Transduction parameter<\/p>\n
We consider the dynamics of a string of N trapped ions perturbed by an external electric field, which results in a force \u03b4Fj(t)\u2009=\u2009\u2212q\u03b4Ej(t) on ion j. Restricting ourselves to a single direction without loss of generality, the Lagrangian of this system is53<\/a><\/p>\n $$\\begin{aligned}{{L}}&=\\frac{{{m}}}{2}\\left(\\sum_{{{p}}=1}^{{{N}}}{(\\dot{Q}_{p}({{t}}))}^{2} – {\\nu}_{{{p}}}^{2}{{{{Q}}}_{{{p}}}}^{2}({{t}})\\right)\\\\ &\\quad{}+{{q}}{{{Q}}}_{{{p}}}({{t}})\\sum_{{{j}}=1}^{{{N}}}{{{b}}}_{{{j}}}^{({{p}})}\\delta {{{E}}}_{{{{j}}}}({{t}}),\\end{aligned}$$<\/p>\n \n (7)\n <\/p>\n where \u03bdp are the normal mode frequencies and \\({{{b}}}_{{{j}}}^{({{p}})}\\) describes how strongly ion j couples to the mode p. The normal modes of motion Qp(t) are related to small displacements of the ion, \u03b4r(t) of equation (1<\/a>), via:<\/p>\n $${{{Q}}}_{{{p}}}({{t}})=\\sum_{{{j}}=1}^{{{N}}}{{{{b}}}_{{{j}}}}^{({{p}})}\\delta {{r}}({{t}}).$$<\/p>\n \n (8)\n <\/p>\n The equation of motion of the pth normal mode is found from the Lagrangian using the relation<\/p>\n $$\\frac{\\rm{d}}{\\rm{d}t}\\left(\\frac{\\partial {{L}}}{\\partial \\dot{Q}_{p}(t)}\\right)=\\frac{\\partial {{L}}}{\\partial Q_{p}(t)},$$<\/p>\n resulting in<\/p>\n $${\\ddot{{{Q}}}}_{{{p}}}({{t}})+{\\nu}_{{{p}}}^{2}{{{Q}}}_{{{p}}}({{t}})=\\frac{{{e}}}{{{m}}}\\sum_{{{j}}=1}^{{{N}}}{{{b}}}_{{{j}}}^{({{p}})}\\delta {{{E}}}_{{{j}}}({{t}}).$$<\/p>\n \n (9)\n <\/p>\n Without loss of generality, we restrict ourselves to a single-ion chain, N\u2009=\u20091, and consider the centre-of-mass motion along the z axis. After setting p\u2009=\u2009z and \\({{{b}}}_{1}^{(1)}=1\\), equation (9<\/a>) becomes<\/p>\n $${\\ddot{{{Q}}}}_{{\\rm{z}}}({{t}})+{\\nu}_{{\\rm{z}}}^{2}{{{Q}}}_{{\\rm{z}}}({{t}})=\\frac{{{e}}}{{{m}}}\\delta {{E}}({{t}}).$$<\/p>\n \n (10)\n <\/p>\n This corresponds to the equation of a driven harmonic oscillator. Taking the Fourier transform, equation (10<\/a>) becomes<\/p>\n $${\\hat{{{Q}}}}_{{{p}}}(\\omega)=\\frac{{{e}}}{{{m}}({\\nu}_{{\\rm{z}}}^{2}-{\\omega}^{2})}\\delta \\hat{{{E}}}(\\omega),$$<\/p>\n \n (11)\n <\/p>\n where \\(\\hat{\\cdot }\\) denotes the Fourier transform. For N\u2009=\u20091 ion, Qp(t)\u2009=\u2009\u03b4r(t) and equation (11<\/a>) becomes<\/p>\n $$\\delta \\hat{{{r}}}(\\omega)=\\frac{{{e}}}{{{m}}({\\nu}_{{\\rm{z}}}^{2}-{\\omega}^{2})}\\delta \\hat{{{E}}}(\\omega).$$<\/p>\n \n (12)\n <\/p>\n In the limit \u03bdz\u2009\u226b\u2009\u03c9, equation (12<\/a>) reduces to<\/p>\n $$\\delta \\hat{{{r}}}(\\omega)=\\frac{{{e}}}{{{m}}{\\nu}_{{\\rm{z}}}^{2}}\\delta \\hat{{{E}}}(\\omega),$$<\/p>\n \n (13)\n <\/p>\n from which one can retrieve the expression of equation (1<\/a>). From equation (12<\/a>), we also find that the coupling of radial micromotion into the spin states is negligible. These oscillations occur at the RF trap frequency, \u03a9RF\/2\u03c0\u2009=\u200919.22\u2009MHz, and the resulting amplitude of the radial oscillation is negligible because \u03a9RF\u2009\u226b\u2009\u03bdx,y.<\/p>\n Experimental set-up<\/p>\n Extended Data Fig. 5<\/a> shows a schematic of the experimental set-up used in this work. The ion trap was mounted inside a vacuum chamber maintained at an average pressure of 2.4\u2009\u00d7\u200910\u221211\u2009mbar. The ion is Doppler cooled using a 369.52\u2009nm laser that is red-detuned from the 2S1\/2 |F\u2009=\u20091\u3009 to the 2P1\/2 |F\u2009=\u20090\u3009 transition. The laser beam is double-passed through an acousto-optic modulator to allow for fine frequency and amplitude control by a field-programmable gate array. An electro-acoustic modulator (EOM) is used to generate 2.11\u2009GHz sidebands for state preparation. These sidebands allow the population to be driven into the 2P1\/2 |F\u2009=\u20091\u3009 state via optical pumping, after which it decays into the |\u2193\u3009\u2009=\u20092S1\/2 |F\u2009=\u20090\u3009 ground state. The population that is off-resonantly driven into the 2S1\/2 |F\u2009=\u20090\u3009 state during Doppler cooling