Coherent transfer between low-angular momentum and circular Rydberg states
Adrien Signoles, Eva-Katharina Dietsche, Adrien Facon, Dorian Grosso, Serge Haroche, Jean-Michel Raimond, Michel Brune, Sébastien Gleyzes
CCoherent transfer between low-angular momentum and circular Rydberg states
A. Signoles, ∗ E.K. Dietsche, A. Facon, D. Grosso, S. Haroche, J.M. Raimond, M. Brune, and S. Gleyzes † Laboratoire Kastler Brossel, Coll`ege de France, CNRS,ENS-PSL Research University, UPMC-Sorbonne Universit´es,11, place Marcelin Berthelot, 75231 Paris Cedex 05, France (Dated: March 20, 2017)We realize a coherent transfer between a laser-accessible low-angular-momentum Rydberg stateand the circular Rydberg level with maximal angular momentum. This transfer is induced by aradiofrequency field with a high-purity σ + polarization tuned at resonance on Stark transitionsinside the hydrogenic Rydberg manifold. We observe over a few microseconds more than twentycoherent Rabi oscillations between the initial Rydberg state and the circular Rydberg level. Wecharacterize in details these complex oscillations involving many Rydberg levels and find them tobe in perfect agreement with a simple theoretical model. This coherent transfer procedure opensthe way to hybrid quantum gates bridging the gap between optical communication and quantuminformation manipulations based on microwave Cavity and Circuit Quantum Electrodynamics. The long-lived Circular Rydberg Levels (CRLs) areideal tools for quantum manipulation of microwave (mw)fields stored in ultra-high-Q 3D superconducting res-onators. They led to early demonstrations of basic quan-tum information processing operations [1] and to thegeneration of non-classical field state superpositions [2].More recently, they have been instrumental in the explo-ration of fundamental Cavity Quantum Electrodynamics(Cavity-QED) effects, such as QND measurements of thephoton number [3] and quantum feedback [4]. Recent ad-vances on the manipulation of Rydberg atoms near atomchips [5] indicate that they could also be interfaced withthe resonant structures used in the flourishing field ofCircuit-QED [6, 7].However, the photons used in mw Cavity- and Circuit-QED are unable to propagate over long-range transmis-sion lines [8]. Optical to mw interfaces are thus the fo-cus of an intense activity [9, 10]. A new realm for mwquantum information manipulation would open if theCRLs could be coherently interfaced with optical pho-tons, which are ideal quantum information carriers overfiber and free-space communication networks [11]. Un-fortunately, the CRLs, with their large orbital quantumnumber (cid:96) = n − n is the principal quantum number),do not couple directly to optical photons.In contrast, low-angular momentum Rydberg statesare accessible from the ground state by coherent laserexcitation [12–14]. They were recently used for op-tical quantum information processing operations suchas photon-photon gates relying on single-photon opti-cal non-linearities induced by the dipole blockade mech-anism [15–18]. They could also lead to quantum gatesentangling a mw photon with a collective hyperfine exci-tation in a ground state atomic ensemble [19] and hence,through the DLCZ protocol [20], to gates entangling opti-cal and mw photons. However, the short lifetime of theselevels, of the order of 100 µ s, sets limits on the quantumtransfer fidelity and on their use in CQED experiments.The missing link between mw and optical photons is a fast coherent transfer from a laser-accessible low- (cid:96) stateto the CRLs. The most efficient CRL preparation tech-nique so far involves a series of radiofrequency (rf) tran-sitions between Stark levels performed in an rapid adia-batic passage [21, 22]. It thus requires a relatively longtime, much longer than the typical Rabi frequency on therf transitions. Therefore, the adiabatic rapid passage re-sults in the accumulation of large dynamic phases. Theirunavoidable experimental fluctuations affect the coher-ence of the process. The crossed fields CRLs preparationtechnique [23, 24], also based on an adiabatic passage,suffers from similar limitations.In this Letter we demonstrate a fast, coherent trans-fer between a low- (cid:96) state and the CRL. By coupling aRydberg atom to a rf field with a well-defined polariza-tion, we isolate in the Stark manifold with the principalquantum number n , a subset of states behaving as a largeangular momentum J ∼ n/
