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Impulsive optical spin orientation by Zeeman state mixing in ruby
Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
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Impulsive optical spin orientation by Zeeman state mixing in ruby Y. Takagi a, R. Minamihara a, T. Makino b,n a b
Department of Material Science, Graduate School of Material Science, University of Hyogo, 3-2-1 Kohto, Kamigouri-cho, Ako-gun, Hyogo 678-1297, Japan Correlated Electron Research Group (CERG) and Cross-Correlated Materials Research Group (CMRG), RIKEN Advanced Science Institute, Wako 351-0198, Japan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 June 2011 Received in revised form 10 August 2012 Available online 5 September 2012
Optical excitation of electron spin orientation and spin coherence is studied associated with the statemixing due to the coupling of the anisotropic fine-structure and Zeeman energies. An adiabaticallyinduced magnetization is none of the consequence of either the angular-momentum transfer from light to matter or the Boltzmann population difference in the presence of magnetic field. Different types of the state-mixings are described with respect to the combination of the Zeeman sublevels. Experiment has been carried out for a sample of ruby at room temperature. Dependences of the intensity of the induced magnetization on wide ranges of the strength and direction of the magnetic field have shown excellent agreement with density-matrix calculation of the spin expectation values. Consideration of the spin state-mixing also accounts for the polarization-insensitive optical excitation of electron spin orientation previously reported for spin ensembles in poly-crystal and solution. & 2012 Elsevier B.V. All rights reserved.
Keywords: Optical orientation Zeeman state mixing Ruby Polarization state of light Magnetic field Photo-induced magnetization
1. Introduction There have been reported several mechanisms of optical creation of the electron spin orientation. The primary mechanism is direct transfer of angular momentum from light to matter with circularly polarized light, referred to as optical pumping or optical orientation, which has been demonstrated in a variety of materials in gases, [1] liquids, [2] and solids [3]. Non-polarized optical excitation of the spin orientation under an external magnetic field has been demonstrated in anisotropic paramagnetic spin systems such as aromatic hydrocarbon molecules [4] and dimeric 3dmetal complexes [5]. In these reports, optically induced magnetization was assigned to a pumping to the magnetic-field-induced mixed-spin states in the excited triplet state with the assistance of the anisotropic spin-orbit interaction in singlet-to-triplet radiation-less transition. Polarization-insensitive optical excitation as a tool for spin orientation allows its application to a variety of optically anisotropic materials. The optically induced magnetization originating from the Zeeman-level anti-crossing was first observed in ruby by Van der Ziel et al. [6]and later by Kolesov [7] using linearly polarized optical excitation. The ground state of Cr3 þ paramagnetic center in ruby is spin-quartet and has a zero-field splitting (ZFS) due to the trigonal (C3) crystal field and to the spin–orbit interaction. Different pairs in the four Zeeman levels cross at particular strengths of the applied magnetic field along the crystal axis. When the magnetic field has a small
n
Corresponding author. E-mail address: [email protected] (T. Makino).
0304-8853/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2012.08.047
transverse component the degenerated levels couple to each other forming an anti-crossing, a pair of mixed Zeeman states. The mixing with the level having a different magnetic quantum number modifies the depopulation rate of the ground state in the optical transition giving rise to an induced magnetization. The level anti-crossing was explained in terms of a two-level model. An impulsive optical excitation of the precessing magnetization associated with the spin coherence due to the level anti-crossing was also observed by Fukuda et al. [8]. Linearly polarized excitation and optical detection of the spin coherence, instead of the direct observation of oscillating magnetization, were reported by Kolesov [9] in the presence of transverse magnetic field for the ground state of ruby. In this report, we verify experimentally that the non-polarized optical orientation is feasible in more general conditions in terms of the strength and direction of the external magnetic field for spin systems possessing anisotropic fine structures. The optical excitation plays a role of creating a spin alignment that is converted into a mixed state during the Zeeman interaction time, yielding the spin orientation. An angular momentum is not brought by light but created in the spin system as a result of the interaction with the magnetic field. The mechanism holds not only in the ground-state spin system in ruby but also in the excited triplet states in different materials [4,5]. The experiment is performed on ruby at room temperature. The way of admixture between Zeeman states depends on the strength and direction of the applied magnetic field with respect to the C3-axis. The spin-forbidden optical transition is used for the optical orientation except for the experiment of spin coherence (for the convenience of light source) because the selection rule is strictly spin-dependent and well known [10].
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
The present work considers the spin orientation in more general scheme including the off-diagonal part of the effective spin Hamiltonian comparable with the ZFS, giving rise to the coupling of all Zeeman states. In this framework, different cases of the level couplings are described. (1) The coupling between Zeeman states with magnetic quantum numbers m¼3/2 and 1/2 and (2) the coupling between Zeeman states with m¼3/2 and 1/2. These cases appear prominently when the relevant levels have nearly the same Zeeman energies. The optical spin orientation observed in Refs. [6,7] corresponds to this case. (3) A different case appears when the magnetic field is applied perpendicular to the C3-axis. Two couplings arise coexistently in a pair between m¼3/2 and 1/2 and in the other pair between m¼1/2 and 3/2, where the quantization axis is taken along the direction of the magnetic field. The partner levels are energetically far from each other in contrast to the above two cases. It is shown that the spin orientation obtained over a wide range in strength and in direction of the magnetic field is attributed to the coupling of the case (3). Spin coherence is also observed for the case (1) and (2) due to an impulsive creation of the superposed state between the mixed states. The contents of the paper are as follows. In Section 2 the expectation values for the spin orientation and coherence are calculated by the density matrix formalism numerically and then analytically for the individual case of the state-mixing. Section 3 provides experimental procedure. In Section 4, experimental results for the impulsive optical spin orientation and spin coherence are described and compared with the calculation. Also described in Section 4 is a simulation of the spin orientation for spin ensembles in poly-crystal. Appendices A and B describe details in derivation of the spin expectation values argued in Section 2.
2. Spin expectation values The effective spin Hamiltonian of S ¼3/2 ground-state (4A2) under an external magnetic field H tilted with a polar angle y from the C3-axis (z-axis) is given by: 1 ð1Þ H ¼ g J bSz H cosy þ g ? bSx H siny þ D S2z SðS þ 1Þ : 3 Choice of the x–z plane as a plane containing the magnetic field vector does not loose generality because of uniaxial symmetry in the present spin system. Values of the g-factor are given by g J ¼ 1:984 and g? ¼1.987 and 2D ¼ 11.472 GHz for the ZFS parameter. From Eq. (2.1), Zeeman splitting is drawn in Fig. 1 as a function of the magnetic field for y ¼161 and 901 and the eigenstates at H¼213 mT and 459 mT are tabulated in Table 1. It is seen for y ¼161 that the states 3=2 and 1=2 couple strongly at H¼213 mT and 3=2 and 1=2 couple at H¼459 mT. On the other hand for y ¼901 the zero-field eigenstates admix all together. In the present work, we ignore the effects of induced change in spin orientation due to the difference in Boltzmann population and detuning in the optical transition (see Section 4). Contribution from the excited-state population is negligible because the thermal relaxation in the excited-state spin system is considered too fast to be observed at room temperature within the time resolution of experiment. The effect of the ground-state repopulation is not accounted for because of the life time in the excited state much longer (order of millisecond) than the spin relaxation time in the ground state. We first consider, using the eigenvectors shown in Table 1 how the optical transition gives rise to the spin orientation when the
187
Fig. 1. Zeeman splitting of the ground-state (4A2) in ruby as a function of external magnetic field for y ¼ 161 and 901. The energy levels are labeled in order of energy. The vertical scale is shown in unit of the Plank constant.
state-mixing is involved. In this picture, the mixed state has been prepared before the optical excitation. Therefore, the spin level which does not take part in the optical transition contributes to the expectation value through the state-mixing. For a given angle y and magnetic field H, the eigenvectors are written in the form of: 9^iS ¼ ni1 3=2 þ ni2 91=2S þ ni3 91=2S þ ni4 93=2S, ði ¼ 1,2,3,4Þ ð2Þ Time-independent terms in the expectation values of the spin operator S are given by ! 4 pffiffi pffiffi X 2 3 n n 3 n hSx i=_ ¼ pj 9nij 9 2 ni1 ni2 þ ni2 ni3 þ 2 ni3 ni4 þcc , i¼1
Sy ¼ 0, hSz i=_ ¼
4 X
0 @
i¼1
4 X j¼1
1
2A
pj 9nij 9
3 1 2 2 1 2 3 2 9ni1 9 þ 9ni2 9 9ni3 9 9ni4 9 , 2 2 2 2 ð3Þ
R
R
2
where jpj ¼1. A summation jpj9nij9 stands for the rate of depopulation from the eigenstate 9^iS and pj the spin-dependent optical transition probability. Subscripts j (and numerals 1 to 4) denote the magnetic quantum numbers 3/2 to 3/2. The relative transition probabilities are indicated in Fig. 2 for the R1 and R2 transitions [Eð2 EÞ’4 A2 and 2Að2 EÞ’4 A2 , respectively] [10]. Assuming linearly polarized (substantially non-polarized) light incident on the sample along the C3-axis and tuned to the R2 line we take pj ¼0.5 (j ¼2, 3) and pj ¼0 (j ¼1, 4). The result of Sy ¼ 0 is understood from the plane symmetry since the magnetic field lies in x–z plane. Numerical values of hSx i and hSz i are given in Table 1 for different y and H. These values are confirmed by the density-operator formalism as follows.
