SPONTANEOUS DECAY RATES OF THE HYPERFINE STRUCTURE ATOMIC STATES INTO AN OPTICAL NANOFIBER
A. V. Masalov"t V. G. Minoginh**
" l.i In ih r Physical Institute, Russian Academy of Sciences 119991, Moscow, Russia
bInstitute of Spectroscopy, Russian Academy of Sciences 142190, Troitsk, Moscow, Russia
Received November 20, 2013
Spontaneous decay rates of atoms into guided modes of an optical nanofiber are found for atomic transitions between the hyperfine structure sublevels. The decay rates are evaluated for the hyperfine structure transitions in Rb atoms. The efficiency of the guided mode excitation by spontaneous decay of the specific hyperfine atomic states is examined for both the fundamental fiber mode HEn and the higher-order modes TE0i, TM0i, and HE21.
DOI: 10.7868/S0044451014050048
1. INTRODUCTION
Recently, there has been an increasing interest in the interactions between optically excited atoms and dielectric nanobodies, such as optical nanofibers, nanospheres, and nanodisks. This is mostly duo to the potential such systems offer for controlling and manipulating single atoms, and also due to the potential of the systems as regards controlling light propagation inside optical nanostructures. From a more general standpoint, optical nanostructures can be regarded as interfaces that allow connecting the atoms located near nanostructures with the electromagnetic fields propagating inside the nanostructures and accordingly control the states of both the atoms and photons guided inside the nanostructures. Known examples of such interface phenomena are the enhanced spontaneous emission of optically excited atoms into the nanofiber fundamental guided mode fl 4], the red shift of light spontaneously emitted by atoms located near nanofibers or nanospheres [5 10], and excitation of atoms trapped around optical nanofibers by light propagating along the nanofibers fll 14]. New interface phenomena may arise in experiments devoted to entanglement between distant trapped atoms via the
* E-mail: masalov'fflsd.lebedev.ru
E-mail: minogin'flisan.troitsk.ru
guided modes of the nanofiber [15, 16]. In particular, efficient excitation of the higher-order guided modes can be achieved by spontaneous emission of atoms located near the nanofiber interfaces [17].
One of the most efficient techniques that allow coupling the atoms and the electromagnetic field of a nanofiber is the pumping of the guided modes of a nanofiber by spontaneous emission of optically excited atoms. This technique has been experimentally verified for Cs atoms and proved to have very high efficiency [6]. In the research 011 pumping the guided modes of optical nanofibers by spontaneous emission of atoms, a two-level atomic scheme has so far been considered as a basic model. The only extension of this model was given in paper [15], which considered the transition between atomic states degenerate over magnetic sublevels with the example of a Cs atom. The present-day experiments basically use the atoms, such as Rb and Cs, that emit spontaneous radiation into the optical modes of nanofibers performing transitions between the hyperfine structure states. Accordingly, evaluating the spontaneous decay rates for atomic transitions between hyperfine structure states is of importance for the comparison between theoretical pumping rates and experimentally observed ones.
In this paper, we present an analysis of the spontaneous decay rates of atoms into the guided modes of optical nanofibers for a rather general scheme of hyperfine structure sublevels related to an allowed dipole
transition. The rates of spontaneous decays into guided Men ^
nanofiber modes are given both for arbitrary hyper-
fine structure transitions and for the scheme relevant ..........................................................................
for 85Rb atoms. The decays of an 85Rb atom into the
guided modes of a nanofiber are considered for both the ....... . ........... .— .......... \ae2Fe2)
fundamental mode HEn and three higher-order modes
TEqi, TMqi, and HE2i. ........ . ............................... \aeiFei)
2. DECAY RATES AT TRANSITIONS
BETWEEN THE HYPERFINE STRUCTURE STATES
We outline a derivation procedure for the rates of spontaneous decay of an atom into nanofiber guided modes for atomic transitions between the hyperfine structure states. We assume that the atom located near the nanofiber surface is excited by laser light and spontaneously decays, pumping the nanofiber guided modes as shown in Fig. 1. We consider spontaneous decays of the atom at the dipole transitions occuring between the hyperfine structure states |agFg) and |<i, I',) that are defined by the total angular momenta Fg and Fc and degenerate over the magnetic sublevels Mg and Mc as shown in Fig. 2. In such a scheme, the dipole interaction terms are to be defined with respect to the nondegenerate states \agFgMg) and |acFcMc).
The derivation of the spontaneous decay rates follows the Weisskopf Wigner approach based on the
M0.
,Fgm)
M0.
