- Email: [email protected]
Corrections to the phenomenological relaxation models for open quantum systems
Journal of Luminescence 137 (2013) 148–156
Contents lists available at SciVerse ScienceDirect
Journal of Luminescence journal homepage: www.elsevier.com/locate/jlumin
Corrections to the phenomenological relaxation models for open quantum systems Mikhail Tokman, Maria Erukhimova n Institute of Applied Physics, Russian Academy of Sciences, 46, Ul’yanov Street, 603155 Nizhny Novgorod, Russia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 September 2012 Received in revised form 6 December 2012 Accepted 27 December 2012 Available online 4 January 2013
The accurate description of dissipative effects in quantum systems is needed for the development of modern quantum information and metrology techniques. We find a simple method for considerably improving the accuracy of phenomenological relaxation models. This method can be applied for both Von Neumann and Heisenberg–Langevin equations. It is based on some general properties of the relaxation operator which follow from initial exact equations for the unreduced system but can be lost due to various approximations (for example rotating-wave approximation) used in popular models. In particular, we show that use of simple relaxation rate model violates the fundamental relation between the macroscopic current and polarization in the dielectric and can lead to ‘‘unphysical’’ instabilities. We show that the relaxation rate dependence of resonance line of two-level system with the modified relaxation operator differs from that obtained from the standard optical Bloch equation and corresponds strictly to the classical oscillator with friction. This result is important for improving the quantum frequency standards. The analogous result is obtained for the damped three-dimensional quantum oscillator in arbitrary oriented magnetic field, which is a robust model for the theory of quantum dots. & 2013 Elsevier B.V. All rights reserved.
Keywords: Open systems Master equation Relaxation operator Optical Bloch equation
1. Introduction The description of relaxation processes in open quantum systems is important and one of the most complicated problems in atomic physics and quantum optics. (see, for example, Refs. [1–13] and literature cited there). Due to fast progress in quantum information and metrology experimental techniques it becomes increasingly important to describe the dissipation process in different quantum systems more and more accurately [14–16]. The direct numerical simulation ‘‘from the first principles’’ of the dynamics of the open quantum system (used in plasma physics, for example) is unrealistic at this time. So the modern approaches for dissipation process analysis are based on different models and approximate schemes. The consequent approach to the description of dissipative dynamics of an open quantum system is based on the analysis of interaction between the considered quantum object and the environment (or bath system). The task is to find the dynamics of the system by tracing out the many degrees of freedom of the bath. The analysis usually involves a variety of simplifications and model assumptions. As a
result, the solution of the final equation may lose some important fundamental properties. We can illustrate how the physically incorrect solution may be obtained because of too crude dissipation model given by the standard master equations. Consider the equation for the density ^ and the ^ of an electron defined by Hamiltonian H matrix r ^ relaxation operator R:
r_^ ¼
i h ^ i ^ ^ þR r ^ H, r _
ð1Þ
Multiply Eq. (1) by the coordinate operator r^ and apply the averaging operation to the result. The following expression for ^ Þ=@t is time-derivative of averaged coordinate r_ ¼ ½ @Tr r^ r obtained after standard transformations: ^ r_ ¼ v þ Trðr^ RÞ,
ð2Þ
^ r^ is velocity operator, v ¼ Tr v^ r ^ is averaged where v^ ¼ i=_ ½H, velocity. We get from Eq. (2) that averaged derivative of the ^ a0. coordinate is not equal to average velocity r_ av if Trðr^ RÞ
In particular, this situation takes place if the standard model of constant-rate relaxation is used (see, for example, Refs. [1–3]) ^ ¼ gmn rmn ; gmn ¼ gnm ð3Þ Rmn r
n
Corresponding author. Tel.: þ7 832 4164754; fax: þ 7 832 4160616. E-mail addresses: [email protected] (M. Tokman), [email protected] (M. Erukhimova). 0022-2313/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jlumin.2012.12.035
The analogous additional members in the equation for the time-derivative of averaged momentum have physical meaning
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
(it is the loss of momentum due to interaction with the reservoir), but the same situation for coordinate is physically senseless. This problem becomes evident if the solution of Eq. (1) is used _ and evN can be equally in Maxwell equations. Two quantities erN taken as an ‘‘averaged current1 ’’ of the ensemble of particles with the charge e and density N. But these quantities are not equal if the relaxation operator (Eq. (3)) is used. It is evident that this ambiguity may be significant or not depending on the ratio between relaxation rates and the typical frequencies of the problem. As it was shown in [13] in definite conditions this incorrectness may even cause a physically senseless instability (negative dissipation) in quantum medium. In the present paper we use the fundamental property of relaxation operator shown in [17]: the correct derivation of evolution equation in the form (Eq. (1)) provides the fulfillment of the following condition ^ r ^ ÞÞ ¼ 0 Trðg^ Rð
ð4Þ
for any operator g^ commutating with the operator of atom– reservoir interaction. In particular this condition, written for the operator of coordinate, has the form: ^ r ^ ÞÞ ¼ 0 Trðr^ Rð It provides ‘‘natural’’ relation: r_ ¼ v, that guarantees correctness of calculated quantum medium susceptibility. In other form the relation (Eq. (4)) can be written as ^ ^ Þ¼0 TrðdR^ r ð5Þ where d^ ¼ er^ is the operator of dipole moment of quantum system. The condition (Eq. (5)) serves as a validation criterion for relaxation models and simultaneously defines the phenomenological method of reconstruction of the standard relaxation model. In addition another general condition may be used. It may also serve as a test for correctness of phenomenological relaxation models. The question concerns proceeding to the limit of constant external field. The susceptibility of the dielectric system in constant field is determined by the state of thermodynamic equilibrium with the reservoir [18,19], so it cannot depend on relaxation rates2 . Again, the simple relaxation model (Eq. (3)) does not provide such a solution. Here we show that the relaxation operator obeying the condition (Eq. (4)) can be obtained only beyond rotating wave approximation (RWA) which is often used for presentation of ‘‘atom-bath’’ interaction operator (see, for example, Ref. [1]). We obtain, that the modified relaxation operator for two-level system in a photon bath represents the antisymmetrized form of the standard operator. We generalize this technique to the density matrix equation of quantum systems with any number of discrete energy levels as well as to density matrix in coordinate representation. The proposed modified relaxation operator is rather more complicated than the standard operator in the form (Eq. (3)), but it provides simple and noncontradictory equations for dynamics of the averaged dipole moment and the energy of the system. In particular, the relaxation rate dependence of resonance line for two-level system with the modified relaxation operator differs from that obtained from the standard optical Bloch 1 Here we do not consider ‘‘vortex’’ current j ¼ r M, arising in the system with intrinsic magnetic moment M. 2 We are not talking about situations with particles exchange: the flow of constant current throw the dissipative medium, for example.
149
equation and corresponds strictly to the classical oscillator with friction. The analogous result is obtained for the damped threedimensional oscillator in arbitrary oriented magnetic field. The paper is organized as follows. In part II we present our proof of the condition (Eq. (4)). We have also analyzed standard derivation of master equations in Redfield form and revealed that the relaxation super-matrix obtained under the Born and Markovian approximations meets this requirement until the secular approximation is applied. In part III the corrected relaxation operator for two-level system in the field reservoir is obtained. It is shown that the modification consists in an antisymmetrization of the standard relaxation term for the off-diagonal density matrix elements. The phenomenological relaxation model is proposed for any quantum systems with discreet number of basis function. Part IV is devoted to proper proceeding to the limit of constant external field. In part V the oscillation equations for twolevel system in a monochromatic wave field are obtained for modified relaxation operator. In part VI the phenomenological relaxation model for the three-dimensional harmonic oscillator placed in arbitrary oriented constant magnetic field and monochromatic field is constructed. The relaxation operator is written in a gauge-invariant form using the results of paper [13]. The macroscopic polarization is obtained for this system. It is shown that the conductivity of the medium obtained from modified equations does not display paradoxical change of sign in a definite frequency range unlike it does for a standard tau-approximation. In part VII we generalize the results obtained for Von Neumann equation with relaxation operator to the Heisenberg–Langevin equations. In Section 8 the results are summarized and discussed.
