# Lifetime Measurement of the Metastable State of Strontium Atoms

###### Abstract

We have measured the lifetime of the metastable state of strontium atoms by magneto-optically trapping the decayed atoms to the ground state, which allowed sensitive detection of the rare decay events. We found that the blackbody radiation-induced decay was the dominant decay channel for the state at K. The lifetime was determined to be s in the limit of zero temperature.

###### pacs:

32.70.Cs, 32.80.Pj^{†}

^{†}preprint: APS/123-QED

Precision measurements require the isolation of the physical system under study from environmental perturbations. The long coherence time thus obtained can be exploited to carry out ultra-high resolution spectroscopy Bergquist2000 or to create and control the macroscopic quantum coherence in atomic systems, such as Bose-Einstein Condensate (BEC) BECNature and Cooper pairing BCS . Furthermore, the origin of environmental decoherence has recently attracted much attention in the context of quantum computation/communication, in which an entangled quantum system should maintain its internal and/or motional state coherence QC ; Bloch . The lower-lying metastable states KatoriICAP ; Nagel ; Hemmerich ; Loftus ; ErtmerMg ; Udem ; Wilpers of alkaline earth species are intriguing candidates for these studies. A long metastable lifetime may allow an optical spectroscopy at the 1 mHz level Katori_Scottland , enabling one to realize an ultra-precise atomic clock. In addition, the possibilities of evaporatively cooling them to reach BEC are discussed theoretically SrBECDerevianko ; SrBECKokoouline .

Laser-cooled and trapped atoms in ultra-high vacuum condition offer an ideal sample for these studies, as the collisional interactions with container walls or residual gases can be substantially removed. However, in some cases, the radiation from the surrounding walls dramatically affects the evolution of the internal state coherences.The room-temperature blackbody radiation (BBR) has its intensity peak around a wavelength of m. Such BBR manifests itself most in Rydberg states Gallagher ; Farley_Wing ; Hollberg , where infrared transitions with large electric dipole moments can be found. However, because of large energy differences, it hardly excites atoms in the lower-lying states. Its influence, therefore, has been rarely discussed in laser cooling and trapping. On the other hand, it is known that the BBR causes small but non-negligible ac Stark shifts in ultra-precise spectroscopy Itano ; Bauch .

As for the metastable state of heavier alkaline-earth atoms, the upper-lying states are connected to the metastable state by electric dipole transitions with mid-infrared wavelength. Therefore, the excitation of the metastable state by the BBR may significantly alter its effective lifetime or introduce detectable blackbody shifts in precision spectroscopy. In this Letter, we study the influence of the room-temperature BBR on the metastable state lifetime of Sr, which is recently predicted to be 1050 s Derevianko . Because the lifetime is significantly longer than the collision-limited lifetime of tens of seconds that is realized in neutral atom traps at a vacuum pressure of Katori_Shimizu , observing the survival of metastable atoms in such traps will not be practical Katori_Shimizu ; Ertmer . Instead of directly observing the decay of the metastable atoms by emitted photons Walhout , we monitored the occurrence of the rare decay events to the ground state by magneto-optically trapping the atoms on the to detect the decay events with unit quantum efficiency Nagourney ; Bergquist . transition. In this way, we used the magneto-optical trap (MOT) as a photon amplifier with a gain of

Figure 1 shows the relevant energy levels for Sr. Three radiative decay channels from the metastable state have been identified by Derevianko Derevianko : 1) The M2 transition to the ground state, 2) Decay to the state through the M1, E2, and M3 transition, 3) Decay to the state through the E2 transition, where E or M stands for an electric or a magnetic -pole transition. The theory suggests that the contribution of the channel 3) is only 0.1% Derevianko . Therefore, 99.9% of the population in the state finally relaxes to the ground state, as the atoms that decayed to the state further decay to the ground state in . On this basis, we use the state population to detect the decay of the state.