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Table 1 Eigenvectors of the spin Hamiltonian for y ¼ 161 and 901 at H¼ 213 mT and 459 mT. j (j ¼ 3/2, 1/2, 1/2, 3/2) are zero-field eigenfunctions. Time-independent terms in the expectation values of the spin operator are shown with corresponding eigenvectors.
y ¼161
H¼ 213 mT
E ^ 1 ¼ 0:213=2 þ 0:951=2 þ 0:241=2 þ 0:013=2 E ^ 2 ¼ 0:673=2 0:041=2 0:741=2 0:063=2 E ^ 3 ¼ 0:713=2 0:311=2 þ 0:621=2 þ 0:53=2 E ^ 4 ¼ 0:003=2 0:011=2 þ 0:081=2 1:003=2 hSx i ¼ 0:22_, hSz i ¼ 0:48_
H¼ 459 mT
E ^ 1 ¼ 0:703=2 þ 0:701=2 þ 0:151=2 þ 0:013=2 E ^ 2 ¼ 0:713=2 0:661=2 0:231=2 0:023=2 E ^ 3 ¼ 0:063=2 0:271=2 þ 0:951=2 þ 0:123=2 E ^ 4 ¼ 0:003=2 0:011=2 þ 0:131=2 0:993=2 hSx i ¼ 0:02_, hSz i ¼ 0:26_
y ¼901
H¼ 213 mT
E ^ 1 ¼ 0:193=2 þ 0:681=2 þ 0:681=2 þ 0:193=2 E ^ 2 ¼ 0:363=2 þ 0:611=2 0:611=2 0:363=2 E ^ 3 ¼ 0:683=2 0:191=2 0:191=2 þ 0:683=2 E ^ 4 ¼ 0:613=2 0:361=2 þ 0:361=2 0:613=2 hSx i ¼ 0:50_, hSz i ¼ 0
H¼ 459 mT
E ^ 1 ¼ 0:253=2 þ 0:661=2 þ 0:661=2 þ 0:253=2 E ^ 2 ¼ 0:513=2 þ 0:491=2 0:491=2 0:513=2 E ^ 3 ¼ 0:663=2 0:261=2 0:261=2 þ 0:663=2 E ^ 4 ¼ 0:493=2 0:511=2 þ 0:511=2 0:493=2 hSx i ¼ 0:33_, hSz i ¼ 0
where r(0) is the initial condition of the density operator. U is a unitary matrix consisting of the eigenvectors of the spin Hamiltonian. Then the expectation values of the spin operator S are obtained by /SS ¼ Tr½rS:
ð6Þ
The initial condition of the density operator is determined by the impulsive optical transition leading to a spin-selective depopulation from the equally distributed Zeeman states in 4A2 as rð0Þij ¼ 1=4aij dij ði,j ¼ 3=2,1=2,1=2,3=2Þ, ð7Þ Fig. 2. Relative transition probabilities for the R1- and R2-lines in ruby. s þ and s are circularly polarized excitations and p is linearly polarized (along the C3-axis) excitation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Impulsive optical excitation may also cause spin coherence attributed to the excitation of a superposed state between the different eigenstates. Then, for treating the population and coherent effect in comprehensive manner, we start with the equation of motion for the density operator, i_
drðtÞ ¼ rðt Þ,H : dt
ð4Þ
The solution of Eq. (4) is given by iHd t=_
rðtÞ ¼ Ue with
Hd ¼ U 1 HU
U
1
iHd t=_
rð0ÞUe
U
1
ð5Þ
where dij denotes Kronecker’s delta. Spectral bandwidth of the excitation light is assumed to be broad enough to cover the Zeeman splitting. Off-diagonal elements are omitted since the optical excitation does not create superposed states responsible for the spin orientation in any direction. Numerical calculation of Eq. (6) with Eqs. (5) and (7) leads to the following equations for the cases of the applied magnetic field shown in Table 1. Here we take in Eq. (7) a3/2 3/2 ¼a 3/2 3/2 ¼0 and a1/2 1/2 ¼a 1/2 1/2 ¼0.5 for the ground-state population established by the linearly polarized optical excitation tuned to the R2 line. Note that the initial population difference does not create spin orientation but a spin alignment. For y ¼161 and H¼213 mT, /Sx ðt ÞS=_ ¼ 0:22 þ0:02 coso12 t0:02 coso23 t 0:03 coso34 t 0:15 coso13 t0:04 coso24 t, /Sy ðt ÞS=_ ¼ 0:17 sino12 t0:20 sino23 t þ0:03 sino34 t þ 0:04 sino24 t, /Sz ðtÞS=_ ¼ 0:480:04 coso12 t0:45 coso23 t:
ð8Þ
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
For y ¼ 161 and H¼ 459 mT. Sx ðt Þ =_ ¼ 0:02 þ 0:12coso12 t þ 0:03coso23 t0:10coso34 t 0:02coso13 t Sy ðt Þ =_ ¼ 0:44sino12 t0:03sino23 t þ 0:11sino34 t þ0:02sino13 t, Sz ðt Þ =_ ¼ 0:260:27coso12 t0:01coso23 t þ 0:02coso34 t:
ð9Þ
For y ¼ 901 and H¼213 mT /Sx ðt ÞS=_ ¼ 0:500:12 coso13 t0:38 coso24 t ¼ 0:12ð1coso13 t Þ þ 0:38ð1coso24 t Þ /Sy ðt ÞS=_ ¼ /Sz ðt ÞS=_ ¼ 0:
ð10Þ
For y ¼ 901 and H¼459 mT, /Sx ðt ÞS=_ ¼ 0:330:10 coso13 t0:23 coso24 t ¼ 0:10ð1coso13 t Þ
/Sy ðt ÞS=_ ¼ /Sz ðt ÞS=_ ¼ 0:
ð11Þ
oij’s are angular frequencies of the Zeeman splitting between the E E eigenstates ^i and ^j (i,j¼1,2,3,4). Evidently /S(t)S vanishes at y ¼01 because no state-mixing occurs. /SxS and /SzS have in each a time-independent term and oscillatory terms except for /SzS at y ¼901. The time-independent terms reproduce hSi in Eq. (3) (see Table 1). The oscillatory terms stand for the occurrence of spin coherence. The state-mixing forms coherent superposed states whose initial phase gives /S(0)S¼0, and /SS rises within the Zeeman interaction time oij 1 . There exist different state-mixings in the Zeeman splitting as shown in Fig. 1. They are mixings within a particular pair of Zeeman sub-states for relatively small y and a mixing of all substates for y ¼901. We treat them analytically in the following. For a small y, the state-mixing can as a two beapproximated level mixing in the case (1) between at H 0.42 T 3=2 and 1=2 in first order and in the case (2) 3=2 and 1=2 at H 0.21 T in second order. For the former, the mixing can be described by the Hamiltonian for a fictitious spin interacting with an effective magnetic field H0 Heff ¼ g O bS0z H0z þg ? bS0x H0x ,
ð12Þ
where H0z ¼ Hcos y2jDj=ðg_bÞ and H0x ¼ H siny. Unitary transformation to diagonalize Hef f is given by 0
U¼
cosy =2 0
siny =2
0
siny =2 0
cosy =2
! ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 0 2 2 2 0 where pffiffiffi siny 0¼ V= D þ V , cosy ¼ D= D þ V , D ¼ g J bHz , and V ¼ 3g ? bHx . The initial condition for the density operator is
1 0 r0 ð0Þ ¼ , ð14Þ 0 0 giving an initial spin orientation S0z ¼_=2 in the fictitious spin system due to the depopulation from 1=2 , which cancels out with the opposite orientation due to the simultaneous depopula tion from 1=2 by the selection rule of the optical transition. The expectation values of the fictitious spin are: 0 0 0 Sx ðt Þ =_ ¼ 12 sin y cos y ð1cos otÞ, D E 1 0 S0y ðt Þ =_ ¼ sin y sin ot, 2 0 1 1 0 ð15Þ Sz ðt Þ =_ ¼ sin2 y ð1cos otÞ, 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where o ¼ D þ V 2 =_. From Eq. (15), the fictitious spin represents a torque equation d 0 0 S ¼ S o, dt
þ 0:23ð1coso24 t Þ
ð13Þ
189
ð16Þ
where o ¼ ðV,0, DÞ=_ is an angular-velocity vector. /S0 S undergoes a precession about o with an angular frequency o as illustrated in Fig. 3(a). Precessing magnetization associated with the spin /S0 S draws an elliptic locus with an effective g-tensor pffiffiffi ðg 0J g J , g 0? 3g ? Þ. Also shown in Fig. 3(b) are amplitudes of the spin components in Eq. (15) as a function of D=V. 2 For such a small y that the contribution of V=D ðV ¼ g ? b H siny) is negligible, the Zeeman coupling at H 0.21 T between 3=2 and 1=2 with the difference in magnetic quantum number by Dm ¼2 can be treated as a three-level system through the matrix elements /3/29Sx91/2S and /1/ 29Sx9 1/2S. Spin expectation values are calculated analytically in Appendix A at H¼0.207 T where 3=2 and 1=2 would degenerate when y ¼01. They have forms of: 3V , Sx ðt Þ =_ ¼ 4D 6V sinot, Sy ðt Þ =_ ¼ 7D 24 ð1cosotÞ, Sz ðt Þ =_ ¼ 49
ð17Þ
where o 7V 2 =ð4_jDjÞ is the angular to frequency corresponding the level anti-crossing between 3=2 and 1=2 . Eq. (17) reproduce corresponding terms in Eq. (8) (o ¼ o23) when we take D=V ¼ 0:29 for y ¼161. Other oscillatory terms of the order of 9V/ D9 with angular frequencies o jDj=_ are omitted in Eq. (17) because their frequencies are too high (4 5 GHz) to be detected in our experiment. Accordingly, /Sx(t)S leaves a non-vanishing
0 0 0 0 Fig. 3. Vector model of /S0 (t)S (a). Amplitudes of S0 =_ as a function of D/V (b). Solid line: siny cosy , open squares: siny , closed squares: sin2 y .
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Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
Table 2
x Eigenvectors of the spin Hamiltonian for y ¼ 901 at H¼ 213 mT and 459 mT. j ðj ¼ 7 3=2, 7 1=2) are eigenfunctions of Sx.
y ¼ 901
H ¼213 mT
H ¼459 mT
initial orientation. This originates from the other state-mixing between 3=2 and 1=2 at H 0.42 T. Actually the magnitude of /SxS in Eq. (15) at H 0.21 T is of the order of 9V/D9 as 0 0 siny V=D ¼ V=jDj and cosy 1. Thus, the locus of the vector /SS draws an ellipse with its major and minor axes of length 1 and 9V/D9, respectively. For y ¼ 901 (HJx-axis), the state-mixing can be divided into two pairs of two-level mixings. For this consideration, it is more favorable to take the quantization axis along the external magnetic field for Sx to be diagonalized. The off-diagonal elements of the spin Hamiltonian are brought by the zero-field interaction in this representation [cf. Appendix (B2)]. Eigenfunctions of the Hamiltonian at two values of the magnetic field represenx in this x tation are shown in Table 2, where 73=2 and 7 1=2 are eigenfunctions of Sx. It is readily seen that the state-mixing arises 3=2 x and 1=2 x and in the other pair of separately in a pair of x x 1=2 and 3=2 each with the difference in magnetic quantum number by Dm ¼ 2. Consequently, the spin expectation value is given as a sum of two components derived from the individual state-mixing: pffiffiffi x 1 2 sin y þ 3sin y cos y ð1cos o t Þ Sx ðt Þ =_ ¼ 4 pffiffiffi 1 2 sin y þ 3sin y þ cos y þ ð1cos o þ t Þ, þ ð18Þ 4 pffiffiffi where siny 7 ¼ 3D=ðo 7 _Þ, cosy 7 ¼ ð2D 7DÞ=ðo 7 _Þ, o 7 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 D 7 DD þ D2 =_, and D ¼g?bH. /Sy(t)Sx and /Sz(t)Sx are zero because the responsible offdiagonal elements of the density matrix rðtÞx i i þ 1 and rðtÞx i þ 1 i ði ¼ 1, 2, 3Þ are zero as seen in Table 2. Derivation of Eq. (18) is given in Appendix B. A profile of the time-independent terms (equal to the coefficient of the time-dependent terms) in Eq. (18) is shown in Fig. 4 as a function of H, together with the profiles corresponding to the individual mixing. Eq. (18) reproduces the results of numerical calculation in Eq. (10) and Eq. (11), where o þ and o are equivalent to o13 and o24, respectively. The value of the magnetic field where the amplitude takes its maximum is given by Eq. (18) to be H¼ 79D9/(g?b). Dependence of /SzS and /SxS upon the state-mixing is shown by stereographic plots in Figs. 5 and 6 over entire ranges of the strength and of direction (y) of the applied magnetic field. For relatively small y, sharp bumps and swings are visible around the fields of 0.2 T and 0.4 T originating from the state-mixing in particular pairs with nearly the same energies. For large y in /SxS, on the other hand, all Zeeman states participate in the mixing. The field dependence shows a gentle peak around H¼9D9/g ? b, where the Zeeman energy becomes equal to the ZFS parameter. The amplitude
E x ^ 1 ¼ 0:9793=2Sx þ 0:091=2Sx 0:261=2 0:093=2Sx E x ^ 2 ¼ 0:093=2Sx 0:8791=2Sx 0:01=2 þ 0:4993=2Sx E x ^ x x 3 ¼ 0:2693=2S þ 0:091=2S þ 0:971=2 þ 0:093=2Sx E x x ^ 4 ¼ 0:093=2Sx 0:491=2 þ 0:01=2 0:8793=2Sx E x ^ 1 ¼ 0:9993=2Sx þ 0:091=2Sx 0:151=2 0:093=2Sx E x ^ 2 ¼ 0:093=2Sx 0:9791=2Sx 0:01=2 þ 0:2393=2Sx E x ^ 3 ¼ 0:1593=2Sx þ 0:091=2Sx þ 0:991=2 þ 0:093=2Sx E x x ^ 4 ¼ 0:093=2Sx 0:231=2 þ 0:01=2 0:9793=2Sx
x Fig. 4. Amplitudes of the time-independent part of Sx ðt Þ =_ as a function of magnetic field. Dashed line: first term of Eq. (18), dots: second term, and solid line: sum of the first and second terms.
