Zeeman splitting of nuclear spin energy levels in Tm3+

By Cathy Patterson,2014-08-12 10:43
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Zeeman splitting of nuclear spin energy levels in Tm3+ ...

    Experimental Tailoring of a

    3+Three-Level System in Tm:YAG

    F. de Seze, A. Louchet, V. Crozatier, I. Lorgeré, F. Bretenaker, J.-L. Le Gouët

    Laboratoire Aimé Cotton, CNRS UPR 3321

    Bâtiment 505, campus universitaire, 91405 Orsay Cedex, France

    O. Guillot-Noël, Ph. Goldner

    Laboratoire de Chimie Appliquée de l’Etat Solide, CNRS-UMR 7574,

    ENSCP, 11 rue Pierre et Marie Curie 75231 Paris Cedex 05, France

Abstract : Quantum information transfer from light to atom ensembles and vice versa has

    both basic and practical importance. Among the relevant topics let us mention entanglement and decoherence of macroscopic systems, together with applications to quantum memory for long distance quantum cryptography. Although the first experimental demonstrations have been performed in atomic vapors and clouds, rare earth ion doped crystals are also interesting

    3+media for such processes. In this paper we address Tm ions capability to behave as three-

    level systems, a key ingredient to convert optical excitation into a spontaneous- emission-

    3+free spin wave. Indeed Tm falls within reach of light sources that can be stabilized easily to the required degree. In the absence of zero-field hyperfine structure we apply an external

    3+magnetic field to lift the nuclear spin degeneracy in Tm:YAG. We experimentally determine

    the gyromagnetic tensor components with the help of spectral hole-burning techniques. Then appropriate orientation of the applied field enables us to optimize the transition probability ratio along the two legs of the . The resulting three-level system should suit quantum

    information processing requirements.

    PACS numbers: 42.50.Gy, 42.50.Md, 71.70.Ej


1 Introduction

    Although quantum physics offers the paradigm for microscopic world description, there has been growing interest in observing macroscopic size quantum processes. The idea of mapping a quantum excitation onto a large ensemble of elementary systems, that can be traced back to the early days of quantum mechanics [1], has been successfully revisited in the frame of atomic physics [2]-[8]. In these works an optical collective excitation entangles the macroscopic ensemble state. The activity in the field is stimulated by the potential application to the quantum memories needed for long distance quantum cryptography. In most schemes, the optical excitation is stored as a spin wave in the atomic ensemble [9]. The basic ingredient to convert incident photons into spin excitation is known as a three- system. The spin

    state is built from two nearby sublevels |1> and |2> of the atomic ground state that are connected to a common upper state |3> by optical transitions along the two legs of the .

    This way one avoids decoherence by spontaneous emission. One recalls the stored information to life by turning back the spin coherence |1><2| into the optical coherence |1><3|. So far, ensemble entanglement has been demonstrated only in atomic vapors [[4]-[6]] and beams [3] or in laser cooled atom clouds [2], [7], [8]. However rare earth ion doped crystals (REIC) also appear as promising candidates in the quest for macroscopic quantum effects. They offer properties similar to atomic vapors with the advantage of no atomic diffusion. At low temperature (< 4K) the optical coherence lifetime may reach several ms in these

    3+materials and a hyperfine coherence lifetime of tens of seconds has been reported in Pr:

    YSiO [10], [10]. Given the absence of atomic motion, extremely long population lifetime 25

    can be observed. Storage of light, with quantum memory in prospect, has given rise to Electromagnetically Induced Transparency (EIT) investigations [12], [13]. A storage time greater than one second has been observed in the most recent work [13]. Among all the rare earth ions, non-Kramers ions with an even number of 4f electrons present the longest


    coherence lifetime [14]. With the additional condition of a few tens of MHz hyperfine

    3+3+ and Pr splitting in the electronic ground state, one is practically left only with Eucompounds as good rare earth ion candidates for systems. However, only dye lasers are

    3+3+available at operation wavelengths in Eu: YSiO (580nm) and Pr: YSiO (606nm). 2525

    Because of the high frequency noise generated by the dye jet, this is a challenging task to reach the sub-kHz line width and jitter that can match the long optical coherence lifetime offered by REIC. Actually few dye laser systems throughout the world offer such a high degree of stability [13], [15], [16], [17].

    Some other non-Kramers REIC fall within reach of more tractable lasers. For instance, the 333+H(0) H(0) transition of Tm in YAlO(YAG) can be driven by semiconductor 643512

    3+lasers. Such lasers can be stabilized easily to sub-kHz linewidth and jitter [18]. The Tm:

    YAG compound has been widely used for coherent transient-based signal-processing

    33applications [19], [20]. The atomic coherence associated with the H(0) H(0) transition 64

    at 793nm exhibits a lifetime of 70µs that has been observed to grow to 105 µs under moderate

    3+magnetic field [21]. Although Tm: YAG exhibits no hyperfine structure, Thulium possesses

    a I= ? nuclear spin. Lifting the nuclear spin degeneracy with an external magnetic field may

    (!M0offer a way for building a system, provided one is able to relax the nuclear spin I

    selection rule. Indeed electronic excitation cannot flip the nuclear spin and this can forbid

    3+optical transition along one leg. Observation of long lifetime spectral hole burning in Tm:

    YAG under applied magnetic field proved the nuclear spin selection rule can indeed be relaxed [22] but the optical transition probability ratio was not measured. In recent papers [23], [24] we theoretically investigated the nuclear state mixing induced by an external magnetic field and determined the best field orientation for optimizing the relative strength of the optical transitions along the two legs. In the present paper we measure the transition

    3+probability ratio and show that Tm: YAG can actually operate as a system.


