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Atomic Quadrupole Moment Measurement Using Dynamic Decoupling R. Shaniv,1 N. Akerman,1 and R. Ozeri1 1Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel We present a method that uses dynamic decoupling of a multi-level quantum probe to distinguish 2 2 small frequency shifts that depend on mj , where mj is the angular momentum of level jji along the quantization axis, from large noisy shifts that are linear in mj , such as those due to magnetic field noise. Using this method we measured the electric quadrupole moment of the 4D 5 level in 2 88 + +0:026 2 Sr to be 2:973−0:033 ea0. Our measurement improves the uncertainty of this value by an order of magnitude and thus helps mitigate an important systematic uncertainty in 88Sr+ based optical atomic clocks and verifies complicated many-body quantum calculations. Increasing the coherence time of quantum superposi- sure of its deviation from perfect spherical symmetry. tions is an ongoing research topic. From the quantum While ground state S orbitals are spherically symmet- information point of view, prolonging the coherence time ric, higher levels, such as states in the D manifold allows for the performance of a larger number of coherent have a finite electric quadrupole moment. This electric operations, making more complex quantum computation quadrupole moment couples to an electric field gradient algorithms possible [1] as well as longer-living quantum across the atomic wavefunction. The resulting energy memory [2]. From a metrology perspective, decoherence shift is usually small in comparison to other shifts, e.g. poses a time limit on the coherent evolution of a quan- Zeeman shift, due to the small atomic length scale over tum probe when used to measure some physical quantity, which the electric field has to change. However small, this thereby limiting the accuracy with which this quantity is is nevertheless an important systematic shift in ion-trap evaluated. optical atomic clocks [16]. Here, electric field gradients Several methods were developed and demonstrated to are inherent to the trap and induce typical shifts to op- increase the coherence time of quantum systems. Exam- tical electric-quadrupole clock transitions on the order of ples include decoherence free subspaces (DFS) [3, 4], in 10's-100's Hz which constitute fractional frequency shifts which states are invariant under the effect of the environ- on the order of 10−13. While the electric field gradient at ment, quantum error correction codes (QECC) [5, 6], in the ion position can be estimated by measuring the trap which an error can be detected by a measurement that spring constant, evaluating the resulting shift requires a does not reveal information on the encoded superposi- reliable knowledge of the quadrupole moment of the lev- tion and can be subsequently corrected, and dynamic els involved. decoupling (DD) methods in which a quantum super- We performed our measurement on a single 88Sr+ ion position is spectrally separated from its environment (a trapped in a linear Paul trap. The S1=2 ! D5=2 electric review of DD methods can be found in [7]). The optimal quadrupole optical transition of this ion is investigated spectrum of qubit modulation depends on the details of as a possible atomic time standard in several experi- noise. While DD methods are not as general as QECC ments [17, 18]. The quadrupole moment of the 4D 5 level and cannot mitigate the effect of any type of noise, their 2 in 88Sr+ was previously measured [19]. Measurements implementation is much simpler as it does not require of the D level quadrupole moment of 199Hg+ [20] and the entanglement of multi-qubit arrays and relies solely 171Y b+ [21] were also published. In addition, a method on single qubit operations. All the methods above have to measure the quadrupole moment of 40Ca+ using an been suggested or used to improve quantum metrology as entangled superposition of two ions was experimentally well [8{15]. In the context of metrology a method is cho- demonstrated [22]. Our new method yields a measured sen such that on top of noise suppression the measured value for the quadrupole moment in the 4D 5 level in an signal is coherently accumulated. 