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Separable nature of nonlocal susceptibilities for general cases of interacting radiation-matter systems
JOURNAL
OF
LUMINESCENCE ELSEVIER
Journal of Luminescence 66&67 (1996) 94-96
Separable nature of nonlocal susceptibilities for general cases of interacting radiation-matter systems Yasushi Ohfuti *, Kikuo Cho Faculty of’ Engineering
Science. Osaka UniversitJl, Machikaneyama 1-3, Toyonaka, 560 Japan
Abstract A revision and an extension has been made to our recent work on the separable nature of nonlocal electric susceptibilities of matter. The separability holds always for resonant processes. It is valid also for nonresonant processes if the spatial variation of the (transverse) vector potential is negligible, which holds approximately in many cases approximately and even exactly for certain cases. This argument can be extended to nonlocal magnetic susceptibilities by including the spin Zeeman term and the spin+rbit interaction in the matter Hamiltonian.
1. Introduction Mesoscopic or nanoscale systems have been attracting much interest because of the observable effects of the quantum mechanical coherence of the electronic wave functions. This coherence naturally gives rise to the nonlocal response of the matter system to the external fields, as obtained by the standard recipe of quantum mechanics. In this sense, all the linear and nonlinear susceptibilities have nonlocal forms. In a semiclassical framework of optical response theory, one solves Maxwell equations with the induced current density as a source term, which is related nonlocally to the field to be solved. The self-consistent motion of radiation and matter is thus determined. It seems formidable at first glance to solve the integro-differential equation which is reduced from the Maxwell equations with such susceptibilities in the source term. However, it is known that when we consider the resonant structures of the matter so that we can neglect the nonresonant terms * Corresponding author. 0022-23 13196/$15.000 1996 Elsevier Science B.V. All rights reserved
SSDZOO22-2313(95)00115-8
the current density, the susceptibility kernels are separable with respect to the integral variables and by this fact the equation can be reduced to a set of polynomial equations [I]. Using this formalism, we have investigated linear and nonlinear resonant optical responses of mesoscopic systems and demonstrated size-, shape- and internal structure-dependence which are the consequences of nonlocality [2-41. The separable nature is a key to taking account of nonlocality within a tractable scheme. Recently, we have generalized the argument for separability to include all the nonresonant cases, claiming its validity for general cases described by the nonrelativistic Hamiltonian of matter [5]. However, it has been found that the proof was not exact [6]. The revised proof says that the susceptibilities are separable as a good approximation in the following sense. For resonant processes, the nonresonant terms causing difficulty can be neglected. For a nonresonant condition, the spatial variation of the light field is small, and therefore the derivative terms of the field can be neglected compared with the field itself. This of
Y. Ohfiti, K. Cho / Journal of‘ Luminescence M&67
approximation becomes exact for zero frequency ceptibility of finite gap materials.
sus-
In this paper, we show the logical steps leading to the above statements. In addition, we will show that the argument can be extended to the nonlocal magnetic susceptibilities by including the relativistic correction terms such as spin Zeeman and spin-orbit interaction terms in the matter Hamiltonian.
2. Separability The fundamental equations of nonlocal response theory [ 1, 21 consist of two functional equations for the temporal Fourier components of the vector potential A and current density j as
(1996)
95
94-96
a dynamical field variable, while A gives the transverse electromagnetic fields which are independent variables. We choose the matter Hamiltonian HO as the A-independent part of Eq. (3): Ho = HI&O.
(4)
This choice fits well with the usual understanding of matter in the nonrelativistic regime. We refer to the eigenenergies and the corresponding eigenstates of Ho by {EiL} and {IL)}, which will be used to derive some relations between matrix elements. The Hamiltonian of the radiation-matter interaction is defined as the A-dependent terms in H, which may read t) . A(r, t)
+(r,
Hint(t) = ./(
k(r, 0) = A&, w) + %[j:],
(1)
jiu, 0) = P[i].
(2)
+&Ii’@,
(5)
t)A(u, t)2 dr, >
where The first equation is the solution of the (microscopic) Maxwell equations with given source terms, charge and current density. & is a free field, which usually represents incident light. We choose the Coulomb gauge, div A = 0, from which the scalar potential is determined by the instantaneous charge density. By the use of the continuity equation, the charge density can be eliminated, and the solution is written in terms of the current density alone. The functional 9 is simply expressed by the dyadic Green’s function. It is on the second equation, Eq. (2), that we want to put the main emphasis in this paper. The current density induced by a given field A(r, t) is calculated from first principles in the following way. The nonrelativistic Hamiltonian of a system of charged particles in given vector potential A(r, t) and scalar potential &Y, t) can be written as a sum of the following one-particle Hamiltonian:
&
H = c
(PI - :4rlrO)2
+ Q&U)},
(3)
I where (r,,pi) and (el,m/) are the (coordinate, momentum) and the (charge, mass) of the Ith particle, respectively, and c is the light velocity in vacuum. It should be noted that in the Coulomb gauge, the sum of 4’s represents the instantaneous Coulomb interaction among all the charged particles and is not
Z(r, t) = c
5
, 2w
4 N(r, t) = C &6(r I
[p&r
- Y/) + 6(r -
- rf).