2. The first criterion of correct relaxation operator Let us show that the proper derivation of the relaxation operator from initial equations leads to fulfillment of condition (Eq. (5)). Somewhat different way to derive this condition was presented in paper [17]. We consider interaction of two quantum subsystems (‘‘the atom’’ and ‘‘the reservoir’’), described by the total Hamiltonian: ^ S ðr,qÞ ¼ H ^ A ðrÞ þ H ^ Q ðqÞ þ ^Iðr,qÞ, H ^ Q ðqÞ act on the functions of ^ A ðrÞ and H where the operators H coordinate of atom r and reservoir q, correspondingly, ^Iðr,qÞ is the interaction operator. We use Von Neumann equation for the ^ S ¼ rmn;nm (Latin indices density matrix of complete system r correspond to atom, and Greek to reservoir):
r_^ S ¼
i i h^ ^S HS , r _
Represent the common density matrix in the form:
r^ S ¼ r^ A r^ Q þ r^ I ,
ð6Þ
^ A ¼ TrQ r ^ Q ¼ TrA r ^ S, r ^ S ; operations TrA,Q denote averaging where r P over corresponding subsystems: TrA ð. . .Þ ¼ dmn ð. . .Þ, TrQ ð. . .Þ ¼ n,m P dn, m ð. . .Þ, TrS ð. . .Þ ¼ TrA TrQ ð. . .Þ ¼ TrQ TrA ð. . .Þ. Suppose that there n, m
is no averaged ‘‘force’’ action of the reservoir on the ‘‘atom’’ to ^ Q Þ ¼ 0 [20,4,13]. Taking first order in interaction energy, i.e. TrQ ð^Ir into account normalization requirements and following relations: ^ A,r ^ Q ,r ^ A,I ¼ TrQ ½H ^ Q ,I ¼ 0, we obtain an equation for the TrA ½H ^ A: ‘‘atomic’’ density matrix r
r_^ A ¼
i i h^ ^A , ^ A þ R^ r HA , r _
150
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
where the relaxation operator has the form: i ^ I Þ R^ ¼ TrQ ð½^I, r _
ð7Þ
^ we use the operator identity To find the convolution TrA ðd^ RÞ ^ ^ ^ ^ ^ ^ TrðA½B, CÞ ¼ Trð½A, BCÞ and take into account that the operator d^ does not act on the functions of reservoir coordinates. As a result we obtain:
This equation can be written in the basis of eigenstates of the ^ A: Hamiltonian H X r_ Aij ¼ ioij rAij þ Rijkm rAkm ð10Þ km
Here oij ¼ 1=_ HAii HAjj . For the coefficients Rijkm the following expressions are obtained: X Rijkm ¼ dmj Gippk þ dik Gnjppm þ Gmjik þ Gnkijm , ð11Þ p
^ ^Ir ^ ¼ i TrA TrQ ð½d, ^ IÞ TrA ðd^ RÞ _
ð8Þ
Gijkm ¼
It follows from Eq. (8) that the condition (Eq. (5)) is automatically satisfied if the operator of atom–reservoir interaction commutates with the operator of dipole moment. For example this commutator is zero if the operator ^I is just a function of coordinates: ^I ¼ U ðr,qÞ. In particular, the relaxation caused by collisions or interaction with lattice oscillations corresponds to ^ ^I ¼ 0 is also fulfilled for the the last case. The condition ½d, interaction with the photon reservoir, if the interaction operator is defined in the electric dipole approximation ^I ¼ d^ E^ (here E^ is the quantum operator of the electric field of the reservoir). If another form of interaction operator with the photon reservoir is used [1–3]: ^I ¼ e a^ p^ þ p^ a^ 2mc
1 2
_
Z
1 0
X HQ aa HQ bb okm t dt rQ aa Iiajb Ikbma exp i _ ab ð12Þ
^ Q and r ^ Q is Here the basis fjaig diagonalizing the operators H used; dij is the Kronecker delta. The following step is so called secular approximation [4,24] in frame of which the relaxation terms of Eq. (10) with okm a oij are omitted. It is implied that the corresponding amendments are small if the following inequality is fulfilled: Rijkm oij okm 5 1,
oij a okm
ð13Þ
Under secular approximation the kinetic equations take the simple standard form, where the off-diagonal density matrix elements relax independently, while the diagonal elements obey the balance equations:
^ (Here a^ is the operator of the vector potential: a_^ ¼ cE, p^ ¼ i_r is the momentum operator), we can apply the commu^ ¼ i_ to Eq. (8). As a result, in a typical case tation relation ½r^ , p when the spatial variation of the field at atomic scales can be neglected, we get3 :
r_ Aij ¼ ioij rAij gij rAij , i a j
2 e2 ^ ¼ e TrA TrQ ð^a r ^ IÞ ¼ ^ IÞ ¼ 0 Tr Q ð^a TrA r TrA ðd^ RÞ mc mc
gij ¼ Rijij
^ I ¼ 0 by the definition Eq. (6)). (TrA r So, the relaxation operator obtained in a proper way from initial relations should always satisfy the condition (Eq. (5)). A more general statement can be made: the relaxation operator R^ should satisfy the condition Eq. (4):
It can be easily shown that for the relaxation operator in the form Eqs. (10)–(12) obtained under Born–Markovian approximation and for any operator g^ obeying commutation relation ½g^ ^I ¼ 0 the following equality is valid: X g ij Rjikm ¼ 0
W ik rAkk W ki rAii
ð14Þ ð15Þ
kai
where W ik ¼ 2ReGkiik
ð16Þ
ij
^ ¼0 Trðg^ RÞ for any operator g^ commutating with operator of atom–reservoir interaction. ^A To obtain the relaxation operator as explicit dependence R^ r it is necessary to solve the equation for ‘‘interaction density ^I ¼r ^ I ðr ^ A,r ^ Q Þ in frame of given reservoir density matrix matrix’’ r approximation (so called Born approximation) and substitute it in Eq. (7). The standard procedure usually means to use the Markovian approximation as well. Then the equations for density matrix of the system take the form of so called kinetic equation: differential equation with constant coefficients (see, for example Refs. [3–6,23,24]). i 1 d ih r^ ¼ H^ A , r^ A 2 dt A _ _
r_ Aii ¼
X
Z
1 0
H^ þ H^ ^ þH ^ H Q Q A A ^ A ðt Þr ^Q dt TrQ ^I ei _ t ^Ie þ i _ t , r ð9Þ
3 The question is more complicated if the variation of the spatial structure of electromagnetic field at the atomic scales is significant and the field is not potential. In any case in frame of classical motion equations the following effect is known: the appearance of additional component of the averaged current density in inhomogeneous rapidly oscillating or noise field [21,22], but this current is pure vortex and disappears after averaging over volume occupied by particles.
^ ¼0 It in turn provides the fulfillment of condition Trðg^ RÞ (Eq. (4)). The dissipative operator in the standard form defined by Eqs. (14)–(16) which is obtained after additional secular approximation does not fulfill this condition. The same conclusion was reached in paper [17]. Note that for the bath representing an ensemble of harmonic oscillators the corresponding ‘‘unsecular’’ terms occur only if nonrotating wave (NRW) form for the interaction operator ^I is used. The kinetic (master) equations out of the frame of RWA (rotating wave approximations) were investigated in papers [25,16,26].
3. The relaxation operator for two-level system in a photon bath Using of relations Eqs. (10)–(12) for constructing the relaxation operator in a model system with finite number of energy levels may not lead to the desired result meeting a requirement (Eq. (5)). That is because the basis is not complete, and as a consequence, commutativity of the dipole moment operator and atom–reservoir interaction operator may violate. The solution is likely to use phenomenological approaches, specific for each system.
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
We consider the simple system, where the desired relaxation operator can be obtained on the basis of relations Eqs. (10)–(12). It is two-level system in a photon reservoir with interaction ^ Here d^ ¼ dðj2ih1jþ j1ih2jÞ is the operator of operator ^I ¼ d^ E. dipole moment of the system (the matrix element is real for definiteness, that can be supposed without loss of generality), P pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ n E^ ¼ i ½ð_on Þ=ð2Þ a^ E^ n ðr a^ n E^ ðrÞÞ is the operator of electric n
n
n
field, on is the frequency of mode with number n, E^ n ðrÞ ¼ E^ n ðrÞen ðrÞ is normalized amplitude of electric field in mode n, a^ nþ , a^ n are the creation and annihilation operators correspondingly. We apply Eqs. (10)–(12) to define the relaxation operator of two-level system. Taking into account, for simplicity sake, only real part of relaxation constants, we get the following expressions for the off-diagonal elements of relaxation operator: ^ 21 ¼ R2121 r21 þ R2112 r12 R r ^ 12 ¼ R1212 r12 þ R1221 r21 , R r
vector of electric field in mode n, z ¼ r, y, j , r is the atom coordinate, y, j are the angles that define the orientation of d. So, for the two-level system in a photon bath we get from initial relations after applying Born–Markovian approximation the relaxation operator Eqs. (19), (20) which meets the criterion (Eq. (5)). It provides the correct relations for averaged observable derivatives in this dissipative system. This relaxation operator is defined by standard relaxation rates; the only modification consists in the antisymmetrized structure of its off-diagonal element. Note once again that the modified relaxation equation (Eqs. (19), (20)) is obtained beyond the framework of secular approximation, when the counter-rotating terms in the interaction operator are taken into account (without employing RWA for interaction operator). At that, as can be seen, this relaxation operator being reduced to frequently used form: R^ ¼
þ þ þ ^ V^ m V^ n , ^ V^ m V^ m V^ n r ^ r Dnm 2V^ n r
2 X
ð22Þ
n,m ¼ 1
where R2121 ¼ R1212 ¼ R1221 ¼ R2112 ¼ g,
1 2
g ¼ ðW 12 þ W 21 Þ
ð17Þ
Consequently, we get that for the matrix elements of relaxation operator R^ the following relations are fulfilled ð18Þ R21 ¼ R12 ¼ g r21 r12 , that automatically leads to fulfillment of condition (Eq. (5)). The equation for matrix element r21 takes the following form (instead of Eq. (14)): r_ 21 þiO21 r21 ¼ g r21 r12 ð19Þ The equations for diagonal elements of density matrix are the same as in Eq. (15): X r_ ii ¼ W ik rkk W ki rii ð20Þ kai
W 12 , W 21 are the standard coefficients, characterizing the rates of atomic transitions j2i-j1i and j1i-j2i, correspondingly: W 12 ¼
2p X _on ^ 2 2 9E n ðr Þ9 9en d9 ðN n þ1Þdðon O21 Þ _2 n 2
W 21 ¼
2p X _on ^ 2 2 9E n ðr Þ9 9en d9 ðN n Þdðon O21 Þ _2 n 2
ð21Þ
Here N n is the averaged number of photons in mode ‘‘n’’. It is assumed that frequency of atomic transition O21 4 0.4 The coefficients given by Eq. (21) can be presented in a usual form: W 12 ¼ A21 ZðO21 , zÞ NðO21 Þ þ 1Þ, W 21 ¼ A21 ZðO21 , zÞNðO21 Þ, 2 where A21 ¼ ½ 4d O321 = 3_c3 is the Einstein coefficient, corresponding to the spontaneous emission rate of an atom in a free space, ZðO21 , zÞ ¼ ½ sðO21 , zÞÞ=ðsfs ðO21 ÞÞ is dimensionless parameter, characterizing the ratio of effective spectral mode density of the reservoir to the corresponding quantity in a free space sfs ðO21 Þ ¼ ½ _O321 = p2 c3 ,
sðO21 , zÞ ¼
X n
2
_on
9E n ðrÞ9 dðon O21 ÞkðzÞ, 4p
2
n d9 kðzÞ ¼ 3 9e9d9 2 is dimensionless parameter characterizing the orien-
tation of the dipole moment of atomic transition relatively to the 4
151
9E ðrÞ9
2
In Eq. (21) dðon O21 Þ is d-function, at that _on n4p dðon O21 Þ is the field spectral energy density at frequency O21 at the atom’s position, if the fiend represents one photon in the mode with frequency on .
