Long-lived circular Rydberg states of laser-cooled Rubidium atoms in a cryostat
Tigrane Cantat-Moltrecht, Rodrigo Cortiñas, Brice Ravon, Paul Méhaignerie, Serge Haroche, Jean-Michel Raimond, Maxime Favier, Michel Brune, Clément Sayrin
LLong-lived circular Rydberg states of laser-cooled Rubidium atoms in a cryostat
T. Cantat-Moltrecht, ∗ R. Cortiñas, ∗ B. Ravon, P. Méhaignerie, S. Haroche, J. M. Raimond, M. Favier, M. Brune, and C. Sayrin † Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne-Université11 place Marcelin Berthelot, 75005 Paris, France (Dated: March 13, 2020)The exquisite properties of Rydberg levels make them particularly appealing for emerging quan-tum technologies. The lifetime of low-angular-momentum laser-accessible levels is however limitedto a few µ s by optical transitions and microwave blackbody radiation (BBR) induced transfersat room temperature. A considerable improvement would be obtained with the few
10 ms lifetimeof circular Rydberg levels in a cryogenic environment reducing the BBR temperature. We demon-strate the preparation of long-lived circular Rydberg levels of laser-cooled Rubidium atoms in acryostat. We observe a . lifetime for the circular level of principal quantum number n = 52 .By monitoring the transfers between adjacent circular levels, we estimate in situ the microwaveBBR temperature to be (11 ±
2) K . The measured atomic coherence time ( µ s ) is limited hereonly by technical magnetic field fluctuations. This work opens interesting perspectives for quantumsimulation and sensing with cold circular Rydberg atoms. Rydberg atoms, i.e. atoms prepared in levels with ahigh principal quantum number n , are the focus of a re-newed interest [1, 2]. Their large coupling to electromag-netic fields, their large mutual dipole-dipole interactionsand their long lifetimes make them ideally suited to cav-ity quantum electrodynamics [3, 4], quantum sensing [5–7], quantum information [1, 8] and quantum simulationsof spin systems [2, 9–12]. However, the laser-accessiblelow-orbital-angular-momentum ( (cid:96) ) Rydberg levels life-time, mainly determined by optical transitions, limitsevolution and measurement times to a few µ s .The circular Rydberg levels | n C (cid:105) , with maximum or-bital angular momentum ( (cid:96) = | m | = n − ), have amuch longer natural lifetime, of
30 ms for n ≈ . Itcan even be increased by several orders of magnitudeinside a spontaneous-emission inhibiting structure [13].Moreover, circular Rydberg atoms (CRAs) are immuneto photoionization, in contrast to low- (cid:96) Rydberg lev-els [14]. Thus, they can be efficiently laser-trapped overlong times [15]. All these features make it possible toreach long interaction times with microwave cavities [16],enabling powerful manipulations of the field quantumstate in cavity QED [17]. Longer interrogation times inquantum sensing can provide higher sensitivities to theprobed fields [18]. Longer simulation times would allowquantum simulators to explore slow dynamics, e.g. , ther-malization [19].All experiments with Rydberg atoms, however, mustface the problem of population transfer induced by res-onant microwave blackbody radiation (BBR). At roomtemperature, the high number of BBR photons per mode(100 photons at
50 GHz ) significantly reduces the Ryd-berg levels lifetimes, particularly for the circular ones (by ∗ These authors equally contributed to this work. † Corresponding author; Electronic address:[email protected] two orders of magnitude for | (cid:105) ). Moreover, for low- (cid:96) levels, spurious transfers to nearby states create impu-rities in the atomic system, detrimental to experimentsdressing ground-state atoms with Rydberg levels [20, 21].Fully harnessing the long lifetimes of Rydberg levels thusrequires a cryogenic environment.Here, we demonstrate the preparation in a cryostatof the n = 52 circular Rydberg state from laser-cooledRubidium atoms at a µ K temperature. We measurea (3 . ± .
