Universal gate-set for trapped-ion qubits using a narrow linewidth diode laser

Light is generated by an external cavity diode laser (ECDL, Toptica DL100) that is locked to a high-finesse Fabry–Perot cavity which is made of ultra-low expansion glass (ULE). The ECDL, with a 10 cm long cavity, produces 10 mW of power at 674 nm and is enclosed in an acoustically-isolated box. The laser frequency is stabilized to a cavity resonance by means of the Pound–Drever–Hall (PDH) locking technique [24]. A single mode polarization maintaining (PM) fiber guides the light from within the box to the optical table where sidebands at 20 MHz are added to the laser spectrum by phase modulation using an electro-optic modulator. Roughly 500 μW of light is sent by a PM fiber to the ULE cavity which is mounted on a separate breadboard. A polarizing beam splitter (PBS) and a $\lambda /4$ retardation plate direct the reflected light from the cavity toward a 100 MHz bandwidth photo-detector to produce the PDH beat signal. The beat signal is demodulated and fed into a fast analog controller (Toptica PDD and FALC modules) to optimize the transfer function of the servo loop. The output from the controller is fed-back into a field-effect transistor (FET) connected in parallel with the diode laser inducing current changes, which in turn steer the laser frequency. Due to the limited dynamic range of the FET, another feedback loop, implemented on a field programmable gate array, keeps the average voltage of the FET fixed by controlling the voltage on the piezoelectric transducer that controls the ECDL grating angle.

The ULE reference cavity (Advanced Thin Films Inc.) has a finesse of ${\mathcal{F}}$=10⁵ measured using ring-down spectroscopy. The cavity is placed in a vacuum chamber at a pressure of 10⁻⁷ Torr. Viton rubber pads are used to reduce acoustic vibrations. The free spectral range of the cavity longitudinal modes is 1.93 GHz and the cavity linewidth is ${{\rm{\Gamma }}}_{{\rm{c}}}=22\;{\rm{kHz}}.$ The cavity is thermally isolated by the vacuum environment and a thermal isolation material which is wrapped around the vacuum chamber. The resulting peak-to-peak amplitude of the cavity frequency drift at 674 nm is about 1 MHz over a time scale of hours and the maximal slope is about 2 kHz per minute. Since the laser has to be tuned to the atomic transition frequency with sub kHz accuracy, we scan the atomic transition every three minutes and by linear interpolation evaluate the cavity resonance frequency and adjust the laser frequency accordingly at the beginning of each experiment.

The various operations on the optical qubit set a few requirements on the laser frequency control setup which is based on acousto-optic modulators (AOM):

• (i)  
  Frequency detuning range up to the trap driving frequency of 21.75 MHz.
• (ii)  
  Driving two frequencies simultaneously for the MS-gate.
• (iii)  
  Shifting all the above output frequencies to compensate for slow ULE cavity drifts.
• (iv)  
  Keeping a controlled phase relation between the various frequencies.

To satisfy all demands the use of two AOMs is unavoidable. One is in a double-pass configuration and the other is in a single-pass configuration. While the double-pass allows for a wider range of detuning it can not be used to produce a bichromatic beam. This is because when the beams are retro-reflected in the second pass they will diffract from both frequencies that drive the AOM, leading to a third beam with frequency shift of ${f}_{1}+{f}_{2}$ which is unwanted. Therefore, the bichromatic beams are generated with the single-pass AOM. We use a separate frequency source for almost each frequency in the experiment in order to control the phase relation between the various frequencies. The alternative of using a single frequency source and varying its frequency will introduce phase noise due to time jitter in the frequency updating time. The schematic diagram of our frequency sources is presented in figure 3. The double pass AOM has three frequency sources which are all based on direct digital synthesizers (DDS, Novatech 409). An rf switch is used to choose between the carrier frequency and the micro-motion (MM) sideband frequency (used for single qubit addressing which is explained in the next section) separated by 21.75/2 MHz. The two frequencies are shifted by the base frequency electronically with a mixer where the fundamental frequencies are filtered by a bandpass filter. We use the base frequency to correct for the ULE drifts without the need to update both the carrier and MM-sideband DDSs which would introduce relative phase noise between them. The single pass AOM has four frequency sources. The red sideband (RSB) and blue sideband (BSB) generate the bichromatic beam for the two-qubit entangling gate. F1 is used to drive carrier transition with a well defined phase with respect to the RSB and BSB. The last frequency source F2 is a DDS that updates its output frequency in real-time to drive operations with no phase relation requirement such as electron shelving for state detection and ground-state cooling.