2. We resonantly drive thisangular momentum and observe more than 20 coherentRabi oscillations between the lowest (low- m ) and upper-most (circular) energy states.The experiment is performed on Rb atoms in a staticelectric field F = F z , which partly lifts the degeneracyof the Rydberg manifold. Its levels can be sorted bytheir magnetic quantum number m (restricted to posi-tive values here) [25]. For a Hydrogen atom (see Fig. 1a),the Stark eigenstates are the parabolic states | n, n , m (cid:105) ,where 0 ≤ n ≤ n − m − m from bottom to top in a regularly spaced lad-der. The lowest level of each m ladder ( n = 0) is mid-way between the lowest ones in the m − σ + rf transitions between these lowest levels are thus allat the same angular frequency ω n = 3 / · nF ea / (cid:126) [26]( e : elementary charge, a : Bohr radius; ω n / π ≈ F = 1 V/cm and n ≈ σ + rf field cou-ples all the n = 0 states from m = 0 up to the CRL.The subspace spanned by these levels is that of an an-gular momentum ˆ J , with J = ( n − / | J, m J = J (cid:105) (“angular mo- a r X i v : . [ qu a n t - ph ] M a r FIG. 1. (color online) (a) Scheme of the Hydrogen Rydberglevels in the n -th manifold in the presence of a static electricfield (only states with m ≥ σ + driving fieldcouples all the n = 0 levels (in red) from m = 0 to theCRL. (b) Scheme of Rb Rydberg levels. A ∼ / cm electricfield lifts the degeneracy of the n -th Stark manifold. Stateswith m ≥ m levels are affectedby quantum defects. The ground state atoms are excited tothe lower m = 2 state of the manifold. A σ + polarized rffield couples | n, i (cid:105) to an angular momentum-like subspace D (red levels) via ∆ m = +1 transitions, leading to a transfertoward the CRL | n, c (cid:105) (blue arrow). The population of eachStark level can be probed by selective mw transitions towardscorresponding level in the n − ∼ µ s time-of-flight the atoms are detectedoutside the structure by field ionization. mentum up”, north pole on a generalized Bloch sphere)and the lowest- m state | n, , (cid:105) is | J, m J = − J (cid:105) (“angu-lar momentum down”, south pole) [28, 29]. A resonantrf-drive at angular frequency ω rf = ω n induces a rotationof ˆ J between the north and south poles at a frequencyΩ rf / π = 3 / nea E + rf /h , where E + rf is the σ + polarized rfelectric field amplitude. This rotation maps the angularmomentum down state into the CRL in a time π/ Ω rf .For Rubidium (Fig. 1b), the levels with m < m ≥ σ + transitions (blue arrow on Fig. 1b). For m = 2, the | n, n = 0 , m = 2 (cid:105) state is shifted far awayfrom the hydrogenic multiplicity. However, a fortunatecoincidence makes the rf transition between the laser-accessible | n, n = 1 , m = 2 (cid:105) = | n, i (cid:105) state and the firsthydrogenic | n, n = 0 , m = 3 (cid:105) = | n, j (cid:105) level nearly degen- FIG. 2. (color online) (a) Rabi oscillation induced by the σ − polarization component of the rf field on the | , i (cid:105) →| , i (cid:48) (cid:105) transition. Population P i (cid:48) of | , i (cid:48) (cid:105) versus the rf pulseduration t (dots). We fit the frequency of the oscillation (redline) to Ω − ii (cid:48) / π = 107 ±
25 kHz. (b) Autler-Townes doubletof the | , i (cid:105) → | , i (cid:48) (cid:105) transition dressed by the σ + rf field.Points are experimental and the red lines are Gaussian fitsleading to Ω + ii (cid:48) / π = 30 . erate with all other transitions toward | n, c (cid:105) . Hence, onlythe two lowest states m = 0 , | n, i (cid:105) to the CRL is a nearly π -rotation ofthe hydrogenic angular momentum ˆ J , through a subset D of its levels (in red on Fig. 1b).The experiment takes place in a structure made up ofa capacitor producing F , surrounded by four ring elec-trodes, on which we apply the rf signal at ω rf / π = 230MHz and dc potentials (Fig. 1c). It is crossed by a Ru-bidium thermal beam (atomic velocity 250 m/s). Atomsare excited from the 5 S ground state by three laser beamsresonant on the transitions 5 S / → P / (780 nm),5 P / → D / (776 nm) and 5 D / → nF, m = 2(1258 nm). The 780 nm and 776 nm laser beams are σ + -polarized and collinear with the quantization axis,defined by a dc electric field (0 .