is returned to the cooling cycle by continuously applied microwaves near 12.64\u2009GHz. Population can also leak out of the Doppler cooling cycle by decaying into the 2D3\/2 manifold, where a 935.18\u2009nm re-pump laser applied on the 2D3\/2 to 3D[3\/2]1\/2 transition returns population to the 2S1\/2 |F\u2009=\u20091\u3009 state. The re-pump laser is also modulated by an EOM at 3.07\u2009GHz to improve the re-pumping efficiency. Microwaves are generated by a vector signal generator (Keysight E8267D PSG), which produces a carrier signal of 12.54\u2009GHz. This is then mixed with RF pulses near 100\u2009MHz generated by a two-channel AWG (Keysight M8190A), which is then amplified and emitted by an external microwave emitter to allow for coherent manipulation of the spin state. The spin state is measured using a state-dependent fluorescence scheme as described in ref. 50<\/a>. The average SPAM error was found to be \u03b7\u2009=\u20091.8\u2009\u00d7\u200910\u22122. The voltage signals used to measure the a.c. and d.c. sensitivities are applied directly to the capacitor from the second channel of the AWG. To measure the electric field noise, a white-noise waveform is generated using a separate AWG (Agilent 33522A). The white-noise signal is attenuated by two 30\u2009dB RF attenuators, and its output controlled with an external RF switch.<\/p>\n Gradient measurement<\/p>\n The strength of the magnetic field gradient along the axial direction was calculated by measuring the transition frequencies of two co-trapped 171Yb+ ions. As the splitting of the 171Yb+ spin states is dependent on the strength of the magnetic field at the position of the ion, the magnetic field gradient in the axial direction is given by<\/p>\n $$\\frac{\\partial {{B}}}{\\partial {{z}}}=\\frac{{{{B}}}_{2}-{{{B}}}_{1}}{\\delta {{Z}}},$$<\/p>\n \n (14)\n <\/p>\n where B1 and B2 are the magnetic field strengths at the location of each ion, and \u03b4Z is the ion separation (Extended Data Fig. 1<\/a>). The ion separation is a result of the mutual Coulomb repulsion between the ions and the oppositely acting axial confinement force. \u03b4Z is given by53<\/a><\/p>\n $$\\delta {{Z}}={\\left(\\frac{{e}^{2}}{4\\uppi {\\epsilon}_{0}{{m}}{\\nu}_{\\rm{z}}^{2}}\\right)}^{1\/3}\\frac{2.018}{{{{N}}}^{0.559}},$$<\/p>\n \n (15)\n <\/p>\n where \u03bdz is the axial vibrational centre-of-mass frequency, m is the mass of a single charged particle and N is the number of ions in the crystal. We measured \u03bdz\/2\u03c0\u2009=\u2009161.191(8)\u2009kHz via the \u2018tickling\u2019 method. An a.c. electric field was applied to the trap using an external RF coil, which excites the axial motion of the ion crystal when the applied frequency is resonant with the axial vibrational frequency, leading to a measurable decrease in ion fluorescence due to the Doppler shift. We then compute \u03b4Z\u2009=\u200912.64(1)\u2009\u03bcm from equation (15<\/a>).<\/p>\n The magnetic field at each ion was calculated by measuring the magnetic field-dependent transition frequency of each ion, as shown in the inset plots of Extended Data Fig. 1<\/a>. From these measurements, B1\u2009=\u20097.1328(8)\u2009G and B2\u2009=\u20099.9655(5)\u2009G. Finally, from equation (14<\/a>), the magnetic field gradient strength was \u2202B\/\u2202z\u2009=\u200922.41(1)\u2009T\u2009m\u22121.<\/p>\n Calibrating \u03b1 and \u03b3<\/p>\n The geometric factor of an electrode, \u03b1, relates the electric field at the position of the ion to the voltage applied to the electrode, and is defined as<\/p>\n $$\\alpha =\\frac{\\partial {{E}}}{\\partial {{V}}}=\\frac{\\partial \\omega}{\\partial {{V}}}\\frac{\\partial {{E}}}{\\partial {{z}}}{\\left(\\frac{\\partial {{B}}}{\\partial {{z}}}\\frac{\\partial \\omega}{\\partial {{B}}}\\right)}^{-1},$$<\/p>\n \n (16)\n <\/p>\n where \u2202E\/\u2202z\u2009=\u2009m\u03bdz2\/e. We calibrate \u03b1 by first measuring the change in magnetic field at the ion due to a change in the voltage applied to the E1 electrode (\u2202B\/\u2202V) using the second-order sensitive