Table Of ContentRubidium Rydberg linear macrotrimers
Nolan Samboy1,2 and Robin Cˆot´e1
1Physics Department, University of Connecticut, 2152 Hillside Rd., Storrs, CT 06269-3046 and
2Physics Department, College of the Holy Cross, 1 College St., Worcester, MA 01610
(Dated: January 14, 2013)
We investigate the interaction between three rubidium atoms in highly excited (58p) Rydberg
states lying along a common axis and calculate the potential energy surfaces (PES) between the
three atoms. We find that three-body long-range potential wells exist in some of these surfaces,
indicating the existence of very extended bound states that we label macrotrimers. We calculate
the lowest vibrational eigenmodes and the resulting energy levels and show that the corresponding
vibrational periods are rapid enough to be detected spectroscopically.
3
1 PACSnumbers: 03.65.Sq,31.50.Df,32.80.Ee,34.20.Cf
0
2
n I. INTRODUCTION II. THREE BODY INTERACTIONS
a
J
In [8] and [9], we predicted the existence of long-range
1 Ultracold Rydberg systems are a particularly inter-
rubidium Rydberg dimers by analyzing potential energy
1 esting avenue of study. Translationally, the atoms are
curves corresponding to the interaction energies between
very slow, yet their internal energies are very high. The
] the two Rydberg atoms. In these works, we diagonal-
h large excitation of a single electron leads to exaggerated ized an interaction Hamiltonian consisting of the long-
p atomic properties, such as long lifetimes, large cross sec-
range Rydberg-Rydberg interaction energy and atomic
- tions,andverylargepolarizabilities[1],whichcanleadto
m fine structure in the Hund’s case (c) basis set. Each
strong interactions between Rydberg atoms [2, 3]. Such
molecular state in the basis was symmetrized with re-
o interactions have led to various applications in quantum
spect to the D symmetry of the homonuclear dimer.
t information processes over the past decade (see [4] for a ∞h
a
In general, adding a third atom to the interaction pic-
. comprehensive review).
s ture will change the physical symmetry of the system.
c
i Another active area of research with Rydberg atoms However, to simplify our calculations, we assume that
s
is in the area of long-range “exotic molecules”. Such three identical Rydberg atoms lie along a common (z-)
y
examples include the trilobite and butterfly states, so- axis (see Fig. 1(a)). This preserves the D symmetry
h ∞h
p calledbecauseoftheresemblenceoftheirrespectivewave and permits the use of much of the two-body physics
[ functions to these creatures. First predicted in [5], these on the three-body system. We analyzed this symmetry
quantum states were detected more recently in [6]. Also state for three Rb 58s atoms and three Rb 58p atoms
1
ofinterestaretheformationanddetectionofmacroscopic andfoundthatthe58pcaseexhibitedexamplesofthree-
v
8 Rydberg molecules. In [7], it was first predicted that dimensional wells, indicating that the three atoms are
3 weaklyboundmacrodimers couldbeformedfromthein- bound in a linear chain.
5 ducedVanderWaalsinteractionsoftwoRydbergatoms.
2 However, we have shown more recently [8, 9] that larger,
.
1 morestabledimerscanbeformedvia thestrong(cid:96)-mixing
0 of various Rydberg states. Recent measurements [10] A. Basis States
3 have shown signatures of such macrodimers using an ul-
1 tracold sample of cesium Rydberg atoms.