    The paper is arranged as follows. Section 2 sets the theoretical framework. Relying on

    3+: YAG symmetry properties, we propose an effective magnetic field geometrical model Tm

    to describe the level mixing effect and demonstrate the connection between the transition probability ratio and the relative level splitting in upper and lower electronic state. The site diversity is taken into account. Section 3 is devoted to the experimental characterization of the

     system. The relevant parameters are deduced from spectral hole-burning spectra, the

    experiment conditions being specified by the magnetic field orientation and the light beam direction of polarization. We conclude in section 4.


2 Theoretical framework

    2.1 Non-Kramers ions and low symmetry sites

    3+With an even number of electrons, Tm is a non-Kramers ion whose electronic level

    degeneracy is totally lifted by the crystal field at sites with less than threefold rotational symmetry [25]. In such low symmetry substitution sites, the resulting singlet states, typically spaced by energy intervals of several hundreds of GHz, are little affected by spin fluctuation. At low temperature, such ions lying in the lower energy level of Stark multiplets then behave in a way similar to atoms in a low pressure vapor. They can be prepared in long lifetime superposition states.

    169The only natural isotope of Thulium (Tm) exhibits a ? nuclear spin. In low symmetry

    sites the two nuclear states are degenerate in the absence of external magnetic field. This is connected with time reversal symmetry [25]. The electronic singlet states are time reversal

    Jeigenstates, and the total electronic angular momentum is an odd operator with respect to

    ,,Jtime-reversal. Therefore the quantum expectation value vanishes in any Stark sub-level,

    Jwhich is known as spin quenching. Hence, interactions expressed in terms of , such as

    3+hyperfine coupling, also vanish to first order of perturbation. In Tm the hyperfine coupling

    also fails to lift nuclear spin degeneracy to second order because of the ? nuclear spin value [26].

    3+2.2 Building a three-level system in Tm

    The nuclear degeneracy reflects the fact that the nuclear spin commutes with the electronic

    3Hamiltonian. As a result the two nuclear spin substates in the ground state H(0) do not 6

    compose a three level system with a single common nuclear spin substate in upper level 3H(0). Indeed optical excitation cannot flip the nuclear spin. Interaction with an external 4

    magnetic field B, if restricted to the nuclear Zeeman effect, does lift the nuclear spin degeneracy but is unable to overcome this selection rule. Fortunately, the cross-coupling of


    electronic Zeeman effect and hyperfine interaction occurs to second order of perturbation and provides us with the needed nuclear spin state mixing. Let the nuclear Zeeman, the electronic Zeeman and the hyperfine Hamiltonians be respectively given by the following expressions:

    , , (1) HAIJ.HgBI!?~.HgBJ!?~.hyp.nZnneZ

    where A represents the hyperfine coupling constant, g and g, ~ and ~ respectively stand for nn

    the nuclear and electronic Lande factors and for the nuclear and Bohr magnetons. Then, in the basis of the zero-magnetic field eigenstates, to second order of perturbation, the nuclear Zeeman effect, as exacerbated by cross electronic Zeeman and hyperfine interactions, can be expressed by the effective Hamiltonian [26]:

     (2) HBI'??nZijij,,,ijxyz


    ?~~!??,gg (3) ijnnij


    00JnnJij (4) ,!A?ijEE0?0nn

    The lower energy substate of the electronic Stark multiplet is denoted and the sum runs 0

    over the other multiplet sublevels . Sites with D symmetry are specially attractive. Indeed n2

    the tensor is diagonal in the site frame that is built along the three orthogonal two-fold ij

    axes of symmetry Ox, Oy and Oz [27]. Indeed, J, J and J operators transform as the electric xyz

    dipole operators x, y, z. In a D point symmetry, the electric dipole transitions are only 2

    allowed along the Ox, the Oy or the Oz direction. The components with thus vanish ij?ij

    and the tensor is diagonal in the site frame. Therefore the effective nuclear Zeeman Hamiltonian reduces to:


    HBIBIBI'!????? (5) nZxxxyyyzzz


    ??????!!!,, (6) xxxyyyzzz

    In ground and excited states the gyromagnetic factors take on different values. They are

    ()g()erespectively denoted as and . These parameters have been determined theoretically ??ii

    3+for a Tm ion doped YAG matrix that offers D symmetry substitutions sites [24]. The results 2

    can be summarized qualitatively in the following way:

    i) the gyromagnetic tensor is strongly anisotropic in both ground and upper levels,

    33()()()eee()()()gggH(0) and H(0), i.e. and ???,,,???,,,64yxzyxz

    ii) the anisotropy in the (x, z) plane is similar in ground and upper levels, i.e.