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The bandwidth of the laser-cavity locking servo was found to be limited by the response of the laser diode to current modulation. Increasing the servo gain to reduce the laser linewidth is impeded by the growth of oscillations around the frequency of inverse feedback phase. Even when the gain at this frequency is kept below unity to maintain stability, noise around this frequency is amplified. These amplified noise bands around the unity gain frequency are known also as 'servo-bumps'. In our system these servo-bumps are located at about 1 to 2 MHz and have FWHM of about 1 MHz. We observed servo-bumps in the spectral content of the laser experimentally in two different ways. The first is a direct measurement of the PDH error signal spectrum with an rf spectrum analyzer. This measurement result is plotted in figure 4(a). Secondly, we measured the ion response to an excitation of the quadrupole transition. Here, the laser frequency was scanned across the transition and the population in the ${D}_{5/2}$ state was measured. The duration of the excitation pulse was set to 100 μs, and the laser propagation direction was perpendicular to the axial direction of the trap to avoid coupling to the axial motion of the ion in the trap. The result of this measurement is presented by the blue solid line in figure 4(b). The fast oscillations around the central resonant peak are due to the 100 kHz Rabi frequency which results in many oscillations during the 100 μs pulse. The additional narrow peaks at around 0.5 MHz are due to second harmonic of a radial sideband of a transition between different Zeeman states. Both the Rabi spectroscopy and the spectrum analyzer are showing a similar feature of a 1 MHz wide response around ±1.1 MHz from resonance. This wide response is attributed to incoherent excitation of the carrier transition due to the servo-bumps in laser spectrum. This spectral feature has undesired consequences when driving vibrational sidebands transitions that are typically detuned by a 1 to 2 MHz from the carrier. In this case the servo-bump will drive incoherently the carrier transition and lead to decoherence. For example, we have observed that the fidelity of the MS entangling gate is severely compromised by this effect.

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One way to eliminate the harmful effect of the servo-bump is to improve the performance of the servo loop by increasing its bandwidth. In this case the unity-gain frequency, and hence the servo-bumps, will be shifted towards a higher frequency and away from the motional sidebands. This can be done, for example, by introducing an intra-cavity EOM into the ECDL [25]. Here, We have taken another approach in which we optically filter those servo-sidebands from the laser spectrum [23]. Fortunately, the cavity itself acts as the narrow bandpass optical filter with a 22 kHz width Lorentzian transfer-function. Hence, the light that is transmitted through the cavity possesses high spectral purity with, in principle, three orders of magnitude reduction in the noise spectral power density around the 1 MHz servo-bumps.

The price for relying on the transmitted light is the attenuation in the available light intensity. In our setup , the cavity transmission is only about 5%. While a decrease by a factor of two can be accounted for spatial mode-mismatch, most of the loss is due to intra-cavity losses. In addition, the intensity of the light that is sent to the cavity is limited to the mW range in order to prevent substantial thermal effects on the cavity mirrors (ULE substrate) that would in turn lead to thermal shifts of the cavity resonance frequency. Overall the transmitted light intensity is limited to few 10's of μW, which is well below the required intensity for ion-qubit operations, and hence needs to be amplified. To this end, we incorporate another slave diode-laser which is injection-locked to the filtered master-laser light. A slave diode rather than a tapered amplifier is chosen due to the much smaller intensity that is needed to injection lock a diode as compared with the intensity that is required to saturate a tapered amplifier.

The complete master-slave setup is shown in figure 2. The transmitted light polarization is restored to linear by another $\lambda /4$ plate and coupled to a PM-fiber. Two optical isolators are placed at the input and output of the cavity to prevent optical interference. Between 5 to 10 $\mu {\rm{W}}$ of filtered light is injected to the slave diode laser using mode-matching optics. This intensity is sufficient for a stable operation of the slave laser with an injected current window of 0.1 mA.

