Versatile control of $^9$Be$^+$ ions using a spectrally tailored UV frequency comb

TL;DR

This paper presents a method for controlling $^9$Be$^+$ ions using a spectrally tailored UV frequency comb, enabling efficient quantum state manipulation for quantum computing applications.

Contribution

It introduces a novel technique for generating a spectrally tailored narrow-bandwidth optical frequency comb near 313 nm and demonstrates its use in quantum control of $^9$Be$^+$ ions.

Findings

Successful internal state control of $^9$Be$^+$ ions.

Efficient internal-motional state coupling demonstrated.

Spectrally optimized comb enables quantum logic operations.

Abstract

We demonstrate quantum control of $^9$Be$^+$ ions directly implemented by an optical frequency comb. Based on numerical simulations of the relevant processes in $^9$Be$^+$ for different magnetic field regimes, we demonstrate a wide applicability when controlling the comb's spectral properties. We introduce a novel technique for the selective and efficient generation of a spectrally tailored narrow-bandwidth optical frequency comb near 313 nm. We experimentally demonstrate internal state control and internal-motional state coupling of $^9$Be$^+$ ions implemented by stimulated-Raman manipulation using a spectrally optimized optical frequency comb. Our pulsed laser approach is a key enabling step for the implementation of quantum logic and quantum information experiments in Penning traps.

Figures

Full text

Laser cooling and manipulation have enabled many groundbreaking experiments in quantum control of atoms and molecules. Prominent applications include precision measurements, frequency metrology and quantum information processing. Whereas continuous wave (CW) lasers are widely established in this context, direct manipulation through mode-locked lasers Hayes et al. (2010) offers unique advantages, enabling a new class of experiments requiring coherent sources spanning large frequency differences. The broad spectrum of regularly spaced, phase-coherent comb modes of pulsed lasers allows to directly bridge large frequency gaps in a controllable, flexible and technically simple way. This scenario is of particular relevance for the control of molecular quantum states Ding and Matsukevich (2012) as well as for trapped-ion experiments in Penning traps Brown and Gabrielse (1986); Thompson et al. (2009), where the characteristic high magnetic fields of up to several Tesla cause large Zeeman shifts between atomic states on the order of several hundreds of GHz. Here, the standard approach of stimulated-Raman laser manipulation for quantum control Wineland et al. (2003) becomes very difficult to implement using CW lasers. Full control over all quantized degrees of freedom has thus not yet been implemented in Penning traps. The use of pulsed lasers in this context promises to overcome this limitation and enable a new class of quantum logic and quantum information experiments in Penning traps.

Most of the laser coolable atomic ions have optical resonances in the ultraviolet (UV) spectral region, whereas mode-locked lasers Udem et al. (2009) typically emit in the infrared spectral domain. Their high intensity, however, enables efficient nonlinear conversion processes, so that a wide range of atomic species can be eventually covered. Due to their light mass and the absence of metastable electronically excited states, 9Be+ ions in particular are considered as auxiliary or “logic” ions in protocols for sympathetic laser cooling and quantum logic spectroscopy with (anti-)protons in Penning traps Heinzen and Wineland (1990); Wineland et al. (1998) and ultra-precise g-factor based tests of fundamental symmetries. Towards this end, here we show numerical simulations of direct frequency comb control of 9Be+ ions for a wide range of magnetic field regimes. We introduce novel techniques for the selective and efficient generation of a narrow-bandwidth UV frequency comb with tunable spectral properties and experimentally demonstrate spin-motional control of 9Be+ ions using a spectrally optimized optical frequency comb.

