ОПТИКА И СПЕКТРОСКОПИЯ, 2014, том 117, № 6, с. 902-906
СПЕКТРОСКОПИЯ ^^^^^^^^
КОНДЕНСИРОВАННОГО СОСТОЯНИЯ
УДК 535.37
COHERENT COOPERATIVE FLUORESCENCE RESONANCE ENERGY TRANSFER © 2014 г. S. K. Sekatskii* and K. K. Pukhov**, ***
* Laboratoire de Physique de la Matière Vivante, IPSB, Ecole Polytechnique Fédérale de Lausanne,
BSP, CH1015 Lausanne, Switzerland ** Laser Materials and Technology Centre of General Physics Institute of Russian Academy of Sciences,
119991 Moscow, GSP-1, Russia *** National Research University "Moscow Power Engineering Institute", 111250 Moscow, Russia
E-mail: serguei.sekatski@epfl.ch Received June 10, 2014
Cooperative fluorescence resonance energy transfer effect is experimentally demonstrated for a few crystals doped with rare-earth ions. We show that, at the liquid helium temperatures in similar crystals, coherent cooperative fluorescence resonance energy transfer, as well as an inverse coherent up-conversion process, could be observed, and briefly discuss possible applications of these effects.
DOI: 10.7868/S0030403414120216
INTRODUCTION
Cooperative fluorescence resonance energy transfer (cooperative FRET) has been predicted in 1957 by Dexter [1]. More than forty years after, this effect has been discovered experimentally first by Basiev et al. for
the transitions from Nd3+ 4 F3/2 ^ 15/2 (donor) to
Ce3+ 2F5/2 ^ 2F7/2 (acceptor) in La1-xCeXF3 : Nd3+ crystals [2—4] and later on by Vergeer et al. for the transitions from Tb3+ 5 D4 ^ 7F6 (donor) to Yb3+ 2F7/2 ^ 2F5/2 (acceptor) in YbxY1-xPO4 : Tb3+ crystals [5]. The above notation means that an excitation energy, which is initially located at the 4 F3/2 state of neody-mium ion, instead of being irradiated in the 4 F3/2 ^ 4I15/2 transition, is transferred to two neighboring cerium ions which initially occupied F5/2 state
and occupy the F7/2 after the transfer, and similarly for the Tb3+—Yb3+ system; see Fig. 1. Observation of cooperative FRET is not only of a purely fundamental interest but such an effect is considered as a practically important case of a quantum cutting phenomenon which exploration can lead to the increase of the efficiency of luminescent materials and solar cells [5—7].
In this paper we predict that, for similar crystals, the coherent cooperative FRET process at low (liquid helium) temperatures could be observed: an electronic excitation, which is initially localized at one rare-earth ion, in appropriate conditions is fully or partially transferred to two neighboring rare-earth ions without the loss of coherency thus forming entangled quantum state of three ions. An inverse process, that is coherent up-conversion, also could be observed for the same
crystals. We believe that, again, these effects are not only of a fundamental interest but that they will find use in a rapidly emerging field of the rare-earth ions-based quantum informatics, where a number of approaches were recently proposed, see e.g. [8—17] as just a few important initial papers, which all were followed by numerous subsequent publications.
COHERENT COOPERATIVE FRET
A necessary prerequisite for existence of the coherent cooperative fluorescence resonance energy trans-
Fig. 1. Schematic diagram of cooperative FRET for the transitions from one initially excited Tb3+ 5D4 ^ 7F6 ion
3 2 2
(donor) to two Yb ions F7/2 ^ F5/2 (acceptors).
fer effect is the really long decoherence times observed for different rare-earth ions in crystals at the liquid helium temperatures. In the most favorable cases, such times can attain a few milliseconds as this was shown,
for example, for 7F0 ^ 5D0 transition of Eu3+ ion in Y2SiO5 crystal [18], while the decoherence times lying in the microsecond range are quite common for the different crystals, especially in the presence of magnetic field. Numerous examples can be found in the literature, see e.g. [19, 20] and [21] where a kind of a mini-review what optical transitions in rare earth ions look especially attractive for quantum computing can be found. For our purposes, here this is enough to cite
the 27 ^s decoherence time measured for the 4 F3/2 excited state of Nd3+ ion in YVO4 crystal [22], the value of the true homogeneous width of 28 kHz, that is the decoherence time around 10 ^s, observed for the transition 5 D4 ^ 1F6 in Tb3+ ion in LiYF4 crystal in the presence of 42 kG magnetic field [23], and so on. Due to the long decoherence times coherent FRET has not only been observed for rare-earth ions in crystals (pair-and quartet-centers of Nd3+ ions in fluoride crystals [24]) but has been already used to prepare different quantum states interesting for the quantum informatics [25].