of the coherence term in /Sz(t)S and its angular frequency o23 in Eq. (8) [or o in Eq. (17)] are shown in Fig. 6(a) and Fig. 6(b), respectively. It is seen that the amplitude profile of the coherence term resembles that of /SzS in Fig. 5(a) except for the opposite polarity. This is consistent with the form of /Sz(t)S in Eq. (17). Theoretical evaluation summarized in Figs. 5 and 6 is examined by the measurement of the optically-induced magnetization.
3. Experiment The light source used for the experiment is a tunable pulsed dye laser pumped by the second-harmonic of a Q-switched neodimium-doped yttrium aluminum garnet (Nd:YAG) laser with a pulse duration of 10 ns and energy of 1 mJ. The wavelength was tuned to the peak of R2 absorption line (692.9 nm) and the spectral width was nominally 0.01 cm 1. For the observation of the spin coherence a frequency-doubled pulse of 20 ps duration from an active-passive-mode-locked Nd:YAG laser at the wavelength of 532 nm was used. The sample is a cylinder-cut ruby (4-mmdiameter and 10-mm-length) with Cr3 þ concentration of 0.05 wt%. A linearly polarized light was focused slightly off the sample along the direction of the C3-axis of the sample. A care was taken for the beam direction with respect to the C3-axis to avoid unintentional conversion into elliptic polarization due to the uniaxial birefringence in ruby. Irradiation with a non-polarized light was also attempted by inserting a quartz depolarizer into the excitation beam. There was no significant difference in its result.
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
191
Fig. 5. Amplitudes of the time-independent term in (a): /SzS and (b): /SxS.
E E Fig. 6. Amplitude (a) and angular frequency (b) of the time-dependent term in /SzS relevant to the Zeeman splitting between the eigenstates 2^ and 3^ .
An external magnetic field up to 0.6 T was applied along arbitrary directions by a rotatable electromagnet. Signal of electromotive force (emf) of the order of 0.1 mV due to the induced magnetization from a ten-turn pick-up coil was amplified with cascaded 0.7-GHz-bandwidth amplifiers with a total gain of 40 dB and its waveform was averaged over 150 laser shots on a digital oscilloscope.
4. Results and discussion 4.1. Optical excitation of spin orientation and coherence The amplitude of the induced magnetization was measured as a function of magnetic-field strength for different polar angles (y) when the pick-up coil is directed along (z component) and perpendicular (x component) to the C3-axis as shown in Figs. 7 and 8, respectively. The vertical scale is commonly taken in all panels in Figs. 7 and 8. In all measurements, the polarity of the magnetization signal changed by inversion of the magnetic field, which verifies that the spin orientation is not brought by the incident light but originates from the external magnetic field. Solid curves show the calculated plots for the time-independent terms in /SzS and /SxS. The experimental results show excellent agreement with the calculation. As expected for relatively small y for z component, the induced magnetization is prominent at H¼0.21 T originating from 3=2 and 1=2 and at H 0.42 T from the state-mixing between the mixing between 3=2 and 1=2 . The peak height in the lower field is about twice larger than the one in the higher field. We should take two origins of the photo-induced magnetization into account in the presence of magnetic field besides the state-mixing [6]. One is the different absorption coefficients for Zeeman sublevels due to the Boltzmann factor and the other
comes from the Zeeman shift in the absorption spectrum for a narrow-band optical excitation. Both contributions should appear as an induced magnetization nearly proportional to the applied magnetic field. The signal intensity due to the former reason is at least two orders-of-magnitude smaller at room temperature than the one given by the state-mixing. The latter origin would give a contribution, to some extent, to the induced magnetization if the excitation is detuned from the center of the spectrum. We have tuned the laser to the center of R2 line in advance to the measurement. However, a slight deviation from the calculation in higher fields commonly seen in Fig. 7 represents that the effect of detuning still remains. The profile of x component in Fig. 8 shows a large swing with opposite polarities for relatively small y around H¼0.4 T. On the other hand, only a small jag is seen around H¼ 0.2 T on the tail of the swing at y ¼2.51. These features are consequences of the firstorder and second-order Zeeman interactions resulting in Eqs. (15) and (17), respectively. With increase in y, the separate swings become blurred and the shape approaches a half wing of the broad dispersion-shaped form, which is coincident with Fig. 4 as well as the numerical plots for y ¼901. The spin coherence associated with the state-mixing has been observed as an oscillating magnetization following an impact optical excitation. For this aim, the light source was replaced to the one with a higher pulse bandwidth described above, and a pick-up coil of single-turn and cascaded amplifiers with a 7-GHzbandwidth were employed. The overall detection bandwidth is limited to 2.6 GHz. Optical excitation with this light does not induce the R2 transition but the broad spin-allowed ‘‘U-band’’ transition 4T2’4A2. However the spin-orbit interaction contributes partly to the oscillator strength inducing a spin-selectivity as in the spin-forbidden transition [11].
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Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
Fig. 7. Amplitudes of the emf signals of optically induced magnetization in ruby at room temperature as a function of magnetic field strength for different directions (y). Pick-up coil is directed along the C3 axis. Dots: experimental results. Solid curves: numerical calculation for the time-independent terms in /SzS. Inset above the profile of y ¼ 481 is a typical signal waveform. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Typical waveforms of the oscillating magnetization, i.e., the free induction decay (FID), are shown in Fig. for the z 9(c) component relevant to the mixing between 3=2 and 1=2 . Appearance of a sharp spike at the leading portion of the FID waveform with a higher frequency and not with a lower frequency is reproduced by an equation for the emf signal of the induced magnetization given by: IðtÞp
Z
t
1
d 1coso23 ðtt 0 Þ jðt 0 Þdt 0 dt 0
ð19Þ
[see Eqs. (8) and (17) for the form of 1coso23 t], where j(t0 ) is a convolution of the excitation intensity with the response function
of the detection system. Assuming as j(t0 ) a Gaussian time profile with a FWHM of 0.4 ns and appropriate decay times, the FID waveforms are simulated as shown in Fig. 9(d). When 1/o23 is much longer than the response time, j(t0 ) plays a role of deltafunction giving rise toIðtÞpsino23 t. On the other hand, when 1/ o23 is close to the response time, the signal is accumulated during the response time and a steep rise appears in the leading edge of the signal. The FID frequencies shown in Fig. 9(a) are in good agreement E E with the splitting frequency between the eigenstates 2^ and 3^ [see Fig. 6(b)]. The range of the magnetic field in which the signal was observed is not limited by the detection frequency
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
193
Fig. 8. Amplitudes of the emf signals of the optically induced magnetization in ruby at room temperature as a function of magnetic field strength for different directions (y). Pick-up coil is directed perpendicular to the C3 axis. Dots: experimental results. Solid curves: numerical calculation for the time-independent terms in /SxS. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
bandwidth but rather limited by the expected amplitude around the level anti-crossing as shown in Fig. 6(a). As expected from Eq. (17), the amplitude of the signal for x- and for y-components must be smaller than the z-component shown in Fig. 9 by a factor of 9V/
D9¼0.2–0.4. Consequently, the signal could not be distinguished from the leak of z-component. FID signals originating from the state-mixing between The 3=2 and 1=2 were also observed around H¼0.42 T. Magnetic-
194
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
Fig. 9. FID frequencies of the z component of induced magnetization vs. magnetic field around (a) H¼0.2 T for different polar angles and (b) around H¼ 0.4 T at y ¼ 51. Dots: experimental results. Solid curves: calculated plots. (c): typical waveforms of the FID signals. (d): simulation of the waveforms in (c). tres is a FWHM of the Gaussian response function. t is the time constant of the exponential decay.
Fig. 10. (a): Amplitude of hSz icosy þ hSx isiny as a function of x- and z-components of the applied field. (b): Ensemble average of hSz icosy þ hSx isiny as a function of the applied field.
field dependence of their frequencies is shown in Fig. 9(b). The signal intensity is expected to be about a half lower than the FID signal around H¼0.2 T for a small y. However, the intensity was much lower and was not enough to measure the magnetic-field dependence of the FID frequency for different y’s. Concerning the decay time of FID, two profiles in Fig. 9(c) look different. Dephasing of the spin coherence is dominated mainly by the local magnetic field from the neighboring Al nuclei in ruby [12]. The local field may affect spin dephasing via the effective g-value defined as do/(bdH), [13] which varies in the range 0–4 in the vicinity of the level anti-crossing. The upper profile should, accordingly, have roughly five-times higher effective g-value, representing a faster decay. The induced oscillation in /SxS for y ¼901 with frequencies o13 and o24 expected in Eqs. (10) and (11) [or o 7 in Eq. (18)] is very
interesting because it represents an oscillating magnetization along the applied magnetic field irrelevant to the Larmor precession. Accordingly the local field could not influence the decay of oscillation. We have not been able to observe it because of our limited detection bandwidth for such high frequencies (410 GHz). 4.2. Simulation of spin orientation in poly-crystal We found previously non-polarized optical orientation for a variety of materials of spin-triplet and spin-sextet systems possessing the ZFS structure [4,5,14]. Most of them were examined in polycrystal and solution. Their magnetic-field dependence equally showed structure-less profiles of the form with a rise to a peak around the magnetic field where Zeeman energy becomes equal to the ZFS energy and then a gentle decrease with increase in magnetic field.