23 V / cm) applied acrossthe ring electrodes during the 1 µ s pulsed laser excitation.The 1258 nm laser beam is π -polarized and perpendicularto the others. After laser excitation, a potential appliedto the capacitor is switched-on in 1 µ s, resulting in afield F = F z with F ∼ | n, i (cid:105) [22].At the end of the sequence, the atoms drift out of thecapacitor towards a field-ionization detector, which re-solves the initial and circular Rydberg states but doesn’tresolve them from levels with similar energies. In or-der to selectively detect neighboring levels | n, p (cid:105) with p = i, i (cid:48) , j, k, l, c, d, e, f, g (Fig. 1b) we use selective mwprobe pulses, which transfer | n, p (cid:105) into another manifold,easily resolved by the field ionization detector.The rf transfer from | n, i (cid:105) to | n, c (cid:105) requires a pure rf σ + polarization, in order to avoid transitions with ∆ m = − D . By applying onthe four ring electrodes 230 MHz signals with the properphases and amplitudes, we can in principle generate apure σ + field in the center of the structure.Due to the difference in the electrode driving linetransmission and the capacitive coupling between theelectrodes, these phases and amplitudes cannot be pre-determined. We therefore use an optimization pro-cedure, based on the direct measurement of the σ + and σ − amplitude. The σ − rf component is measuredby tuning the electric field to F = 1 .
76 V/cm, sothat the transition between | n = 52 , n = 1 , m = 2 (cid:105) and | n = 52 , n = 3 , m = 1 (cid:105) ( | , i (cid:105) and | , i (cid:48) (cid:105) on figure 1b),coupled to the σ − field, is resonant at 230 MHz. All othertransitions from | , i (cid:105) or | , i (cid:48) (cid:105) are then out of resonancedue to the quantum defects. The rf thus induces Rabi os-cillation at a frequency Ω − ii (cid:48) / π = √ d ii (cid:48) E − rf /h where E − rf is the amplitude of the σ − component and d ii (cid:48) the dipolematrix element of this transition. In order to measurethe σ + field amplitude, we align the electric field alongthe − z direction instead of + z . The atom is then pre-pared in the m = +2 state with respect to the reversed Oz axis (noted − Oz ). The transition between | , i (cid:105) and | , i (cid:48) (cid:105) is driven by the σ − polarization w.r.t. − Oz , i.e.the σ + polarization w.r.t. Oz .The full optimization procedure is described in [30].Fig. 2a presents the slow Rabi oscillation between | , i (cid:105) and | , i (cid:48) (cid:105) due to the weak residual σ − component. Afit (solid red line) yields a Rabi oscillation frequencyΩ − ii (cid:48) / π = 107 ±
25 kHz. Fig. 2b presents the Autler-Townes doublet induced by the strong σ + componentprobed on the | , i (cid:105) to | , n = 0 , m = 3 (cid:105) mw tran-sition. We get Ω + ii (cid:48) / π = √ d ii (cid:48) E + rf /h = 30 . E − rf / (( E + rf ) + ( E − rf ) ) / = 0 . ± .
08 %. It is consistentwith the spatial inhomogeneity of the rf field over the ex-tension of the atomic sample (about one mm) calculatedusing the SIMION software package.As a first test, we use this optimized 230 MHz rf fieldto prepare the CRL by the standard adiabatic transfermethod [22]. To allow us to use calibrated mw probes[30], we perform this experiment in the 51 manifold. Weinitially prepare the atoms in | , i (cid:105) and set the elec-tric field to F = 2 .