v,)p/l,
(6) (7)
(Since we treat A as classical variables, Hint depends on t explicitly even in the Schrodinger picture for the matter system.) Correspondingly, the current density operator is given by J(r, t) = Z(r, t) - ffi(r,
t)A(r,t).
(8)
The expectation value of the current density at (r, t) is given by j(r, t) = Tr{p(t)J(r)}, where p is the matter density matrix which obeys the Liouville equation. The solution is obtained by the perturbation expansion with respect to Hi”, [7]. The A dependence ofj comes from Z . A and fiA2 in H,,t and Y?A in J . The last two terms seem to break the separability while the first one keeps the separability because it contains only one A and is integrated over the space. Thus our task is to rewrite the two terms in approximate forms which are separable. To this end, we examine the operator Q defined by Q(o)
= [ c
e,rd(r - q). A”(r,o) dr.
(9)
96
Y. Oh&ii, K. Cho J Journal of‘ Luminescence 66 & 67 (I 996) 94-96
Q satisfies the following [Q,Ho] = ih [Q, Z(r)1=
s
identities:
dr(Z - J(r,u)
WN(r)A(r, 0) +
+ Q’(r, 0)) ,
(10)
Q”(r, o)),
(11)
where
(12) Q”(r, 0)
=-cs dr’@(r
- rj)V&r’,o)6(r
- q).
(13)
in the treatments of linear and nonlinear optical processes in mesoscopic systems. For matter with finite excitation energies, the current density due to the time evolution of polarization becomes zero as o + 0. Thus Eq. ( 1) leads to i(r,w) + &(r,o) as cc)-+ 0. Since 2s is a free field, its spatial derivatives go to zero for w + 0. In this case, Q’ and Q” in Eqs. (10) and (11) are zero, and therefore the separability argument holds exactly. Finally, we consider the case when the spin Zeeman term, _
I<
The Q’ and Q” include the spatial derivatives of 2 which are negligibly small in the nonresonant condition as mentioned in Section 1. Using these identities with Q’ and Q” omitted, we have
J’N,,,.i(u) -k(d)
+ &(eW,~~(~‘)
E rp
En
=
drZ,,,(r) - j(r, o), J’
* rot A(r/
),
(17)
is added to the Hamiltonian, where ,& is the Bohr magneton and s/ is the spin operator of the Ith particle. The spin-orbit interaction may also be included in HO. If we note that the added term is linear in A and thereby the magnetic current density, 2c cI pg rot S&V - VI) is independent of A, it is clear that the separability nature also holds for (nonlocal) magnetic susceptibilities.
Acknowledgements >’
(15)
where F,,,(o)
$Bs/
dr
F&0)F&u’) =I( 7
c
(16)
This work was supported in part by Grants-in-Aid for Scientific Research for Priority Areas “Mutual quantum manipulation of radiation field and matter” from the Ministry of Education, Science and Culture of Japan.
and E,,, = E, - E,. Thus j is written in terms of Zpv and the polynomials of F,,. Its k-dependence appears only through the F’s. In other words, the susceptibility functions are separable. This argument is valid approximately in the sense that we neglected Q’ and Q” terms in Eqs. (10) and (11). Putting j in Eq. ( 1), one obtains A as a polynomial series of F’s, Substitution of this expression in Eq. (16) yields a selfconsistent set of polynomial equations for the F’s with various combinations of (11,v) and frequencies. By this reformulation, considerable calculations become feasible
References [I] K. Cho, Prog. Theor. Phys. Suppl. 106 (1991) 225. [2] [3] [4] [5] [6]
K. Cho, H. lshihara and Y. Ohfuti, J. Lumin. 58 (1993) 95. H. Ishihara and K. Cho, Phys. Rev. B 48 (i993) 7960. Y. Ohfuti and K. Cho, Phys. Rev. B 51 (1995-11) 14379. Y. Ohfuti and K. Cho, Phys. Rev. B 52 (1995-I) 4828. We are indebted to Mr. T. Garm for making us aware of a logical defect. [7] T.K. Yee and T.K. Gustafson, Phys. Rev. A 18 (1978) 1597.