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
It is interesting here to reproduce such a general feature in non-polarized optical orientation by simulating spin expectation values in an ensemble of randomly oriented crystals. We suppose a hypothetic sample of ‘powdered’ ruby composed of fine particles whose optic axes are isotropically directed. The amplitude of the spin component along the direction of the applied magnetic field is written as hSi ¼ hSz icosy þ hSx isiny for a fine particle tilted by y from the magnetic field. /SzS and /SxS are the timeindependent terms given in Section 2 Fig. 10(a) shows the amplitude of /SS as a function of the magnetic field applied in x–z plane. If we take a spin orientation in ensemble average for the powdered ruby it is expected from Fig. 10(a) that the contribution from the four-level mixing would be predominant over the local rises brought by the coupling between nearly degenerated levels. Integration over a stereo-angle up to an angle y0 from the direction of the magnetic field as: R y0 ðhSz icos y þ hSx isinyÞsiny dy hhSiispace ¼ 0 , ð20Þ R y0 0 siny dy gives profiles shown in Fig. 10(b) for different y0 as a function of magnetic field. The profile becomes structure-less when the inte gration range approaches the full stereo-angle hhSiiy0 ¼ p ¼ 2hhSiiy0 ¼ p=2 Þ. The profile for y0 ¼ p/2 resembles that of /SxS in single crystal at y ¼901 (see Figs. 4 and 8) with respect to the peak position and the slope of decrease except for a sharp peak and straight rise toward the peak due mainly to the influence of the higher one of the two bumps in Fig. 10(a). Thus, for randomly oriented spin ensemble, the effect of state-mixing appears dominantly through the coupling between distantly separated energy levels (four-level mixing) rather than the coupling between nearly degenerated levels. Also, the magnetic field where the amplitude takes a peak tells us the magnitude of ZFS parameter. In summary, we have simulated the magnetic-field dependence of non-polarized optical orientation previously reported for different materials in which the molecules are isotropically directed. A spin alignment is created either in the ground or in the excited state as a result of the spin-selectivity in optical transition or in radiation-less transition (intersystem crossing). In the presence of magnetic field a coupling between Zeeman energy and anisotropic fine-structure yields the state-mixing giving rise to the creation of spin orientation.
5. Conclusion Polarization-insensitive optical excitation of electron spin orientation has been observed in ruby under the regime that the angular momentum is not directly transferred from light to the spin system. A spin alignment is created by the optical excitation and converted to the spin orientation in the interaction with the external magnetic field. The experimental result was in good agreement with the calculated spin expectation values based on the spin-state mixing due to the coupling between the fine-structure and Zeeman energy. Spin precessions have also been observed by an impulsive optical excitation creating a superposed state of the mixed states.
the field lifts the degeneracy between 3=2 and 1=2 levels through a partial mixing with 1=2 . In this range of the field state the energy of 3=2 is far below and the contribution of the mixing to 1=2 can be neglected. The spin Hamiltonian is written as: pffiffi 0 1 3 0 12 D 2 g ? bH x pffiffi B C 3 H0 ¼ B ðA1Þ 32 D g ? bH x C @ 2 g ? bH x A: 0 g ? bH x 12 D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Three eigenvalues are D/2 andð2D 7 D2 þ 7V 2 Þ=2, where V¼g?bHx. Their eigenvectors compose a unitary matrix qffiffi 1 0 pffiffi 3V 3 p2ffiffi 2D 7C 7 B pffiffi C B 7V C B 1 ðA2Þ U¼B 0 2D C @ qffiffi A p2ffiffi 37 VD 7
which diagonalizes H0 . Relatively small terms of the order of (V/ D)2 are omitted. The initial condition for the density operator is a22 ¼a33 ¼0.5 and other elements are zero in Eq.(7). Then r(t)ij are r11 ¼ 12 U 11 2 U 31 2 þ U 13 2 U 33 2 þ 2U 11 U 31 U 33 U 13 coso23 t
r22 ¼ 12 U 22 4
U 31 4 þ U 33 4 þ 2U 31 2 U 33 2 coso23 t 0 1 U 12 U 22 3 þ U 11 U 31 2 U 21 þ U 13 U 33 2 U 23 C 2 r12 ¼ 12 B @ þ U 13 U 23 U 22 expðio12 t Þ þU 11 U 31 U 32 U 22 expðio13 t Þ A þ U 32 U 22 U 13 U 33 expðio12 t Þ þU 33 U 23 U 11 U 31 expðio23 t Þ 0 1 U 22 3 U 32 þ U 33 3 U 23 B 2 2 r23 ¼ 12 @ þ U 23 U 33 U 22 expðio12 tÞ þU 31 U 22 U 32 expðio13 tÞ C A þ U 33 2 U 22 U 32 expðio12 t Þ þU 31 2 U 23 U 33 expðio23 t Þ
r33 ¼
1 2
ðA3Þ where oij are of the same notation as in the text. Terms of the order of V/9D9 in the diagonal elements and those of the order of (V/D)2 in the off-diagonal elements are omitted. We eliminate the terms of o12 and o13 in Eq. (A3) for the calculation of the spin expectation values because their frequencies are too high to be detected in our experiment. Then putting o23 o, Eq. (17) is derived from Eqs. (6), (A2), and (A3). B. Derivation of spin expectation values in a different quantization axis. A unitary transformation to change the quantization axis from z-axis to x-axis for spin S¼ 3/2 is given as: 0 1 pffiffiffi pffiffiffi 1 3 3 1 pffiffiffi C B pffiffiffi 3 1 1 3 C 1 B B pffiffiffi C T ¼ pffiffiffi B pffiffiffi ðB1Þ C: 1 3 C 2 2B @ 3 1 A pffiffiffi pffiffiffi 1 3 3 1 The spin Hamiltonian for HD x-axis is given in this representation as 0
Appendix A. Spin expectation values for the level anti-crossing between 3=2 and 1=2 . Spin expectation values are derived for a magnetic field given by H¼9D9/g the Zeeman level-anti-crossing arises ?b, where between 3=2 and 1=2 . A slight perpendicular component of
195
3
B2 B B Hx T 1 HT ¼ B B B @
1
0
pffiffi 3 2 D
0
1 D 2 g ? bH þ 2
0
0
12 g ? bH þ D2
0
0
pffiffi 3 2 D
0
32 g ? bH D2
g ? bH D2 pffiffi 3 2 D
0
pffiffi 3 2 D
C C C C C C A
ðB2Þ Zero-field interaction contributes to the coupling in a pair x x x between 3=2 and 1=2 and in the other pair between 1=2
196
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
and / 3/2x9. /SySx and /SzSx vanish because the off-diagonal elements adjacent to the diagonal elements in Eq. (B2) are zero. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Eigenvalues of Eq. (B2) is D=2 7 D DD þD2 and D=2 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 D þ DD þ D , where V ¼g?bH. Their eigenfunctions give a unitary matrix U which diagonalizes the Hamiltonian: pffiffi 0 pffiffi3D 1 0 2K3þD 0 2K B pffiffi pffiffi C B 0 3D 0 2M3Dþ C B C 2M B C, U¼B L ðB3Þ C Lþ 0 0 B K C Kþ @ A Nþ N 0 0 M Mþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 2 D 4 2 3 D þ L 7 , L 7 ¼ 7 D 8 2 þ P , M 7 ¼ 3 D þN7 , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 7 ¼ 7 D 7 D2 þ P þ , and P 7 ¼ D2 7 DD þ D2 : The density matrix for the initial condition given by the optical transition is rewritten in this representation as 0 1 pffiffiffi 3 0 3 0 p ffiffiffi B C 1 0 3C 1B 0 C, pffiffiffi ðB4Þ rð0Þx T 1 rð0ÞT ¼ B B 8@ 3 0 1 0 C A pffiffiffi 0 3 0 3 where K 7 ¼
where 0
0
B B0
0
0
0
1
0
0
0 1 0
0 0 0
1 rð0Þ ¼ B 2B @0 0
1 C C C C A
The density matrix r(t)xis given according to Eq. (5) in this representation as o pffiffiffi 1n 2 3 sin y þ 3siny cosy ð1coso t Þ rðtÞx 11 ¼ 8 o pffiffiffi 1n 1þ sin2 y þ 3siny þ cosy þ ð1coso þ t Þ rðtÞx 22 ¼ 8
o pffiffiffi 1n 2 1 þ sin y þ 3siny cosy ð1coso t Þ 8 o pffiffiffi 1n 2 3- sin y þ 3siny þ cosy þ ð1coso þ t Þ rðtÞx 44 ¼ ðB5Þ 8 pffiffiffi for the diagonal elements, where siny 7 ¼ 3D=ð2P Þ, cosy 7 ¼ ðD þ D=2Þ=ð2P Þ, and o 7 ¼ 2P 7 =_. Non-zero off-diagonal elements are rðtÞx i i þ 2 and rðtÞx i þ 2 i (i¼ 1, 2) which do not contribute to creation of the spin expectation values. Eq. (6) with the spin operator 0 1 3=2 0 0 0 B C 1=2 0 0 C B 0 C Sx ¼ B B 0 0 1=2 0 C @ A 0 0 0 3=2
rðtÞx 33 ¼
and Eq. (B5) leads to /SxSx as in Eq. (18). References [1] W. Happer, Reviews of Modern Physics 44 (1972) 169. [2] Y. Takagi, Optics Communications 59 (1986) 122. [3] F. Meier, H.P. Zakharchenuya, Optical Orientation, North-Holand, Amsterdam, 1984. [4] Y. Takagi, Chemical Physics Letters 119 (1985) 5. [5] Y. Takagi, S. Miyazaki, K. Ishikawa, Chemical Physics Letters 393 (2004) 314. [6] J.P. van der Ziel, N. Bloembergen, Physical Review 138A (1965) 1287. [7] R. Kolesov, Physical Review A 76 (2006) 043808. [8] Y. Fukuda, Y. Takagi, K. Yamada, T. Hashi, Journal of the Physical Society of Japan 42 (1977) 1061. [9] R. Kolesov, Physical Review A 72 (2005) 051801(R). [10] S. Sugano, Y. Tanabe, Journal of the Physical Society of Japan 13 (1958) 880. [11] Y. Takagi, Y. Fukuda, T. Hashi, Optics Communications 55 (1985) 115. [12] A. Compaan, L.Q. Lambert, I.D. Abella, Optics Communications 3 (1971) 236. [13] Y. Takagi, Y. Fukuda, K. Yamada, T. Hashi, Journal of the Physical Society of Japan 50 (1981) 2672. [14] Y. Takagi, Solid-State Physics (in Japanese) 27 (1992) 197.