Intensity fluctuations of the 674 nm light lead to fluctuations in the Rabi frequency, and therefore affect any coherent operation. The origin of intensity noise is mainly due to fiber phase noise that is translated to intensity noise when passing through a PBS on the way to the ion. To prevent this, the intensity of the laser is monitored close to the input port of the vacuum camber and stabilized by adjusting the rf power that drives the AOM.

Figure 4(b) compares between the excitation spectrum of the ion when excited with the slave laser when it is injection locked to the unfiltered master laser (blue curve) and to the cavity filtered light (red curve). The significant improvement in the spectral purity is clearly observed. The inset shows a measurement of the excitation probability as function of the pulse duration at the maximum of the servo-bump in the two cases. In the unfiltered light case already at a pulse duration of 400 μs the population is saturated close to 50% while in the filtered light case as little as 1% to 2% are observed. The ratio between the slopes of the two curves in the linear regime is about 50 which should be regarded only as a lower bound due to the small signal in the filtered case which could be a result of power broadening together with other decoherence mechanisms such as magnetic field fluctuations or leakage of the 1033 nm repump light.

In QIP the qubit coherence time is determined by the stability of the relative phase between the qubit and the local oscillator. In the case of an optical qubit, the local oscillator that keeps track of the qubit phase, is a narrow linewidth laser. The physical mechanisms that cause phase fluctuations to the qubit and the laser are different and uncorrelated. Therefore, reaching a long coherence time requires that the phases of both the qubit and the oscillator are independently stable. While commercially available oscillators at radio and microwave frequencies can easily reach coherence times of many seconds (requiring fractional frequency uncertainty of 10⁻⁶–10⁻⁹), obtaining a laser with a coherence time of one second, i.e. 1 Hz linewidth, is a much more challenging task and requires fractional frequency uncertainty of 10⁻¹⁴–10⁻¹⁵. On the qubit side, the dominant source of dephasing are fluctuations of the ambient magnetic field .

One method of characterizing the spectral properties of a laser is a delayed self-heterodyne interferometer [26]. Here the laser light interferes with a delayed fraction of itself by using a long optical fiber to create a substantial time delay τ. However, as the spectral resolution of this method is given by $1/\tau ,$ to measure the sub-kHz frequency fluctuations requires hundreds of kilometers of fiber length which render this method impractical. The alternative is to compare against another stable reference. Examples for such a reference include another independent laser, an ultra stable Fabry–Perot cavity or a narrow atomic transition.