Ma
I Q92 F g 2 }
lQ91 Fg 1 }
Fig. 2. The atomic level scheme describing hyperfine structure sublevels for the allowed dipole transition
Hamiltonian for a "multilevel atom—vacuum field" system taken in the rotating wave approximation:
H = ^w-AVv + r,l0x (ua«a + ^ J -
A
4) - (D
9,C:
where uicg is the frequency of the atomic transition between the ground-state sublevel |g) = \agFgMg) and the excited-state sublevel |e) = |<i, I', M,). uix are the photon frequencies, a^ and u\ are the photon creation and annihilation operators, b\g and bgc are the atomic excitation and de-excitation operators, and d(:g and dg(. are matrix elements of the atomic dipole moment. The vector £\ defines the "electric field amplitude of a single photon" with a unit polarization vector e^ and the wave vector k.
Fig. 1. Schematic of spontaneous emission of an atom into an optical nanofiber. An atom A is optically excited by near-resonant laser light LL. Spontaneous emission can excite four guided modes GM, two with the cr± polarization and the propagation direction +2, and two with the cr± polarization and the propagation direction
£x = I
fiio X jk.r
—eX( . 2fnV
(2)
where V is the quantization volume and eo is the electric constant.
For Hamiltonian (1), equations for the probability amplitudes in the simplest case of a vacuum field taken initially in the vacuum state are
4 >K9T<J>, libiii. 5
817
<V л) = d' / • /A\.f ^/Jc^.l ,
(-a,u = -¿дОхрС/АЛ.,,/)<•,.„.
(3)
where сддА are the probability amplitudes of the states that include the ground atomic substato |g) = = |agFgMg) and the vacuum field state with one photon in the mode Л, and cC;0 is the probability amplitude of the state that includes the excited atomic substato |e) = I'i, I', M,) and the vacuum field state with zero photon numbers in all modes. In the above equation, Лд.,;/ = uix — uJcg is the detuning of the vacuum mode frequency uix with respect to the atomic transition frequency uicg.
Taking a formal solution of the second equation in the above set,
i.
caAx = Jtd!i' ■ £\ j ?4>(i^x,eat')ce,o(t') dt', (4) ¿0
and substituting it in the first equation, we can obtain an equation that describes the spontaneous decay of the upper atomic substato |e) = |<i, I', M,) to the lower substates |g) = \ngFgMg):
¿e,o = ^ ^ l^cjr ' £лГ П" A
I.
x j охр [¿Ад;СЙ(#' — t)] r,.,) (t')dt'. (5)
¿0
This equation is used below to determine the rates of spontaneous decays of an atom into guided modes of a nanofiber for a rather general atomic hyperfine structure scheme shown in Fig. 2.
2.1. Decay into free space
Applying Eq. (5) to the vacuum modes of free space regarded as periodic with the quantization lengths Lj,
kjLj = 2 nnj, j = x.y,
(6)
and standardly summing over all possible photon states and accordingly over proper ground substates gives a well-known decay equation for the upper atomic substate le) = In, I', M,):
Cc, 0 — —IspCc, 0-
(7)
Here, 7sp = Wsp(acFc)/2 is half the total spontaneous decay rate from the hyperfine structure state |<i, /•",) to
all the hyperfine structure states |agFg) belonging to the ground state,
H;,.(>!, /•", ) = i. 2 Y^ "•<>. i..n,i,- (8)
agFg
The partial spontaneous decay rate from the excited state i, /■",) to the ground state |agFg) is
И ",,,(> l, F, -ï agFg) = 2"iacFC}agFg =
1 4 I(ncFc \\d\\agFg)\2 ujf:g
4тге0 3 (2FC + l)he?'
0)
where {<1, /•", ||<i|| is a reduced dipole matrix ele-
ment for a hyperfine structure transition.
The reduced dipole matrix element (<i, /•", ||<i|| agFg) can be expressed through the reduced dipole matrix element {a:c 11ii|| ag) for the fine structure transition defined by the quantum numbers a = nLS-JI [18]. This gives an expression for the spontaneous decay rate between two hyperfine structure states in terms of the 6j-symbols:
И"..,.(>I, F, agFg) = i.n,i, =
r \ 2
Jr F( I
= (2Jf.+l)(2Fy + l)| Wgp(ag ac) =
Fy Jy 1
Wsp(og m,
1 4 I {ас \\d\ \ Q'g)|2 '*of:g
4тге0 3 (2JC + I)he?'
where Wgp(ag ae) is the spontaneous decay rate at the fine structure transition |a:c) —> and Ji is the quantum number of the atomic momentum for the excited fine structure state |a:f.).
2.2. Decay into nanofiber guided modes
We represent the electric field operator of the guided modes of a nanofiber in the standard form
E = + H.c„
(10)
where £\ is the electric field of a single vacuum guided mode and the index A describes the direction of propagation and polarization of a single vacuum guided mode. The electric field of a single guided mode is represented similarly to Eq. (2) as
£x = I
hui x
£\ охр(//?дг + invf),
(H)
where uix is the mode frequency, B\ is the propagation constant, m is the quantum number of the mode angular momentum, ¿x = ¿x(r) is tho normalized
amplitude of the electric field, which depends on the transverse coordinate r, and L is the length of a one-dimensional "quantization box" introduced for counting the allowed values of the propagation
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