is not of the ‘‘Lindblad-form’’ [23,27,28]. (Here the basis operators ^ þ ¼ j2ih1j, V^ 2 ¼ s ^ ¼ j1ih2j). It is characterized by ‘‘disare V^ 1 ¼ s sipator’’ super-operator D, with elements Dnm , which is not a positive-definite matrix: 0 1 W 21 2
D ¼ @ W 12 þ W 21 4
W 12 þ W 21 4 W 12 2
A
Here W 21 and W 12 are given by Eq. (21). This result can be obtained directly from Eq. (9) where the non-RWA form for the ^ þ þs ^ ÞE^ ¼ dðs ^ þ þs ^ Þ interaction operator is used: ^I ¼ dðs ðE^ a þ E^ a þ Þ. If the interaction operator is taken in the RWA-form ^I ¼ dðs ^ þ E^ a þ s ^ E^ a þ Þ and/or secular approximation is applied then usual positive-definite Lindblad form (Eq. (22)) for relaxation operator with D21 ¼ D12 ¼ 0 is obtained. The question arising here does the modified operator define trace-preserving and positive density matrix transformation or not. The detailed analysis has shown that the positive definiteness of density matrix, as solution of this relaxation equation, may be violated only for very narrow range of initial conditions, when the initial state of the system is nearly pure coherent superposition of energy states with definite initial phase. But what is the most important is that the density matrix positive definiteness may be violated only during very short initial time interval, less than the period of oscillations at the transition frequency. But, it is evident that during this transient period the time averaging procedure over reservoir modes that separates the resonance frequency cannot be applied. We note that similar ‘‘antisymmetrized’’ kind of relaxation operator for two-level system was obtained also in papers [25,26] where the interaction of quantum system with reservoir was analyzed beyond the framework of RWA. The same conclusion that the Non Rotating Wave (NRW) master equation is not of the Lindblad form was pointed in Ref. [26]. It was also shown that in the NRW master equation there is little validity to considering time scales shorter than the thermal correlation time because the coarse-grain time-scale approach used in its derivation. By analogy with the two-level system the phenomenological model of relaxation operator for quantum system with any number of discreet levels can be constructed, if there is a basis of real eigenfunctions5 and correspondingly the matrix elements of the dipole moment are real: dmn ¼ dnm . In this case we can assume that 5 It is so, for example, for finite motion in the absence of external magnetic field [29]. In magnetic field this approach can be applied only if the effect of magnetic field on the relaxation processes is neglected.
152
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
the relaxation processes at the transition 9m 439n 4 are defined by off-diagonal elements of density matrix at this transition only. By analogy with two-level system the standard expression for operator of transverse relaxation with given relaxation rates gmn (Eq.(3)) should be ‘‘antisymmetrized’’: ^ ¼ gmn rmn rnm ð23Þ Rm a n r
4. Correctness of relaxation operator in the limit of constant electric field Consider the quantum system with discreet number of eigen^ Cm . functions and eigenvalues of the energy operator: W m Cm ¼ H Suppose that system has definite symmetry due to which the diagonal matrix elements of the dipole moment are equal to zero: dmm ¼ 0. In the external electric field E the equations (Eq. (1)) for the off-diagonal elements of density matrix take the form: i X ^ , r_ mn þ iOmn rmn ¼ dmp rpn rmp dpn E þ Rmn r ð24Þ _ p where, Omn ¼ ½ðW m W n Þ=ð_Þ denote transition frequencies and Rmn matrix elements of the relaxation operator. Consider the stationary solution of the linearized equation (Eq.(24)) in the constant electric field:
rðsÞ mn ¼ ðdmn E Þ
rmm rnn _Omn
i
Rmn
Omn
The averaged dipole moment corresponding to this solution is: d ¼ 2
X
dnm ðdmn EÞ
m4n
rmm rnn _Omn
i
X dnm Rmn dmn Rnm
m4n
Omn
ð25Þ
Find the corresponding expression for the system assuming that it is in the state of thermodynamic equilibrium with the reservoir. Suppose that the constant homogeneous electric field E is rather small so that the perturbation method may be used: 9Edmn 9 59W m W n 9. In the linear approximation by the field amplitude E the spectrum of energies W m is not changed since dmm ¼ 0, but eigenfunctions are perturbed in this way [29]: ~ m ¼ Cm C
X
Cn
nam
dmn E W m W n
In this new basis the diagonal elements of the dipole moment are ^ C ~ m 9d9 ~ m S ¼ 2 P ½dnm ðdmn EÞÞ=ðW m W n Þ. not zero: d~ mm ¼ /C nam
In stationary case the averaged dipole moment d ¼
P~ d mm rmm is m
equal to: X dmn ðdmn EÞ X r rnn rmm ¼ 2 dnm ðdmn EÞ mm , W W _Omn m n m,n a m m4n
d ¼ 2
ð26Þ
where rmm are the corresponding populations. As far as the alteration of the energy spectrum is not essential in this case, the stationary distribution of populations over levels rmm can be taken unchanged (i.e. defined only by condition of thermodynamic equilibrium with reservoir) in the linear approximation by external field E. Further, take into account the fact that the dipole moment of the system in thermodynamic equilibrium state with the reservoir cannot depend ^ ðsÞ Þ. on relaxation rates, which are defined by the quantities Rmn ðr Consequently it should always be defined by expression Eq. (26). Matching expressions Eq. (25) and Eq. (26), we get the following condition: ^ ðsÞ dmn Rnm r ^ ðsÞ X dnm Rmn r ¼0 ð27Þ m4n
Omn
Note that for the relaxation operator in the form Eq. (23) we ^ ðsÞ Þ ¼ 0, so that the condition Eq. (27) is fulfilled. always get Rmn ðr But it is violated for a standard relaxation operator Eq. (3). It is worth noting that the standard relaxation operator given by Eq. (3) by itself describes tending of the system to the fixed equilibrium state that does not corresponds to the equilibrium state in the field E. While the modified relaxation operator, meeting a requirement Eq. (27), corresponds to tending of the system at every instant to some ‘‘instantaneous equilibrium state’’, that is defined by ‘‘instantaneous’’ field value. The same approach is known in physics of magnetic phenomena [30,31].
5. Optical Bloch equations for two-level system We consider an open two-level system (electro-dipole transition) interacting with classic external field EðtÞ. The equations for the density matrix elements with relaxation operator defined by Eqs. (19),(20) have the following form: 9 r_ 21 þiO21 r21 ¼ _i dEðtÞDng r21 r12 , > > = r12 ¼ rn21 , ð28Þ > ; Dn_ þ GðDnDn0 Þ ¼ 2i dEðtÞ r21 r12 , > _
where Dn ¼ r11 r22 , d ¼ d21 ¼ d12 , G ¼ W 21 þ W 12 , Dn0 ¼ ½ðW 21 W 12 Þ=ðW 21 þ W 12 Þ, the transverse relaxation rate g is given by the relation Eq. (17). The expressions for the averaged dipole moment and population difference of the system follow from Eq. (28): 2O21 Dn € _ , dþ 2gdþ O221 d ¼ dðEðt ÞdÞ _
ð29Þ
_ _ þ GWW ¼ EðtÞd, W 0
ð30Þ
where
d ¼ d r21 þ r12
is
the
averaged
dipole
moment,
W ¼ ½ð_O21 Þ=ð2Þð1DnÞ is the averaged energy (for the normalization r11 þ r22 ¼ 1), W 0 ¼ ½ð_O21 Þ=ð2Þð1Dn0 Þ. We compare Eqs. (29),(30) with well-known relations corresponding to the standard relaxation model (Eq. (3)) [2,3]: 2O21 Dn _ € , ð31Þ dþ 2gdþ O221 þ g2 d ¼ dðEðt ÞdÞ _ _ þ GWW ¼ EðtÞ d_ þ gd W 0
ð32Þ
Compare the equations for the energy Eqs. (30),(32). There is a physically meaningless term gdE in the right-hand side of Eq. (32), which has the same order of smallness pg=O21 with respect to the ‘‘main’’ terms of equation as crucially important relaxation _ term 2gd in the equation for the dipole moment. The system of Eqs. (31),(32) was obtained in the well-known monograph [2] where the term gdE was just ignored. However it is obvious that it is incorrect to take into account and ignore the terms of the same order in different equations of the compatible system. Another approach was developed in paper [13]. As far as the quantity _ dþ gd, appearing in the system (Eqs. (31),(32)), coincides with the ^ namely ^ d, averaged value j of the current operator ^j ¼ ½ðiÞ=ð_Þ½H, _ this quantity instead of d should be inserted in the Maxwell _ þ GWW ¼ jE equation as a current. In that case the relation W 0
(with the Pointing theorem taken into account) provides the correct formulation of the energy relations of the system ‘‘atomþelectromagnetic field þdissipative reservoir’’. It is worth to note that if the correct relaxation operator (meeting the requirement Eq. (5)) is used Eq. (28) is reduced to the physically
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
meaningful system (Eqs. (29),(30)) corresponding to the natural _ relation d j. As for the comparison of equations for the dipole moment (Eqs. (29) and (30)), the different dependence of the oscillator eigenfrequency on the relaxation rate g should be pointed out. If for the standard optical Bloch equation (Eq. (31)) the eigenfrequency o0 , defined by characteristic equation: o20 þ 2ig O221 þ g2 ¼ 0 is equal to
For the modified equation (Eq. (29)), which is equivalent to the classical oscillator with the friction, the characteristic equation and its solution are the following:
o20 þ 2igO221 ¼ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o0 ¼ ig þ O221 g2 ig þ O21
~^ denotes ^ 990 is some given ‘‘equilibrium’’ density matrix, r where r 8 ^ transposed density matrix r. The relaxation operator (Eq. (33)) obeys the following conditions: p^ R^ ¼ 2gp,
^ 0 R^ ¼ G H0 W 0 , H
^ 0r ^ 990 . where W 0 ¼ H Now we add the external fields: magnetostatic field B0 and HF electric filed EðtÞ. We define these fields by a vector potential function Aðr,t Þ: Z t A ¼ c EðtÞdt þ AB ðrÞ, 1
1 g2 2 O21 6
The obtained frequency shift effect , which is lost if the simplest relaxation model is used, may be important in such actual problems as development of quantum frequency standards. The analogous effect is known for interaction of two-level atom with one-mode optical field as the Bloch–Siegert shift [32,33]. It is the correction to the strict resonance with atomic transition due to taking into account counter-rotating terms in atom–field interaction operator.