The implementation of quantum control is carried out using two-photon stimulated Raman transitions. The coupling allows full control over internal and motional degrees of freedom and thereby enables sideband (ground state) cooling, quantum information processing and metrology applications of atoms and molecules. In the standard approach, these transitions are driven by two CW laser beams with a frequency detuning equal to the energy splitting between the states to be coupled. The beams are derived either from a single laser source using frequency modulators or from two phase-locked lasers. With increasing energy splitting, this approach becomes intractable due to the lack of efficient high frequency modulators and technical complexities of phase locking two CW lasers with large frequency differences. In contrast, driving the transitions directly using an optical frequency comb allows a coupling across large energy differences without needing a second laser or high frequency modulators, if the comb’s spectral bandwidth is broader than the level splitting. In this scenario, the atom undergoes coherent transitions by absorption of a photon from one comb tooth and stimulated emission into another tooth, as illustrated in Fig. 1(a). Due to the regularity of comb mode spacing, multiple pairs of comb teeth coherently add up and contribute to the process Hayes et al. (2010). If only internal spin states of the atom are to be coupled (“carrier transitions”), the comb teeth of each pair can be provided by a single laser beam, if the internal-state transition frequency $\omega_{0}$ is an integer multiple of the laser’s repetition rate $\omega_{\text{rep}}$, $\omega_{0}=q\cdot\omega_{\text{rep}}$ with $q\in\mathbb{Z}$. If changes in the motional state of the ion are simultaneously pursued (“motional sideband transitions”), the comb teeth of each pair must originate from separate beams, directed onto the ion from different directions and synchronized in time. In this case, the resonance condition for $n$-th order sideband transitions with motional frequency $\omega_{m}$, $\omega_{0}\pm n\cdot\omega_{m}=\lvert j\cdot\omega_{\text{rep}}\pm\Delta{\omega}\rvert$ with $n,j\in\mathbb{Z}$ is fulfilled by a relative frequency shift between the beam’s frequencies $\Delta\omega$, implemented such that teeth from the one beam’s spectrum in combination with teeth from the other beam’s spectrum can bridge the level splitting, as illustrated in Fig. 1(b). In contrast to the CW-laser approach, the required frequency shift between the beams does not need to cover the entire level splitting and the optimum shift instead is on the order of only one third of the laser’s repetition rate Mizrahi et al. (2014). This is a clear advantage of this type of direct frequency comb control, which previously has been demonstrated with ${}^{171}\text{Yb}^{+}$ ions for a level splitting near 10 GHz Hayes et al. (2010) and recently been employed for spectroscopy on carrier transitions in ${}^{40}\mathrm{Ca}^{+}$ Solaro et al. (2018).

In our scenario Smorra et al. (2015); Paschke (2017) we intend to apply the pulsed laser approach for full quantum control of 9Be+ ions in the environment of a Penning trap at a magnetic field of 5 T leading to a Zeeman level splitting near 140 GHz. A particular challenge for the implementation arises from beryllium’s atomic structure. The atomic levels for implementing quantum control are chosen within the sublevels of beryllium’s single ground state $S$-level, $1s^{2}2s$, whereas the stimulated-Raman coupling is facilitated by off-resonant coupling to beryllium’s electronically excited $P$-state, $1s^{2}2p$. Here, the small excited state fine-structure splitting of only 198 GHz between the ${}^{2}P_{1/2}$ and ${}^{2}P_{3/2}$ levels requires a Raman detuning outside the $P$-level manifold in order to avoid resonant excitation of the $P$-levels, which would be followed by spontaneous emission decay. For this atomic configuration the comb’s spectral properties, such as the spectral envelope shape and its width, need to be precisely controlled in order to achieve high Raman coupling strengths, while simultaneously guaranteeing reasonably low scattering rates.