1) ms lifetime for | (cid:105) . By monitoring thepopulation transfers between adjacent circular levels, weestimate in situ the microwave BBR temperature to be (11 ±
2) K . We also measure the coherence time of amicrowave transition between circular levels and identifyits current purely technical limitations.The experimental setup [15, 22], sketched on Fig. 1,is enclosed in a wet He optical cryostat. Room-temperature microwave radiation enters the cryostatthrough SF-57 optical ports (total surface
16 cm ). Inorder to keep the microwave BBR temperature as low aspossible, we have installed
355 cm of RAM (radar ab-sorption material) plates [23] inside the copper ther-mal shield. From a simple balance of the respective sur-faces of RAM and optical ports, we get a rough a prioriestimate of the BBR temperature of
17 K .Rubidium-87 atoms are laser-cooled and trapped in a3D mirror magneto-optical trap (MOT) created in frontof a Rubidium-coated metallic mirror [24]. An atomicbeam that streams out of a 2D MOT along the z -axis(axes definition in Fig. 1) loads the 3D MOT. Atomsare further cooled down to about µ K via an opti-cal molasses stage that optically pumps them into the (cid:12)(cid:12) / , F = 2 (cid:11) ground state.We excite the atoms from (cid:12)(cid:12) / , F = 2 , m F = 2 (cid:11) into (cid:12)(cid:12) / , m J = 5 / (cid:11) at time t = 0 by a two-photonlaser-excitation (780 and
480 nm wavelengths). The
780 nm -wavelength red laser beam, blue-detuned from (cid:12)(cid:12) / , F (cid:48) = 3 , m F = 3 (cid:11) by
560 MHz , has a waist of a r X i v : . [ phy s i c s . a t o m - ph ] M a r C FIG. 1: a. Experimental setup with axes definition. Theblue and red excitation laser beams cross in the cold atomcloud (green) from the surface of the MOT mirror. Theelectrode V S applies the electric field F and the ionizationfield. The Rb + ions (violet dashed line) are guided to thechanneltron C by the deflection electrodes V d . Four addi-tional compensation electrodes (only two, RF1 and RF2, areshown) control electric field gradients and apply the circularstate preparation RF field. b. Simplified level scheme. Left:two-photon laser excitation (solid arrows) and circular statepreparation (dotted arrow). Right: Partial diagram of theStark levels sorted by m values. The green arrows indicatethe one- and two-photon transitions used in the coherencemeasurements. µ m and is perpendicular to the mirror surface. Itcrosses the
480 nm -wavelength blue laser beam ( µ m waist), that propagates along the x -axis, about away from the mirror surface. Both lasers are pulsed( µ s pulse duration).The excitation to the low- (cid:96) Rydberg level (cid:12)(cid:12) / , m J = 5 / (cid:11) takes place in a F = 0 .
36 V / cm electric field lifting the degeneracy between the m J levels. The electric field, F , is aligned along the y -axisand is defined by the electrode V S and the groundedMOT metallic mirror. Four compensation electrodes,arranged on a -side square surrounding the Rydberg-excitation region, are used to minimize the electric fieldgradients, measured by microwave spectroscopy.We then efficiently transfer the atoms into the | , m = 2 (cid:105) level in an adiabatic process, starting at t = 14 µ s . We shine a µ s -long .
85 GHz -frequencymicrowave pulse while raising the electric field up to F = 1 . / cm . The microwave is on resonance withthe (cid:12)(cid:12) / , m J = 5 / (cid:11) → | , m = 2 (cid:105) transition when F = 1 .
