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We investigate the effect of van der Waals interactions and dipole-dipole interactions on collisional loss rate coefficients of
*Cs* Rydberg
*nP* states, and in detail analyze the variation of collisional loss coefficients under the initial Rydberg atomic velocity and van der Waals interactions. We obtain the total collisional loss coefficients for different nP states, and provide the possible ionization mechanism for the results of experimental observation using our analysis model.

The ultracold Rydberg atoms have been extensively studied due to their strong and controllable long-rang interactions. The interaction-induced dipole blockade effect has been proposed as an ideal candidate for realizing a scalable quantum logic gate applying in quantum information processing [

Blackbody radiation (BBR) is known to strongly affect the populations of Rydberg atoms by transitions from initial state to nearby dipole allowed states or BBR-induced direct ionization [

In this paper, we in detail analyze the collisional loss mechanisms induced by Rydberg atoms interactions for different Cs nP states using the theoretical model used before. The validity of this theoretical model has been proved in our previous article by comparing the theoretical and experimental values of collisional loss rate coefficient of Cs 63S state [

The collisional characteristics of cold atoms are conventionally determined from the analysis of the decay curve. We can obtain the analytic solution of the time evolution of the Rydberg atoms number [

N R y ( t ) = ( 4 π w R y 2 ) 3 / 2 α ′ N R y ( 0 ) [ ( 4 π w R y 2 ) 3 / 2 α ′ + β N R y ( 0 ) ] e α ′ t − β N R y ( 0 ) (1)

where

α ′ = α + β R y - G N G [ 1 2 π ( w R y 2 + w G 2 ) ] 3 / 2 + β R y - e N e [ 1 2 π ( w R y 2 + w e 2 ) ] 3 / 2 (2)

Here, α is the loss rate coefficient including the spontaneous radiation rate G_{Spon} and blackbody radiation rate G_{BBR}. The blackbody radiation rate include not only the blackbody radiation transition rates to all nearby Rydberg states through absorption and stimulated emission, but also the direct blackbody radiation photoionization rate. β(β_{Ry}_{-G}) is the loss rate coefficient caused by collisions between ultracold Rydberg-Rydberg (Rydberg-ground) atoms; β_{Ry}_{-e} is the loss rate coefficient caused by collisions between ultracold Rydberg atoms and electrons. N is the atomic number; subscript Ry (G) denotes Rydberg (ground) state; w_{Ry}, w_{G} and w_{e} are the waist radius of Rydberg atoms, cold atoms sample and ionized electrons, respectively.

The value of dipole elements between Rydberg-ground atoms is two orders of magnitude smaller than that of between Rydberg atoms, and decrease with increasing of n, so the collisional loss rate induced by the weak interactions between Rydberg-ground atoms has negligible effect on the total collisional loss rate. Although the large geometrical cross section of Rydberg atoms will lead to the large quenching collisional loss for higher Rydberg states, but has weak contribution to the total collisional loss rate for n < 50 [

α ′ = α + β R y - e N e [ 1 2 π ( w R y 2 + w e 2 ) ] 3 / 2 (3)

In order to investigate the collisional loss induced by Rydberg atomic interactions, we detailed analyze the relation between collisional loss rate coefficient and interactions. The collisional loss rate coefficient can be written as β = 〈 σ R y ν R y 〉 , the s_{Ry} and v_{Ry} are the collisional cross section and relative collisional velocity of ultracold Rydberg atoms, both depend on their van der Waals interaction or dipole-dipole interaction. The collisional cross section induced by interactions given by [

σ i = π ( f ( n ) C n m C s v e s c v c ) 2 / n (4)

where m_{Cs} is Cs atom mass; v_{c} ≈ (8k_{B}T/pm_{Cs})^{1/2} (k_{B} and T are the Boltzmann constant and the temperature of cold atoms) is the average velocity of cold atoms; C_{n} is the interaction coefficient, which is given by theoretical calculations [_{vdW} ≈ 6C_{6}/(m_{Cs}R^{7}) and a_{DD} ≈ 3C_{3}/(m_{Cs}R^{4}), respectively. v_{esc} is a minimum escape velocity of Rydberg atomic from center to edge of average interaction region under corresponding acceleration, i.e., v_{esc} = (2ar)^{1/2}, the average interaction region is approximately equal to the Rydberg excitation volume with the average radius r, in actual experiment, the overlap between trapping laser beam and the Rydberg excitation laser beam defines an effective Rydberg excitation volume. We also obtain the relative collisional velocity between Rydberg atoms under corresponding acceleration, v_{Ry} = (2aR)^{1/2}, the detailed analysis process has been described in our previous work [_{vdW} and β_{DD} induced by van der Waals interaction and dipole-dipole interaction can be written as