Figure 2 illustrates the influence of the comb’s spectral properties on achievable Raman coupling strengths for representative magnetic fields $B$ from 1 T to 10 T with corresponding transition frequencies $\omega_{0}(B)$ from $2\pi\cdot$28 GHz to $2\pi\cdot$280 GHz. For each magnetic field the comb’s spectral properties in general set the minimum Raman detuning required in order to control the scattering rate, which strongly influences and limits the maximum achievable Raman coupling strength. For an unmodified spectrum (solid lines) the optimal spectral bandwidth is only slightly broader than the respective level splitting at a given magnetic field, as demonstrated in Fig. 2(a). For narrower spectra too few pairs of comb teeth exist that contribute to the Raman process, while for broader spectra a strong loss of achievable Raman coupling occurs because the detuning required to suppress spontaneous scattering below a certain threshold significantly increases, as shown in Fig. 2(b). This loss of achievable coupling strength due to the increasing required detuning strongly depends on the comb’s specific spectral envelope. In particular, any power contained in spectral wings close to resonance strongly enhances decoherence through spontaneous emission without equally contributing to the coupling strength. A significant improvement of the achievable coupling strength (for a given target scattering rate) can hence be implemented through spectral pulse shaping by blocking the inefficient spectral tails of the UV pulse close to resonance. The absence of these outer-lying frequency components allows for much smaller detunings, which has a strong impact on maximizing achievable Raman coupling strengths, as shown in Fig. 2 for exemplary modified spectra (dashed and dotted lines). In addition the comb’s spectral bandwidth can be chosen in a much wider range than for unmodified spectra and achievable couplings become almost identical over a wide range of magnetic fields; both properties strongly improve the flexibility of this approach.

In order to obtain an efficient Raman coupling for 9Be+ ions it is thus mandatory to combine a narrow-bandwidth frequency comb generation with a technique for spectral control allowing to block the comb’s spectral wings. The resulting comb then provides enormous flexibility enabling efficient quantum control of 9Be+ ions at nearly any experimentally relevant conditions.

For the experimental implementation of efficient and versatile quantum control we have developed a narrow-bandwidth UV frequency comb with tunable spectral properties (details in Paschke (2017)). The system is based on a custom-built femtosecond frequency comb operating near 626 nm with a tunable repetition rate near 100 MHz. The output is then frequency doubled to the UV in order to generate the desired wavelength for beryllium’s optical resonance near 313 nm. Control of the spectral properties is realized by the combination of a nonlinear spectral compression technique implemented during second harmonic generation (SHG) into the UV and a subsequent blocking of the comb’s spectral wings.

The nonlinear spectral compression technique applied to our system (cf. supplementar material) uses the SHG process to transfer the energy of broadband fundamental frequency (FF) femtosecond pulses into narrow-bandwidth second harmonic (SH) ones. Due to different group velocities of the FF and the SH pulses inside the crystal, the SH pulse is temporally broadened during propagation, which is directly accompanied by a spectral narrowing. In the spectral domain this process is related to the limited phase-matching bandwidth of the nonlinear crystal in terms of the SHG process expressed as $\Delta\omega_{\text{SH}}\approx 2\pi\cdot\frac{0.886}{\lvert GVM\rvert\cdot L}$ Marangoni et al. (2007), with $GVM$ denoting the group velocity mismatch between the FF and the SH pulse and $L$ the crystal length. In case the pulses remain spatially overlapped during propagation within the nonlinear crystal, the resulting SH bandwidth can thus be controlled by the crystal length, as shown in previous work for the generation of tunable narrow-bandwidth pulses in the visible spectral region Marangoni et al. (2007). Here we include the effect of spatial walkoff, where the FF and SH pulses travel in different directions while propagating through the crystal. We have theoretically and experimentally investigated spatio-temporal coupling effects onto the nonlinear conversion and spectral compression process and show the UV pulse output bandwidth for a given nonlinear crystal length to be selectively tunable by varying the focusing conditions of the FF pulse. For implementation, the nonlinear crystal bismuth triborate Hellwig et al. (1998, 2000); Ghotbi and Ebrahim-Zadeh (2004), BiBO, is chosen as it combines a large $GVM$ (984 fs/mm Smith (2013)) for efficient spectral compression with a high nonlinearity ($d_{\text{eff}}=3.36$ pm/V Smith (2013)) for high conversion efficiencies. Results for an exemplary crystal length of 4.5 mm are presented in Fig. 3. The spectral properties of the SH pulse are shown to strongly depend on the focal beam dimension along the direction of spatial walkoff and can be tuned over a wide range of bandwidths for a given crystal length. With increasing focal radii the effective crystal length over which the fundamental and the SH beams are overlapped increases and thus a stronger spectral compression is achieved, while the peak intensities and the corresponding conversion efficiencies drop. A selective and efficient generation of narrow-bandwidth UV pulses with a wide range of bandwidths is thus achieved by choosing appropriate combinations of crystal lengths and focusing conditions.