spin state transition frequency and \u03bdz\/2\u03c0\u2009=\u2009161.191(8)\u2009kHz (Extended Data Fig. 2<\/a>). The measurement was performed with a single 171Yb+ ion by applying a voltage V0+\u03b4V to the electrode, where V0\u2009=\u20091.75\u2009V is the static voltage contributing to the axial confining potential and \u03b4V is an offset that is varied from \u221250 to +50\u2009mV. We extract the value of \u2202B\/\u2202V from a least squares fit to a straight line of the magnetic field measurements for each voltage offset. From this, we then determine<\/p>\n $$\\frac{\\partial \\omega}{\\partial {{V}}}=\\frac{\\partial {{B}}}{\\partial {{V}}}\\frac{\\partial \\omega}{\\partial {{B}}}=-382\\times 1{0}^{3}\\,{\\rm{rad}}\\,{{\\rm{V}}}^{-1}.$$<\/p>\n The geometric factor is then calculated from equation (16<\/a>), giving \u03b1\u2009=\u2009\u221295.64(4)\u2009m\u22121.<\/p>\n The transduction parameter is found using<\/p>\n $$\\gamma =\\frac{1}{\\alpha }\\frac{\\partial \\omega }{\\partial {{V}}}=\\left(\\frac{\\partial {{V}}}{\\partial {{E}}}\\frac{\\partial \\omega }{\\partial {{V}}}\\right).$$<\/p>\n For the second-order magnetic field sensitive transition, we measure \u03b3\u2009=\u20093,998(2)\u2009rad\u2009m\u2009V\u22121.<\/p>\n Our scheme measures the electric field component along the z axis, as the sensitivities to electric fields in the x and y axes are negligible. To see this, we calculate the ratio between the transduction parameter in the z direction, \u03b3\u2009=\u2009\u03b3z, and the transduction parameter in the x and y directions, \u03b3x,y, using equation (2<\/a>). The magnetic field gradient along the z axis was measured to be \u2202B\/\u2202z\u2009=\u200922.41(1)\u2009T\u2009m\u22121, whereas the gradient along the x and y axes was estimated through numerical simulations to be \u2202B\/\u2202rx,y\u2009\u2248\u200911\u2009T\u2009m\u22121. With the motional frequencies \u03bdz\/2\u03c0\u2009=\u2009161.191(8)\u2009kHz and \u03bdx,y\/2\u03c0\u2009\u2248\u20091.5\u2009MHz, the ratio of the transduction parameters is<\/p>\n $${\\gamma}_{{\\rm{z}}}\/{\\gamma}_{{\\rm{x}},{\\rm{y}}}=\\frac{\\partial {{B}}}{\\partial {{z}}}\\frac{\\partial {{z}}}{\\partial {{{E}}}_{{\\rm{z}}}}\\left\/\\frac{\\partial {{B}}}{\\partial {{{r}}}_{{{\\rm{x}}},{\\rm{y}}}}\\frac{\\partial {{{r}}}_{{{\\rm{x}}},{\\rm{y}}}}{\\partial {{{E}}}_{{{\\rm{x}}},{\\rm{y}}}}\\approx 180,\\right.$$<\/p>\n which indicates that the sensitivity to electric fields in the radial direction is over two orders of magnitude weaker.<\/p>\n Electric field sensing protocol<\/p>\n For the sensing of a.c. fields, we follow the pulse sequence protocol outlined in ref. 35<\/a> and illustrated in Extended Data Fig. 3<\/a>. The a.c. sensing sequence is realized by first initializing the two-level system into the \\(\\vert +\\rangle =({1}\/{\\sqrt{2}})(\\vert \\downarrow \\rangle +\\vert \\uparrow \\rangle )\\) state using a \u03c0\/2 pulse. The superposition state then evolves under an electric field perturbation for a time \u03c4\/2. A \u03c0 pulse reorients the spin along the equator of the Bloch sphere, before the quantum state again evolves under the electric field perturbation for a time \u03c4\/2. A final \u03c0\/2 pulse mapps the state population into the \u03c3z basis for detection. Using this pulse sequence, the sensitivity of the spin state transition frequency is maximized for a.c. signals oscillating at a frequency of \u03c4\u22121.<\/p>\n The d.c. sensing experiments also use a Hahn echo type pulse sequence, whose benefits are twofold. First, the coherence time of the sensor is greatly extended when compared to that of the Ramsey-type sequence, which allows for increased sensitivities. Second, the refocusing \u03c0 pulse also compensates for detuning errors in the microwave pulses. The pulse sequence is illustrated in Extended Data Fig. 3<\/a>, and begins with a \u03c0\/2 pulse to initialize the spin into the \\(\\vert +\\rangle =({1}\/{\\sqrt{2}})(\\vert \\downarrow \\rangle +\\vert \\uparrow \\rangle )\\) state. d.c. signals cannot be applied through a capacitor. The low-pass filter signal chain of the d.c. electrode is also not suitable for fast application of d.c. square pulses during the sensing pulse sequence, as the low-pass filter would significantly attenuate and distort the signal. Therefore, to quantify the sensor\u2019s response to d.c. signals, we apply an a.c. signal of frequency \u03c4\u22121 for the duration of the first \u03c4\/2 delay time. This corresponded to an equivalent d.c. voltage on the electrode of Vd.c.