: Obtaining properly symmetrized basis functions for
v
More recently, the focus of study has moved toward the three-atom case is very similar to that of the two-
i
X few-body interactions, such as between atom-diatom in- atom system (see [20]), but much more technically de-
r teractions [11–13] and diatom-diatom interactions [14, manding. We construct the molecular wave functions
a 15]. Coinciding with this shift, there have been pro- from three free Rydberg atoms in respective states
posals [16–19] for many-body long-range interactions in- |a1(cid:105)≡|n1,(cid:96)1,j1,mj1(cid:105), |a2(cid:105)≡|n2,(cid:96)2,j2,mj2(cid:105), and |a3(cid:105)≡
volving Rydberg atoms. However, these works focus on |a3,(cid:96)3,j3,mj3(cid:105), where ni is the principal quantum num-
the interactions between one Rydberg atom and ground ber of atom i, (cid:96)i is the orbital angular momentum of
state atoms or molecules. In this paper, we describe the atom i, and mji is the projection of the total angular
long-rangeinteractionsbetweenthreeRydbergatomsar- momentum (cid:126)j = (cid:126)(cid:96) +(cid:126)s of atom i onto a quantization
i i i
ranged along a common axis and provide calculations, axis (chosen in the z-direction). As in the two-atom
whichpredicttheexistenceofboundtrimerstates. Here, case, we assume that the three Rydberg atoms interact
we also present the lowest vibrational energies of these via long-range dipole-dipole and quadrupole-quadrupole
bound states, calculated via the oscillation eigenmodes couplings. Here, long-range indicates that the distance
of the bound system. between each Rydberg atom is greater than the Le-Roy
2
Hamiltonian consists of a three-body long-range Ryd-
berg interaction energy and atomic fine structure, i.e.
H =V +H . Using the wave functions defined
int 3−body fs
R R by Equation (2), we write the matrix elements of the
1 2
Hamiltonian as the sums of multiple interactions. Each
a a a
1 2 3 matrix element is defined as:
2 (a)
(cid:104)a ;a ;a |V |b ;b ;b (cid:105)=
1 2 3 3−body 1 2 3
a a a 1 (cid:88) (cid:104)a(1)a(2)a(3)|V |b(1)b(2)b(3)(cid:105) (3)
1 2 3 6 i j k 3−body i(cid:48) j(cid:48) k(cid:48)
i,j,k
(b) i(cid:48),j(cid:48),k(cid:48)
2 ×(Θ +P Θ )(Θ +P Θ ) ,
C a A C b A
a a a where each summation index is over the total number of
1 2 3
atoms, i.e. from 1 to 3, P is as before, we have defined
(c)
(cid:26)
−1 for cyclic permutations
Θ =
FIG. 1. (Color online) (a) Three Rydberg atoms lie along C 0 for anti-cyclic permutations
a common z-axis. The distance between atoms 1 and 2 is
represented by R and the distance between atoms 2 and 3 and
1
is represented by R . Each atom is in state |a (cid:105), defined in
2 i
the text. Each bound macrotrimer has two eigenmodes of (cid:26) 0 for cyclic permutations
oscillation: (b) ω+2 and (c) ω−2 . (See text) ΘA = −1 for anti-cyclic permutations
radius [21]: and we have defined |a(1)a(2)a(3)(cid:105) ≡ |a (cid:105) |a (cid:105) |a (cid:105) , etc.
i j k i 1 j 2 k 3
R =2[(cid:104)n (cid:96) |r2|n (cid:96) (cid:105)1/2+(cid:104)n (cid:96) |r2|n (cid:96) (cid:105)1/2] , (1) In the case that |a1;a2;a3(cid:105) = |b1;b2;b3(cid:105) (i.e. along the
LR 1 1 1 1 2 2 2 2
diagonal of the matrix), the matrix element is given by:
such that there is no overlapping of the electron clouds.
The properly symmetrized long-range three-atom wave-
(cid:104)a ;a ;a |H |a ;a ;a (cid:105)=
functions take the form: 1 2 3 int 1 2 3
(cid:104)a ;a ;a |V |a ;a ;a (cid:105)+E , (4)
1 2 3 3−body 1 2 3 123
|a ;a ;a (cid:105)=
1 2 3
√1 [(|a1(cid:105)1|a2(cid:105)2|a3(cid:105)3 +|a2(cid:105)1|a3(cid:105)2|a1(cid:105)3 with E123 =E1+E2+E3, where Ek =−21(nk−δ(cid:96)k)−2
6 aretheatomicRydbergenergieswithrespectivequantum
+ |a (cid:105) |a (cid:105) |a (cid:105) ) (2) defects δ .