We measured the number of atoms in the ground state by applying the MOT on the transition. The change of the atom number in the MOT is given by the rate equation,

(1) |

The atoms in the ground state are collisionally lost at a rate or leaked out of the MOT transition via the state with the typical rate of . The atoms are supplied by the metastable state with its atom number of , which either radiatively decay () or collisionally quench () to the ground state, where holds as discussed later. Ignoring both of the collisional loss terms of and , the steady-state solution of Eq. (1) for is given by,

(2) |

where we assumed to be constant as the decay of the population in the state with the collisional decay rate of is small enough in the time scale of interest. The metastable decay rate , therefore, can be determined by the ratio of population distributed in both of the states and the MOT decay rate . The number of atoms in the ground state can be derived by observing the MOT fluorescence intensity of , where is the photon counting rate per atom trapped in the MOT. Similarly, can be obtained by transferring the metastable state population into the ground state and measuring the MOT fluorescence intensity of . By the ratio of these fluorescence intensities, is accurately determined regardless of the coefficient .

The apparatus for magneto-optically trapping strontium atoms is similar to that described in Ref. Katori . The MOT fluorescence was collected by a lens with a solid angle of and then sent to a photomultiplier tube (PMT). An interference filter was placed in front of the PMT to block except the 461 nm light. The output signal was then sent to a multi-channel scaler. The two transitions, 1), were used to pump the metastable state population into the ground state. (Fig. and

We first loaded the MOT from an atomic beam on the metastable state was populated via the weak branching decay to the state from the state, which is estimated to be branch . About atoms in the low-field-seeking state were thus accumulated in the magnetic trap, which is formed by the quadrupole magnetic field used for the MOT. The field gradient was 100 G/cm along its axis of symmetry. At the typical peak density of , two-body collisional loss rate is estimated to be much smaller than the collisional loss term in Eq. (1), where we assumed an inelastic collisional loss rate of SrBECDerevianko ; SrBECKokoouline . We then turned off the MOT lasers and closed the shutter that blocked both the atomic beam and the thermal radiation from the oven. We waited for 0.3 s so that all atoms except magnetically trapped metastable atoms diffused out of the trap region. After that, we turned on the MOT lasers again to capture atoms that were radiatively decayed from the state and recorded the fluorescence intensity. At , we irradiated both of the pumping lasers to transfer the metastable state population into the ground state and determined the number of atoms trapped in the state by the MOT fluorescence intensity. transition for 0.3 s. In the mean time, the

Figure 2 shows a change of fluorescence intensity averaged over measurements, where the background level was subtracted by alternating the same procedure with and without loading atoms into the magnetic trap. The number of atoms in the state at s was determined by exponentially extrapolate the whole decay curve of the MOT fluorescence to obtain with better statistics. The fluorescence decay in s was mainly caused by the collisional atom loss in the magnetic trap with the decay rate at the background gas pressure of . By transferring the metastable state population into the ground state at , the fluorescence intensity sharply rose up to its maximum in about 5 ms. We approximated by the peak fluorescence intensity shown in the inset of Fig. 2. The signal then decayed double-exponentially, consisting of the MOT decay with due to the branching loss and the much slower collisional decay of the metastable atoms recaptured in the magnetic trap. The metastable state lifetime is calculated by applying the ratio and the measured MOT decay rate in Eq. (2). The measurement shown in Fig. 2 gave an effective radiative lifetime of s, which is only one tenth of the theoretical lifetime Derevianko . This shortening can be attributed to the BBR-induced decay via the state as discussed later. However, before discussing the BBR-induced decay, we checked the other decay channels.

The reduction of the lifetime may be caused by the fine structure mixing of the metastable state with the or the state in the presence of the trapping magnetic field. Assuming a magnetic field of 10 G, the magnetically induced decay rates are estimated to be 0.1% of the natural decay rate of the state. Actually, we measured the metastable lifetime under various magnetic field gradient, however, the change of the lifetime was within the statistical errors of 5 % as expected. Second, due to collisions with background gases, atoms in the metastable state may be 1) kicked out of the magnetic trap with the rate or 2) quenched into the ground state with . For the former issue, since we compared the fluorescence intensity and just before and after the population transfer, the collisional atom loss in the transferring period of ms may cause an error. However its fraction is estimated to be as small as . For the latter issue, the collisional quench to the ground state may cause pressure-dependent shortening of the metastable state lifetime. To check this influence, we increased the background gas pressure up to and measured the lifetime. However, the change in the decay rate was well within statistical uncertainties.