\u2009=\u2009(2\/\u03c0)VPK, where VPK is the amplitude of the applied signal. Here, (2\/\u03c0)VPK is the average voltage over the half-oscillation of the a.c. waveform. The applied time-varying pulse therefore causes the spin state to accumulate the same amount of phase \u03d5 as a square d.c. pulse of amplitude (2\/\u03c0)VPK applied for a duration \u03c4\/2 based on the equation relating phase accumulation to the detuning of the spin transition: \\(\\phi =\\int_{0}^{{\\tau}\/{2}}\\gamma \\alpha \\delta {{V}}({{t}})\\,\\mathrm{d}t\\). The refocusing \u03c0 pulse is then applied, followed by the second \u03c4\/2 delay time, during which no other voltage signals are applied to the electrode, followed by a final \u03c0\/2 pulse.<\/p>\n In addition to the electric field interaction time \u03c4, the second relevant time parameter from equation (4<\/a>) is tm, which breaks down as follows for our experimental implementation: (1) d.c. offset application delay time, 50\u2009ms (see next section), (2) Doppler cooling and detection, 14.599\u2009ms, (3) state preparation and microwave pulses, 2.155\u2009ms and (4) data processing and field-programmable gate array delays, 85\u2009\u03bcs. The total tm\u2009=\u200966.839\u2009ms.<\/p>\n Capacitive coupling of a.c. signals<\/p>\n Due to the absence of an in-vacuum antenna, the electric field signals measured by the trapped ion were emitted from an in-vacuum end-cap electrode, which also generated a d.c. confinement electric field. Voltage waveforms were generated by an AWG and capacitively coupled onto the electrode across a 220\u2009pF capacitor. Due to their frequency-dependent impedance, capacitors act as high-pass filters, thereby attenuating the lower-frequency signals more strongly. The fixed response time of a capacitor will also shift the phase of a.c. signals that are applied across it. This shift in phase of the a.c. signal can, if unaccounted for, affect the total coherent phase \u03d5 that is accumulated by the spin states. To achieve an optimal measurement of the sensitivity of our experimental system, it is necessary for the electric field signal at the ion to be in phase with the Hahn-echo sensing pulse sequence. This is because \u03d5 is the difference between the coherent phase accrued during the first and second interaction times \u03c4\/2. An electric field signal that is not in phase with the Hahn-echo sequence will, therefore, reduce the measured sensitivity. References 11<\/a> and 35<\/a> provide further information about this effect.<\/p>\n We measure the phase shift on signals applied across the capacitor for the span of frequencies used in the a.c. and d.c. sensing experiments using an oscilloscope. Based on these measurements, we then pre-compensate the signal applied across the capacitor by applying an inverse phase shift, negating the effect of the capacitor on the phase of the voltage waveform. This ensures that the voltage on the electrode and, therefore, the electric field signal at the ion, are in phase with the Hahn-echo sequence.