined by a bias magnetic field. Circularly polarized $393\text{\,}\mathrm{nm}$ laser pulses that are sent through the holes in the trap’s endcap electrodes therefore couple only pairs of Zeeman states of the ground and excited state.

3.2.1 Pumping into a dark state with many pulses.

We send pulse trains of between 500 and 5000 pulses to the Doppler-cooled and optically pumped ion and collect fluorescence photons on the 4S1/2 $\leftrightarrow$ 4P1/2 transition in order to determine whether the ion has decayed into the dark 3D5/2 state. We repeat the measurement 100 times to statistically determine the probability that the ion has decayed into the 3D5/2 state. Due to the large number of pulses, the pulse switching effects described in section 2.5 and which are affecting only the first one or two pulses, can be disregarded. For the same reason the experiment does not necessitate the use of a pulse picker that is faster than the fundamental repetition rate.

We measure the 3D5/2 state population $P_{\text{D}}$ as a function of the laser detuning for different pulse train lengths. Figure 7 shows data sets for repetition rates ranging from $1.25\text{\,}\mathrm{GHz}5\text{\,}\mathrm{GHz}$. Each data set consists of four data sub-sets which differ only in the number of pulses. In order to extract $\theta$ we fit the simulation in equation (13) simultaneously to each of the four sub-sets such that we get a single value for $\theta$. The only other fit parameter is a detuning offset which is eliminated in the figure. For the repetition rate of $5\text{\,}\mathrm{GHz}$ ($2.5\text{\,}\mathrm{GHz}$, $5\text{/}3\text{\,}\mathrm{GHz}$, $1.25\text{\,}\mathrm{GHz}$) we obtain $\theta=$0.227(12)\text{,}\pi$$ ($0.323(5)\text{\,}\pi$, $0.363(5)\text{\,}\pi$, $0.345(5)\text{\,}\pi$).

3.2.2 Pumping into a dark state with single pulses.

In order to implement our phase gate scheme we need to ensure that every single pulse is a $\pi$-pulse and it does not suffice to characterize an ensemble of hundreds of pulses. By instead sending only $n\leq 12$ pulses at a time to the ion we can gain crucial insights into single pulse dynamics and characteristics. In order to prevent the problem of having to measure very small 3D5/2 state populations, we amplify the signal by repeating the pulse train $m=20$ times as shown in the experimental sequence in figure 8. Between repetitions, a waiting time $t_{\text{w}}=$20\text{,}\mathrm{\SIUnitSymbolMicro s} much larger than the 4P*3/2* state’s lifetime $\tau=$6.924\text{\,}\mathrm{ns} ensures that any population in that state has decayed and we accumulate population in the 3D5/2 state [20].

Figure 9 shows the probability for being in the 3D5/2 state, inferred from our experimental data and fit/simulation side-by-side. We vary the detuning $\Delta$ and the number of pulses for different repetition rates. Free fit parameters are $\theta$, the phase offset of the first pulse $\delta\phi_{1,4}$ and a detuning offset. Since we know from earlier measurements described in section 2.5 that the first pulse in the case of $5\text{\,}\mathrm{GHz}$ repetition rate has a different area than the other pulses, we allow an additional fit parameter in that case: the rotation angle of the first pulse $\theta^{\text{1st}}$. All fit values are within three standard deviations of those acquired previously with long pulse trains (section 3.2.1), as well as those acquired with the Michelson interferometer (section 2.5). Contrary to measuring $\delta\phi_{i,j}$ with the interferometer, this measurement is able to also determine the sign of $\delta\phi_{i,j}$ and therefore to distinguish between phases $\delta\phi_{i,j}=\Phi$ and $\delta\phi_{i,j}=-\Phi$ (the later being the same as $\delta\phi_{i,j}=2\pi-\Phi$). We therefore assume that $\delta\phi_{1,4}$ at $5\text{\,}\mathrm{GHz}$ is about $1.25\text{\,}\pi$ and not $0.75\text{\,}\pi$ as the ellipse fit suggested. Furthermore, $\theta^{\text{1st}}$ is found to be a factor of about $\sqrt{3}$ larger than $\theta$ for the $5\text{\,}\mathrm{GHz}$ repetition rate, which is the expected amount for a three times larger pulse area (also compare with section 2.5).

3.2.3 Single pulse with area $\pi$.

In order to check if a single pulse can act as a $\pi$-pulse we repeat the previous experiment with only a single pulse while varying the $393\text{\,}\mathrm{nm}$ light power ($n=1$, $m=15$). From the experimentally determined 3D5/2 state probability $P_{\text{D}}$ after the 15 repetitions of a single pulse we calculate the 4P3/2 state probability $P_{\text{P}}$ after only one single pulse using

[TABLE]

Due to measurement fluctuations, this can sometimes lead to unphysical values of $P_{\text{P}}>1$ if we measure a $P_{\text{D}}$ that happens to be larger than the maximum expectation value of $P_{\text{D, max}}=1-(1-0.0587)^{15}\approx 0.60$.