β v d W = 2 π ( 45 π 4 m C s 2 v c ( 2 r ) 1 / 2 R 5 / 2 ) 1 / 3 C 6 2 / 3 (5)

β D D = 2 π ( 48 2 r v c 2 m C s 3 R ) 1 / 6 C 3 5 / 6 (6)

Due to the existence of BBR, some Rydberg atoms in initial nP state will be gradually transferred to nearby dipole-allowed n’D and nS states, so the initial weak van der Waals interaction between Rydberg atoms also gradually evolve into strong dipole-dipole interaction between initial nP state and nearby dipole-allowed states, these strong attractive dipole-dipole interactions will give rise to faster collision and even ionization between Rydberg atoms. In order to obtain the effective collisional loss rate coefficient β D D e f f including the contributions of all larger dipole-allowed states, we use an effective coefficient C 3 e f f = ∑ p i μ i 2 instead of C_{3} in Equation (6), where μ_{i} is dipole matrix elements between initial nP state and nearby dipole-allowed states, p_{i} is the transition probability from the initial nP Rydberg atoms to these dipole-allowed states, which is proportional to the strength of transition dipole moments. The effective collisional loss rate coefficient β_{DD} and β_{vdW} of nP_{3/2} at 3 μm atoms separation as a function of n are shown in _{3/2} and some nearby larger dipole-allowed states for nP_{3/2} − nS_{1/2}, nP_{3/2} − (n + 1)S_{1/2} and nP_{3/2} − (n − 1)D_{5/2} are also shown in

are the dominant collisional loss mechanism. Because the values of dipole matrix element between initial nP and nearby dipole-allowed states increase with increasing of n, and BBR transitions rates gradually equal to even exceed the corresponding spontaneous decay rates with n increasing, which lead to the effective collisional loss rate coefficient slightly nonlinear increase.

In

In order to obtain the moving distance of atoms compare to the initial atoms separation, we need to consider the relative acceleration of Rydberg atoms pair under initial interaction force and initial velocity of ultracold Rydberg atoms, which is given by ΔR = v_{c}t + 1/2at^{2}. _{c}, which is about 18 cm/s at cold atoms temperature of 100 μK. It clear seen that the ΔR rapidly increase with delay time and even much greater than their initial atoms separation for higher n. For lower Rydberg states, the weak interactions give rise to relatively small distance even if long delay time. But the initial velocity of ultracold Rydberg atoms plays an important role, which will give rise to the relative moving distance up to about 1 μm at typical 5 μs delay time, the similar behaviors are experimental investigated for Rb Rydberg atoms [_{vdW} with n are shown in _{6} is the average value of all molecular states. The Rydberg atomic separation after 5 μs delay time will gradually increase to about 4 μm for n < 42 and gradually decrease to about 2 μm for n > 42 because their different interaction characters. In contrast to the results of the fixed initial atoms separation R = 3 μm, we can see that the collisional loss coefficient β_{vdW} increases with the decreasing of R for n > 42, and have slight decrease for n < 42 after the same change of R, which are shown in _{DD} compare to the initial atoms separation if the BBR has been occurred. Analogous to the

above results and analysis, the variation of the β_{DD} with n due to the change of interatomic separation induced by initial velocity are also shown in

When we simultaneously consider the initial Rydberg atomic velocity and the van der Waals interactions at the same delay time, because the relative acceleration between ultracold Rydberg atoms will rapidly increase with n scaling as n^{11}, the relative moving distances of ultracold Rydberg atoms for high states are much more than that of the low states at the same delay time, which will lead to more obviously changes of interactions with n. The variation of the β_{vdW} considering the above two effects for n > 42 are shown in _{DD} for n > 42 under the effect of the above two factors are also shown in

The total collisional loss rates induced by ultracold Rydberg-Rydberg interactions not only include the attractive (repulsive) interactions between initial nP states, but also the attractive dipole-dipole interactions between nP and nearby dipole-allowed states. Whether Rydberg atomic pairs initially excited on repulsive or attractive potential, the BBR redistribution effect always exists and must

be taken into account, especially for the repulsive potential case. In order to comprehensive consider all Rydberg interactions contributions to collisional loss rate, we define a ratio such that F = G_{BBR}/α, which is the transition ratio from initial nP state to nearby dipole-allowed states induced by BBR, compare to the total decay rates, the corresponding values of G_{BBR} and α are obtained by theoretical calculations from reference [

β T = η M ( β v d W + F β D D e f f ) (7)

where η_{M} is enhancement factor of many-body effects because Rydberg atoms in many atoms system maybe collide each other several times [_{M} = 10 in the following calculations; The van der Waals interactions will introduce gradually increasing contribution for the total collisional loss rate coefficient, so the attractive van der Waals interactions between initial nP states for n > 42 have non-negligible effect on the total collisional loss compared to the strong attractive dipole-dipole interactions between initial nP and nearby dipole-allowed states.