<\/p>\n Shifting the phase of the voltage waveform introduces a discontinuity into the signal. This manifests as a sudden change in the voltage across the capacitor from 0 to V\u03a6\u2009=\u2009VA\u2009sin\u2009\u03a6, where \u03a6 is the phase of the a.c. voltage signal. Given that the current across a capacitor is defined as I\u2009=\u2009C\u2009dV\/dt, where C is the capacitance of the capacitor, the high rate of change of voltage induces a large current flow across the capacitor, which introduces additional coherent phase offsets of the superposition state. To suppress this unwanted perturbation, we apply a d.c. voltage offset of V\u03a6 into the capacitor in the time before the initialization of the |+\u3009 state, which minimized the sudden voltage spike across the capacitor from the phase-shifted a.c. voltage waveform. To ensure that the sensor reaches a steady state before the application of the a.c. electric field signal, an extra 50\u2009ms delay is added between the application time of the d.c. offset and the first resonant microwave pulse. This made up most of the tm time, which was broken down in the previous section. The pre-compensation technique for the a.c. and d.c. sensing pulse sequences is visualized in Extended Data Fig. 3<\/a>, which illustrates both the AWG and in-vacuum electrode voltage evolution throughout the experimental pulse sequence.<\/p>\n We also measure the frequency-dependent attenuation of the capacitor using an oscilloscope. We determine the transfer function of the capacitor by fitting a Butterworth high-pass filter function to these data. We then find the total attenuation of the electric field signal for a given frequency \u03c4\u22121.<\/p>\n Determination of the coherence time<\/p>\n We measure the coherence time of the two-level system using a Hahn-echo experiment. The spin is initialized in the |\u2193\u3009 state, after which a \u03c0\/2 pulse rotates the spin into the |+X\u3009 eigenstate. A refocusing \u03c0 pulse is applied between the two free evolution periods of duration \u03c4\/2. A final \u03c0\/2 pulse maps the state into the \u03c3z basis for detection. Varying the phase of the final pulse from \u22122\u03c0 to 2\u03c0 results in sinusoidal fringes in the probability of measuring |\u2191\u3009. As the free evolution time is increased, decoherence leads to a reduction in the amplitude of these fringes. The coherence time T2 is given by the point at which the fringe contrast reaches e\u22121. As the a.c. and d.c. sensing experiments were also based on the Hahn-echo sequence, the fringe amplitudes from these experiments can also be used for the coherence time measurement. The fringe amplitudes in these three experiments are shown against the free evolution time in Extended Data Fig. 4<\/a>. These data are aggregated and fitted to a Gaussian decay function of the form \u03c7\u22121(t)\u2009=\u2009exp(\u2212t2\/T22) using a least squares fit, yielding a coherence time of T2\u2009=\u2009304(3)\u2009ms.<\/p>\n","protected":false},"excerpt":{"rendered":"Transduction parameter We consider the dynamics of a string of N trapped ions perturbed by an external electric…\n","protected":false},"author":2,"featured_media":185761,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3845],"tags":[11701,76176,11700,11705,11704,3968,11699,11702,11703,74,17844,15190,11112,70,11698,16,15],"class_list":{"0":"post-185760","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-atomic","9":"tag-atomic-and-molecular-physics","10":"tag-classical-and-continuum-physics","11":"tag-complex-systems","12":"tag-condensed-matter-physics","13":"tag-general","14":"tag-mathematical-and-computational-physics","15":"tag-molecular","16":"tag-optical-and-plasma-physics","17":"tag-physics","18":"tag-quantum-mechanics","19":"tag-quantum-metrology","20":"tag-quantum-physics","21":"tag-science","22":"tag-theoretical","23":"tag-uk","24":"tag-united-kingdom"},"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@uk\/114686121959753665","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts\/185760","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/comments?post=185760"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts\/185760\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/media\/185761"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/media?parent=185760"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/categories?post=185760"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/tags?post=185760"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}