Contents lists available at SciVerse ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Impulsive optical spin orientation by Zeeman state mixing in ruby Y. Takagi a, R. Minamihara a, T. Makino b,n a b
Department of Material Science, Graduate School of Material Science, University of Hyogo, 3-2-1 Kohto, Kamigouri-cho, Ako-gun, Hyogo 678-1297, Japan Correlated Electron Research Group (CERG) and Cross-Correlated Materials Research Group (CMRG), RIKEN Advanced Science Institute, Wako 351-0198, Japan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 June 2011 Received in revised form 10 August 2012 Available online 5 September 2012
Optical excitation of electron spin orientation and spin coherence is studied associated with the statemixing due to the coupling of the anisotropic fine-structure and Zeeman energies. An adiabaticallyinduced magnetization is none of the consequence of either the angular-momentum transfer from light to matter or the Boltzmann population difference in the presence of magnetic field. Different types of the state-mixings are described with respect to the combination of the Zeeman sublevels. Experiment has been carried out for a sample of ruby at room temperature. Dependences of the intensity of the induced magnetization on wide ranges of the strength and direction of the magnetic field have shown excellent agreement with density-matrix calculation of the spin expectation values. Consideration of the spin state-mixing also accounts for the polarization-insensitive optical excitation of electron spin orientation previously reported for spin ensembles in poly-crystal and solution. & 2012 Elsevier B.V. All rights reserved.
Keywords: Optical orientation Zeeman state mixing Ruby Polarization state of light Magnetic field Photo-induced magnetization
1. Introduction There have been reported several mechanisms of optical creation of the electron spin orientation. The primary mechanism is direct transfer of angular momentum from light to matter with circularly polarized light, referred to as optical pumping or optical orientation, which has been demonstrated in a variety of materials in gases, [1] liquids, [2] and solids [3]. Non-polarized optical excitation of the spin orientation under an external magnetic field has been demonstrated in anisotropic paramagnetic spin systems such as aromatic hydrocarbon molecules [4] and dimeric 3dmetal complexes [5]. In these reports, optically induced magnetization was assigned to a pumping to the magnetic-field-induced mixed-spin states in the excited triplet state with the assistance of the anisotropic spin-orbit interaction in singlet-to-triplet radiation-less transition. Polarization-insensitive optical excitation as a tool for spin orientation allows its application to a variety of optically anisotropic materials. The optically induced magnetization originating from the Zeeman-level anti-crossing was first observed in ruby by Van der Ziel et al. [6]and later by Kolesov [7] using linearly polarized optical excitation. The ground state of Cr3 þ paramagnetic center in ruby is spin-quartet and has a zero-field splitting (ZFS) due to the trigonal (C3) crystal field and to the spin–orbit interaction. Different pairs in the four Zeeman levels cross at particular strengths of the applied magnetic field along the crystal axis. When the magnetic field has a small
n
Corresponding author. E-mail address: [email protected] (T. Makino).
0304-8853/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2012.08.047
transverse component the degenerated levels couple to each other forming an anti-crossing, a pair of mixed Zeeman states. The mixing with the level having a different magnetic quantum number modifies the depopulation rate of the ground state in the optical transition giving rise to an induced magnetization. The level anti-crossing was explained in terms of a two-level model. An impulsive optical excitation of the precessing magnetization associated with the spin coherence due to the level anti-crossing was also observed by Fukuda et al. [8]. Linearly polarized excitation and optical detection of the spin coherence, instead of the direct observation of oscillating magnetization, were reported by Kolesov [9] in the presence of transverse magnetic field for the ground state of ruby. In this report, we verify experimentally that the non-polarized optical orientation is feasible in more general conditions in terms of the strength and direction of the external magnetic field for spin systems possessing anisotropic fine structures. The optical excitation plays a role of creating a spin alignment that is converted into a mixed state during the Zeeman interaction time, yielding the spin orientation. An angular momentum is not brought by light but created in the spin system as a result of the interaction with the magnetic field. The mechanism holds not only in the ground-state spin system in ruby but also in the excited triplet states in different materials [4,5]. The experiment is performed on ruby at room temperature. The way of admixture between Zeeman states depends on the strength and direction of the applied magnetic field with respect to the C3-axis. The spin-forbidden optical transition is used for the optical orientation except for the experiment of spin coherence (for the convenience of light source) because the selection rule is strictly spin-dependent and well known [10].
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
The present work considers the spin orientation in more general scheme including the off-diagonal part of the effective spin Hamiltonian comparable with the ZFS, giving rise to the coupling of all Zeeman states. In this framework, different cases of the level couplings are described. (1) The coupling between Zeeman states with magnetic quantum numbers m¼3/2 and 1/2 and (2) the coupling between Zeeman states with m¼3/2 and 1/2. These cases appear prominently when the relevant levels have nearly the same Zeeman energies. The optical spin orientation observed in Refs. [6,7] corresponds to this case. (3) A different case appears when the magnetic field is applied perpendicular to the C3-axis. Two couplings arise coexistently in a pair between m¼3/2 and 1/2 and in the other pair between m¼1/2 and 3/2, where the quantization axis is taken along the direction of the magnetic field. The partner levels are energetically far from each other in contrast to the above two cases. It is shown that the spin orientation obtained over a wide range in strength and in direction of the magnetic field is attributed to the coupling of the case (3). Spin coherence is also observed for the case (1) and (2) due to an impulsive creation of the superposed state between the mixed states. The contents of the paper are as follows. In Section 2 the expectation values for the spin orientation and coherence are calculated by the density matrix formalism numerically and then analytically for the individual case of the state-mixing. Section 3 provides experimental procedure. In Section 4, experimental results for the impulsive optical spin orientation and spin coherence are described and compared with the calculation. Also described in Section 4 is a simulation of the spin orientation for spin ensembles in poly-crystal. Appendices A and B describe details in derivation of the spin expectation values argued in Section 2.
2. Spin expectation values The effective spin Hamiltonian of S ¼3/2 ground-state (4A2) under an external magnetic field H tilted with a polar angle y from the C3-axis (z-axis) is given by: 1 ð1Þ H ¼ g J bSz H cosy þ g ? bSx H siny þ D S2z SðS þ 1Þ : 3 Choice of the x–z plane as a plane containing the magnetic field vector does not loose generality because of uniaxial symmetry in the present spin system. Values of the g-factor are given by g J ¼ 1:984 and g? ¼1.987 and 2D ¼ 11.472 GHz for the ZFS parameter. From Eq. (2.1), Zeeman splitting is drawn in Fig. 1 as a function of the magnetic field for y ¼161 and 901 and the eigenstates at H¼213 mT and 459 mT are tabulated in Table 1. It is seen for y ¼161 that the states 3=2 and 1=2 couple strongly at H¼213 mT and 3=2 and 1=2 couple at H¼459 mT. On the other hand for y ¼901 the zero-field eigenstates admix all together. In the present work, we ignore the effects of induced change in spin orientation due to the difference in Boltzmann population and detuning in the optical transition (see Section 4). Contribution from the excited-state population is negligible because the thermal relaxation in the excited-state spin system is considered too fast to be observed at room temperature within the time resolution of experiment. The effect of the ground-state repopulation is not accounted for because of the life time in the excited state much longer (order of millisecond) than the spin relaxation time in the ground state. We first consider, using the eigenvectors shown in Table 1 how the optical transition gives rise to the spin orientation when the
187
Fig. 1. Zeeman splitting of the ground-state (4A2) in ruby as a function of external magnetic field for y ¼ 161 and 901. The energy levels are labeled in order of energy. The vertical scale is shown in unit of the Plank constant.
state-mixing is involved. In this picture, the mixed state has been prepared before the optical excitation. Therefore, the spin level which does not take part in the optical transition contributes to the expectation value through the state-mixing. For a given angle y and magnetic field H, the eigenvectors are written in the form of: 9^iS ¼ ni1 3=2 þ ni2 91=2S þ ni3 91=2S þ ni4 93=2S, ði ¼ 1,2,3,4Þ ð2Þ Time-independent terms in the expectation values of the spin operator S are given by ! 4 pffiffi pffiffi X 2 3 n n 3 n hSx i=_ ¼ pj 9nij 9 2 ni1 ni2 þ ni2 ni3 þ 2 ni3 ni4 þcc , i¼1
Sy ¼ 0, hSz i=_ ¼
4 X
0 @
i¼1
4 X j¼1
1
2A
pj 9nij 9
3 1 2 2 1 2 3 2 9ni1 9 þ 9ni2 9 9ni3 9 9ni4 9 , 2 2 2 2 ð3Þ
R
R
2
where jpj ¼1. A summation jpj9nij9 stands for the rate of depopulation from the eigenstate 9^iS and pj the spin-dependent optical transition probability. Subscripts j (and numerals 1 to 4) denote the magnetic quantum numbers 3/2 to 3/2. The relative transition probabilities are indicated in Fig. 2 for the R1 and R2 transitions [Eð2 EÞ’4 A2 and 2Að2 EÞ’4 A2 , respectively] [10]. Assuming linearly polarized (substantially non-polarized) light incident on the sample along the C3-axis and tuned to the R2 line we take pj ¼0.5 (j ¼2, 3) and pj ¼0 (j ¼1, 4). The result of Sy ¼ 0 is understood from the plane symmetry since the magnetic field lies in x–z plane. Numerical values of hSx i and hSz i are given in Table 1 for different y and H. These values are confirmed by the density-operator formalism as follows.