Since the purpose of our laser is to drive the ${S}_{1/2}\to {D}_{5/2}$ transition in ${}^{88}{\mathrm{Sr}}^{+}$ and with no other tool at hand, we inferred our laser linewidth by performing a Ramsey spectroscopy measurements on the ion. One complication that arises in our case is that the relevant quadrupole transition is first order Zeeman sensitive so a conventional Ramsey experiment can not discriminate between the magnetic field noise and the laser frequency noise. Previously, custom-designed entangled state were used to differentiate laser phase noise from magnetic field noise [27] Here we demonstrate a custom-designed dynamic decoupling sequence to this end. The advantage of using dynamic decoupling for this purpose is the simplicity of its implementation. Our magnetic field dynamic decoupling (MFDD) eliminates only phase fluctuations due to slow magnetic field noise, thus rendering the optical transition to an effective clock transition. After initializing to the superposition state $\psi =1/\sqrt{2}(| {S}_{1/2,-1/2}\rangle +| {D}_{5/2,-1/2}\rangle )$ the scheme follows conventional dynamic decoupling methods [28] in the form of echo pulses with two main differences. The first is that the pulses are performed in the ${S}_{1/2}$ spin manifold and not on the optical transition as this will reverse the effect of the laser frequency noise as well as the magnetic field noise. The basic idea is illustrated in figure 5 and relies on the opposite sign of the Zeeman shift of the ${S}_{1/2,-1/2}\leftrightarrow {D}_{5/2,-1/2}$ transition as compared with the ${S}_{1/\mathrm{2,1}/2}\leftrightarrow {D}_{5/2,-1/2}$ transition. As long as the magnetic field does not vary between echo pulses, the phase that is accumulated when the ground state population is in the ${S}_{1/2,-1/2}$ due to an arbitrary Zeeman detuning, can be reversed by transferring the population to the ${S}_{1/\mathrm{2,1}/2}$ state where the phase is accumulated with the opposite sign. The second difference from a regular echo sequence is that the time which is spent in each state is not equal. This is because the magnitude of the Zeeman shift in the two states is different with a ratio of 1:4. The experimental sequence is illustrated in figure 5(b). For a Ramsey experiment of a total time T and N rf echo sequences, each echo sequence is of duration ${T}_{N}=T/N$ and is composed of two rf π pulses separated by a time ${T}_{N}/5$ at the middle of the sequence. Notice that in terms of the optical transition this scheme is identical to a conventional Ramsey sequence having only two $\pi /2$ pulse with a wait time in between where it is only the ground state part that is manipulated with rf pulses. We have shown, in previous work [29], that the coherence of ground-state superpositions can be maintained for as long as a second using several hundred echo pulses. In general any dynamic decoupling scheme can be adapted for the manipulation of the ground state part. The experimental results are presented in figure 6(a). We plot the Ramsey fringe contrast as a function of the total experiment time for a plain Ramsey sequence (blue dots) and MFDD sequence with N = 4 (red dots). The contrast is obtained by scanning the phase of the second $\pi /2$ optical pulse and fitting the data to a sine function with the amplitude and phase as fit parameters. Solid lines are theoretical fits to a Bessel function in the plain Ramsey case and a Gaussian in the MFDD case. In the regular Ramsey experiment the contrast dropped to 50% after 1.7 ms. Here, the 50 Hz monotone magnetic field noise component dominated and the contrast followed a Bessel shape [30]. Applying the MFDD sequence resulted in a coherence time of 6.7 ms (more than a four-fold improvement) and in addition the contrast follows a Gaussian function shape rather than a Bessel, which implies that there is no single frequency component that dominated the noise spectrum. Applying MFDD sequences with $N\gt 4$ did not extend the coherence time. Hence, we infer that at this point the laser frequency noise, rather than the magnetic field noise, is the limiting factor. Assuming a Gaussian line-shape, we infer a laser linewidth of 65(5) Hz at integration time of 10 s. More information on the laser spectral content can be obtained by a standard single echo sequence on the optical transition. Here, we did not apply the MFDD sequence but rather incorporated an active stabilization system that reduced the ambient magnetic field noise to a negligible level. As shown in figure 6(b). the contrast extended to 20 ms following exponential decay. Since the echo pulse acts as a high-pass filter we infer that the coherence time measured without the echo is limited by low frequency noise which we assume to has a 1/f characteristic. If we consider the echo measurement to represent the laser fast-linewidth (white noise contribution) then from the exponential fit we obtained11(1) Hz linewidth [31]. Finally, The inset in figure 6(b) shows a Rabi spectroscopy scan using a weak and long excitation pulse with a duration of 11 ms. The spectral feature has a width of is 106 ± 20 (Hz) which is consistent with a transform limited result and roughly consistent with the laser linewidth measured using MFDD. In this case the low frequency noise impeded our ability to increase the pulse time and obtain much a narrower feature.

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Our experimental implementation of the MS-interaction is similar to that outlined in [38]. A bichromatic beam was produced by single-passing an AOM (AOM-S) which was driven by two rf frequencies (RSB and BSB DDS sources). The beam then propagated to the ions through a PM single mode fiber after which it was focused to a waist of about 20 μm. At the trap center each spectral component had up to 1 mW of power at 674 nm. At each experiment the two ions were initialized to the $| {SS}\rangle$ state and their stretched mode which was used for the MS-interaction, was cooled to the ground state $(\bar{n}\approx 0.05).$

As a first step, we examined the ions state population for a wide interval of the interaction parameters: symmetric detuning δ, asymmetric detuning Δ and interaction duration t. Here the center of mass vibrational mode with frequency of $\nu =0.98\;{\rm{MHz}}$ was used. These scans provide a visual insight to the population dynamics and its dependence of the various parameters of the interaction. In addition, some experimental uncertainties in the gate parameters can substantially effect the gate fidelity. As an example, uncertainties in the intensities of each component in the bichromatic beam, the vibrational mode frequency and light shift from other levels affect the required gate-time and detuning. As shown in [38], the experimental calibration of all these parameters using the gate itself is possible. A different approach, which is more straightforward, however time consuming, is to scan the relevant parameters: symmetric detuning δ, asymmetric detuning Δ and time T_gate and measure the two-ion response. Once the experimental maps are obtained the real parameters can be by extracted by comparing the measured maps to the calculated ones. Figures 10–12 present the states population maps, both measured and calculated, for three 2D scans: asymmetric versus symmetric detuning, asymmetric detuning versus time and symmetric detuning versus time.