OF
LUMINESCENCE ELSEVIER
Journal of Luminescence 66&67 (1996) 94-96
Separable nature of nonlocal susceptibilities for general cases of interacting radiation-matter systems Yasushi Ohfuti *, Kikuo Cho Faculty of’ Engineering
Science. Osaka UniversitJl, Machikaneyama 1-3, Toyonaka, 560 Japan
Abstract A revision and an extension has been made to our recent work on the separable nature of nonlocal electric susceptibilities of matter. The separability holds always for resonant processes. It is valid also for nonresonant processes if the spatial variation of the (transverse) vector potential is negligible, which holds approximately in many cases approximately and even exactly for certain cases. This argument can be extended to nonlocal magnetic susceptibilities by including the spin Zeeman term and the spin+rbit interaction in the matter Hamiltonian.
1. Introduction Mesoscopic or nanoscale systems have been attracting much interest because of the observable effects of the quantum mechanical coherence of the electronic wave functions. This coherence naturally gives rise to the nonlocal response of the matter system to the external fields, as obtained by the standard recipe of quantum mechanics. In this sense, all the linear and nonlinear susceptibilities have nonlocal forms. In a semiclassical framework of optical response theory, one solves Maxwell equations with the induced current density as a source term, which is related nonlocally to the field to be solved. The self-consistent motion of radiation and matter is thus determined. It seems formidable at first glance to solve the integro-differential equation which is reduced from the Maxwell equations with such susceptibilities in the source term. However, it is known that when we consider the resonant structures of the matter so that we can neglect the nonresonant terms * Corresponding author. 0022-23 13196/$15.000 1996 Elsevier Science B.V. All rights reserved
SSDZOO22-2313(95)00115-8
the current density, the susceptibility kernels are separable with respect to the integral variables and by this fact the equation can be reduced to a set of polynomial equations [I]. Using this formalism, we have investigated linear and nonlinear resonant optical responses of mesoscopic systems and demonstrated size-, shape- and internal structure-dependence which are the consequences of nonlocality [2-41. The separable nature is a key to taking account of nonlocality within a tractable scheme. Recently, we have generalized the argument for separability to include all the nonresonant cases, claiming its validity for general cases described by the nonrelativistic Hamiltonian of matter [5]. However, it has been found that the proof was not exact [6]. The revised proof says that the susceptibilities are separable as a good approximation in the following sense. For resonant processes, the nonresonant terms causing difficulty can be neglected. For a nonresonant condition, the spatial variation of the light field is small, and therefore the derivative terms of the field can be neglected compared with the field itself. This of
Y. Ohfiti, K. Cho / Journal of‘ Luminescence M&67
approximation becomes exact for zero frequency ceptibility of finite gap materials.
sus-
In this paper, we show the logical steps leading to the above statements. In addition, we will show that the argument can be extended to the nonlocal magnetic susceptibilities by including the relativistic correction terms such as spin Zeeman and spin-orbit interaction terms in the matter Hamiltonian.
2. Separability The fundamental equations of nonlocal response theory [ 1, 21 consist of two functional equations for the temporal Fourier components of the vector potential A and current density j as
(1996)
95
94-96
a dynamical field variable, while A gives the transverse electromagnetic fields which are independent variables. We choose the matter Hamiltonian HO as the A-independent part of Eq. (3): Ho = HI&O.
(4)
This choice fits well with the usual understanding of matter in the nonrelativistic regime. We refer to the eigenenergies and the corresponding eigenstates of Ho by {EiL} and {IL)}, which will be used to derive some relations between matrix elements. The Hamiltonian of the radiation-matter interaction is defined as the A-dependent terms in H, which may read t) . A(r, t)
+(r,
Hint(t) = ./(
k(r, 0) = A&, w) + %[j:],
(1)
jiu, 0) = P[i].
(2)
+&Ii’@,
(5)
t)A(u, t)2 dr, >
where The first equation is the solution of the (microscopic) Maxwell equations with given source terms, charge and current density. & is a free field, which usually represents incident light. We choose the Coulomb gauge, div A = 0, from which the scalar potential is determined by the instantaneous charge density. By the use of the continuity equation, the charge density can be eliminated, and the solution is written in terms of the current density alone. The functional 9 is simply expressed by the dyadic Green’s function. It is on the second equation, Eq. (2), that we want to put the main emphasis in this paper. The current density induced by a given field A(r, t) is calculated from first principles in the following way. The nonrelativistic Hamiltonian of a system of charged particles in given vector potential A(r, t) and scalar potential &Y, t) can be written as a sum of the following one-particle Hamiltonian:
&
H = c
(PI - :4rlrO)2
+ Q&U)},
(3)
I where (r,,pi) and (el,m/) are the (coordinate, momentum) and the (charge, mass) of the Ith particle, respectively, and c is the light velocity in vacuum. It should be noted that in the Coulomb gauge, the sum of 4’s represents the instantaneous Coulomb interaction among all the charged particles and is not
Z(r, t) = c
5
, 2w
4 N(r, t) = C &6(r I
[p&r
- Y/) + 6(r -
- rf).