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Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
Table 1 Eigenvectors of the spin Hamiltonian for y ¼ 161 and 901 at H¼ 213 mT and 459 mT. j (j ¼ 3/2, 1/2, 1/2, 3/2) are zero-field eigenfunctions. Time-independent terms in the expectation values of the spin operator are shown with corresponding eigenvectors.
y ¼161
H¼ 213 mT
E ^ 1 ¼ 0:213=2 þ 0:951=2 þ 0:241=2 þ 0:013=2 E ^ 2 ¼ 0:673=2 0:041=2 0:741=2 0:063=2 E ^ 3 ¼ 0:713=2 0:311=2 þ 0:621=2 þ 0:53=2 E ^ 4 ¼ 0:003=2 0:011=2 þ 0:081=2 1:003=2 hSx i ¼ 0:22_, hSz i ¼ 0:48_
H¼ 459 mT
E ^ 1 ¼ 0:703=2 þ 0:701=2 þ 0:151=2 þ 0:013=2 E ^ 2 ¼ 0:713=2 0:661=2 0:231=2 0:023=2 E ^ 3 ¼ 0:063=2 0:271=2 þ 0:951=2 þ 0:123=2 E ^ 4 ¼ 0:003=2 0:011=2 þ 0:131=2 0:993=2 hSx i ¼ 0:02_, hSz i ¼ 0:26_
y ¼901
H¼ 213 mT
E ^ 1 ¼ 0:193=2 þ 0:681=2 þ 0:681=2 þ 0:193=2 E ^ 2 ¼ 0:363=2 þ 0:611=2 0:611=2 0:363=2 E ^ 3 ¼ 0:683=2 0:191=2 0:191=2 þ 0:683=2 E ^ 4 ¼ 0:613=2 0:361=2 þ 0:361=2 0:613=2 hSx i ¼ 0:50_, hSz i ¼ 0
H¼ 459 mT
E ^ 1 ¼ 0:253=2 þ 0:661=2 þ 0:661=2 þ 0:253=2 E ^ 2 ¼ 0:513=2 þ 0:491=2 0:491=2 0:513=2 E ^ 3 ¼ 0:663=2 0:261=2 0:261=2 þ 0:663=2 E ^ 4 ¼ 0:493=2 0:511=2 þ 0:511=2 0:493=2 hSx i ¼ 0:33_, hSz i ¼ 0
where r(0) is the initial condition of the density operator. U is a unitary matrix consisting of the eigenvectors of the spin Hamiltonian. Then the expectation values of the spin operator S are obtained by /SS ¼ Tr½rS:
ð6Þ
The initial condition of the density operator is determined by the impulsive optical transition leading to a spin-selective depopulation from the equally distributed Zeeman states in 4A2 as rð0Þij ¼ 1=4aij dij ði,j ¼ 3=2,1=2,1=2,3=2Þ, ð7Þ Fig. 2. Relative transition probabilities for the R1- and R2-lines in ruby. s þ and s are circularly polarized excitations and p is linearly polarized (along the C3-axis) excitation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Impulsive optical excitation may also cause spin coherence attributed to the excitation of a superposed state between the different eigenstates. Then, for treating the population and coherent effect in comprehensive manner, we start with the equation of motion for the density operator, i_
drðtÞ ¼ rðt Þ,H : dt
ð4Þ
The solution of Eq. (4) is given by iHd t=_
rðtÞ ¼ Ue with
Hd ¼ U 1 HU
U
1
iHd t=_
rð0ÞUe
U
1
ð5Þ
where dij denotes Kronecker’s delta. Spectral bandwidth of the excitation light is assumed to be broad enough to cover the Zeeman splitting. Off-diagonal elements are omitted since the optical excitation does not create superposed states responsible for the spin orientation in any direction. Numerical calculation of Eq. (6) with Eqs. (5) and (7) leads to the following equations for the cases of the applied magnetic field shown in Table 1. Here we take in Eq. (7) a3/2 3/2 ¼a 3/2 3/2 ¼0 and a1/2 1/2 ¼a 1/2 1/2 ¼0.5 for the ground-state population established by the linearly polarized optical excitation tuned to the R2 line. Note that the initial population difference does not create spin orientation but a spin alignment. For y ¼161 and H¼213 mT, /Sx ðt ÞS=_ ¼ 0:22 þ0:02 coso12 t0:02 coso23 t 0:03 coso34 t 0:15 coso13 t0:04 coso24 t, /Sy ðt ÞS=_ ¼ 0:17 sino12 t0:20 sino23 t þ0:03 sino34 t þ 0:04 sino24 t, /Sz ðtÞS=_ ¼ 0:480:04 coso12 t0:45 coso23 t:
ð8Þ
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
For y ¼ 161 and H¼ 459 mT. Sx ðt Þ =_ ¼ 0:02 þ 0:12coso12 t þ 0:03coso23 t0:10coso34 t 0:02coso13 t Sy ðt Þ =_ ¼ 0:44sino12 t0:03sino23 t þ 0:11sino34 t þ0:02sino13 t, Sz ðt Þ =_ ¼ 0:260:27coso12 t0:01coso23 t þ 0:02coso34 t:
ð9Þ
For y ¼ 901 and H¼213 mT /Sx ðt ÞS=_ ¼ 0:500:12 coso13 t0:38 coso24 t ¼ 0:12ð1coso13 t Þ þ 0:38ð1coso24 t Þ /Sy ðt ÞS=_ ¼ /Sz ðt ÞS=_ ¼ 0:
ð10Þ
For y ¼ 901 and H¼459 mT, /Sx ðt ÞS=_ ¼ 0:330:10 coso13 t0:23 coso24 t ¼ 0:10ð1coso13 t Þ
/Sy ðt ÞS=_ ¼ /Sz ðt ÞS=_ ¼ 0:
ð11Þ
oij’s are angular frequencies of the Zeeman splitting between the E E eigenstates ^i and ^j (i,j¼1,2,3,4). Evidently /S(t)S vanishes at y ¼01 because no state-mixing occurs. /SxS and /SzS have in each a time-independent term and oscillatory terms except for /SzS at y ¼901. The time-independent terms reproduce hSi in Eq. (3) (see Table 1). The oscillatory terms stand for the occurrence of spin coherence. The state-mixing forms coherent superposed states whose initial phase gives /S(0)S¼0, and /SS rises within the Zeeman interaction time oij 1 . There exist different state-mixings in the Zeeman splitting as shown in Fig. 1. They are mixings within a particular pair of Zeeman sub-states for relatively small y and a mixing of all substates for y ¼901. We treat them analytically in the following. For a small y, the state-mixing can as a two beapproximated level mixing in the case (1) between at H 0.42 T 3=2 and 1=2 in first order and in the case (2) 3=2 and 1=2 at H 0.21 T in second order. For the former, the mixing can be described by the Hamiltonian for a fictitious spin interacting with an effective magnetic field H0 Heff ¼ g O bS0z H0z þg ? bS0x H0x ,
ð12Þ
where H0z ¼ Hcos y2jDj=ðg_bÞ and H0x ¼ H siny. Unitary transformation to diagonalize Hef f is given by 0
U¼
cosy =2 0
siny =2
0
siny =2 0
cosy =2
! ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 0 2 2 2 0 where pffiffiffi siny 0¼ V= D þ V , cosy ¼ D= D þ V , D ¼ g J bHz , and V ¼ 3g ? bHx . The initial condition for the density operator is
1 0 r0 ð0Þ ¼ , ð14Þ 0 0 giving an initial spin orientation S0z ¼_=2 in the fictitious spin system due to the depopulation from 1=2 , which cancels out with the opposite orientation due to the simultaneous depopula tion from 1=2 by the selection rule of the optical transition. The expectation values of the fictitious spin are: 0 0 0 Sx ðt Þ =_ ¼ 12 sin y cos y ð1cos otÞ, D E 1 0 S0y ðt Þ =_ ¼ sin y sin ot, 2 0 1 1 0 ð15Þ Sz ðt Þ =_ ¼ sin2 y ð1cos otÞ, 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where o ¼ D þ V 2 =_. From Eq. (15), the fictitious spin represents a torque equation d 0 0 S ¼ S o, dt
þ 0:23ð1coso24 t Þ
ð13Þ
189
ð16Þ
where o ¼ ðV,0, DÞ=_ is an angular-velocity vector. /S0 S undergoes a precession about o with an angular frequency o as illustrated in Fig. 3(a). Precessing magnetization associated with the spin /S0 S draws an elliptic locus with an effective g-tensor pffiffiffi ðg 0J g J , g 0? 3g ? Þ. Also shown in Fig. 3(b) are amplitudes of the spin components in Eq. (15) as a function of D=V. 2 For such a small y that the contribution of V=D ðV ¼ g ? b H siny) is negligible, the Zeeman coupling at H 0.21 T between 3=2 and 1=2 with the difference in magnetic quantum number by Dm ¼2 can be treated as a three-level system through the matrix elements /3/29Sx91/2S and /1/ 29Sx9 1/2S. Spin expectation values are calculated analytically in Appendix A at H¼0.207 T where 3=2 and 1=2 would degenerate when y ¼01. They have forms of: 3V , Sx ðt Þ =_ ¼ 4D 6V sinot, Sy ðt Þ =_ ¼ 7D 24 ð1cosotÞ, Sz ðt Þ =_ ¼ 49
ð17Þ
where o 7V 2 =ð4_jDjÞ is the angular to frequency corresponding the level anti-crossing between 3=2 and 1=2 . Eq. (17) reproduce corresponding terms in Eq. (8) (o ¼ o23) when we take D=V ¼ 0:29 for y ¼161. Other oscillatory terms of the order of 9V/ D9 with angular frequencies o jDj=_ are omitted in Eq. (17) because their frequencies are too high (4 5 GHz) to be detected in our experiment. Accordingly, /Sx(t)S leaves a non-vanishing
0 0 0 0 Fig. 3. Vector model of /S0 (t)S (a). Amplitudes of S0 =_ as a function of D/V (b). Solid line: siny cosy , open squares: siny , closed squares: sin2 y .
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Table 2
x Eigenvectors of the spin Hamiltonian for y ¼ 901 at H¼ 213 mT and 459 mT. j ðj ¼ 7 3=2, 7 1=2) are eigenfunctions of Sx.
y ¼ 901
H ¼213 mT
H ¼459 mT
initial orientation. This originates from the other state-mixing between 3=2 and 1=2 at H 0.42 T. Actually the magnitude of /SxS in Eq. (15) at H 0.21 T is of the order of 9V/D9 as 0 0 siny V=D ¼ V=jDj and cosy 1. Thus, the locus of the vector /SS draws an ellipse with its major and minor axes of length 1 and 9V/D9, respectively. For y ¼ 901 (HJx-axis), the state-mixing can be divided into two pairs of two-level mixings. For this consideration, it is more favorable to take the quantization axis along the external magnetic field for Sx to be diagonalized. The off-diagonal elements of the spin Hamiltonian are brought by the zero-field interaction in this representation [cf. Appendix (B2)]. Eigenfunctions of the Hamiltonian at two values of the magnetic field represenx in this x tation are shown in Table 2, where 73=2 and 7 1=2 are eigenfunctions of Sx. It is readily seen that the state-mixing arises 3=2 x and 1=2 x and in the other pair of separately in a pair of x x 1=2 and 3=2 each with the difference in magnetic quantum number by Dm ¼ 2. Consequently, the spin expectation value is given as a sum of two components derived from the individual state-mixing: pffiffiffi x 1 2 sin y þ 3sin y cos y ð1cos o t Þ Sx ðt Þ =_ ¼ 4 pffiffiffi 1 2 sin y þ 3sin y þ cos y þ ð1cos o þ t Þ, þ ð18Þ 4 pffiffiffi where siny 7 ¼ 3D=ðo 7 _Þ, cosy 7 ¼ ð2D 7DÞ=ðo 7 _Þ, o 7 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 D 7 DD þ D2 =_, and D ¼g?bH. /Sy(t)Sx and /Sz(t)Sx are zero because the responsible offdiagonal elements of the density matrix rðtÞx i i þ 1 and rðtÞx i þ 1 i ði ¼ 1, 2, 3Þ are zero as seen in Table 2. Derivation of Eq. (18) is given in Appendix B. A profile of the time-independent terms (equal to the coefficient of the time-dependent terms) in Eq. (18) is shown in Fig. 4 as a function of H, together with the profiles corresponding to the individual mixing. Eq. (18) reproduces the results of numerical calculation in Eq. (10) and Eq. (11), where o þ and o are equivalent to o13 and o24, respectively. The value of the magnetic field where the amplitude takes its maximum is given by Eq. (18) to be H¼ 79D9/(g?b). Dependence of /SzS and /SxS upon the state-mixing is shown by stereographic plots in Figs. 5 and 6 over entire ranges of the strength and of direction (y) of the applied magnetic field. For relatively small y, sharp bumps and swings are visible around the fields of 0.2 T and 0.4 T originating from the state-mixing in particular pairs with nearly the same energies. For large y in /SxS, on the other hand, all Zeeman states participate in the mixing. The field dependence shows a gentle peak around H¼9D9/g ? b, where the Zeeman energy becomes equal to the ZFS parameter. The amplitude
E x ^ 1 ¼ 0:9793=2Sx þ 0:091=2Sx 0:261=2 0:093=2Sx E x ^ 2 ¼ 0:093=2Sx 0:8791=2Sx 0:01=2 þ 0:4993=2Sx E x ^ x x 3 ¼ 0:2693=2S þ 0:091=2S þ 0:971=2 þ 0:093=2Sx E x x ^ 4 ¼ 0:093=2Sx 0:491=2 þ 0:01=2 0:8793=2Sx E x ^ 1 ¼ 0:9993=2Sx þ 0:091=2Sx 0:151=2 0:093=2Sx E x ^ 2 ¼ 0:093=2Sx 0:9791=2Sx 0:01=2 þ 0:2393=2Sx E x ^ 3 ¼ 0:1593=2Sx þ 0:091=2Sx þ 0:991=2 þ 0:093=2Sx E x x ^ 4 ¼ 0:093=2Sx 0:231=2 þ 0:01=2 0:9793=2Sx
x Fig. 4. Amplitudes of the time-independent part of Sx ðt Þ =_ as a function of magnetic field. Dashed line: first term of Eq. (18), dots: second term, and solid line: sum of the first and second terms.