Since the excitation probability is a function of the sine squared of the Rabi frequency $\Omega$, we plot $P_{\text{P}}$ versus the square root of the $393\text{\,}\mathrm{nm}$ light power, which is proportional to $\Omega$ (and $\theta$) in figure 10. The data points should therefore follow the curve

[TABLE]

where $\omega$ is a proportionality factor of dimension $\sqrt{\mathrm{W}}$ and $P_{\text{P, max}}$ is the maximum probability of a single pulse exciting the ion to the 4P3/2 state. A fit of this curve to our data yields $P_{\text{P, max}}=$96.4(19)\text{,}%$$, showing how close we are to achieving single pulse $\pi$-pulses.

Please note that the maximum $393\text{\,}\mathrm{nm}$ light power plotted in figure 4 (e) ($110\text{\,}\mathrm{mW}$) and figure 10 ($($7.6\text{,}\sqrt{\mathrm{m}\mathrm{W}}$)^{2}\approx$58\text{,}\mathrm{mW}$$) was measured at the same fundamental light power. The discrepancy is due to coupling losses in the fiber guiding the $393\text{\,}\mathrm{nm}$ light to the ion.

3.2.4 Ramsey contrast decay and revival.

In experiments using coherent manipulations of the qubit it is important to know the effect of the manipulation on the relative phase of the two qubit states. The experiments of the previous three subsections used only one of the two qubit states and the relative phase appeared only as an unobservable global phase. Here however, the second qubit state serves as a phase reference, giving us access to the desired information.

We let single pulses interact with the ion during the waiting time of a Ramsey experiment and monitor the coherence at the end of the Ramsey sequence in the following way: We start by bringing the ion into a coherent superposition of two states by a $\frac{\pi}{2}$-pulse on the 4S1/2 $\leftrightarrow$ 3D5/2 transition with a $729\text{\,}\mathrm{nm}$ laser as illustrated in figure 11:

[TABLE]

We then use single pulses of our $393\text{\,}\mathrm{nm}$ laser system to coherently drive the 4S1/2 $\leftrightarrow$ 4P3/2 transition. If there is any population remaining in the 4P3/2 state after the pulses, this part of the population will undergo spontaneous decay and thus destroy the coherence. We finally analyze how much coherence of the original state remains and also extract the phase between the two states.

In mathematical terms, let $\rho$ be the density matrix describing our system after the $393\text{\,}\mathrm{nm}$ pulses and $\sigma_{x}^{\text{SD}}$, $\sigma_{y}^{\text{SD}}$ the Pauli matrices acting only on the states \left|\text{4S{}{1/2}}\right\rangle and \left|\text{3D{}{5/2}}\right\rangle of the three level system. The expected contrast $C$ and phase $\Phi$ can then be written as

[TABLE]

[TABLE]

Measuring P_{\text{D{}_{5/2}}} as a function of the phase $\phi^{\prime}$ of the second $\frac{\pi}{2}$-pulse allows us to fit a sinusoidal curve to the data and extract contrast and phase from the fit. If the contrast is equal to one the ion was still in a fully coherent superposition of the 4S1/2 and 3D5/2 states and there was no spontaneous decay of the 4P3/2 state. If the contrast is zero all the initial 4S1/2 state population had been transferred to the 4P3/2 state and the inevitable spontaneous decay had destroyed all coherence. The phase on the other hand is expected to undergo jumps of value $\pi$ each time the ion’s state population is transferred to the 4P3/2 state and back again to the 4S1/2 state by the pulse train. This is due to the fact that a two-level system picks up a sign after a $2\pi$ rotation $\left|0\right\rangle\binrel@{\longrightarrow}\binrel@@{\mathop{\kern 0.0pt\longrightarrow}\limits^{2\pi}}-\left|0\right\rangle$.

We repeat the measurement as a function of the number $n$ of $393\text{\,}\mathrm{nm}$ laser pulses. Figure 12 (a) shows a data set where we plot the remaining coherence and phase against the number of pulses. For this set it took about 15 optical pulses at $5\text{\,}\mathrm{GHz}$ repetition rate to complete a $\pi$-pulse, which corresponds to $\theta\approx$0.067\text{,}\pi$$. Nevertheless, one can already see that we can drive the 4S1/2 $\leftrightarrow$ 4P3/2 transition coherently and that each time we return to the 4S1/2 state, the data is consistent with the observation of phase jumps by $\pi$.

In figure 12 (b) we plot the same quantities of another data set taken at the same laser power as the measurements presented in the other subsections. We can observe the same structure as in the first data set but we need only about 2.5 optical pulses to complete one $\pi$-pulse on the transition. A fit of the simulation of the ion-light interaction to the experimental data allows us to extract $\theta=$0.389(5)\text{,}\pi$$.