45 V / cm so that δ = ω − ω rf =2 π ·
10 MHz. The rf field power is ramped up in 1 µ s. Theelectric field F is then linearly decreased to 2 .
24 V / cm( δ = − π ·
10 MHz) in 1 . µ s, so that ω crosses the res-onance. The rf is finally switched off in 1 µ s. We recordthe number of detected atoms as a function of the ion-ization field (Fig. 3a). The cyan dashed line presents theionization signal of the initial | , i (cid:105) level. The black lineshows the signal when the adiabatic passage is performed.It is now centered around the ionization field of the CRL | , c (cid:105) . From the residual number of atom detected atthe ionization field of the | , i (cid:105) level, we estimate thatthe transfer efficiency is larger than 98%. The differenceof height between the cyan and black signals providesthe relative detection efficiency, η = 0 .
23, between the | , i (cid:105) and | , c (cid:105) levels. This low value is mainly due tothe difference of lifetimes between the levels.We investigate the adiabatic transfer as a function of ... ... . . .
40 60 80 100 120 140 1600.00.81.6 0 1 2 3 40.010.11
FIG. 3. (color online) Adiabatic passage in the n = 51manifold. (a) Field ionization spectra (signal versus thefield applied in the ionization detector) of the initial state | , i (cid:105) (dashed cyan line) and at the end of the adiabaticpassage sequence (solid black line). (b) Populations P p for p = i, c, d, e, f, g of the states | , p (cid:105) (color code for the levelsidentification in the inset) as a function of the Rabi frequencyΩ rf / π . Dots are experimental with statistical error bars.The solid lines are the result of a numerical simulation of thefull Hamiltonian evolution. Ω rf . The populations of the levels c, d, e, f, g (points inFig. 3 with statistical error bars) are measured with cal-ibrated mw probes [30] as a function of the applied rfamplitude, proportional to E + rf . The conversion factor isdetermined from the measurement of Ω + ii (cid:48) (Fig. 2). Thedata are in excellent agreement with the result of a nu-merical simulation of the ideal evolution in a pure σ + rffield (solid lines). For Ω rf > d, e, f, g being at most 5% (note that a partof this spurious population is overestimated due to mwprobe imperfections). This is another demonstration ofthe excellent rf polarization control achieved here.We now investigate the circular state preparation inthe resonant rf regime, which provides the coherent trans-fer between a low- (cid:96) state and the CRL. The field F isadjusted to 2 .
35 V / cm corresponding to ω ≈ ω rf . Weselect a fixed rf amplitude. We scan the pulse duration∆ t rf and measure the state populations as above. Thecircular state population P c (dots in Fig. 4a with sta-tistical error bars) exhibits periodic peaks. They revealmore than 20 full Rabi oscillations over 6 µ s, demonstrat-ing the coherence of the atomic evolution. A fit of thesedata with a numerical simulation of the ideal Hamilto-nian evolution in a pure σ + rf field (solid line) providesa calibration of the rf Rabi frequency Ω rf = 3 .
52 MHzand of the effective rf pulse duration, which is slightly
FIG. 4. (color online) Dynamics of the atomic state for a rf pulse duration ∆ t rf . (a) Long timescale evolution of the CRLpopulation, P c (dots). The solid line is the result of a numerical simulation, with Ω rf / π = 3 .