6. Harmonic dissipative oscillator in a magnetic field We consider interaction of dissipative quantum oscillator with monochromatic electric field in the presence of external constant magnetic field. In particular this system is of interest in the theory of interaction of radiation and quantum dots [34]. It is convenient to analyze this problem using the coordinate representation of ^ ¼ rðr,r 0 Þ (see Ref. [29]). the density matrix r The averaged value of the physical quantity defined by the ^ ^ operator gðrÞ and the commutation operator of gðrÞ with density matrix are calculated in the coordinate representation as follows [29]: Z n 0 ^ rðr,r 0 ÞÞdðrr 0 Þd3 r 0 d3 r, ½g, ^ r ^ g^ ðr ÞÞrðr,r 0 Þ ^ ¼ ^ ¼ gðr g ¼ g^ r gðr ^ where the operator gðrÞ defined in the r-space acts only on the coordinate r of the function rðr,r 0 Þ. It is convenient to separate so called ‘‘longitudinal’’ and ‘‘trans^ ? , obeying ^ 99 and r verse’’ components of the density matrix7 r R 3 0 0 0 3 ^ ^ ^ ^ ^ conditions r r99 ¼ pr99 ¼ 0, r? ¼ r? ðr,r Þdðrr Þd rd r ¼ 0. Let this system in the absence of external monochromatic electric and constant magnetic fields be determined by Eq. (1), ^ ¼H ^ 0: where H ^2 ^ 0 ¼ p þU ðr Þ, H 2m 6
and the potential UðrÞ corresponds to the symmetric well: UðrÞ U x2 ,y2 ,z2 . The simplest relaxation operator obeying condition Eq. (5) in the coordinate representation can be chosen in the following way: ~^ G r ^ 990 , ^ ? r ^ 99 r ð33Þ R^ ¼ g r ?
d^ R^ ¼ 0,
o0 ¼ ig þ O21
153
We do not discuss here another frequency shift that is also induced by the interaction with the reservoir. It is defined by the imaginary part of the relaxation constant (ignored in the present analysis) which is defined by the Cauchy principal value of the integral over bath modes (see Refs. [4,24]). 7 The meaning of operators r99 ðr,r0 Þ and r? ðr,r0 Þ becomes clear if the coordinate representation is compared with matrix representation rmn [29]: rðr,r0 Þ ¼ Sm,n rmn cm ðrÞcnn ðr0 Þ. For ‘‘symmetric’’ basis, where r mm ¼ pmm ¼ 0, the components r99 ðr,r0 Þ and r? ðr,r0 Þ are defined by diagonal (m ¼ n) and offdiagonal (ma n) matrix elements, correspondingly.
where rUA ¼ 0, r AB ¼ B0 . In this case, firstly, it is necessary to add the interaction operator with an external field to the ^ 0: Hamiltonian H ^ ^ ¼H ^ 0 þ h, H
e e2 ^ þ Ap^ þ pA h^ ¼ A2 2mc 2mc2
ð34Þ
Secondly, as it was obtained in paper [13], the gauge invariance condition of initial equations of quantum mechanics requires that the relaxation operator be modified if supplementing the Hamiltonian by terms with vector potential9: ^ ¼ eiu R^ eiu r ^ , ^ ) R^ A r ð35Þ R^ r R e 0 where uðr,r 0 ,t Þ ¼ _c F Aðx,t Þdx. F Fðr,r Þ is a contour that connects the points r and r 0 , it is arbitrary if r A ¼ 0. If r A a 0, as it is shown in paper [13], the transformation (Eq. (35)) can also be used for construction of the relaxation operator in magnetic field to provide an invariance of the solution of von Neumann equations with respect to the electromagnetic potentials gauge transformation. At that the result does not depend on choosing the contour Fðr,r 0 Þ if the class of admissible contours is limited by condition of contour length vanishing for r-r 0 . Now we use the results of paper [13], where the properties of transformation (Eq. (35)) were analyzed for the ‘‘standard’’ ðstÞ ^ 990 . For an arbitrary ^ ? G r ^ 99 r relaxation operator R^ ¼ gr contour Fðr,r 0 Þ, belonging to the specified class, the following relations were proved: e e_ ^ ¼ i A, lim p^ 2 u ¼ div A, ðeiu Þr0 ¼ r ¼ 1, lim0 pu r-r r-r 0 c c As a consequence the following formulas were obtained for the ðstÞ ðstÞ ^ ¼ eiu R^ ^ . operator R^ A r eiu r ! eA ^ ðstÞ ^dR^ ðstÞ ¼ d^ R^ ðstÞ ¼ gd, p ^ RA A c ! eA ^ ^ ðstÞ ¼ g p , HRA ¼ G WW 0 , c ^r ^ . Repeating calculations described in paper [13] for where W ¼ H the operators R^ and R^ A defined by relations Eq. (33) and Eq.(35) we get somewhat different formulas: 8 In the coordinate representation the transposing operation is reduced to the ~^ ¼ r ~^ ¼ r ^ 99 , r ^ n? . interchange of arguments r and r 0 , i.e. r~ ðr,r0 Þ ¼ rðr 0 ,rÞ, at that r 99 ? 9 ^ can be used only if the interaction operator with electric The operator R^ r ^ and the vector potential is field is defined in the dipole approximation h^ E ¼ dE given by AB ¼ ½ð1Þ=ð2Þ½B0 ,r.
154
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
d^ R^ A ¼ d^ R^ ¼ 0,
^ p
! eA ^ eA RA ¼ 2g p , c c
^ R^ A ¼ G WW 0 , H ð36Þ
(only the last equality remained the same). Note that the gauge-invariant relaxation operator is constructed by transformation of the relaxation operator given in the absence of magnetic field. This method cannot take into account possible dependence of relaxation constants g and G on magnetic field. This task is beyond the capabilities of the phenomenological approach. As for ‘‘equilibrium’’ energy W 0 , it is defined only by reservoir temperature, so its preservation under transformation (Eq. (35)) is additional check of this approach robustness. Consider the equation for density matrix i h^ i ^ r_^ ¼ H, r^ þ RA r^ _ ^ and R^ A r ^ defined by Eqs. (34),(35) and (33). with operators H ^ (Eq. (36)), we get equations Using relations for the operator R^ A r for
averaged
values
of
dipole r and energy W ¼ H^ r^ :
^, d ¼ d^ r
current
^ ¼ j ¼ ^jr
e ^ ^ m ½p½ðeAÞ=ðcÞ
e e _ € ^ þ oH b d_ ¼ d þ2gd þ rU r r^ EðtÞ, m m
ð37Þ
_ j ¼ d,
ð38Þ
_ þ G WW ¼ EðtÞd_ W 0
ð39Þ
azz z2 Þ the term me rU r^ in expression Eq. (37) is reduced to e ^ ^ ^ m rU r ¼ Yd, where Y is a diagonal tensor: Yxx ¼ axx =m, Yyy ¼ ayy =m, Yzz ¼ azz =m. So the system becomes closed. So, we obtained equation, which are absolutely equivalent to the equations for classical anisotropic oscillator with a friction placed in constant magnetic field and HF field. It is interesting to compare Eqs. ((37)–(39)) with corresponding formulas obtained in paper [13] for the standard relaxation ðstÞ model R^ A :
_ þ GWW ¼ EðtÞ d_ þ gd W 0
e2 1 m Y? o2 þ ooH 2igo
ð42Þ
^ are supposed to be Here the transversal components of tensor Y equal Y? for simplicity. At the same time from Eq. (40) we get:
wðstÞ þ ¼
e2 1 m Y? o2 þ ooH þ g2 2ig o oH =2 Þ
ð43Þ
The frequency profiles of real and imaginary parts of susceptibilities defined by Eqs. (42) (43) are depicted in Fig. 1. It is obvious that for different relaxation operators the antiHermitian components of susceptibilities differ by the factor ½1 oH =2o . In the resonant region where 9O þ oH =2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi o9, g 5 oH (here O ¼ Y? þ o2H =4 ) there is only quantitative difference:
wþ
e2 1 2mO O þ oH =2 oi 1 þ oH =2O Þg,
wðstÞ þ
e2 1 2mO O þ oH =2 oig
wðstÞ þ wþ
where B0 ¼ bB0 , b is a unit vector along magnetic field, R oH ¼ eB0 =mc is a gyrofrequency, r^ ¼ rðr,r0 Þdðrr0 Þd3 r0 d3 r ¼ 1. For a three-dimensional oscillator U ¼ ½1=2 axx x2 þ ayy y2 þ
e2 e _ € ^ þ oH b d_ þ gd ¼ d þ2gd þ g2 d þ rU r r^ Eðt Þ m m
wþ ¼
There is no difference at all in the region of high frequency, where o b O, oH , g:
2
from Eq. (37) d ¼ Rew þ E þ eiot , where
ð40Þ
ð41Þ
As is easily seen there are the same differences in equations with standard and modified relaxation operator as for two-level system discussed in part IV. In particular, the stationary solution of Eq. (37) for the dipole moment in a constant electric field ^ 1 E does not depend on relaxation constants. The d ¼ ½ e2 =ðmÞY energy exchange between HF field and the medium (Eq. (39)) is _ described by usual term Ed (without any additional terms), as it should be for the correct relaxation description. It is essential that not only terms quadratic in relaxation constant but also additional terms of the order pgoH d b g2 d appear in the system with magnetic field and standard relaxation operator. The last terms affect the anti-Hermitian component of the electric susceptibility in the same order p½oH =o as magnetic field affects the Hermitian component. Indeed, let as consider the circularly polarized monochromatic field EðtÞ ¼ ReE þ eiot , orthogonal with respect to the constant magnetic field, and rotating with frequency o in the direction of dipole rotation in magnetic field. In that case we get
e2 1 m o2 2igo
However, the difference becomes significant as the frequency decreases. The expression (Eq. (42)) defines conductivity of constant sign in all frequency range, whereas expression Eq. (43) predicts paradoxical sign change of conductivity when frequency goes through the value o ¼ oH =2.