75 V / cm . The preparation of | (cid:105) ( n = 52 , m =
50 75 100 125 150 17523 47 70 94 117 1410.000.020.040.06 F D Ionization field (V/cm)Ionization time ( μ s) C o un t s / μ s C C C C C C C FIG. 2: Ionization spectra of two low- (cid:96) levels( (cid:12)(cid:12) / , m J = 5 / (cid:11) and | , m F = 2 (cid:105) ) and circularlevels | n C , ≤ n ≤ (cid:105) . The number of detected counts per µ s is plotted versus the ionization field (bottom horizontalaxis). The upper horizontal axis gives the time delay fromthe beginning of the ionization ramp. The shaded areasdepict the integration windows used for the measurements ofthe N n s. Each curve is the result of the average over ≈ experimental runs. +51 ) then requires the transfer of 49 orbital momen-tum quanta. At t = 25 µ s , we raise the electric fieldto F = 2 . / cm and turn on a σ + -polarized radio-frequency (RF) field at ν RF = 230 MHz . It is on res-onance with the m → m + 1 transitions in the Starkmanifold when F = 2 . / cm . By scanning the electricfield down to F = 2 . / cm in µ s , we perform a secondadiabatic transfer from | , m = 2 (cid:105) to | (cid:105) [25]. TheRF field is produced by applying electric potentials at ν RF on two adjacent compensation electrodes. The pre-cise tuning of the amplitudes and relative phase of thesepotentials makes it possible to cancel the σ − RF polariza-tion component at the position of the atoms. Once | (cid:105) is prepared, we reduce electric field to F = 1 . / cm at t = 34 µ s .We finally measure the population of individual Ry-dberg levels by state-selective field ionization. We ap-ply with electrode V S a µ s -long electric field rampthat successively ionizes the Rydberg levels. The Rb + ions are then guided to a channeltron using the deflec-tion electrode V d (Fig.1) [22]. By recording the arrivaltimes of the ions, we recover the ionization spectrum ofthe Rydberg atom cloud. The complete sequence is re-peated several hundred times on the same initial coldatom cloud, for a total duration of ≈ during whichthe optical molasses is kept on.Figure 2 shows the ionization spectra recorded afterthe preparation of (cid:12)(cid:12) / , m J = 5 / (cid:11) , | , m = 2 (cid:105) and | (cid:105) . The ionization ramp is triggered at t = t ion =200 µ s . It is apparent that the field ionization distin-guishes well low- from high- (cid:96) Rydberg levels. We alsoprepared all circular levels with ≤ n ≤ by apply-ing µ s -long single- or multi-photon microwave π -pulses (a) (b) FIG. 3: (a) Decay of circular Rydberg atoms initially prepared in | (cid:105) . Ionization spectra are plotted as a function of the delaytime τ . The color shadings indicate the integration windows for the measurement of the circular level normalized populations, Π n s ( ≤ n ≤ ). Population transfer on the | n C (cid:105) ↔ | ( n − (cid:105) transitions (frequencies ν n = 40 . , . , . , . , . , . and . for n = 55 to , respectively) is apparent. (b) Time evolution of the populations Π n s, from the data in (a), as a function of τ . Circles are experimental, solid lines are the results of their best fit to therate equation model, with T MW = 11 . µ K . The dashed line is a fit of Π to an exponential decay of time constant . .Statistical error bars are smaller than the data points. In (a) and (b), each curve and data point is the result of the averageover ≈ experimental runs. Panels (a) and (b) and Fig. 2 share the same color code. on the | (cid:105) → | n C (cid:105) transitions, at frequencies ν MW =52 . , . , . , . and . for n = 48 to , respectively. The ionization spectra inFig. 