3 1 1 2 2 3 (cid:96)k
−P (|a (cid:105) |a (cid:105) |a (cid:105) +|a (cid:105) |a (cid:105) |a (cid:105) The long-range assumption assures that these are
1 1 3 2 2 3 2 1 1 2 3 3
three free atoms interacting via long-range two-
+ |a (cid:105) |a (cid:105) |a (cid:105) )] ,
3 1 2 2 1 3
body potentials. That is, the transition element
where P = p(−1)(cid:96)1+(cid:96)2+(cid:96)3, with p = +1(−1) for gerade (cid:104)a(1)a(2)a(3)|V |b(1)b(2)b(3)(cid:105)giveninEquation(3)is
i j k 3−body i(cid:48) j(cid:48) k(cid:48)
(ungerade) molecular states. defined as a sum of two-body interactions:
The basis set consists of the Rydberg molecular level
beingprobed(e.g. 58p+58p+58p), aswellasallnearby
(cid:104)a(1)a(2)a(3)|V |b(1)b(2)b(3)(cid:105)=
asymptotes with significant coupling to this level and to i j k 3−body i(cid:48) j(cid:48) k(cid:48)
each other. All of these states are properly symmetrized (cid:104)a(1)a(2)|V (R )|b(1)b(2)(cid:105)
i j L 12 i(cid:48) j(cid:48)
via Equation (2) according to their molecular symmetry
Ω = mj1 +mj2 +mj3. In this paper, we consider the +(cid:104)a(j2)a(k3)|VL(R23)|b(j2(cid:48))b(k3(cid:48))(cid:105)
Ω=1/2 symmetry.
+(cid:104)a(1)a(3)|V (R )|b(1)b(3)(cid:105) , (5)
i k L 13 i(cid:48) k(cid:48)
B. Interaction Hamiltonian where R is the nuclear separation between atoms α
αβ
and β, and L = 1(2) for dipolar (quadrupolar) interac-
As in the two-atom case, we construct the interac- tions. Since we are assuming that the three atoms lie
tion picture for the three-atom system by diagonaliz- along a common axis, each two-body interaction term
ing an interaction Hamiltonian in the properly sym- (cid:104)a(α)a(β)|V (R )|b(α)b(β)(cid:105)definesthelong-rangetransi-
i j L αβ i(cid:48) j(cid:48)
metrizedbasisdescribedintheprevioussubsection. The tionelementbetweenthetworespectiveRydbergatoms.
3
Each transition element is given by [9, 20]:
-3230
-3231
(cid:104)12|V (R)|34(cid:105)= Energy (GHz) -3232
L
(cid:113) RL RL -3230 -3233
(−1)L−1−mjtot+jtot (cid:96)ˆ1(cid:96)ˆ2(cid:96)ˆ3(cid:96)ˆ4ˆj1ˆj2ˆj3ˆj4 R123L+124 --33223342 --33223354
(cid:18)(cid:96) L (cid:96) (cid:19)(cid:18)(cid:96) L (cid:96) (cid:19) -3236 -3236
× 1 3 2 4
0 0 0 0 0 0 -3238 -3237
(cid:26) (cid:27)(cid:26) (cid:27)
j L j j L j
× 1 3 2 4
(cid:96) 1 (cid:96) (cid:96) 1 (cid:96) 90000
3 2 1 4 2 2
75000
L (cid:18) (cid:19)
× (cid:88) BL+m j1 L j3 R2 (a0) 60000
m=−L 2L −mj1 m mj3 450 0300000 45000 60000 75000 90000
×(cid:18) j2 L j4 (cid:19) , (6) 30000 R1 (a0)
−m −m m
j2 j4
FIG. 2. (Color online) Potential energy surface (PES) cor-
wmhj4e,re(cid:96)ˆijt=ot2=(cid:96)ij+1+1,jˆj2i+=j32+jij+4,1m, ajtnodt =RmLijj=1+(cid:104)im|rLj2|j+(cid:105)misjt3h+e rTehlaistesdurtfoacteheis|5a6npa21lo,g12o;u5s8pto23,a−r12e;p6s0upls32iv,e12(cid:105)poatseynmtipatloctiucrvsetaftoer.
radial matrix element. the two-body case: As the distance of either the first or last
atom in the linear chain is increased, the two local atoms re-
main repulsed. The color scheme denotes the energy values
giveninGHz,withthescalepresentedtotherightoftheplot.