From now on, we will restrict our consideration
with the transition from Tb3+ 5 D4 ^ 1F6 (donor) to
Yb3+ 2Fi/2 ^ 2F5/2 (acceptor) in YbxY1_xPO4 : Tb3+ crystals. As has been indicated above, incoherent cooperative FRET as well as an inverse up-conversion process (for the latter excitations from two Yb ions sum up onto one Tb ion) were observed for this system earlier, see [5] and references cited therein. We failed to find in the literature the decoherence data pertaining exactly to such crystal but its similarity with many other analyzed systems enables to propose comparable, that is of the order of a few microseconds, decoherence times for it.
The theory of non-coherent cooperative FRET is relatively well established (its foundations have been laid out already by Dexter [1], and important refinements were done lately; see [5, 26, 27]). For our current purposes the exposition given by Kushida [26] seems the most appropriate, hence below we follow the main lines and notation of his papers, whose straightforward modification enables to analyze the case of coherent cooperative FRET.
Let us consider an electromagnetic interaction involving three particles A, B, C whose initial state | abe) changes to | a' b' e'). If, say, particle A is initially excited while particles B and C are not, we have a case of coherent cooperative FRET. For the case of an exact energy resonance between the transitions involved (see
below for the discussion of this assumption), the transfer rate can be written as
P = abc\H\a ' b ' c ').
(1)
Here H is a three-particle interaction Hamiltonian which can be expressed as a sum of appropriate combinations of two-particle interaction Hamiltonians HAB, HAC, HBC thus giving in the second order of the perturbation theory the following general expression suitable to be used for both coherent cooperative FRET and coherent up-conversion:
(abc\H\db' c) = I
1
1
E _e
\ ac pc
Eab Epb'J
x (ab\HAB\|ab') (vc\HAc\a'c)
/ \
I
1
1
\Eab' Epb
Eac E.
(2)
pcj
x (ac |HAC| pc') (pb\HAB\d b') +
+ [HaB][HBC ] + [HBC ][HAB] +
+ [HAC ][HBC] + [HBC][HAC ].
Here the notation Ea. c ■, etc. means the energy of the state | a'd) and p designates some energy level of an appropriate particle (for example, particle A for the case of the term (ab\ HAB | ^b')(^c| HAC \a'c') above), and the expressions in square brackets abbreviate the terms similar to those two which are explicitly written. Hamiltonians HAB, HAC, HBC are "standard" Hamil-
tonians describing the two-particle multipolar electromagnetic interaction. Kushida performed the detailed analysis of the contributions to the matrix element (2) coming from dipole-dipole (dd), dipole-quadrupole (dq), and quadrupole-quadrupole (qq) two-particle interactions, and one of examples given by him deals exactly with the interactions between ions of Yb3+ and Tb3+ which are of interest now. In particular, he demonstrated that the main contribution to the value of
(abc|H a'b'c')2 comes from the lowest-order parity-al-
owed dq-dq process. (Note, that for this dq-dq process an intermediate level ^ should be located in the d-zone of either donor or acceptor ions; correspondingly one can speak about a cooperative energy transfer pathway for the former case and an accrective energy transfer pathway for the latter case; cf. [5, 27, 28]). Kushida also found that the dq-dd (or dd-dq) and dd-dd processes give contributions which are respectively one and two orders of magnitude smaller than that due to the dq-dq mechanism, and hence they can be neglected for our current purposes.
Based essentially on the theoretical estimations of different parameters important for the problem at hand, in paper [26] Kushida presented numerical esti-
mations for the rate of the non-coherent cooperative FRET process for the system of Tb—Yb—Yb ions for the interionic distances RAB = RAC = RBC = 7 a.u. = 3. 7 Â, that is for the value very close to 3.8 Â which corresponds to the ion positions found in the meridian
plane of an elementary cell of YPO4 crystal containing four yttrium ions; see Fig. 2 in [29], cf. also [5]. From his estimations, the following contribution of the dq-dq process to the squared matrix element can be straightforwardly inferred:
|(2^7/2(Yb3+), 2F7/2(Yb3+), 5A(Tb3+)H 2Fs/2(Yb3+), 2FW2(Yb3+), 7F6(Tb3+))|2 /Й2 = 7.2 x 1012 s-2. (3)
As for the pure intraconfiguration qq-qq processes, their consideration is evidently difficult due to the large number of levels of the 4fN electron configuration participating in such a process as an intermediate state ц, hence the precision of the relevant estimations is limiting. Nevertheless, Kushida succeeded to demonstrate that both the interconfigurational and intracon-figurational contributions are of the same order of magnitude.
Using the aforementioned results of Kushida and taking into account the dependence of the parameters of the problem on the characteristics of the concrete crystal as well as the uncertainty with the contribution of the qq-qq process, from (1)—(3) one can give reasonably reliable estimation of the coherent cooperative FRET rate as about P = 3 x 106 s-1. This means that the characteristic time necessary, for example, for the complete entanglement of three interacting ions, T = n/2P, is around 500 ns (or better to
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