7. The relaxation operator in the Heisenberg–Langevin equation The Heisenberg–Langevin approach is popular for different applications on a par with density matrix approach. In frame of secondary quantization representation this approach leads to the following equation for the system dynamics: d þ i h^ þ i ^ þ þ þ a^ a^ m ¼ H, a^ n a^ m Rmn a^ 1 a^ 1 , a^ 1 a^ 2 ,. . .a^ n a^ m . . . þ F^ mn , dt n _ ð44Þ þ
where a^ n n a^ m are the creation and annihilation operators of the quantum–mechanical states 9n 4 and 9m 4 correspondingly, R^ mn is the relaxation operator, F^ mn is the d-correlated Langevin source describing the system fluctuations (see, for example Refs. [1,35]). As is easily seen, the standard commutation relations for the þ operators a^ n and a^ m lead to the following commutation relations þ for the products a^ n a^ m : þ þ þ þ ð45Þ ½a^ p a^ q , a^ n a^ m ¼ a^ p a^ m dnq dmp a^ n a^ q , and this result does not depend on either bosons or fermions are considered. The presence of the Langevin source should provide the fulfillment of commutation relation (Eq. (45)) even if the relaxation processes are taken into account (see, for example Ref. [1]). Although the evolution equation for the operator (Eq. (44)) formally differs from equation for the density matrix (Eq. (1)) by the sign before the commutator term (see Refs. [29,19]), this equation can be reduced to the von Neumann equation form if we ^ in the secondary quantization represenwrite the Hamiltonian H ^ ¼ PH ^ pq a^ þ a^ q into tation. Indeed, substituting the expression10 H p pq
equation (Eq. (44)) and taking into account commutation
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
m 2 Im e2
155
m 2 Re e2
Fig. 1. The anti-Hermitian (a) and Hermitian (b) parts of susceptibilities as dependence of the field frequency. The solid lines correspond to w þ defined by Eq. (42), the dashed lines correspond to wðstÞ þ defined by Eq. (43). The parameters: oH =O ¼ 0:5, g=O ¼ 0:1.
relations (Eq. (45)), we get an equation of the same form as usual equation for the density matrix: i ih r_^ mn ¼ H^ mp r^ pn r^ mp H^ pn þ R^ mn r^ 11 , r^ 21 ,. . .r^ mn . . . þ F^ mn , ð46Þ _ ^ mn corresponds ^ mn ¼ a^ nþ a^ m . We emphasize that symbol r where r to matrix where each element is an operator (unlike standard density matrix). In such an approach the averaged value of physical quantity, which is given in the secondary quantization P þ representation g^ ¼ g pq a^ p a^ q , is calculated by formula g¼
pq
oC 9 ð0Þ
P
^ nm 9Cð0Þ 4 , where 9Cð0Þ 4 is the initial quang mn r
m,n
tum state of the system, symbol /. . .S denotes averaging over Langevin source statistics. The presence of Langevin source does not affect the equation for the averaged values of dynamic variables (see, for example, Ref. [1]). So it follows from identity of forms of equations (Eqs. (1) (46)) that the proof procedure of relaxation operator properties in von Neumann equation can be carried over to the properties of ^ 21 ,. . .r ^ mn . . . . In particular, for the basis ^ 11 , r the operator R^ mn r with real elements 9m 4 the standard expression ^ mn (see Refs. [1,35]) should be ‘‘antisymmetrized’’: R^ m a n ¼ gmn r ^ nm . ^ mn r R^ m a n ¼ gmn r It is worth to note that such a modification of relaxation term in Eq. (46) does not affect commutation properties of Langevin source, which provide conservation of commutation properties (Eq. (45)) with an allowance for relaxation processes: j k ^ mm dðtt 0 Þ: ^ nn r F^ mn tÞF^ nm ðt 0 Þ ¼ 2gmn r
8. Conclusion In this paper we propose a new and simple method for improving accuracy of description of relaxation processes in quantum systems. We have investigated the problem of violation of two fundamental properties of relaxation operators in master equation that follow from initial relation for the unreduced quantum system, but can be lost due to various approximations. The first condition provides the correct relation between macroscopic current and polarization of the quantum medium; the other one is defined by thermodynamic arguments. We have shown that widely used 10 ^ pq , if It makes sense to preserve the operator sign over matrix elements H ^ takes into account interaction of the atom with any nonthe Hamiltonian H classical system.
conventional constant-rate relaxation model does not meet these requirements and the use of this model under some conditions can lead even to ‘‘unphysical’’ instability. The proposed correction lies in the antisymmetrization of standard relaxation term for the off-diagonal density matrix elements. This phenomenological method is described for any quantum systems with discreet number of basis function, as well as for a quantum system described by density matrix in coordinate representation. The dissipative three-dimensional oscillator in arbitrary oriented magnetic field is considered in detail. The proposed modification of relaxation operator leads eventually to more simple and noncontradictory equations for dynamics of averaged values of such physical quantities as current, dipole moment and energy of the system. This method can be applied for both Von Neumann and Heisenberg–Langevin equations. Properly introduced relaxation operator does not significantly modify relaxation time scales but can modify the detailed properties of quantum states. The proposed method can be used for the accurate description of quantum system evolution, including estimation of asymptotical entanglement dynamics (see, for example Refs. [25,36,37]), investigation of the systems with strong qubits–environment interaction [33,15,38], development of optical frequency standards [39,40] and many other actual problems of quantum optics and atomic physics.
Acknowledgments This work was supported by Russian Foundation for Basic Research Grant no. 11-02-97079, Federal Target Program ‘‘Research and Development in Priority Directions of Development of Russia Scientific-Technological Complex for 2007-2013’’ (Grant contract no. 07.514.11.4162). The authors are grateful to I.D. Tokman, Vl.V. Kocharovsky and V.Ye. Semenov for the helpful discussions.
References [1] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, England, 1997. [2] R.H. Pantell, H.E. Puthoff, Fundamentals of Quantum Electronics, Wiley, New York, 1969. [3] V.M. Fain, Y.I. Khanin, Quantum Electronics. Basic Theory, 1, MIT, Cambridge, 1969. [4] K. Blum, Density Matrix Theory and Applications, Plenum, New York, 1981. [5] H. Haken, in: L. Genzel (Ed.), Laser Theory in Encyclopedia of Physics, Vol. XXV/2c, Springer, Heidelberg, 1970.
156
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
[6] G.S. Agarwal, Quantum Statistics Theories of Spontaneous Emission and Their Relation to Other Approaches, in: G. Hohler (Ed.), Springer Tracts in Modern Physics, Vol. 70, Springer-Verlag, Heidelberg, 1974. [7] R.P. Feynman, F.L. Vernon Jr., Ann. Phys. 24 (1963) 118. [8] S. Khademi, S. Nasiri, Int. J. Mod. Phys. B 18 (1993) 7. [9] H. Dekker, Phys. Rep. 80 (1981) 1. [10] Y. Subouti, S. Nasiri, e-print arXiv:quant-ph/051124. [11] M.S. El Naschie, Chaos Solitons Fractals 32 (2007) 271. [12] M. Blasone, E. Gelegheni, P. Jizba, G. Vitello, Phys. Lett. A 310 (2003) 393. [13] M.D. Tokman, Phys. Rev. A 79 (2009) 053415. [14] A. Kurcz, A. Capolupo, A. Beige, E. Del Giudice, G. Vitiello, Phys. Rev. A 81 (2010) 063821. [15] F. Beaudoin, J.M. Gambetta, A. Blais, Phys. Rev. A 84 (2011) 043832. [16] A. Stokes, A. Kurcz, T.P. Spiller, A. Beige, Phys. Rev. A 85 (2012) 053805. ¨ onen, ¨ [17] J. Salmilahto, P. Solinas, M. Mott Phys. Rev. A 85 (2012) 032110. [18] L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Course of Theoretical Physics. Electrodynamics of Continous Madia, Vol. 8, Butterworth-Heinemann, Oxford, 1984. [19] L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics. Statistical Physics, Vol. 5, Pergamon, London, 1980. [20] R. Ravinder, Puri, Mathematical Methods of Quantum Optics, Springer Series in Optical Sciences (2001). Vol. 79; Springer-Verlag Berlin and Heidelberg GmbH and Co. K; W.T. Rhodes. [21] A.G. Litvak, V.Yu Trakhtengerts, Sov. Phys. JETP 33 (1971) 921. [22] M.D. Tokman, Sov. J. Plasma Phys. 10 (1984) 331–335. [23] H.-P. Breuer, F. Petruccione, The theory of Open Quantum Systems, Oxford University Press, New York, 2002. [24] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom–Photon Interactions, Wiley, New York, 1992.
[25] Chris Fleming, N.I. Cummings, Charis Anastopoulos, B.L. Hu, J. Phys. A Math. Theor. 43 (2010) 405304. [26] W.J. Munro, C.W. Gardiner, Phys. Rev. A 53 (1996) 2633. [27] G. Lindblad, Commun. Math. Phys. 40 (1975) 147. [28] V. Gorini, A. Kossakowski, E.C.G. Sudarsham, J. Math. Phys. 17 (1976) 821. [29] L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Nonrelativistic Theory, Oxford, Vol. 3, Pergamon, 1977. [30] R.K. Wangness, Phys. Rev. 98 (1955) 927. [31] A. Abragam, The Principles of Nuclear Magnetism, Clarendon, Oxford, 1961. [32] J.J. Sakurai, Modern. Quantum Mechanics., Revised Edition (1994), AddisonWesley, San Fu Tuan. [33] P. Forn-Diaz, J. Lisenfeld, D. Marcos, J.J. Garcia-Ripoll, E. Solano, C.J.P.M. Harmans, J.E. Mooij, Phys. Rev. Lett. 105 (2010) 237001. [34] T. Chakraborty, Comment. Condens. Matter Phys. 16 (1992) 35. [35] M.D. Lukin, Rev. Mod. Phys. 75 (2003) 457–472. [36] R.C. Drumond, M.O. Terra Cunha, J. Phys. A Math. Theor. 42 (2009) 285308. [37] J. Novotny´, G. Alber, I. Jex, Phys. Rev. Lett. 107 (2011) 090501. [38] J. Casanova, G. Romero, I. Lizuain, J.J. Garcia-Ripoll, E. Solano, Phys. Rev. Lett. 105 (2010) 263603. [39] C.W. Chou, D.B. Hume, J.C.J. Koelemeij, D.J. Wineland, T. Rosenband, Phys. Rev. Lett. 104 (2010) 070802. [40] A.D. Ludlow, et al., Science 319 (2008) 1805.