III. POTENTIAL ENERGY SURFACES
-3190
A. General Cases -3192
-3194
-3196
We diagonalize the three-body Hamiltonian at succes- -3198
-3200
sivevaluesofR1 andR2,resultinginaseriesofpotential Energy (GHz) -3202
-3204
energy surfaces (PES), where each surface corresponds -3190 -3206
to a different molecular asymptote in the basis. In each -3195 -3208
-3210
of the plots shown, R1 represents the distance between -3200
atom 1 and atom 2, R represents the distance between
2 -3205
atom 2 and atom 3, both in a (see Fig. 1(a)) and the
0 -3210 90000
energy is measured in GHz. The color scheme for the
75000
energy values is given in the scales to the right of each 60000
plot. 30000 45000 45000 R2 (a0)
As a result of the large (cid:96)-mixing that occurs be- 60000 75000 30000
R1 (a0) 90000
tween the Rydberg atoms, these surfaces have in-
teresting topographies. For example, Figures 2
and 3 illustrate potential surfaces analogous to two- FIG. 3. (Color online) Potential energy surface (PES) cor-
dimensional repulsive and attractive curves, respec- related to the |58s1,1;59s1,−1;57d5,1(cid:105) asymptotic state.
2 2 2 2 2 2
tively. The repulsive PES shown in Fig. 2 corre- This surface is analogous to an attractive potential curve for
sponds to the |56p1,1;58p3,−1;60p3,1(cid:105) state, while the two-body case: As the distance of either the first or last
2 2 2 2 2 2
the attractive PES shown in Fig. 3 corresponds to the atom in the linear chain is increased, the two local atoms re-
|58s1,1;59s1,−1;57d5,1(cid:105) state. We see that in both main attracted. The color scheme denotes the energy values
2 2 2 2 2 2 giveninGHz,withthescalepresentedtotherightoftheplot.
cases, the distance of the third atom has very little ef-
fect on the other two atoms: as either R or R is
1 2
increased (while keeping the other distance fixed), the
twostationaryatomsconsistentlydemonstrateanattrac- B. Potential Wells
tive/repulsive behavior.
Figure 4 illustrates another type of surface, in which Although the surfaces highlighted in the previous
there is a significant “ridge” running along one of the section are interesting, ultimately we seek surfaces
axes (in this case along the R axis). Such a ridge that illustrate potential wells, as these indicate bound
2
indicates that the two local atoms (e.g. atom 1 and three-atom systems. Upon separately investigating
atom 2) form a bonded pair, existing even as atom 3 three excited 58s rubidium atoms and three excited
is moved away. This particular surface corresponds to 58p rubidium atoms, we found that in the case of the
the |59s1,−1;55d3,3;58d3,−1(cid:105) asymptotic state. three excited 58p atoms, surface plots corresponding
2 2 2 2 2 2
4
-3221.8 -3222.46
-3221.85 -3222.48
Energy (GHz) -3221.9 -3222.5
Energy (GHz)
-3221.8 -3221.95 -3222.44 -3222.52
-3221.9 -3222 -3222.48 -3222.54
-3222.05 -3222.56
-3222 -3222.1 -3222.52 -3222.58
-3222.1 -3222.15 -3222.56 -3222.6
-3222.2 -3222.2 -3222.6 -3222.62
-3222.64
90000
75000
R2 (a0) 600 4050000 R2 (a0 )36000 32000
30000 30000 45000 600R010 (a0) 75000 90000 28000 22000 23000 2400R01 (a0 2)5000 26000
FIG. 4. (Color online) Potential energy surface (PES) corre-
lated to the |59s1,−1;55d3,3;58d3,−1(cid:105) asymptotic state.