2 88 + In this work we present a new DD method which decou- Sr ion with about ten times smaller uncertainty in ples a superposition of atomic levels in a single trapped comparison to the previous measurement, and requires a 88 + single ion only. arXiv:1511.07277v1 [quant-ph] 23 Nov 2015 Sr ion from external magnetic field noise while mea- suring the electric quadrupole shift induced by an electric The relevant levels for our experiment are the two- field gradient. In contrast with usual two-level quantum fold 5S 1 ground level jS; mji, and the six-fold 4D 5 level 2 2 probes, here we take advantage of the six-fold 4D 5 Zee- 2 jD; mji shown schematically on Fig. 1. A constant mag- man manifold of equidistant levels. Using this method netic field of typically B = 3 × 10−4 T, splits the dif- we measured the quadrupole moment of the 4D 5 level 2 ferent Zeeman states in both levels. This magnetic field 88 + +0:026 2 in Sr to be 2:973−0:033 ea0, where e is the electron is generated by three Helmholtz coils in three orthogo- charge and a0 is Bohr radius. nal axes. The relative current between the coils deter- The quadrupole moment of an atomic state is a mea- mines the magnetic field direction with respect to the 2 trap quadrupole axis. The Zeeman frequency splitting between two adjacent levels in the 4D 5 manifold ∆fZ is 2 typically around 5 MHz. The ion 5S 1 ! 4D 5 transition a 2 2 is addressed with a 674 nm narrow linewidth laser, with spectral fast line-width of 100 Hz over several minutes [23]. An RF electrode located two mm away from the 휋 휋 ion trap center generates oscillating magnetic fields at 2 휋 휋 2 the Zeeman splitting frequency leading to spin rotations. The orientation of the trap electrodes, the laser ~k vec- b tor and the quantization magnetic field are shown in Fig. 휏 휏 휏 휏 1a. Ion state detection is performed using state selective fluorescence on a strong dipole allowed transition at 422 RF 5/2 nm [24]. More details about the experimental apparatus 3/2 1/2 can be found in [25]. c While the 5S 1 electronic wavefunction is spherically -1/2 2 symmetric, and therefore has no electric quadrupole -3/2 moment, states in the 4D 5 level have a quadrupole -5/2 2 charge distribution, and therefore an electric field gra- dient across these wavefunctions will shift their energy. Since the ion is held in a linear Paul trap, it is held in a constant (DC) electric field gradient induced by the trap electrodes. The electric DC potential generated by these d electrodes is well approximated by, population [%] population - 1 dE 1 U = z (x2 + y2 − 2z2); (1) 4 dz where z is the direction of the trap axis, and the resulting shift of state jD; mji level is given by [26], 2 1 dEz 5 35 − 12mj 2 ∆νQ = Θ D; 3cos (β) − 1 : 4h dz 2 40 e (2) dE [%] population Here, z is the electric field gradient along the - dz 1 5 quadrupole axis, Θ D; is the 4D 5 level quadrupole 2 2 moment and β is the angle between the quantization axis set by the magnetic field and the quadrupole axis set by the trap geometry. In our trap this shift is on the order FIG. 1: Layout of the experiment and pulse sequence scheme. (a) Scheme of the trap structure and orientation with respect of 50 Hz resulting from typical dEz ∼ 100 V . This dz mm2 ~ small shift is usually overshadowed by the much larger to the laser k vector and the magnetic field directions. The trap axis is shown by the dashed line passing through both time variation in Zeeman shifts, typically on the order endcaps. The laser direction is at a right angle with respect of a few kHz in several msec. In order to measure the to the trap axis, the magnetic field direction can be scanned quadrupole shift, we developed a DD sequence that em- and is shown here to be at 45o angle with respect to both ploys the equidistant 4D 5 Zeeman sub-levels to decouple the laser and the trap axis. (b) Pulse sequence for the mea- 2 our ion from effects linear with mJ , while leaving it sus- surement of the quadrupole shift. The dashed frame marks 2 the sequence building-block which is being repeated. (c) The ceptible to shifts that depend on mJ . Unlike most pre- vious measurements of clock transition quadrupole mo- definition of an RF π pulse. The blue and white circles mark the two amplitudes, and the numbers on the right column ments, our measurement protocol involves only the D 5 2 denote the different mJ values. The black two-sided arrow levels and radio frequency (RF) transitions and no opti- marks the RF frequency used to drive the transition. (d),(e) cal transitions. Therefore, many systematic effects that Rabi oscillations on the six-fold 4D 5 energy level for initial 2 5 1 5 1 affected previous quadrupole shift measurements [19{21] states j i = D; − and j i = p D; − + D; − i 2 i 2 2 2 such as ac Stark shifts, second-order Doppler effects and respectively. The vertical axis shows the population outside their dependence on micromotion, as well as limitations the detected sub-level.

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