Contents lists available at SciVerse ScienceDirect
Journal of Luminescence journal homepage: www.elsevier.com/locate/jlumin
Corrections to the phenomenological relaxation models for open quantum systems Mikhail Tokman, Maria Erukhimova n Institute of Applied Physics, Russian Academy of Sciences, 46, Ul’yanov Street, 603155 Nizhny Novgorod, Russia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 September 2012 Received in revised form 6 December 2012 Accepted 27 December 2012 Available online 4 January 2013
The accurate description of dissipative effects in quantum systems is needed for the development of modern quantum information and metrology techniques. We find a simple method for considerably improving the accuracy of phenomenological relaxation models. This method can be applied for both Von Neumann and Heisenberg–Langevin equations. It is based on some general properties of the relaxation operator which follow from initial exact equations for the unreduced system but can be lost due to various approximations (for example rotating-wave approximation) used in popular models. In particular, we show that use of simple relaxation rate model violates the fundamental relation between the macroscopic current and polarization in the dielectric and can lead to ‘‘unphysical’’ instabilities. We show that the relaxation rate dependence of resonance line of two-level system with the modified relaxation operator differs from that obtained from the standard optical Bloch equation and corresponds strictly to the classical oscillator with friction. This result is important for improving the quantum frequency standards. The analogous result is obtained for the damped three-dimensional quantum oscillator in arbitrary oriented magnetic field, which is a robust model for the theory of quantum dots. & 2013 Elsevier B.V. All rights reserved.
Keywords: Open systems Master equation Relaxation operator Optical Bloch equation
1. Introduction The description of relaxation processes in open quantum systems is important and one of the most complicated problems in atomic physics and quantum optics. (see, for example, Refs. [1–13] and literature cited there). Due to fast progress in quantum information and metrology experimental techniques it becomes increasingly important to describe the dissipation process in different quantum systems more and more accurately [14–16]. The direct numerical simulation ‘‘from the first principles’’ of the dynamics of the open quantum system (used in plasma physics, for example) is unrealistic at this time. So the modern approaches for dissipation process analysis are based on different models and approximate schemes. The consequent approach to the description of dissipative dynamics of an open quantum system is based on the analysis of interaction between the considered quantum object and the environment (or bath system). The task is to find the dynamics of the system by tracing out the many degrees of freedom of the bath. The analysis usually involves a variety of simplifications and model assumptions. As a
result, the solution of the final equation may lose some important fundamental properties. We can illustrate how the physically incorrect solution may be obtained because of too crude dissipation model given by the standard master equations. Consider the equation for the density ^ and the ^ of an electron defined by Hamiltonian H matrix r ^ relaxation operator R:
r_^ ¼
i h ^ i ^ ^ þR r ^ H, r _
ð1Þ
Multiply Eq. (1) by the coordinate operator r^ and apply the averaging operation to the result. The following expression for ^ Þ=@t is time-derivative of averaged coordinate r_ ¼ ½ @Tr r^ r obtained after standard transformations: ^ r_ ¼ v þ Trðr^ RÞ,
ð2Þ
^ r^ is velocity operator, v ¼ Tr v^ r ^ is averaged where v^ ¼ i=_ ½H, velocity. We get from Eq. (2) that averaged derivative of the ^ a0. coordinate is not equal to average velocity r_ av if Trðr^ RÞ
In particular, this situation takes place if the standard model of constant-rate relaxation is used (see, for example, Refs. [1–3]) ^ ¼ gmn rmn ; gmn ¼ gnm ð3Þ Rmn r
n
Corresponding author. Tel.: þ7 832 4164754; fax: þ 7 832 4160616. E-mail addresses: [email protected] (M. Tokman), [email protected] (M. Erukhimova). 0022-2313/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jlumin.2012.12.035
The analogous additional members in the equation for the time-derivative of averaged momentum have physical meaning
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
(it is the loss of momentum due to interaction with the reservoir), but the same situation for coordinate is physically senseless. This problem becomes evident if the solution of Eq. (1) is used _ and evN can be equally in Maxwell equations. Two quantities erN taken as an ‘‘averaged current1 ’’ of the ensemble of particles with the charge e and density N. But these quantities are not equal if the relaxation operator (Eq. (3)) is used. It is evident that this ambiguity may be significant or not depending on the ratio between relaxation rates and the typical frequencies of the problem. As it was shown in [13] in definite conditions this incorrectness may even cause a physically senseless instability (negative dissipation) in quantum medium. In the present paper we use the fundamental property of relaxation operator shown in [17]: the correct derivation of evolution equation in the form (Eq. (1)) provides the fulfillment of the following condition ^ r ^ ÞÞ ¼ 0 Trðg^ Rð
ð4Þ
for any operator g^ commutating with the operator of atom– reservoir interaction. In particular this condition, written for the operator of coordinate, has the form: ^ r ^ ÞÞ ¼ 0 Trðr^ Rð It provides ‘‘natural’’ relation: r_ ¼ v, that guarantees correctness of calculated quantum medium susceptibility. In other form the relation (Eq. (4)) can be written as ^ ^ Þ¼0 TrðdR^ r ð5Þ where d^ ¼ er^ is the operator of dipole moment of quantum system. The condition (Eq. (5)) serves as a validation criterion for relaxation models and simultaneously defines the phenomenological method of reconstruction of the standard relaxation model. In addition another general condition may be used. It may also serve as a test for correctness of phenomenological relaxation models. The question concerns proceeding to the limit of constant external field. The susceptibility of the dielectric system in constant field is determined by the state of thermodynamic equilibrium with the reservoir [18,19], so it cannot depend on relaxation rates2 . Again, the simple relaxation model (Eq. (3)) does not provide such a solution. Here we show that the relaxation operator obeying the condition (Eq. (4)) can be obtained only beyond rotating wave approximation (RWA) which is often used for presentation of ‘‘atom-bath’’ interaction operator (see, for example, Ref. [1]). We obtain, that the modified relaxation operator for two-level system in a photon bath represents the antisymmetrized form of the standard operator. We generalize this technique to the density matrix equation of quantum systems with any number of discrete energy levels as well as to density matrix in coordinate representation. The proposed modified relaxation operator is rather more complicated than the standard operator in the form (Eq. (3)), but it provides simple and noncontradictory equations for dynamics of the averaged dipole moment and the energy of the system. In particular, the relaxation rate dependence of resonance line for two-level system with the modified relaxation operator differs from that obtained from the standard optical Bloch 1 Here we do not consider ‘‘vortex’’ current j ¼ r M, arising in the system with intrinsic magnetic moment M. 2 We are not talking about situations with particles exchange: the flow of constant current throw the dissipative medium, for example.
149
equation and corresponds strictly to the classical oscillator with friction. The analogous result is obtained for the damped threedimensional oscillator in arbitrary oriented magnetic field. The paper is organized as follows. In part II we present our proof of the condition (Eq. (4)). We have also analyzed standard derivation of master equations in Redfield form and revealed that the relaxation super-matrix obtained under the Born and Markovian approximations meets this requirement until the secular approximation is applied. In part III the corrected relaxation operator for two-level system in the field reservoir is obtained. It is shown that the modification consists in an antisymmetrization of the standard relaxation term for the off-diagonal density matrix elements. The phenomenological relaxation model is proposed for any quantum systems with discreet number of basis function. Part IV is devoted to proper proceeding to the limit of constant external field. In part V the oscillation equations for twolevel system in a monochromatic wave field are obtained for modified relaxation operator. In part VI the phenomenological relaxation model for the three-dimensional harmonic oscillator placed in arbitrary oriented constant magnetic field and monochromatic field is constructed. The relaxation operator is written in a gauge-invariant form using the results of paper [13]. The macroscopic polarization is obtained for this system. It is shown that the conductivity of the medium obtained from modified equations does not display paradoxical change of sign in a definite frequency range unlike it does for a standard tau-approximation. In part VII we generalize the results obtained for Von Neumann equation with relaxation operator to the Heisenberg–Langevin equations. In Section 8 the results are summarized and discussed.