v,)p/l,
(6) (7)
(Since we treat A as classical variables, Hint depends on t explicitly even in the Schrodinger picture for the matter system.) Correspondingly, the current density operator is given by J(r, t) = Z(r, t) - ffi(r,
t)A(r,t).
(8)
The expectation value of the current density at (r, t) is given by j(r, t) = Tr{p(t)J(r)}, where p is the matter density matrix which obeys the Liouville equation. The solution is obtained by the perturbation expansion with respect to Hi”, [7]. The A dependence ofj comes from Z . A and fiA2 in H,,t and Y?A in J . The last two terms seem to break the separability while the first one keeps the separability because it contains only one A and is integrated over the space. Thus our task is to rewrite the two terms in approximate forms which are separable. To this end, we examine the operator Q defined by Q(o)
= [ c
e,rd(r - q). A”(r,o) dr.
(9)
96
Y. Oh&ii, K. Cho J Journal of‘ Luminescence 66 & 67 (I 996) 94-96
Q satisfies the following [Q,Ho] = ih [Q, Z(r)1=
s
identities:
dr(Z - J(r,u)
WN(r)A(r, 0) +
+ Q’(r, 0)) ,
(10)
Q”(r, o)),
(11)
where
(12) Q”(r, 0)
=-cs dr’@(r
- rj)V&r’,o)6(r
- q).
(13)
in the treatments of linear and nonlinear optical processes in mesoscopic systems. For matter with finite excitation energies, the current density due to the time evolution of polarization becomes zero as o + 0. Thus Eq. ( 1) leads to i(r,w) + &(r,o) as cc)-+ 0. Since 2s is a free field, its spatial derivatives go to zero for w + 0. In this case, Q’ and Q” in Eqs. (10) and (11) are zero, and therefore the separability argument holds exactly. Finally, we consider the case when the spin Zeeman term, _
I<
The Q’ and Q” include the spatial derivatives of 2 which are negligibly small in the nonresonant condition as mentioned in Section 1. Using these identities with Q’ and Q” omitted, we have
J’N,,,.i(u) -k(d)
+ &(eW,~~(~‘)
E rp
En
=
drZ,,,(r) - j(r, o), J’
* rot A(r/
),
(17)
is added to the Hamiltonian, where ,& is the Bohr magneton and s/ is the spin operator of the Ith particle. The spin-orbit interaction may also be included in HO. If we note that the added term is linear in A and thereby the magnetic current density, 2c cI pg rot S&V - VI) is independent of A, it is clear that the separability nature also holds for (nonlocal) magnetic susceptibilities.
Acknowledgements >’
(15)
where F,,,(o)
$Bs/
dr
F&0)F&u’) =I( 7
c
(16)
This work was supported in part by Grants-in-Aid for Scientific Research for Priority Areas “Mutual quantum manipulation of radiation field and matter” from the Ministry of Education, Science and Culture of Japan.
and E,,, = E, - E,. Thus j is written in terms of Zpv and the polynomials of F,,. Its k-dependence appears only through the F’s. In other words, the susceptibility functions are separable. This argument is valid approximately in the sense that we neglected Q’ and Q” terms in Eqs. (10) and (11). Putting j in Eq. ( 1), one obtains A as a polynomial series of F’s, Substitution of this expression in Eq. (16) yields a selfconsistent set of polynomial equations for the F’s with various combinations of (11,v) and frequencies. By this reformulation, considerable calculations become feasible
References [I] K. Cho, Prog. Theor. Phys. Suppl. 106 (1991) 225. [2] [3] [4] [5] [6]
K. Cho, H. lshihara and Y. Ohfuti, J. Lumin. 58 (1993) 95. H. Ishihara and K. Cho, Phys. Rev. B 48 (i993) 7960. Y. Ohfuti and K. Cho, Phys. Rev. B 51 (1995-11) 14379. Y. Ohfuti and K. Cho, Phys. Rev. B 52 (1995-I) 4828. We are indebted to Mr. T. Garm for making us aware of a logical defect. [7] T.K. Yee and T.K. Gustafson, Phys. Rev. A 18 (1978) 1597.