For n < 42 repulsive potential case, Rydberg atoms will move away from each other under the initial repulsive van der Waals interactions, the collisional ionization does not occur. The collisional probability of ultracold Rydberg atoms initially excited on attractive potential is two orders of magnitude higher than that of repulsive potential case even consider the BBR redistribution effect [

β T = η M F β D D e f f (8)

At 3 μm initial atoms separation and typical 5 μs delay time, the total collisional loss rate coefficient β_{T} of nP_{3/2} versus n is shown in

enhance the interactions between initial nP and nearby dipole-allowed states at the longer delay time that the BBR occurred. Because the BBR has occurred before collision induced by attractive van der Waals interactions and stronger dipole-dipole interaction between nP states, the above analysis is suitable to larger Rydberg atomic separation or the lower Rydberg density case. For high Rydberg density, the smaller initial Rydberg atomic separation will lead to strong attractive van der Waals interactions between initial nP state and the rapid decrease of Rydberg atomic separation, then the attractive van der Waals interactions will quickly evolve into stronger dipole-dipole interactions before BBR transition occurrence, we do not consider the contribution of interactions between initial nP and nearby dipole-allowed states, because the Rydberg atoms have been collision and ionization under initial attractive van der Waals interactions and latter stronger dipole-dipole interactions between initial nP state at shorter delay time than that of BBR transition occurrence.

The attractive interaction case have higher ionization probabilities than that of the repulsive case due to their different collisional mechanisms, in actual experiment, the collision ionization characteristics have significantly difference for n > 42 and n < 42 at longer delay time [^{−}^{1} [^{−}^{1} [^{−}^{1}. Second, we consider the interaction induced the collisional ionization rate, the initial atomic separation value R is about 3 μm for the Rydberg density

r = 2.5 × 10^{9} cm^{−}^{3} by R = ( 1 ( 2 π ) 3 / 2 ρ ) 1 / 3 , which will give rise to about 10% of the

Rydberg atoms close together [^{−}^{5} cm^{3}/s using our model, and the corresponding collisional loss rate is about 3.5 × 10^{4} s^{−}^{1}. We can extract the ionization rate is about 700 s^{−}^{1} with ionization ratio of 2%, even for higher Rydberg density of 4.0 × 10^{9} cm^{−}^{3}, the ionization rate is only about 1120 s^{−1}, which is also smaller than the experimental value. For n < 42, we only consider the contribution of the initial Rydberg atomic velocity to the atomic separation, and think that the weak interactions between Rydberg atoms make atoms separation almost unchanged before BBR transition occurrence. When some close atoms pairs exist at the beginning, they will repeatedly approach each other on long delay time, but the atoms redistributed to nearby dipole-allowed states by BBR will have the chance to collide with these repelled and coming closer atoms, the smaller interatomic distance will give rise to faster collisions due to their stronger dipole-dipole interactions [^{9} cm^{3}/s, the distance between 40P Rydberg atoms under initial interatomic interaction will increase 2 μm on 10 μs delay time, and a fraction of these repelled atoms may closer to redistributed atoms in nearby dipole-allowed states, compared to their initial atomic separation, which will lead to the faster collisions and ionizations under their stronger dipole-dipole interactions, the calculated ionization rate is about 2240 s^{−}^{1} with ionization ratio of 2%. In additional, the stray fields from ions can transfer more initial Rydberg atoms to nearby attractive dipole-coupled Rydberg state, and tune their interaction strengths by potential curve determining atomic collisional properties; the present of electric field also can accelerate the impact ionization of Rydberg-electron.

We in detail analyze the different collisional loss mechanisms of Cs nP Rydberg atoms, present simple analytical formula to estimate these collisional loss rate coefficients for the van der Waals and dipole-dipole case, respectively, estimate Cs 40P collisional ionization rate using our above model, and compare it with the previous measured results, which have the important guiding significance for analyzing the experimental results.

This work was supported by the Fundamental Research Fund Project of NIM (Grant No. 27-AKY1705) and the Exploration and Innovation Project of NIM (Grant No. 27-AKYCX1604).

Feng, Z.G., Zhou, X., Li, B., Gao, H.Y. and Liu, Z. (2018) Collision Characteristics between Ultracold Cesium nP Rydberg Atoms. Journal of Modern Physics, 9, 820-831. https://doi.org/10.4236/jmp.2018.95053