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The practical value of such maps is best demonstrated in the time and symmetric detuning scan which is shown in figure 12. The calculated populations as function of the gate-time and symmetric detuning fully describe the measurement result up to a shift in the detuning of 35 kHz. This large shift is attributed to a bias in our vibrational frequency measurement scheme. Our sideband frequency was measured via sideband spectroscopy using intense beams which generated light shift on the carrier transition frequency which was different in the MS gate beam configuration. Therefore, the measured sideband frequency included a significant light shift of 35 kHz.

The process of finding the optimal values for the gate parameters begins by inspecting the 2D maps in figure 10 which is comprised of two prominent features. One is the central vertical line which arises from $| {SS}\rangle \to | {DD}\rangle$ two-photon coupling, whereas the diagonal line is a result of a BSB excitation.The opposite diagonal line which correspond to a RSB transition is absent here since the initial state is the vibrational ground state. From the crossing of the two lines we obtain a rough estimate for both the symmetric and asymmetric detunings. Note that the interaction time in this scan is much longer than the presumed gate-time. In the next step we determine the asymmetric detuning with higher precision. To this end, we scan the asymmetric detuning at a fixed interaction time which is equal to twice the presumed gate-time. This corresponds to the dashed vertical line in the 2D map of P0 that is presented in figure 11. The asymmetric detuning value is then set by maximizing P0. The result hardly depends on the interaction time for relatively large interval as can be seen in the figure. The gate-time and symmetric detuning can be found by performing 2D scan similar to that of figure 12 but with higher resolution and only in the region of the presumed values. The two parameters are fixed at the point where P1 is minimal (ideally zero) and P0 = P2 ≈ 50%. The correct values of these two parameters can be obtained more efficiently by an iterative process that consists of two 1D scans. The procedure is illustrated in figure 13. First, we scan the gate symmetric detuning at the presumed gate-time and set it to the value where the populations P0 = P2 which is represented by the blue curve in the figure. Next, the gate-time is scanned with the obtained gate detuning. Here the gate-time is fixed at the point where P1 is minimal (represented by the red curve). After iteratively repeating the two scans (black arrows) the process converges and the correct values (black rectangle) are obtained as long as the starting value of the gate-time does not deviate from it actual value by more than about 25% (the plotted region). Having a residual asymmetric detuning of less than 1 kHz modified the two curves but still allows the process to converge although to slightly different values.

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Figure 14(a) presents the populations as a function of the interaction time. Here, we used the stretch vibrational mode . At a gate-time of ${T}_{\mathrm{gate}}=140\;\mu {\rm{s}}$ the ions superposition consists of only the $| {SS}\rangle$ and $| {DD}\rangle$ states. To verify that the resulting state is indeed a Bell state, the off-diagonal elements of the density matrix must be measured as well. This was done by applying another $\pi /2$ pulse with a phase ϕ and measuring the state parity $P=P0+P2-P1.$ The amplitude of parity oscillations, A_p, as function of ϕ is equal to $2| {\rho }_{{SS},{DD}}| .$ The gate fidelity for producing the Bell state $| {\rm{\Phi }}\rangle$ is given by ${F}_{{\rm{\Phi }}}=(P0+P2)/2+{A}_{{\rm{p}}}.$ Using the maximum likelihood method to fit the data in figure 14(b), we conclude gate fidelity³ of F = 0.985(10). While part (≈5×10⁻³) of the gate error can be attributed to imperfection in the preparation and detection operations, the rest is assumed to arise from error in the gate itself. Although the exact source of error has not been fully identified, the nature of the error has been studied in [35] where we have performed process tomography on five consecutive gates. The results were consistence with a collective depolarization channel. A possible source of error is off-resonance $| S\rangle \leftrightarrow | D\rangle$ incoherent transition as indicated by the measurement in the inset of figure 4(b).

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