3.3 Discussion

As shown above, we were able to deduce the rotation angle per pulse $\theta$ by letting an ion interact with hundreds of consecutive pulses and afterwards measuring the probability of finding the ion in the 3D5/2 state. Using only single pulses allowed us to measure not only $\theta$ again, but also the phase offset of the first pulse after a dark time, i.e. a pause in the pulse train. By injecting single pulses into the waiting time between the two $\pi/2$-pulses of a Ramsey experiment, we were able to gain information on how the pulses influence the phase of the \left|\text{4S{}_{1/2}}\right\rangle state as well as measure $\theta$.

We now have three reliable ways to measure $\theta$ and the phase offset of the first pulses after a dark time. The methods produce the same results within their respective error margins but each have their advantages and disadvantages as described below. Table 1 summarizes our results.

Using many pulses to pump into a dark state allows us to measure the laser-ion system’s rotation angle per pulse $\theta$ and requires neither the ability to pick single pulses nor a laser resonant to the 4S1/2 $\leftrightarrow$ 3D5/2, $729\text{\,}\mathrm{nm}$, $1\text{\,}\mathrm{Hz}$ linewidth, quadrupole transition. We therefore only need additional lasers that are readily available and only need to be stabilized to a linewidth of $\lesssim$1\text{,}\mathrm{MHz}$$, which can be achieved easily. The only parameters that need to be known to fit the data are the number of pulses $n$ and the well-known decay rates $\Gamma_{\text{PS}}$ and $\Gamma_{\text{PD}}$. From the fit we can extract $\theta$ and also the detuning between the transition frequency and a laser mode. This allows us to tune the laser into resonance with the transition.

The method works well for any laser repetition rate $f_{\text{rep}}$ as long as it is much larger than the 4S1/2 $\leftrightarrow$ 4P3/2 transitions linewidth. Also, the rotation angle should satisfy $\theta\leq\pi$, since rotation angles $\theta^{\prime}=\pi+\delta$ can not be discerned from $\theta=\pi-\delta$.

By using only single pulses to pump into a dark state (method 2) we can observe single pulse dynamics and gain additional information about the pulse characteristics, such as changing phase shifts and pulse powers. With these measurements we are able to reproduce results obtained both with long pulse trains and with the Michelson interferometer.

The Ramsey contrast technique (method 3) is experimentally more challenging than the previous procedures. We need to be able to create a coherent superposition of the 4S1/2 and 3D5/2 states which requires a few-$\mathrm{kHz}$ linewidth laser. It is also necessary that any AC Stark shift has been canceled, otherwise it is not possible to transfer all 4S1/2 state population to the 4P3/2 state as required. The advantage is that we can immediately see if we have reached our goal and if one pulse suffices to do a $\pi$-pulse on the ion. By fitting an empirical model to the data we can also learn how many pulses are currently needed to do a $\pi$-pulse, if the power is not sufficient, yet. Furthermore, the method allows one to track the phase of the population in the 4S1/2 state because the 3D5/2 state population serves as a phase reference.

A fourth method [34] exists that we have not pursued systematically. It is based on measuring the fluorescence rate of the ion while it is interacting with a continuous pulse train. While it is experimentally easy to conduct, it requires precise knowledge of the detector’s total efficiency.

As stated before, almost all experiments presented in this work were conducted at an average output power of the high power EDFA of $\approx 600\text{\,}\mathrm{mW}$ which corresponds to $\approx 10\text{\,}\mathrm{mW}$ $393\text{\,}\mathrm{nm}$ light power. Nevertheless, we are able to increase the $393\text{\,}\mathrm{nm}$ light power to $\gtrsim$100\text{,}\mathrm{mW}$$ as seen in figure 4, which suffices to create pulses that act as $\pi$-pulses with $96.4(19)\text{\,}\%$ probability.

4 Conclusion

In summary, we have designed, set up and characterized a high repetition rate laser, derived from a stabilized optical frequency comb, which is suitable for the implementation of ultrafast quantum gate operations with trapped 40Ca+ ions. We amplify the light at $1572\text{\,}\mathrm{nm}$ and shift the wavelength via cascaded SHG to $393\text{\,}\mathrm{nm}$, resonant to the 4S1/2 $\leftrightarrow$ 4P3/2 transition in 40Ca+. We have demonstrated that we can pick arbitrary pulse sequences out of the $5\text{\,}\mathrm{GHz}$ pulse train and that our laser can coherently drive the 4S1/2 $\leftrightarrow$ 4P3/2 transition in 40Ca+.

We have developed and applied three different techniques to measure the rotation angle per pulse $\theta=\Omega\,\delta t$ of a pulsed laser-ion system and shown that we can create approximate $\pi$-pulses on the 4S1/2 $\leftrightarrow$ 4P3/2 transition with only a single optical pulse: a key-requisite to implementing a resonant, ultrafast, two-qubit phase gate operation.