of the coherence term in /Sz(t)S and its angular frequency o23 in Eq. (8) [or o in Eq. (17)] are shown in Fig. 6(a) and Fig. 6(b), respectively. It is seen that the amplitude profile of the coherence term resembles that of /SzS in Fig. 5(a) except for the opposite polarity. This is consistent with the form of /Sz(t)S in Eq. (17). Theoretical evaluation summarized in Figs. 5 and 6 is examined by the measurement of the optically-induced magnetization.
3. Experiment The light source used for the experiment is a tunable pulsed dye laser pumped by the second-harmonic of a Q-switched neodimium-doped yttrium aluminum garnet (Nd:YAG) laser with a pulse duration of 10 ns and energy of 1 mJ. The wavelength was tuned to the peak of R2 absorption line (692.9 nm) and the spectral width was nominally 0.01 cm 1. For the observation of the spin coherence a frequency-doubled pulse of 20 ps duration from an active-passive-mode-locked Nd:YAG laser at the wavelength of 532 nm was used. The sample is a cylinder-cut ruby (4-mmdiameter and 10-mm-length) with Cr3 þ concentration of 0.05 wt%. A linearly polarized light was focused slightly off the sample along the direction of the C3-axis of the sample. A care was taken for the beam direction with respect to the C3-axis to avoid unintentional conversion into elliptic polarization due to the uniaxial birefringence in ruby. Irradiation with a non-polarized light was also attempted by inserting a quartz depolarizer into the excitation beam. There was no significant difference in its result.
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191
Fig. 5. Amplitudes of the time-independent term in (a): /SzS and (b): /SxS.
E E Fig. 6. Amplitude (a) and angular frequency (b) of the time-dependent term in /SzS relevant to the Zeeman splitting between the eigenstates 2^ and 3^ .
An external magnetic field up to 0.6 T was applied along arbitrary directions by a rotatable electromagnet. Signal of electromotive force (emf) of the order of 0.1 mV due to the induced magnetization from a ten-turn pick-up coil was amplified with cascaded 0.7-GHz-bandwidth amplifiers with a total gain of 40 dB and its waveform was averaged over 150 laser shots on a digital oscilloscope.
4. Results and discussion 4.1. Optical excitation of spin orientation and coherence The amplitude of the induced magnetization was measured as a function of magnetic-field strength for different polar angles (y) when the pick-up coil is directed along (z component) and perpendicular (x component) to the C3-axis as shown in Figs. 7 and 8, respectively. The vertical scale is commonly taken in all panels in Figs. 7 and 8. In all measurements, the polarity of the magnetization signal changed by inversion of the magnetic field, which verifies that the spin orientation is not brought by the incident light but originates from the external magnetic field. Solid curves show the calculated plots for the time-independent terms in /SzS and /SxS. The experimental results show excellent agreement with the calculation. As expected for relatively small y for z component, the induced magnetization is prominent at H¼0.21 T originating from 3=2 and 1=2 and at H 0.42 T from the state-mixing between the mixing between 3=2 and 1=2 . The peak height in the lower field is about twice larger than the one in the higher field. We should take two origins of the photo-induced magnetization into account in the presence of magnetic field besides the state-mixing [6]. One is the different absorption coefficients for Zeeman sublevels due to the Boltzmann factor and the other
comes from the Zeeman shift in the absorption spectrum for a narrow-band optical excitation. Both contributions should appear as an induced magnetization nearly proportional to the applied magnetic field. The signal intensity due to the former reason is at least two orders-of-magnitude smaller at room temperature than the one given by the state-mixing. The latter origin would give a contribution, to some extent, to the induced magnetization if the excitation is detuned from the center of the spectrum. We have tuned the laser to the center of R2 line in advance to the measurement. However, a slight deviation from the calculation in higher fields commonly seen in Fig. 7 represents that the effect of detuning still remains. The profile of x component in Fig. 8 shows a large swing with opposite polarities for relatively small y around H¼0.4 T. On the other hand, only a small jag is seen around H¼ 0.2 T on the tail of the swing at y ¼2.51. These features are consequences of the firstorder and second-order Zeeman interactions resulting in Eqs. (15) and (17), respectively. With increase in y, the separate swings become blurred and the shape approaches a half wing of the broad dispersion-shaped form, which is coincident with Fig. 4 as well as the numerical plots for y ¼901. The spin coherence associated with the state-mixing has been observed as an oscillating magnetization following an impact optical excitation. For this aim, the light source was replaced to the one with a higher pulse bandwidth described above, and a pick-up coil of single-turn and cascaded amplifiers with a 7-GHzbandwidth were employed. The overall detection bandwidth is limited to 2.6 GHz. Optical excitation with this light does not induce the R2 transition but the broad spin-allowed ‘‘U-band’’ transition 4T2’4A2. However the spin-orbit interaction contributes partly to the oscillator strength inducing a spin-selectivity as in the spin-forbidden transition [11].
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Fig. 7. Amplitudes of the emf signals of optically induced magnetization in ruby at room temperature as a function of magnetic field strength for different directions (y). Pick-up coil is directed along the C3 axis. Dots: experimental results. Solid curves: numerical calculation for the time-independent terms in /SzS. Inset above the profile of y ¼ 481 is a typical signal waveform. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Typical waveforms of the oscillating magnetization, i.e., the free induction decay (FID), are shown in Fig. for the z 9(c) component relevant to the mixing between 3=2 and 1=2 . Appearance of a sharp spike at the leading portion of the FID waveform with a higher frequency and not with a lower frequency is reproduced by an equation for the emf signal of the induced magnetization given by: IðtÞp
Z
t
1
d 1coso23 ðtt 0 Þ jðt 0 Þdt 0 dt 0
ð19Þ
[see Eqs. (8) and (17) for the form of 1coso23 t], where j(t0 ) is a convolution of the excitation intensity with the response function
of the detection system. Assuming as j(t0 ) a Gaussian time profile with a FWHM of 0.4 ns and appropriate decay times, the FID waveforms are simulated as shown in Fig. 9(d). When 1/o23 is much longer than the response time, j(t0 ) plays a role of deltafunction giving rise toIðtÞpsino23 t. On the other hand, when 1/ o23 is close to the response time, the signal is accumulated during the response time and a steep rise appears in the leading edge of the signal. The FID frequencies shown in Fig. 9(a) are in good agreement E E with the splitting frequency between the eigenstates 2^ and 3^ [see Fig. 6(b)]. The range of the magnetic field in which the signal was observed is not limited by the detection frequency
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
193
Fig. 8. Amplitudes of the emf signals of the optically induced magnetization in ruby at room temperature as a function of magnetic field strength for different directions (y). Pick-up coil is directed perpendicular to the C3 axis. Dots: experimental results. Solid curves: numerical calculation for the time-independent terms in /SxS. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
bandwidth but rather limited by the expected amplitude around the level anti-crossing as shown in Fig. 6(a). As expected from Eq. (17), the amplitude of the signal for x- and for y-components must be smaller than the z-component shown in Fig. 9 by a factor of 9V/
D9¼0.2–0.4. Consequently, the signal could not be distinguished from the leak of z-component. FID signals originating from the state-mixing between The 3=2 and 1=2 were also observed around H¼0.42 T. Magnetic-
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Fig. 9. FID frequencies of the z component of induced magnetization vs. magnetic field around (a) H¼0.2 T for different polar angles and (b) around H¼ 0.4 T at y ¼ 51. Dots: experimental results. Solid curves: calculated plots. (c): typical waveforms of the FID signals. (d): simulation of the waveforms in (c). tres is a FWHM of the Gaussian response function. t is the time constant of the exponential decay.
Fig. 10. (a): Amplitude of hSz icosy þ hSx isiny as a function of x- and z-components of the applied field. (b): Ensemble average of hSz icosy þ hSx isiny as a function of the applied field.
field dependence of their frequencies is shown in Fig. 9(b). The signal intensity is expected to be about a half lower than the FID signal around H¼0.2 T for a small y. However, the intensity was much lower and was not enough to measure the magnetic-field dependence of the FID frequency for different y’s. Concerning the decay time of FID, two profiles in Fig. 9(c) look different. Dephasing of the spin coherence is dominated mainly by the local magnetic field from the neighboring Al nuclei in ruby [12]. The local field may affect spin dephasing via the effective g-value defined as do/(bdH), [13] which varies in the range 0–4 in the vicinity of the level anti-crossing. The upper profile should, accordingly, have roughly five-times higher effective g-value, representing a faster decay. The induced oscillation in /SxS for y ¼901 with frequencies o13 and o24 expected in Eqs. (10) and (11) [or o 7 in Eq. (18)] is very
interesting because it represents an oscillating magnetization along the applied magnetic field irrelevant to the Larmor precession. Accordingly the local field could not influence the decay of oscillation. We have not been able to observe it because of our limited detection bandwidth for such high frequencies (410 GHz). 4.2. Simulation of spin orientation in poly-crystal We found previously non-polarized optical orientation for a variety of materials of spin-triplet and spin-sextet systems possessing the ZFS structure [4,5,14]. Most of them were examined in polycrystal and solution. Their magnetic-field dependence equally showed structure-less profiles of the form with a rise to a peak around the magnetic field where Zeeman energy becomes equal to the ZFS energy and then a gentle decrease with increase in magnetic field.