2 2 2 2 2 2 -3222.46
We note the “ridge” lying along the R axis, which indicates
2 "TriOutTEST2.dat" using 1:2:386
that atom 1 and atom 2 are bound (see text). The color 38000 -3222.48
schemedenotestheenergyvaluesgiveninGHz,withthescale -3222.5
36000
presented to the right of the plot. -3222.52
R (a)20 34000 -3222.54
to various 58p+58p+58p asymptotes illustrated such 32000 -3222.56
(three-dimensional) potential wells. Specifically, wells 30000 -3222.58
were determined for the following states: -3222.6
28000
-3222.62
|1(cid:105)≡|58p12,12;58p12,12;58p12,−12(cid:105) 22000 23000 R 214 0(a000) 25000 26000
|2(cid:105)≡|58p1,−1;58p1,−1;58p3,3(cid:105)
2 2 2 2 2 2
|3(cid:105)≡|58p1,1;58p3,1;58p3,−1(cid:105)
2 2 2 2 2 2
|4(cid:105)≡|58p1,−1;58p3,−1;58p3,3(cid:105) FIG. 5. (Color online) Potential energy surface (top)
2 2 2 2 2 2 and two-dimensional projection (bottom) correlated to the
|5(cid:105)≡|58p3,−1;58p3,−1;58p3,3(cid:105) .
2 2 2 2 2 2 |58p11;58p1,1;58p1,−1(cid:105) asymptotic molecular state. The
22 2 2 2 2
mainfeatureoftheseplotsisthethree-dimensionalwell,cen-
As an example, in Figure 5 we show the PES and tered at R = 22 500 a and R = 31 000 a , indicating
1e 0 2e 0
the corresponding two-dimensional projection corre- that the three Rydberg atoms are bound together in a lin-
lated to the |1(cid:105) state, which demonstrates a potential ear chain (see text). The red (vertical) and blue (horizontal)
well approximately 50-100 MHz deep. Due to the lines imposed on the bottom plot are centered at the well
large equilibrium separations, e.g. R ∼ 22 500 a minimum and indicate the cross-sections for which quadratic
1e 0
and R ∼ 31 000 a , we label these bound states fits are performed (see Figure 6). The color scheme denotes
2e 0
the energy values given in GHz, with the scale presented to
macrotrimers.
the right of each plot.
Forthethree-atomconfigurationshowninFigure1(a),
the (non-zero) oscillation modes can be calculated via:
(k +k )±(cid:112)k2−k k +k2 tential wells along the R1 and R2 axes at each well’s
ω2 = 1 2 1 1 2 2 , (7) minima (see Fig. 6). The deepest portions of each well
(±) m
can be modeled as a simple harmonic oscillator, and it
where k are “effective spring constants” that need to is easily shown that the desired k-value equals the sec-
i
be calculated and m is the mass of a single rubidium ondderivativeofthesequadraticfits(withrespecttothe
atom. In Figure 1(b) and (c), we show the physical de- nuclear separation in that particular direction), i.e.
scription of each eigenmode. Panel (b) corresponds to
theω2 eigenvalueandshowsthatthetwoouterRydberg d2V (cid:12)(cid:12)
atoms+vibrate in the same direction, opposite to the in- ki = dR2(cid:12)(cid:12) , (8)
ner atom’s direction of motion. Panel (c) corresponds i Rie
to the ω2 eigenvalue and shows that the inner Rydberg where R is the equilibrium separation along axis R .
− ie i
atom is stationary while the outer two atoms oscillate in Table I summarizes the results of the polynomial fitting
opposing directions. including the k -values, the goodness-of-fits (r2 values)
eff
To calculate the k for Equation (7), we perform poly- of the harmonic oscillator potentials, and the potential
i
nomial fits to two-dimensional cross sections of the po- energyrangeforwhichthequadraticassumptionisvalid.