For the subsequent spectral pulse modification, we utilize the angular spectral chirp resulting from spatio-temporal couplings occurring during the SHG process. Here, different spectral components of the generated UV signal leave the crystal under different angles. Rather than relying on a pulse shaping arrangement involving UV gratings with high transmission losses, this allows us to block the comb’s undesired spectral wings in a straightforward way by using razor blades located closely behind the crystal, as shown in Fig. 4.

The experimental demonstration of stimulated-Raman laser control of 9Be+ ions using the spectrally modified comb system has been carried out in a surface-electrode experiment at an externally applied magnetic field of 22.3 mT Wahnschaffe et al. (2017); Paschke (2017) because our Penning trap apparatus with a 5 T field is still in commissioning. Furthermore, for $\delta\omega_{\mathrm{FWHM}}/2\pi>300\,\mathrm{GHz}$, achievable coupling strengths are similar for 22.3 mT and for 5 T as shown in Fig. 5 (a), demonstrating the flexibility of our approach. In the 22.3 mT case, in principle even lower power may be required if the spectrum can be made narrower, down to the qubit splitting of $\approx 1\,\mathrm{GHz}$; in that case, however, the coupling strength for the 5 T case would be significantly reduced. Furthermore, the required length of the nonlinear crystal to achieve such extremely small bandwidths would be unrealistic (tens of centimeters).

Internal-state control has been implemented on beryllium’s ground state sublevel transition between the $|F=2,m_{F}=0\rangle$ and the $\lvert F=1,m_{F}=0\rangle$ state. A single train of spectrally tailored pulses was focused onto the ion, switched on and off using a single-pass acousto-optic modulator (AOM). Fulfilling the resonance condition $\omega_{0}=2\pi\cdot 1397.8372\,\text{MHz}=14\cdot\omega_{\text{rep}}$, the interaction with the pulsed Raman beam causes the atom to undergo a coherent evolution resulting in Rabi oscillations, as shown in Fig. 5(c). The achieved $\pi$-time of $t_{\pi}=29.04\,\mu\text{s}\pm 0.07\,\mu\text{s}$ coincides with simulations under given experimental conditions Paschke (2017). The benefit of the pulse shaping is evident when, after choosing laser detuning and bandwidth, by moving in the razor blades, the contrast of Rabi oscillations improves until limited by ambient magnetic field fluctuations (cf. caption and supplementary information) without apparent loss of oscillation speed. This is also evident from shining in the beam such that the two-photon resonance condition is not met. Without pulse shaping, spontaneous scattering leads to a strong depletion of the initial state (cf. supplementary information).

Internal-motional coupling has been demonstrated on the first-order magnetic field-insensitive $|F=2,m_{F}=1\rangle\leftrightarrow\lvert F=1,m_{F}=1\rangle$ transition. The output of the spectrally modified UV comb was divided in two beams which were focused onto the ion from orthogonal directions and synchronized in arrival time, as shown in Fig. 5(b). By sending each beam through a single-pass AOM with adjusted frequency, the resonance condition for a red sideband transition on the axial mode, $\omega_{0}-\omega_{m}=2\pi\cdot 1081.6551\,\text{MHz}=7\cdot\omega_{\text{rep}}+\Delta\omega_{\text{AOM}}$ was fulfilled, as shown in Fig. 5(d). Note that the spectrum was obtained for a Doppler cooled ion, and therefore we do not observe full contrast on the sideband transition.

In summary we have demonstrated internal state control and internal-motional state coupling of 9Be+ ions implemented by stimulated-Raman control using an optical frequency comb. This represents a key enabling step for full quantum control of 9Be+ ions in quantum information and quantum logic inspired precision experiments in Penning traps. Towards this end, we have developed novel techniques for the generation of narrow-bandwidth UV pulses with tunable spectral properties fulfilling all requirements for efficient and flexible quantum control. These techniques might further find use in similar scenarios of interest, such as cooling and manipulation of ultracold molecules or time-resolved spectroscopy.