52 MHz and a rf pulse shortenedby 68 ns to model the rise and fall times of the driving electronics. (b) Populations P p for p = c, d, e, f, g around the secondmaximum of P c (same color conventions as in Fig.2). Dots are experimental and solid lines are the results of the simulation.(c) Calculated (bars) and measured (open circles) populations P p at the top of the second CRL population maximum. (d)Enlargement of (a) around three late CRL population maxima. (e) Calculated (bars) and measured (open circles) populations P p at the top of the first CRL population maximum in (d). For all panels, the experimental error bars correspond to thestatistical fluctuations. different ( ≈
70 ns) from ∆ t rf due to the finite rise andfall times of the electronic drive.On Fig. 4b we plot the populations P p for p = i, c, d, e, f, g around the second CRL population max-imum. We observe for P d and P e the theoreticallyexpected double peak structure, typical of a multi-level Rabi oscillation. Fig. 4c presents the calculated(bars) and measured (open circles) population at ∆ t rf =0 . µ s. The deviations w.r.t the hydrogen atom modelbecome more important with time, since the quantum de-fects affect the evolution each time the atom returns tolow- m states. Fig 4d shows an enlargement of the evolu-tion at long times (dots) together with the numerical pre-diction (solid line) and Fig 4e the calculated (bars) andmeasured (open circles) populations at ∆ t rf = 4 . µ s.The agreement between theory and experiment remainsexcellent up to large time intervals ∆ t rf , demonstratingthe coherence of the oscillation over long time scales.This experiment shows that it is possible to perform anefficient coherent Rabi oscillation from the low- m state | n, i (cid:105) to the CRL | n, c (cid:105) within 200 ns. The limited trans-fer rate ( (cid:39) m = 0 , m = 2. These residual imperfectionscould be reduced by tailoring the amplitude and the fre-quency of the rf field via optimal control [31] .We envision a phase gate between an optical and a mw photon, along the lines proposed in [19] for low-angularmomentum Rydberg states. It relies on a dense ensem-ble of ground state Rb atoms prepared in their F = 1hyperfine ground state, and held by a laser dipole trapinside a high-quality open Fabry Perot mw resonator [1]tuned to the transition between | , c (cid:105) and | , c (cid:105) . Anincoming optical photon is transformed into a collectiveexcitation in the F (cid:48) = 2 ground state [20]. A laser pulsecoherently transfers this excitation into the | , i (cid:105) Ryd-berg level, immediately cast onto the 50 circular state.The atom is then coupled to the cavity for the durationof a resonant 2 π Rabi rotation on the 50 to 51 transi-tion. If the cavity contains one mw photon, this resultsin a conditional π phase shift for the atom [32]. This π shift is brought back to the F (cid:48) ground state by the time-reversed excitation process and converted back into anoptical photon. All the components of this innovative hy-brid photon-photon quantum gate have now been testedseparately, including the transfer between the low- (cid:96) andthe CRL. Assembling these components would bridge thegap between optical quantum communication and quan-tum information manipulations based on mw cavity andcircuit quantum electrodynamics.We acknowledge funding by the EU under the ERCprojet ‘DECLIC’ (Project ID: 246932) and the RIAproject ‘RYSQ’ (Project ID: 640378). ∗ Present address: Physikalisches Instit¨ut, Universit¨at Hei-delberg, Im Neuenheimer Feld 226, 69120, Heidelberg,Germany. † [email protected][1] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod.Phys. , 565 (2001).[2] M. Brune et al. , Phys. Rev. Lett. , 4887 (1996).[3] C. Guerlin et al. , Nature , 889 (2007).[4] C. Sayrin et al. , Nature , 73 (2011).[5] R. C. Teixeira et al. , Phys. Rev. Lett. , 013001 (2015).[6] S. D. Hogan et al. , Phys. Rev. Lett. , 063004 (2012).[7] M. H. Devoret and R. J. Schoelkopf, Science , 1169(2013).[8] H. Wang et al. , Phys. Rev. Lett. , 060401 (2011).[9] D. Maxwell et al. , Phys. Rev. A , 043827 (2014).[10] R. W. Andrews et al. , Nat. Phys. , 321 (2014).[11] J. I. Cirac and H. J. Kimble, Nature Photonics , 18(2017).[12] T. A. Johnson et al. , Phys. Rev. Lett. , 113003(2008).[13] M. Reetz-Lamour, T. Amthor, J. Deiglmayr, and M. Wei-dem¨uller, Phys. Rev. Lett. , 253001 (2008).[14] Y. Miroshnychenko et al. , Phys. Rev. A , 013405(2010).[15] T. Peyronel et al. , Nature , 57 (2012).[16] V. Parigi et al. , Phys. Rev. Lett. , 233602 (2012).[17] S. Baur, D. Tiarks, G. Rempe, and S. D¨urr, Phys. Rev.Lett. , 073901 (2014).[18] D. Paredes-Barato and C. Adams, Phys. Rev. Lett. ,040501 (2014).[19] J. D. Pritchard, J. A. Isaacs, M. A. Beck, R. McDermott,and M. Saffman, Phys. Rev. A - At. Mol. Opt. Phys. ,1 (2014), 1310.3910.[20] L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller,Nature , 413 (2001).[21] R. G. Hulet and D. Kleppner, Phys. Rev. Lett. , 1430(1983).[22] P. Nussenzveig et al. , Phys. Rev. A , 3991 (1993).[23] D. Delande and J. Gay, EPL (Europhysics Letters) ,303 (1988).[24] J. Hare, M. Gross, and P. Goy, Phys. Rev. Lett. , 1938(1988).[25] T. F. Gallagher, Rydberg Atoms