5
energyvaluescorrespondtoµsoscillationperiods, which
TABLEI.Characteristicsofthepolynomialfittingprocedures
are rapid enough to allow for several oscillations during
for each trimer state |i(cid:105) (see text), assumed to be quadratic.
the lifetimes of these Rydberg atoms (roughly 500 µs
A two-dimensional cross-section was taken at the minimum
for n ∼ 60 [22]). The energies corresponding to the ω
of each well for both the R and R axes, where the well is −
1 2
frequency are more closely spaced and demonstrate os-
centered (R , R ). The k -values (in N/m) correspond to
1e 2e eff
thenumericfittingalongeachrespectiveaxis. Thewelldepth cillation periods that are slower, but should still be able
indicates the potential energy range for which the quadratic to be detected experimentally.
assumptionisvalidandforwhichthegivenr2 goodness-of-fit
values are appropriate.
State Axis R (×103a ) k (N/m) r2-value Depth (MHz)
e 0 eff
R 22.50 8.37×10−11 0.9879 17.50
|1(cid:105) 1
R2 31.00 4.40×10−12 0.9926 16.60 TABLEII.Lowestvibrationallevelsandcorrespondingbound
R 23.15 3.85×10−11 0.9861 16.10 state energies for both oscillation frequencies of each trimer
|2(cid:105) 1
R 33.10 1.07×10−12 0.9781 77.50 state |i(cid:105) (see text). Energy± represents the bound energies
2 associated with ω and is measured from the bottom of the
R 22.80 2.55×10−11 0.9908 37.70 ±
|3(cid:105) 1 potential well.
R 34.60 3.02×10−12 0.9892 21.30
2
State v Energy (MHz) Energy (MHz)
R 22.85 4.63×10−11 0.9985 29.30 + −
|4(cid:105) 1 0 2.71 0.62
R 31.50 5.54×10−12 0.9943 5.80
2
1 8.13 1.86
R 22.00 6.87×10−11 0.9982 16.24
|5(cid:105) 1 |1(cid:105) 2 13.55 3.11
R 31.70 1.72×10−12 0.9992 11.94
2
3 − 4.34
. . .
. . .
. . .
12 − 15.53
0 1.84 0.31
-3222.40
1 5.51 0.92
z)-3222.45 2 9.18 1.53
GH-3222.50 |2(cid:105)
ergy (-3222.55 34 12−.86 22..1745
n-3222.60 . . .
E . . .
. . .
-3222.65
21 22 23 24 25 26 25 − 15.60
R separation (1000 a)
1 0 0 1.50 0.51
-3222.54
1 4.48 1.54
Hz)-3222.56 2 7.48 2.57
G 3 10.47 3.60
gy (-3222.58 |3(cid:105) 4 13.47 4.63
er
En 5 16.46 5.66
-3222.60
6 19.45 6.69
28 30 32 34 36 38 40 7 − 7.72
R separation (1000 a)
2 0 . . .
. . .
. . .
FIG. 6. (Color online) Two-dimensional cross sections of the 20 − 21.10
three dimensional surface shown in Figure 5 taken along the
0 2.02 0.70
R (top)andR (bottom)axesatthewell’sminimum. Each
1 2
cross-section is fitted with a harmonic potential centered at |4(cid:105) 1 − 2.09
the minimum. The results of these fits are summarized in 2 − 3.49
Table I. 3 − 4.88
0 2.46 0.39
Basedontheseproperties,weuseEquation(7)andthe 1 7.37 1.17
familiar E =(cid:126)ω(v+1/2) to find the oscillation energies |5(cid:105)
2 − 1.94
in the deepest portions of the highlighted wells. Table II . . .