2 reveal that our measurement distinguishes effi-ciently between circular Rydberg levels with different n values.Elliptical levels with high m ( m (cid:46) ), spuriously pre-pared by the imperfections of the circularization process,are not addressed by the π microwave pulses in the ap-plied electric field. They are left in the n = 52 manifoldwhen we prepare | ( n (cid:54) = 52)C (cid:105) and ionized in nearly thesame field as | (cid:105) ( F = 123 V / cm ). From the area ofthese residual peaks, we estimate a lower bound of the | (cid:105) purity to be . Most of the spurious popula-tion is attributed to elliptical levels with m ≥ n − ,close to the circular state. For all relevant circular Ry-dberg levels | n C (cid:105) , we determine the total number ofdetected atoms N n for each excitation pulse by inte-grating the ionization signal, corrected for “dark” counts( ≈ . · − count / µ s ), over the field windows pictoriallyshown in Fig. 2. We observe that, within experimentaluncertainties, the detection efficiencies of all these circu-lar Rydberg levels are identical.At T = 0 K , the circular state | n C (cid:105) decays by sponta-neous emission on the single | n C (cid:105) → | ( n − (cid:105) transi-tion, at frequency ν n , with a free-space rate γ n ( γ − ≈
35 ms ). At a finite microwave BBR temperature T MW ,stimulated emission on this transition [rate n ph ( ν n ) γ n ]and absorption on the | n C (cid:105) → | ( n + 1)C (cid:105) transition [rate n ph ( ν n +1 ) γ n +1 ] reduce the lifetime. Here n ph ( ν n ) is thenumber of BBR photons at frequency ν n . Note that ab-sorption on transitions towards elliptical levels is negli-gible due to the small corresponding dipole matrix ele-ments [26]. We have checked by microwave spectroscopy in similar experimental conditions that there is no mea-surable transfer from circular to elliptical levels over thetimescale of our experiments [15]. Note also that thesmall fraction of elliptical levels spuriously prepared bythe circularization process decays in a similar way as | n C (cid:105) , with almost the same rates. In the following, wethus analyze the data as if all the atoms were prepared in | (cid:105) , disregarding the small preparation imperfections.In order to determine the lifetime of the CRAs and T MW , we prepare the atoms in | (cid:105) and detect themafter a delay time τ varying from . to . (Fig. 3a).The population in the low- (cid:96) levels decays rapidly (life-time < µ s [27]) and has completely vanished for τ = 1 . . Population transfers from | n C (cid:105) to adjacentcircular levels is conspicuous. The total number of atomsin all detected circular levels, N tot ( τ ) , is constant, up toexperimental drifts of about , over the whole data set.In Fig. 3b, we plot the relative populations Π n ( τ ) = N n ( τ ) /N tot ( τ ) of the | n C , ≤ n ≤ (cid:105) levels as a func-tion of τ . Strikingly, an exponential fit to the decay ofpopulation Π (dashed line) features a time constant of (5 . ± .
2) ms , an exceptionally long time for a Rydbergatom. In a more precise model, the Π n s obey a simplerate equation that reads ˙Π n = n ph ( ν n ) γ n Π n − + [1 + n ph ( ν n +1 )] γ n +1 Π n +1 − { [1 + n ph ( ν n )] γ n + n ph ( ν n +1 ) γ n +1 } Π n . (1)We add to this model a contamination of Π n +1 by of Π n to account for the overlap of the ionization signals.We fit the data points in Fig. 3b with the outcome of themodel (solid lines), with T MW as the only free parameter.The simulation is restricted to values of n between 49and 55, as Π and Π remain much smaller than onefor τ ≤ . . We find a good agreement with our mea-surements for T MW = (11 . ± .
4) K , corresponding to n ph ( ν ) = (4 . ± . (error bars given by the fittingprocedure). The lifetime of | (cid:105) is (3 . ± .
1) ms , givenby the loss rate of Π at τ = 0 . Note that the time con-stant of the simple exponential fit to Π is longer due tothe replenishment of | (cid:105) from the neighboring levels atlater times.The spontaneous-decay rates γ n could be affected bythe modification of the microwave density of modes dueto the surrounding electrodes. By only considering theeffect of the closest electrode (the MOT mirror), we finda modification of the rates, which we consider asan upper-bound of the effect of the farther electrodes.Modifying the γ n s by ± , we find satisfactory fits tothe data with T MW = (11 ±
2) K , compatible with therough estimation given above (
17 K ).We also investigate the coherence properties of the cir-cular Rydberg levels. We record Ramsey interferencefringes and Hahn-echo signals on the | (cid:105) → | (cid:105) (2 × .
2) GHz -frequency two-photon transition. Here,we reduce the electric field to F = 0 .