The rapid decay can be attributed to the metastable state quenching due to the BBR field that transfers atom population in the state to the short-lived state via the states. By solving the coupled rate equations, the steady-state value of the BBR-induced decay rate is expressed as,

(3) |

where is the radiative decay rate of the state, and is the BBR photon occupation number at temperature with the wavelength for the s of the states Miller , the BBR-induced decay rate is calculated to be for K. Therefore the intrinsic decay rate of the state at K is determined as , which corresponds to a lifetime of s. To confirm that this lifetime shortening originates in the BBR excitation, we measured the decay rate as a function of the ambient temperature by heating up the vacuum chamber that enclosed the MOT. We note that this temperature change altered the vacuum pressure in the range of . However, the resultant collisional losses did not affect the measured lifetime as mentioned previously. The measured decay rate is plotted in Fig. 3 by filled squares, where error bars indicate one standard deviation. The monotonic increase of the decay rate as the temperature clearly supports that this lifetime shortening is caused by the BBR. transition. Using the radiative lifetime

The BBR-induced decay rate is estimated as follows. Initially, the chamber that surrounds the atoms is assumed to be in thermal equilibrium. However, in heating up the chamber, care should be taken if it is in thermal equilibrium. The vacuum chamber is made of stainless steel with 150 mm diameter and has six viewing ports made of BK7 glass with 40 mm diameter. It was heated by a ribbon heater wound around the chamber body. We monitored the temperature of the viewing ports (at their rim and center) and the body of the chamber one hour after changing the heating power: Whereas the temperature of the rim was the same as that of the chamber body , that of the center was somewhat lower by . Assuming the linear change of the temperature between the rim and the center, the average temperature for the viewing ports was approximated as . Owing to this temperature inhomogeneity, atoms are not in the BBR field of thermal equilibrium. Assuming a spherical radiant cavity, the effective photon occupation number for the body temperature of is calculated as Chandos ,

(4) |

Here is the solid angle covered by the chamber body and is its spectral emissivity at m. Likewise we defined and for the viewing ports, which fraction in the solid angle was . We used this effective photon occupation number to calculate the BBR-induced decay rate given by Eq. (3). We took the spectral emissivity of both objects as fitting parameters so that the measured decay rate should have a constant offset to the calculated BBR-induced decay rate at each temperature. After the least squares fitting, we obtain the metastable decay rate of and the spectral emissivities of and for stainless steel and BK7 glass, respectively. The obtained emissivity for BK7 glass showed fair agreement with derived from the index of refraction for BK7 glass at m. Reference data for polished stainless steel, however, was not available.

In the above discussion, we have not included the uncertainty of the state radiative lifetime, which gives coupling strength between the and states and thus significantly affects the BBR-induced decay rate. In Ref. Miller , the observed radiative lifetime of the state is given as . This uncertainty brings another statistical error of to the metastable decay rate. Thus, we finally obtain the metastable decay rate , or the metastable lifetime of s.

In summary, we have determined the lifetime of the state of to be s in the limit of zero temperature. Because the room-temperature BBR considerably shortens the metastable lifetime, care should be taken when dealing with the state to form a BEC SrBECDerevianko ; SrBECKokoouline or any other applications that require long coherence time. We, therefore, need to prepare cold environment to suppress thermal photons. For example, by just lowering the ambient temperature down to 275 K, the BBR-induced decay rate becomes comparable to its intrinsic decay rate. A straightforward comparison of the measured lifetime with the theory will be possible by further lowering the temperature down to 217 K, where the BBR quenching rate is expected to be 1% of the natural decay rate.

The authors thank K. Okamura and M. Takamoto for their technical support. This work was supported by the Grant-in-Aid for Scientific Research (B) (12440110) from the Japan Society for the Promotion of Science.

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