. . .
liststhevibrationalenergiesofthefirstfewboundstates . . .
foreachmodalfrequency,ω andω . Weseethatineach 14 − 11.27
+ −
case, the energies defined by the ω frequency illustrate
+
spacings of about 3-6 MHz, which are separated enough
to be detected through spectroscopic means. The MHz
6
C. Other asymptotes
-3222.11
-3222.12
-3222.13
Ofcourse,thepotentialwellsdescribedinsectionIIIB Energy (GHz) -3222.14
-3222.15
-3222.1
are not the only wells that exist. During the course of -3222.16
-3222.12 -3222.17
our analysis, we also found wells corresponding to vari- -3222.14 -3222.18
-3222.19
ous|59s55d58d(cid:105)asymptotes;someexamplesofwhichare -3222.16 -3222.2
presented below. Although in principle these wells can -3222.18 -3222.21
-3222.22
-3222.2
beevaluatedinthesameamountofdetailaswasdonefor
-3222.22
the |58p58p58p(cid:105) wells in section IIIB, we merely present
visual evidence of their existence in this paper. Should
these additional asymptotes lend themselves to specific 39000
experimental probing, then computing their respective 36000
R2 (a0)
energy levels would be of value. At this time, however, 33000 23000 24000 25000 26000
such evaluation is beyond the goals of this paper. R1 (a0)
-3222.15
-3222.16 FIG.8. (Coloronline)Potentialsurfacecorrespondingtothe
Energy (GHz) -3222.17 |59s1 − 1;55d3 − 1;58d33(cid:105) asymptotic state. The color
-3222.14 -3222.18 2 2 2 2 22
-3222.19 schemedenotestheenergyvaluesgiveninGHz,withthescale
-3222.16 -3222.2
presentedtotherightoftheplot. Thisparticularsurfaceex-
-3222.18 -3222.21
-3222.2 -3222.22 hibits two potential wells, each about 30-40 MHz deep.
-3222.23
-3222.22
-3222.24
-3222.24 -3222.25
-3222.26 -3222.08
Energy (GHz) -3222.1
-3222.08 -3222.12
39000
-3222.1 -3222.14
36000 -3222.12
R2 (a0) 33000 26000 -3222.14 -3222.16
30000 23000 24000 25000 -3222.16 -3222.18
R1 (a0) -3222.18 -3222.2
-3222.2
FIG.7. (Coloronline)Potentialsurfacecorrespondingtothe
|59s1−1;55d31;58d31(cid:105)asymptoticstate. Thecolorscheme 39000
deno2tes2the en2e2rgy va2lu2es given in GHz, with the scale pre- R2 (a0) 36000
sentedtotherightoftheplot. Thisparticularsurfaceactually 33000
24000 25000 26000
exhibits a few potential wells, the largest one being between
R1 (a0)
40-50 MHz deep.
FIG.9. (Coloronline)Potentialsurfacecorrespondingtothe
|59s1 − 1;55d33;58d3 − 1(cid:105) asymptotic state. The color
2 2 22 2 2
IV. CONCLUSIONS schemedenotestheenergyvaluesgiveninGHz,withthescale
presented to the right of the plot. The well exhibited in this
The work presented in this letter demonstrates results surface is 60-100 MHz deep.
for the Ω = 1/2 symmetry of 58p+58p+58p rubidium
Rydberg atoms. During the course of our analysis, we
also examined the surfaces corresponding to asymptotic oneRydbergatominteractingwithmultiplegroundstate
states near the Rb 58s+58s+58s asymptote, but no atoms or a ground state molecule.
potential wells were found for this case. The formalism We seek to continue to analyze cases of Rydberg
could also be applied to the 58d+58d+58d case, but trimers for various alkali elements, asymptotes, and lin-
thiswouldcorrespondtoalargeincreaseincomputation ear Ω symmetries, as well as exploring the energy levels
time and thus, is outside the scope of the work shown in corresponding to transverse (bending) modes of oscilla-
this paper. Due to an increased number of basis states tion. Similarly, we seek to extend the theory to differ-
forthetriple58dcase,wewouldexpecttofindadditional entmolecularconfigurations, suchastriangularsystems.