2. The first criterion of correct relaxation operator Let us show that the proper derivation of the relaxation operator from initial equations leads to fulfillment of condition (Eq. (5)). Somewhat different way to derive this condition was presented in paper [17]. We consider interaction of two quantum subsystems (‘‘the atom’’ and ‘‘the reservoir’’), described by the total Hamiltonian: ^ S ðr,qÞ ¼ H ^ A ðrÞ þ H ^ Q ðqÞ þ ^Iðr,qÞ, H ^ Q ðqÞ act on the functions of ^ A ðrÞ and H where the operators H coordinate of atom r and reservoir q, correspondingly, ^Iðr,qÞ is the interaction operator. We use Von Neumann equation for the ^ S ¼ rmn;nm (Latin indices density matrix of complete system r correspond to atom, and Greek to reservoir):
r_^ S ¼
i i h^ ^S HS , r _
Represent the common density matrix in the form:
r^ S ¼ r^ A r^ Q þ r^ I ,
ð6Þ
^ A ¼ TrQ r ^ Q ¼ TrA r ^ S, r ^ S ; operations TrA,Q denote averaging where r P over corresponding subsystems: TrA ð. . .Þ ¼ dmn ð. . .Þ, TrQ ð. . .Þ ¼ n,m P dn, m ð. . .Þ, TrS ð. . .Þ ¼ TrA TrQ ð. . .Þ ¼ TrQ TrA ð. . .Þ. Suppose that there n, m
is no averaged ‘‘force’’ action of the reservoir on the ‘‘atom’’ to ^ Q Þ ¼ 0 [20,4,13]. Taking first order in interaction energy, i.e. TrQ ð^Ir into account normalization requirements and following relations: ^ A,r ^ Q ,r ^ A,I ¼ TrQ ½H ^ Q ,I ¼ 0, we obtain an equation for the TrA ½H ^ A: ‘‘atomic’’ density matrix r
r_^ A ¼
i i h^ ^A , ^ A þ R^ r HA , r _
150
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
where the relaxation operator has the form: i ^ I Þ R^ ¼ TrQ ð½^I, r _
ð7Þ
^ we use the operator identity To find the convolution TrA ðd^ RÞ ^ ^ ^ ^ ^ ^ TrðA½B, CÞ ¼ Trð½A, BCÞ and take into account that the operator d^ does not act on the functions of reservoir coordinates. As a result we obtain:
This equation can be written in the basis of eigenstates of the ^ A: Hamiltonian H X r_ Aij ¼ ioij rAij þ Rijkm rAkm ð10Þ km
Here oij ¼ 1=_ HAii HAjj . For the coefficients Rijkm the following expressions are obtained: X Rijkm ¼ dmj Gippk þ dik Gnjppm þ Gmjik þ Gnkijm , ð11Þ p
^ ^Ir ^ ¼ i TrA TrQ ð½d, ^ IÞ TrA ðd^ RÞ _
ð8Þ
Gijkm ¼
It follows from Eq. (8) that the condition (Eq. (5)) is automatically satisfied if the operator of atom–reservoir interaction commutates with the operator of dipole moment. For example this commutator is zero if the operator ^I is just a function of coordinates: ^I ¼ U ðr,qÞ. In particular, the relaxation caused by collisions or interaction with lattice oscillations corresponds to ^ ^I ¼ 0 is also fulfilled for the the last case. The condition ½d, interaction with the photon reservoir, if the interaction operator is defined in the electric dipole approximation ^I ¼ d^ E^ (here E^ is the quantum operator of the electric field of the reservoir). If another form of interaction operator with the photon reservoir is used [1–3]: ^I ¼ e a^ p^ þ p^ a^ 2mc
1 2
_
Z
1 0
X HQ aa HQ bb okm t dt rQ aa Iiajb Ikbma exp i _ ab ð12Þ
^ Q and r ^ Q is Here the basis fjaig diagonalizing the operators H used; dij is the Kronecker delta. The following step is so called secular approximation [4,24] in frame of which the relaxation terms of Eq. (10) with okm a oij are omitted. It is implied that the corresponding amendments are small if the following inequality is fulfilled: Rijkm oij okm 5 1,
oij a okm
ð13Þ
Under secular approximation the kinetic equations take the simple standard form, where the off-diagonal density matrix elements relax independently, while the diagonal elements obey the balance equations:
^ (Here a^ is the operator of the vector potential: a_^ ¼ cE, p^ ¼ i_r is the momentum operator), we can apply the commu^ ¼ i_ to Eq. (8). As a result, in a typical case tation relation ½r^ , p when the spatial variation of the field at atomic scales can be neglected, we get3 :
r_ Aij ¼ ioij rAij gij rAij , i a j
2 e2 ^ ¼ e TrA TrQ ð^a r ^ IÞ ¼ ^ IÞ ¼ 0 Tr Q ð^a TrA r TrA ðd^ RÞ mc mc
gij ¼ Rijij
^ I ¼ 0 by the definition Eq. (6)). (TrA r So, the relaxation operator obtained in a proper way from initial relations should always satisfy the condition (Eq. (5)). A more general statement can be made: the relaxation operator R^ should satisfy the condition Eq. (4):
It can be easily shown that for the relaxation operator in the form Eqs. (10)–(12) obtained under Born–Markovian approximation and for any operator g^ obeying commutation relation ½g^ ^I ¼ 0 the following equality is valid: X g ij Rjikm ¼ 0
W ik rAkk W ki rAii
ð14Þ ð15Þ
kai
where W ik ¼ 2ReGkiik
ð16Þ
ij
^ ¼0 Trðg^ RÞ for any operator g^ commutating with operator of atom–reservoir interaction. ^A To obtain the relaxation operator as explicit dependence R^ r it is necessary to solve the equation for ‘‘interaction density ^I ¼r ^ I ðr ^ A,r ^ Q Þ in frame of given reservoir density matrix matrix’’ r approximation (so called Born approximation) and substitute it in Eq. (7). The standard procedure usually means to use the Markovian approximation as well. Then the equations for density matrix of the system take the form of so called kinetic equation: differential equation with constant coefficients (see, for example Refs. [3–6,23,24]). i 1 d ih r^ ¼ H^ A , r^ A 2 dt A _ _
r_ Aii ¼
X
Z
1 0
H^ þ H^ ^ þH ^ H Q Q A A ^ A ðt Þr ^Q dt TrQ ^I ei _ t ^Ie þ i _ t , r ð9Þ
3 The question is more complicated if the variation of the spatial structure of electromagnetic field at the atomic scales is significant and the field is not potential. In any case in frame of classical motion equations the following effect is known: the appearance of additional component of the averaged current density in inhomogeneous rapidly oscillating or noise field [21,22], but this current is pure vortex and disappears after averaging over volume occupied by particles.
^ ¼0 It in turn provides the fulfillment of condition Trðg^ RÞ (Eq. (4)). The dissipative operator in the standard form defined by Eqs. (14)–(16) which is obtained after additional secular approximation does not fulfill this condition. The same conclusion was reached in paper [17]. Note that for the bath representing an ensemble of harmonic oscillators the corresponding ‘‘unsecular’’ terms occur only if nonrotating wave (NRW) form for the interaction operator ^I is used. The kinetic (master) equations out of the frame of RWA (rotating wave approximations) were investigated in papers [25,16,26].
3. The relaxation operator for two-level system in a photon bath Using of relations Eqs. (10)–(12) for constructing the relaxation operator in a model system with finite number of energy levels may not lead to the desired result meeting a requirement (Eq. (5)). That is because the basis is not complete, and as a consequence, commutativity of the dipole moment operator and atom–reservoir interaction operator may violate. The solution is likely to use phenomenological approaches, specific for each system.
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
We consider the simple system, where the desired relaxation operator can be obtained on the basis of relations Eqs. (10)–(12). It is two-level system in a photon reservoir with interaction ^ Here d^ ¼ dðj2ih1jþ j1ih2jÞ is the operator of operator ^I ¼ d^ E. dipole moment of the system (the matrix element is real for definiteness, that can be supposed without loss of generality), P pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ n E^ ¼ i ½ð_on Þ=ð2Þ a^ E^ n ðr a^ n E^ ðrÞÞ is the operator of electric n
n
n
field, on is the frequency of mode with number n, E^ n ðrÞ ¼ E^ n ðrÞen ðrÞ is normalized amplitude of electric field in mode n, a^ nþ , a^ n are the creation and annihilation operators correspondingly. We apply Eqs. (10)–(12) to define the relaxation operator of two-level system. Taking into account, for simplicity sake, only real part of relaxation constants, we get the following expressions for the off-diagonal elements of relaxation operator: ^ 21 ¼ R2121 r21 þ R2112 r12 R r ^ 12 ¼ R1212 r12 þ R1221 r21 , R r
vector of electric field in mode n, z ¼ r, y, j , r is the atom coordinate, y, j are the angles that define the orientation of d. So, for the two-level system in a photon bath we get from initial relations after applying Born–Markovian approximation the relaxation operator Eqs. (19), (20) which meets the criterion (Eq. (5)). It provides the correct relations for averaged observable derivatives in this dissipative system. This relaxation operator is defined by standard relaxation rates; the only modification consists in the antisymmetrized structure of its off-diagonal element. Note once again that the modified relaxation equation (Eqs. (19), (20)) is obtained beyond the framework of secular approximation, when the counter-rotating terms in the interaction operator are taken into account (without employing RWA for interaction operator). At that, as can be seen, this relaxation operator being reduced to frequently used form: R^ ¼
þ þ þ ^ V^ m V^ n , ^ V^ m V^ m V^ n r ^ r Dnm 2V^ n r
2 X
ð22Þ
n,m ¼ 1
where R2121 ¼ R1212 ¼ R1221 ¼ R2112 ¼ g,
1 2
g ¼ ðW 12 þ W 21 Þ
ð17Þ
Consequently, we get that for the matrix elements of relaxation operator R^ the following relations are fulfilled ð18Þ R21 ¼ R12 ¼ g r21 r12 , that automatically leads to fulfillment of condition (Eq. (5)). The equation for matrix element r21 takes the following form (instead of Eq. (14)): r_ 21 þiO21 r21 ¼ g r21 r12 ð19Þ The equations for diagonal elements of density matrix are the same as in Eq. (15): X r_ ii ¼ W ik rkk W ki rii ð20Þ kai
W 12 , W 21 are the standard coefficients, characterizing the rates of atomic transitions j2i-j1i and j1i-j2i, correspondingly: W 12 ¼
2p X _on ^ 2 2 9E n ðr Þ9 9en d9 ðN n þ1Þdðon O21 Þ _2 n 2
W 21 ¼
2p X _on ^ 2 2 9E n ðr Þ9 9en d9 ðN n Þdðon O21 Þ _2 n 2
ð21Þ
Here N n is the averaged number of photons in mode ‘‘n’’. It is assumed that frequency of atomic transition O21 4 0.4 The coefficients given by Eq. (21) can be presented in a usual form: W 12 ¼ A21 ZðO21 , zÞ NðO21 Þ þ 1Þ, W 21 ¼ A21 ZðO21 , zÞNðO21 Þ, 2 where A21 ¼ ½ 4d O321 = 3_c3 is the Einstein coefficient, corresponding to the spontaneous emission rate of an atom in a free space, ZðO21 , zÞ ¼ ½ sðO21 , zÞÞ=ðsfs ðO21 ÞÞ is dimensionless parameter, characterizing the ratio of effective spectral mode density of the reservoir to the corresponding quantity in a free space sfs ðO21 Þ ¼ ½ _O321 = p2 c3 ,
sðO21 , zÞ ¼
X n
2
_on
9E n ðrÞ9 dðon O21 ÞkðzÞ, 4p
2
n d9 kðzÞ ¼ 3 9e9d9 2 is dimensionless parameter characterizing the orien-
tation of the dipole moment of atomic transition relatively to the 4
151
9E ðrÞ9
2
In Eq. (21) dðon O21 Þ is d-function, at that _on n4p dðon O21 Þ is the field spectral energy density at frequency O21 at the atom’s position, if the fiend represents one photon in the mode with frequency on .