Y. Takagi et al. / Journal of Magnetism and Magnetic Materials 326 (2013) 186–196
It is interesting here to reproduce such a general feature in non-polarized optical orientation by simulating spin expectation values in an ensemble of randomly oriented crystals. We suppose a hypothetic sample of ‘powdered’ ruby composed of fine particles whose optic axes are isotropically directed. The amplitude of the spin component along the direction of the applied magnetic field is written as hSi ¼ hSz icosy þ hSx isiny for a fine particle tilted by y from the magnetic field. /SzS and /SxS are the timeindependent terms given in Section 2 Fig. 10(a) shows the amplitude of /SS as a function of the magnetic field applied in x–z plane. If we take a spin orientation in ensemble average for the powdered ruby it is expected from Fig. 10(a) that the contribution from the four-level mixing would be predominant over the local rises brought by the coupling between nearly degenerated levels. Integration over a stereo-angle up to an angle y0 from the direction of the magnetic field as: R y0 ðhSz icos y þ hSx isinyÞsiny dy hhSiispace ¼ 0 , ð20Þ R y0 0 siny dy gives profiles shown in Fig. 10(b) for different y0 as a function of magnetic field. The profile becomes structure-less when the inte gration range approaches the full stereo-angle hhSiiy0 ¼ p ¼ 2hhSiiy0 ¼ p=2 Þ. The profile for y0 ¼ p/2 resembles that of /SxS in single crystal at y ¼901 (see Figs. 4 and 8) with respect to the peak position and the slope of decrease except for a sharp peak and straight rise toward the peak due mainly to the influence of the higher one of the two bumps in Fig. 10(a). Thus, for randomly oriented spin ensemble, the effect of state-mixing appears dominantly through the coupling between distantly separated energy levels (four-level mixing) rather than the coupling between nearly degenerated levels. Also, the magnetic field where the amplitude takes a peak tells us the magnitude of ZFS parameter. In summary, we have simulated the magnetic-field dependence of non-polarized optical orientation previously reported for different materials in which the molecules are isotropically directed. A spin alignment is created either in the ground or in the excited state as a result of the spin-selectivity in optical transition or in radiation-less transition (intersystem crossing). In the presence of magnetic field a coupling between Zeeman energy and anisotropic fine-structure yields the state-mixing giving rise to the creation of spin orientation.
5. Conclusion Polarization-insensitive optical excitation of electron spin orientation has been observed in ruby under the regime that the angular momentum is not directly transferred from light to the spin system. A spin alignment is created by the optical excitation and converted to the spin orientation in the interaction with the external magnetic field. The experimental result was in good agreement with the calculated spin expectation values based on the spin-state mixing due to the coupling between the fine-structure and Zeeman energy. Spin precessions have also been observed by an impulsive optical excitation creating a superposed state of the mixed states.
the field lifts the degeneracy between 3=2 and 1=2 levels through a partial mixing with 1=2 . In this range of the field state the energy of 3=2 is far below and the contribution of the mixing to 1=2 can be neglected. The spin Hamiltonian is written as: pffiffi 0 1 3 0 12 D 2 g ? bH x pffiffi B C 3 H0 ¼ B ðA1Þ 32 D g ? bH x C @ 2 g ? bH x A: 0 g ? bH x 12 D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Three eigenvalues are D/2 andð2D 7 D2 þ 7V 2 Þ=2, where V¼g?bHx. Their eigenvectors compose a unitary matrix qffiffi 1 0 pffiffi 3V 3 p2ffiffi 2D 7C 7 B pffiffi C B 7V C B 1 ðA2Þ U¼B 0 2D C @ qffiffi A p2ffiffi 37 VD 7
which diagonalizes H0 . Relatively small terms of the order of (V/ D)2 are omitted. The initial condition for the density operator is a22 ¼a33 ¼0.5 and other elements are zero in Eq.(7). Then r(t)ij are r11 ¼ 12 U 11 2 U 31 2 þ U 13 2 U 33 2 þ 2U 11 U 31 U 33 U 13 coso23 t
r22 ¼ 12 U 22 4
U 31 4 þ U 33 4 þ 2U 31 2 U 33 2 coso23 t 0 1 U 12 U 22 3 þ U 11 U 31 2 U 21 þ U 13 U 33 2 U 23 C 2 r12 ¼ 12 B @ þ U 13 U 23 U 22 expðio12 t Þ þU 11 U 31 U 32 U 22 expðio13 t Þ A þ U 32 U 22 U 13 U 33 expðio12 t Þ þU 33 U 23 U 11 U 31 expðio23 t Þ 0 1 U 22 3 U 32 þ U 33 3 U 23 B 2 2 r23 ¼ 12 @ þ U 23 U 33 U 22 expðio12 tÞ þU 31 U 22 U 32 expðio13 tÞ C A þ U 33 2 U 22 U 32 expðio12 t Þ þU 31 2 U 23 U 33 expðio23 t Þ
r33 ¼
1 2
ðA3Þ where oij are of the same notation as in the text. Terms of the order of V/9D9 in the diagonal elements and those of the order of (V/D)2 in the off-diagonal elements are omitted. We eliminate the terms of o12 and o13 in Eq. (A3) for the calculation of the spin expectation values because their frequencies are too high to be detected in our experiment. Then putting o23 o, Eq. (17) is derived from Eqs. (6), (A2), and (A3). B. Derivation of spin expectation values in a different quantization axis. A unitary transformation to change the quantization axis from z-axis to x-axis for spin S¼ 3/2 is given as: 0 1 pffiffiffi pffiffiffi 1 3 3 1 pffiffiffi C B pffiffiffi 3 1 1 3 C 1 B B pffiffiffi C T ¼ pffiffiffi B pffiffiffi ðB1Þ C: 1 3 C 2 2B @ 3 1 A pffiffiffi pffiffiffi 1 3 3 1 The spin Hamiltonian for HD x-axis is given in this representation as 0
Appendix A. Spin expectation values for the level anti-crossing between 3=2 and 1=2 . Spin expectation values are derived for a magnetic field given by H¼9D9/g the Zeeman level-anti-crossing arises ?b, where between 3=2 and 1=2 . A slight perpendicular component of
195
3
B2 B B Hx T 1 HT ¼ B B B @
1
0
pffiffi 3 2 D
0
1 D 2 g ? bH þ 2
0
0
12 g ? bH þ D2
0
0
pffiffi 3 2 D
0
32 g ? bH D2
g ? bH D2 pffiffi 3 2 D
0
pffiffi 3 2 D
C C C C C C A
ðB2Þ Zero-field interaction contributes to the coupling in a pair x x x between 3=2 and 1=2 and in the other pair between 1=2
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and / 3/2x9. /SySx and /SzSx vanish because the off-diagonal elements adjacent to the diagonal elements in Eq. (B2) are zero. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Eigenvalues of Eq. (B2) is D=2 7 D DD þD2 and D=2 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 D þ DD þ D , where V ¼g?bH. Their eigenfunctions give a unitary matrix U which diagonalizes the Hamiltonian: pffiffi 0 pffiffi3D 1 0 2K3þD 0 2K B pffiffi pffiffi C B 0 3D 0 2M3Dþ C B C 2M B C, U¼B L ðB3Þ C Lþ 0 0 B K C Kþ @ A Nþ N 0 0 M Mþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 2 D 4 2 3 D þ L 7 , L 7 ¼ 7 D 8 2 þ P , M 7 ¼ 3 D þN7 , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 7 ¼ 7 D 7 D2 þ P þ , and P 7 ¼ D2 7 DD þ D2 : The density matrix for the initial condition given by the optical transition is rewritten in this representation as 0 1 pffiffiffi 3 0 3 0 p ffiffiffi B C 1 0 3C 1B 0 C, pffiffiffi ðB4Þ rð0Þx T 1 rð0ÞT ¼ B B 8@ 3 0 1 0 C A pffiffiffi 0 3 0 3 where K 7 ¼
where 0
0
B B0
0
0
0
1
0
0
0 1 0
0 0 0
1 rð0Þ ¼ B 2B @0 0
1 C C C C A
The density matrix r(t)xis given according to Eq. (5) in this representation as o pffiffiffi 1n 2 3 sin y þ 3siny cosy ð1coso t Þ rðtÞx 11 ¼ 8 o pffiffiffi 1n 1þ sin2 y þ 3siny þ cosy þ ð1coso þ t Þ rðtÞx 22 ¼ 8
o pffiffiffi 1n 2 1 þ sin y þ 3siny cosy ð1coso t Þ 8 o pffiffiffi 1n 2 3- sin y þ 3siny þ cosy þ ð1coso þ t Þ rðtÞx 44 ¼ ðB5Þ 8 pffiffiffi for the diagonal elements, where siny 7 ¼ 3D=ð2P Þ, cosy 7 ¼ ðD þ D=2Þ=ð2P Þ, and o 7 ¼ 2P 7 =_. Non-zero off-diagonal elements are rðtÞx i i þ 2 and rðtÞx i þ 2 i (i¼ 1, 2) which do not contribute to creation of the spin expectation values. Eq. (6) with the spin operator 0 1 3=2 0 0 0 B C 1=2 0 0 C B 0 C Sx ¼ B B 0 0 1=2 0 C @ A 0 0 0 3=2
rðtÞx 33 ¼
and Eq. (B5) leads to /SxSx as in Eq. (18). References [1] W. Happer, Reviews of Modern Physics 44 (1972) 169. [2] Y. Takagi, Optics Communications 59 (1986) 122. [3] F. Meier, H.P. Zakharchenuya, Optical Orientation, North-Holand, Amsterdam, 1984. [4] Y. Takagi, Chemical Physics Letters 119 (1985) 5. [5] Y. Takagi, S. Miyazaki, K. Ishikawa, Chemical Physics Letters 393 (2004) 314. [6] J.P. van der Ziel, N. Bloembergen, Physical Review 138A (1965) 1287. [7] R. Kolesov, Physical Review A 76 (2006) 043808. [8] Y. Fukuda, Y. Takagi, K. Yamada, T. Hashi, Journal of the Physical Society of Japan 42 (1977) 1061. [9] R. Kolesov, Physical Review A 72 (2005) 051801(R). [10] S. Sugano, Y. Tanabe, Journal of the Physical Society of Japan 13 (1958) 880. [11] Y. Takagi, Y. Fukuda, T. Hashi, Optics Communications 55 (1985) 115. [12] A. Compaan, L.Q. Lambert, I.D. Abella, Optics Communications 3 (1971) 236. [13] Y. Takagi, Y. Fukuda, K. Yamada, T. Hashi, Journal of the Physical Society of Japan 50 (1981) 2672. [14] Y. Takagi, Solid-State Physics (in Japanese) 27 (1992) 197.