wells, however. The detection of such trimer states could have applica-
In addition, the methodology presented here can be tions in a variety of research areas, including quantum
appliedtoRydbergstatesofotherΩ-symmetryvaluesas information processing and exotic, ultracold chemistry.
well as to other alkali elements. The current literature Furthermore, it might be possible to extend the lin-
regardingultracoldmulti-bodyRydbergphysicsinvolves ear chain to include N-Rydberg atoms, although this
7
is purely speculative at this stage. Such investigations
could prove fruitful in the advancement of ultracold
multibody physics.
ACKNOWLEDGMENTS
This work was partially supported by the CSBG divi-
sion of the Department of Energy.
[1] T. Gallagher, Rydberg Atoms (Cambridge University [12] P.Solda´n,M.T.Cvitaˇs,J.M.Hutson,P.Honvault, and
Press, Cambridge, United Kingdom, 1994). J. M. Launay, Phys. Rev. Lett. 89, 153201 (2002).
[2] W.R.Anderson,J.R.Veale, andT.F.Gallagher,Phys. [13] L. P. Parazzoli, N. J. Fitch, P. S. Z˙uchowski, J. M. Hut-
Rev. Lett. 80, 249 (1998). son, and H. J. Lewandowski, Phys. Rev. Lett. 106,
[3] I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, 193201 (2011).
P. Nosbaum, V. M. Akulin, and P. Pillet, Phys. Rev. [14] W. T. Zemke, J. N. Byrd, H. H. Michels, J. John
Lett. 80, 253 (1998). A. Montgomery, and W. C. Stwalley, J. Chem. Phys.
[4] M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. 132 (2010), 10.1063/1.3454656.
Phys. 82, 2313 (2010). [15] J.N.Byrd,J.A.Montgomery, andR.Coˆt´e,Phys.Rev.
[5] C. H. Greene, A. S. Dickinson, and H. R. Sadeghpour, A 82, 010502 (2010).
Phys. Rev. Lett. 85, 2458 (2000). [16] S. T. Rittenhouse, M. Mayle, P. Schmelcher, and H. R.
[6] V. Bendowsky, B. Butscher, J. Nipper, J. P. Shaffer, Sadeghpour, Journal of Physics B: Atomic, Molecular
R. Lo¨w, and T. Pfau, Nature 458, 1005 (2009). and Optical Physics 44, 184005 (2011).
[7] C.Boisseau,I.Simbotin, andR.Coˆt´e,Phys.Rev.Lett. [17] S. T. Rittenhouse and H. R. Sadeghpour, Phys. Rev.
88, 133004 (2002). Lett. 104, 243002 (2010).
[8] N. Samboy, J. Stanojevic, and R. Coˆt´e, Phys. Rev. A [18] I. C. H. Liu and J. M. Rost, Eur. Phys. J. D 40, 65
83, 050501 (2011). (2006).
[9] N. Samboy and R. Coˆt´e, Journal of Physics B: Atomic, [19] I. C. H. Liu, J. Stanojevic, and J. M. Rost, Phys. Rev.
Molecular and Optical Physics 44, 184006 (2011). Lett. 102, 173001 (2009).
[10] K.R.Overstreet,A.Schwettmann,J.Tallant,D.Booth, [20] J. Stanojevic, R. Coˆt´e, D. Tong, S. Farooqi, E. Eyler,
and J. P. Shaffer, Nature Physics 5, 581 (2009). and P. Gould, Eur. Phys. J. D 40, 3 (2006).
[11] J. N. Byrd, J. A. Montgomery, H. H. Michels, and [21] R. J. LeRoy, Can. J. Phys. 52, 246 (1974).
R. Coˆt´e, International Journal of Quantum Chemistry [22] I.I.Beterov,I.I.Ryabtsev,D.B.Tretyakov, andV.M.
109, 3112 (2009). Entin, Phys. Rev. A 79, 052504 (2009).