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We present here the procedure that we followed tooptimize the polarization of the rf created by the ringelectrodes, and the calibration of the probes that weused to measure the population of the Stark levels | , p (cid:105) ( p = c, d, e, f, g ). A. Polarization optimization procedure
In order to generate a high-purity σ + polarized rf fieldwe optimize the phases and amplitudes V i of the rf drivesapplied on each of the four ring electrodes ( i labeling theelectrodes from 1 to 4, 1 and 3 on the one hand and 2and 4 on the other being diametrically opposite). Thefinal adjustment should minimize the σ − component andproduce a much larger σ + field with a good spatial ho-mogeneity around the centre of the structure. This ad-justment must be impervious to the imperfections dueto the complex standing waves in the transmission lines,to the cross-talks between the capacitively-coupled elec-trodes and to the imperfections of the geometry.The σ ± components are measured as explained in themain text. The Rabi frequencies are measured using ei-ther the spectroscopy of the Autler Townes doublet in-duced on a probe mw transition or the direct temporalmeasurement of the induced Rabi oscillation.We first apply the rf on only one electrode and measurethe amplitude of the σ + field it generates when all theothers are grounded. We then adjust V so that electrode3 generates a σ + field with the same amplitude as elec-trode 1, and V so that electrode 4 generates a σ + fieldwith the same amplitude as electrode 2. We finally setthe relative phase between 1 and 3 (respectively 2 and4), in order to maximize the amplitude of the σ + fieldgenerated when electrodes 1 and 3 (2 and 4) are drivensimultaneously.We now cancel the total σ − component. We first mea-sure the amplitude of the σ − component created by eachpair of electrodes (pair 1-3 or pair 2-4) separately. Wescale V and V together so that pair 2-4 generates thesame σ − amplitude as pair 1-3. Finally, driving the fourelectrodes, we set the relative phases between the twopairs to minimize the global σ − amplitude.From this initial setting, the σ + field amplitude is var-ied by scaling with the same factor all the amplitudesapplied on the four electrodes by means of identical vari-ables attenuators driven by a common signal. B. Calibration of the microwave probes efficiency
In order to measure the populations of the Stark levels | , p (cid:105) ( p = c, d, e, f, g ) we use mw probes tuned to thetwo-photon transitions from | , p (cid:105) to | , p (cid:105) , which areresolved in the applied electric field of 3,8 V/cm. Foreach transition, we perform a π pulse ideally transfer-ring all the population of | , p (cid:105) into | , p (cid:105) . In practiceeach π pulse has a limited efficiency. Moreover, the de-tection efficiency of levels | , p (cid:105) and | , p (cid:105) is different.We denote by η p the overall transfer efficiency between | , p (cid:105) and | , p (cid:105) . We estimate it by preparing selectively | , p (cid:105) and measuring the fraction of atoms detected atthe ionization threshold of | , p (cid:105) after a probe pulse.We prepare | , p (cid:105) by first exciting | , i (cid:105) or | , j (cid:105) witha laser pulse. We then transfer the atoms into the level | , q (cid:105) ( q = i, j, k, l ) using a mw pulse. We finally applya rf adiabatic passage, which ideally maps | , q (cid:105) with q = i, j, k, l into | , p (cid:105) with p = c, d, e, f respectively. Due to imperfections in the preparation process, thestates neighboring | , p (cid:105) are also slightly populated,which leads to underestimate η p . For levels c, d and e ,theses spurious population are small and the | , d (cid:105) and | , e (cid:105) populations in figures 3 and 4 are slightly over-estimated. The preparation of f leads to comparablepopulations in f and g . We use them both to get anunderestimated value of η f and η g .The probability P p to end-up in the state | , p (cid:105) is then P p = η η p N (49 , p ) N (51 , i ) , where N (49 , p ) is the number of atom detected in | , p (cid:105) after the mw probe pulse, and N (51 , i ) is the initial pop-ulation in the laser-excited | , i (cid:105)(cid:105)