46 V / cm in orderto minimize the sensitivity of the transition to electricfield dispersion while keeping a well-defined quantizationaxis. We then wait µ s to let the electric field reach itssteady-state, making its residual drifts negligible. Themicrowave source is set
27 kHz away from the resonancefrequency. We apply two . µ s -long π/ pulses, sepa-rated by a variable waiting time t Ram , and measure thefraction β , = N / ( N + N ) of atoms transferredfrom | (cid:105) to | (cid:105) as a function of t Ram (Fig. 4a). Forthe echo measurements (Fig. 4b), we perform an addi-tional . µ s -long π pulse at a time t E / ≤ t Ram after thefirst π/ pulse and scan t Ram around t E for 8 differentecho times ( µ s ≤ t E ≤ µ s , t E = 150 µ s in Fig. 4b).We fit the Ramsey fringes and the Hahn-echo signals tosines sharing identical carrier frequency and width of aGaussian envelope (red lines in Fig. 4). For every dataset,we get a contrast C , i.e. , the amplitude of the Gaussianenvelope, and a revival time t R , i.e. , the position of themaximum of the envelope ( t R (cid:46) t E ). The shared halfwidth at half maximum of the Gaussian envelopes corre-sponds to a reversible coherence time T ∗ = (38 . ± µ s .We plot in Fig. 4c the contrast of the echo C as a func-tion of t R . To reproduce the observed decoherence, weuse a simple dephasing model. We consider a Gaussiannoise on the energy difference ∆ E between | (cid:105) and | (cid:105) , of variance σ E , with an exponential correlationfunction of characteristic time τ M . Its power spectrumdensity is Lorentzian with a cut-off frequency of ν c = (2 πτ M ) − . We fit the Ramsey and Hahn-echo sig-nals to this model (blue lines in Fig. 4a, b, c), taking as fitparameters ν c and σ E . It agrees remarkably well with allexperimental data, with a cut-off frequency ν c = 76 Hz and σ E = h × . . The irreversible decoherence time, T (cid:48) , defined by C (T (cid:48) ) = C (0) / , is T (cid:48) = 270 µ s .The coherence of the CRAs can be limited by boththeir Stark and Zeeman effects. In order to estimate thecontribution of electric field noise or inhomogeneities to (a)(b)(c)
50 100 150 200 2500.00.20.40.60.8
FIG. 4:
Coherence time measurements. (a) Ramseyfringes and (b) Hahn-echo experiment: β , is plotted w.r.t.the time t Ram between the two π/ pulses. In (b), an ad-ditional π pulse is made at t E / µ s after the first π/ pulse. Each point is the result of the average over ex-perimental runs. (c) Contrast C of the revival of oscillationsof β , as a function of the revival time t R . In all panels,error bars are statistical. Red dashed lines are fits to a sinewith Gaussian damping, blue solid lines are the outcome ofthe stochastic noise model. the measured T ∗ and T (cid:48) times, we recorded the spec-trum of the | (cid:105) → | (cid:105) transition, where | (cid:105) isthe low-lying elliptical level with m = 51 (Fig. 1b).This ∆ m = 0 transition is insensitive to the mag-netic field to first order, but its electric field sensitivity, α CE = 102 MHz / (V / cm) , is large. After an electric-field-gradient minimization, we found a Gaussian line of fullwidth at half maximum (FWHM) δ CE = (175 ±
9) kHz .Correcting for the
121 kHz
Fourier-limited linewidth ofthe µ s -long interrogation pulse, it corresponds to anelectric field variation of δF y = (1 . ± .
1) mV / cm .This electric field variation is too small to accountfor the measured value of T ∗ . The | (cid:105) → | (cid:105) two-photon transition has a linear Zeeman effect ( α B =2 .
80 MHz / G ) but no first order Stark effect. Its quadraticStark shift is
535 kHz / (V / cm) . Small fluctuations δF y around F y = 0 .
46 V / cm electric field thus result in afrequency shift α CC δF y , where α CC = 582 kHz / (V / cm) .The measured value of δF y would then induce a reversiblecoherence time of / ( α CC δF y ) = 5 ms (cid:29) T ∗ .The limited decoherence time is thus likely mainly dueto magnetic field fluctuations. The fitted value of σ E cor-responds to magnetic field fluctuations of σ E / ( hα B ) =1 . . They may result from electric current noise inthe MOT magnetic coils, given the mV noise on the cur-rent supply analog control and the circuit bandwidth ofabout
100 Hz , similar to the estimated ν c .We have prepared cold CRAs in a cryogenic environ-ment and measured their lifetime and coherence time.Much longer coherence times, of the order of the circularstates lifetime, could be obtained by getting rid of thepurely technical magnetic field fluctuations. The . lifetime of | (cid:105) is already ≈ times longer than that oflow- (cid:96) laser-accessible levels. Combined with ponderomo-tive laser trapping [15], it opens bright perspectives for quantum simulation with circular Rydberg states. Thelifetime could even be pushed into the minutes range in-side a spontaneous-emission inhibition structure [19]. In-terestingly, recording the transfer of populations betweencircular Rydberg levels allowed us to estimate in situ the absolute temperature of the microwave BBR. 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