    22eeggeegg()()()()()()()()????????////?,,? ;;;;xzxzxzxz

    iii) the (x, y) anisotropy is much larger in ground state than in upper state, i.e.

    ()()()()eegg ????//,,yxyx

    An effective magnetic field picture helps to predict the magnetic field orientation that optimizes the balance of the transition probabilities along the two branches of the .

    2.3 Effective magnetic field picture

    In either ground or upper level, according to Eq. (5), the frequency splitting reads as:

    1/2222222?????BBB(!?? (7) xxyyzz??

    for ? spin states. Let us define an effective field unit vector as:

    ˆ (8) BBBBXYZ???!(((!/,/,/(,,);;effxxyyzz

    In terms of this unit vector the effective nuclear Zeeman Hamiltonian, as defined in Eq. (5), turns into:


    ˆHBI (9) '.!(;;nZeff

    In either electronic level the wavefunctions can be expressed as the product of a common

    -2-1H'H'electronic part and of the eigenvectors. Indeed , of order <10cm, can be ?nZnZel

    -1H'regarded as a perturbation of the electronic Hamiltonian, of order 10 to 100 cm. The nZ

    eigenvectors read as or , where and stand for I 1!???ab2!???abz1122

    eigenvectors and the coefficients are given by:

    111XiYZ?? (10) ab!!,111/21/22211??ZZ;;;;

    111XiYZ?? (11) ab!!?,221/21/22211??ZZ;;;;

    The ratio of the transition probabilities along the two branches of the , or in other words the

    branching ratio to the forbidden transition, is then easily expressed as:

    2eg()()2eff()BB?,,2|3effeff1cos (12) R!!!eff2()2eg()()1cos,,1|3BB1??;;effeff

    ()effwhere represents the angle of the effective field directions in lower and upper electronic

    ()e()gˆˆlevels. Therefore the branching ratio R vanishes when is parallel to and equals BBeffeff

    ()e()gˆˆunity when is orthogonal to . BBeffeff

    2.4 Optimizing the magnetic field orientation

    ()()()eee()()()gggTheory predicts and . Because of this strong anisotropy, ???,,,???,,,yxzyxz

    should the applied magnetic field have a significant component along Oy, the effective field in

    ()effboth states would be aligned along Oy, resulting in a very small relative angle of the

    effective fields in both states and a small branching ratio. Thus an optimally oriented field

    shall be strongly slanted with respect to Oy. However B should not be orthogonal to Oy.


    ()()()()eegg, which means that upper and lower level Indeed, according to theory, ????//xzxz

    effective field components are nearly collinear in plane xOz. Therefore the branching ratio is

    B0expected to be small when . Theory also predicts that anisotropy in ground state is y

    much larger than in excited state. Let us start with an applied field lying in the xOz plane and

    let us add some field component along Oy. Because of the much larger anisotropy in the ground state the effective field component along Oy will grow much faster in ground state than in upper state, which will result in a large effective field relative angle at small applied

    field component along Oy.

     To be more specific, let us suppose that the applied field is initially directed along Ox

    and we look for optimal direction within the plane xOy. From previous discussion we expect the maximum branching ratio will be obtained by slightly tilting the applied field away from

    direction Ox by an angle . The applied field component along Oy is given by where B((

    . The effective field unit vectors read as: (,,1

    ()22jˆBrr((!?,,0/ (13) ;;jjeff

    ()()jjwhere and ()j stands for the level label (e) or (g). The cross product reads as: r??/jxy

    rr(;;eg()()eg?!!sin (14) BBeffeffeff2222??rr((eg

    and is maximum at the following optimal tilt angle value:

     (15) (rr0eg

    The maximum cross product and branching ratio maximum values respectively read as:

    rr;;eg()()eg?! (16) BBeffeffmaxrr;;eg


    2??rreg??R (17) max??rreg??

    According to the definition of level splitting in Eq. (7), the maximum branching ratio can be written as:

    2ge()()??(?(//??gyey?? (18) Rmaxge()()??(?(//??gyey??

    ((where and represent the ground and excited state level splitting at B =0. In terms of yge

    ?the level splitting and of Eq.(15) reads as: y

    ()()ge(??!((/ (19) 0geyy








    3+Figure 1 Tm substitution sites in the Yttrium Aluminium garnet (YAG) matrix. The crystal cell axes are denoted [100], [010] and [001]. There are six cristallographically equivalent, orientationally inequivalent sites, each of them being defined by a local frame. The local axes x, y, and z for site 1 are represented in the figure.

    The Oy axis of site 1 coincides with the crystal cell diagonal [110]. In each site, the transition dipole moment, represented by an oblong, is directed along the local Oy axis. Experiments were carried out with the light passing

    [110]along the axis and the light polarization was directed along the cell diagonal [111].

    In the Appendix we show that these results can be generalized to situations where B is not contained in plane xOy. We show that, whatever the orientation of the applied magnetic field projection in plane xOz, a lower boundary to the optimal branching ratio R and the optimal max


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