is not of the ‘‘Lindblad-form’’ [23,27,28]. (Here the basis operators ^ þ ¼ j2ih1j, V^ 2 ¼ s ^ ¼ j1ih2j). It is characterized by ‘‘disare V^ 1 ¼ s sipator’’ super-operator D, with elements Dnm , which is not a positive-definite matrix: 0 1 W 21 2
D ¼ @ W 12 þ W 21 4
W 12 þ W 21 4 W 12 2
A
Here W 21 and W 12 are given by Eq. (21). This result can be obtained directly from Eq. (9) where the non-RWA form for the ^ þ þs ^ ÞE^ ¼ dðs ^ þ þs ^ Þ interaction operator is used: ^I ¼ dðs ðE^ a þ E^ a þ Þ. If the interaction operator is taken in the RWA-form ^I ¼ dðs ^ þ E^ a þ s ^ E^ a þ Þ and/or secular approximation is applied then usual positive-definite Lindblad form (Eq. (22)) for relaxation operator with D21 ¼ D12 ¼ 0 is obtained. The question arising here does the modified operator define trace-preserving and positive density matrix transformation or not. The detailed analysis has shown that the positive definiteness of density matrix, as solution of this relaxation equation, may be violated only for very narrow range of initial conditions, when the initial state of the system is nearly pure coherent superposition of energy states with definite initial phase. But what is the most important is that the density matrix positive definiteness may be violated only during very short initial time interval, less than the period of oscillations at the transition frequency. But, it is evident that during this transient period the time averaging procedure over reservoir modes that separates the resonance frequency cannot be applied. We note that similar ‘‘antisymmetrized’’ kind of relaxation operator for two-level system was obtained also in papers [25,26] where the interaction of quantum system with reservoir was analyzed beyond the framework of RWA. The same conclusion that the Non Rotating Wave (NRW) master equation is not of the Lindblad form was pointed in Ref. [26]. It was also shown that in the NRW master equation there is little validity to considering time scales shorter than the thermal correlation time because the coarse-grain time-scale approach used in its derivation. By analogy with the two-level system the phenomenological model of relaxation operator for quantum system with any number of discreet levels can be constructed, if there is a basis of real eigenfunctions5 and correspondingly the matrix elements of the dipole moment are real: dmn ¼ dnm . In this case we can assume that 5 It is so, for example, for finite motion in the absence of external magnetic field [29]. In magnetic field this approach can be applied only if the effect of magnetic field on the relaxation processes is neglected.
152
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
the relaxation processes at the transition 9m 439n 4 are defined by off-diagonal elements of density matrix at this transition only. By analogy with two-level system the standard expression for operator of transverse relaxation with given relaxation rates gmn (Eq.(3)) should be ‘‘antisymmetrized’’: ^ ¼ gmn rmn rnm ð23Þ Rm a n r
4. Correctness of relaxation operator in the limit of constant electric field Consider the quantum system with discreet number of eigen^ Cm . functions and eigenvalues of the energy operator: W m Cm ¼ H Suppose that system has definite symmetry due to which the diagonal matrix elements of the dipole moment are equal to zero: dmm ¼ 0. In the external electric field E the equations (Eq. (1)) for the off-diagonal elements of density matrix take the form: i X ^ , r_ mn þ iOmn rmn ¼ dmp rpn rmp dpn E þ Rmn r ð24Þ _ p where, Omn ¼ ½ðW m W n Þ=ð_Þ denote transition frequencies and Rmn matrix elements of the relaxation operator. Consider the stationary solution of the linearized equation (Eq.(24)) in the constant electric field:
rðsÞ mn ¼ ðdmn E Þ
rmm rnn _Omn
i
Rmn
Omn
The averaged dipole moment corresponding to this solution is: d ¼ 2
X
dnm ðdmn EÞ
m4n
rmm rnn _Omn
i
X dnm Rmn dmn Rnm
m4n
Omn
ð25Þ
Find the corresponding expression for the system assuming that it is in the state of thermodynamic equilibrium with the reservoir. Suppose that the constant homogeneous electric field E is rather small so that the perturbation method may be used: 9Edmn 9 59W m W n 9. In the linear approximation by the field amplitude E the spectrum of energies W m is not changed since dmm ¼ 0, but eigenfunctions are perturbed in this way [29]: ~ m ¼ Cm C
X
Cn
nam
dmn E W m W n
In this new basis the diagonal elements of the dipole moment are ^ C ~ m 9d9 ~ m S ¼ 2 P ½dnm ðdmn EÞÞ=ðW m W n Þ. not zero: d~ mm ¼ /C nam
In stationary case the averaged dipole moment d ¼
P~ d mm rmm is m
equal to: X dmn ðdmn EÞ X r rnn rmm ¼ 2 dnm ðdmn EÞ mm , W W _Omn m n m,n a m m4n
d ¼ 2
ð26Þ
where rmm are the corresponding populations. As far as the alteration of the energy spectrum is not essential in this case, the stationary distribution of populations over levels rmm can be taken unchanged (i.e. defined only by condition of thermodynamic equilibrium with reservoir) in the linear approximation by external field E. Further, take into account the fact that the dipole moment of the system in thermodynamic equilibrium state with the reservoir cannot depend ^ ðsÞ Þ. on relaxation rates, which are defined by the quantities Rmn ðr Consequently it should always be defined by expression Eq. (26). Matching expressions Eq. (25) and Eq. (26), we get the following condition: ^ ðsÞ dmn Rnm r ^ ðsÞ X dnm Rmn r ¼0 ð27Þ m4n
Omn
Note that for the relaxation operator in the form Eq. (23) we ^ ðsÞ Þ ¼ 0, so that the condition Eq. (27) is fulfilled. always get Rmn ðr But it is violated for a standard relaxation operator Eq. (3). It is worth noting that the standard relaxation operator given by Eq. (3) by itself describes tending of the system to the fixed equilibrium state that does not corresponds to the equilibrium state in the field E. While the modified relaxation operator, meeting a requirement Eq. (27), corresponds to tending of the system at every instant to some ‘‘instantaneous equilibrium state’’, that is defined by ‘‘instantaneous’’ field value. The same approach is known in physics of magnetic phenomena [30,31].
5. Optical Bloch equations for two-level system We consider an open two-level system (electro-dipole transition) interacting with classic external field EðtÞ. The equations for the density matrix elements with relaxation operator defined by Eqs. (19),(20) have the following form: 9 r_ 21 þiO21 r21 ¼ _i dEðtÞDng r21 r12 , > > = r12 ¼ rn21 , ð28Þ > ; Dn_ þ GðDnDn0 Þ ¼ 2i dEðtÞ r21 r12 , > _
where Dn ¼ r11 r22 , d ¼ d21 ¼ d12 , G ¼ W 21 þ W 12 , Dn0 ¼ ½ðW 21 W 12 Þ=ðW 21 þ W 12 Þ, the transverse relaxation rate g is given by the relation Eq. (17). The expressions for the averaged dipole moment and population difference of the system follow from Eq. (28): 2O21 Dn € _ , dþ 2gdþ O221 d ¼ dðEðt ÞdÞ _
ð29Þ
_ _ þ GWW ¼ EðtÞd, W 0
ð30Þ
where
d ¼ d r21 þ r12
is
the
averaged
dipole
moment,
W ¼ ½ð_O21 Þ=ð2Þð1DnÞ is the averaged energy (for the normalization r11 þ r22 ¼ 1), W 0 ¼ ½ð_O21 Þ=ð2Þð1Dn0 Þ. We compare Eqs. (29),(30) with well-known relations corresponding to the standard relaxation model (Eq. (3)) [2,3]: 2O21 Dn _ € , ð31Þ dþ 2gdþ O221 þ g2 d ¼ dðEðt ÞdÞ _ _ þ GWW ¼ EðtÞ d_ þ gd W 0
ð32Þ
Compare the equations for the energy Eqs. (30),(32). There is a physically meaningless term gdE in the right-hand side of Eq. (32), which has the same order of smallness pg=O21 with respect to the ‘‘main’’ terms of equation as crucially important relaxation _ term 2gd in the equation for the dipole moment. The system of Eqs. (31),(32) was obtained in the well-known monograph [2] where the term gdE was just ignored. However it is obvious that it is incorrect to take into account and ignore the terms of the same order in different equations of the compatible system. Another approach was developed in paper [13]. As far as the quantity _ dþ gd, appearing in the system (Eqs. (31),(32)), coincides with the ^ namely ^ d, averaged value j of the current operator ^j ¼ ½ðiÞ=ð_Þ½H, _ this quantity instead of d should be inserted in the Maxwell _ þ GWW ¼ jE equation as a current. In that case the relation W 0
(with the Pointing theorem taken into account) provides the correct formulation of the energy relations of the system ‘‘atomþelectromagnetic field þdissipative reservoir’’. It is worth to note that if the correct relaxation operator (meeting the requirement Eq. (5)) is used Eq. (28) is reduced to the physically
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
meaningful system (Eqs. (29),(30)) corresponding to the natural _ relation d j. As for the comparison of equations for the dipole moment (Eqs. (29) and (30)), the different dependence of the oscillator eigenfrequency on the relaxation rate g should be pointed out. If for the standard optical Bloch equation (Eq. (31)) the eigenfrequency o0 , defined by characteristic equation: o20 þ 2ig O221 þ g2 ¼ 0 is equal to
For the modified equation (Eq. (29)), which is equivalent to the classical oscillator with the friction, the characteristic equation and its solution are the following:
o20 þ 2igO221 ¼ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o0 ¼ ig þ O221 g2 ig þ O21
~^ denotes ^ 990 is some given ‘‘equilibrium’’ density matrix, r where r 8 ^ transposed density matrix r. The relaxation operator (Eq. (33)) obeys the following conditions: p^ R^ ¼ 2gp,
^ 0 R^ ¼ G H0 W 0 , H
^ 0r ^ 990 . where W 0 ¼ H Now we add the external fields: magnetostatic field B0 and HF electric filed EðtÞ. We define these fields by a vector potential function Aðr,t Þ: Z t A ¼ c EðtÞdt þ AB ðrÞ, 1
1 g2 2 O21 6
The obtained frequency shift effect , which is lost if the simplest relaxation model is used, may be important in such actual problems as development of quantum frequency standards. The analogous effect is known for interaction of two-level atom with one-mode optical field as the Bloch–Siegert shift [32,33]. It is the correction to the strict resonance with atomic transition due to taking into account counter-rotating terms in atom–field interaction operator.
6. Harmonic dissipative oscillator in a magnetic field We consider interaction of dissipative quantum oscillator with monochromatic electric field in the presence of external constant magnetic field. In particular this system is of interest in the theory of interaction of radiation and quantum dots [34]. It is convenient to analyze this problem using the coordinate representation of ^ ¼ rðr,r 0 Þ (see Ref. [29]). the density matrix r The averaged value of the physical quantity defined by the ^ ^ operator gðrÞ and the commutation operator of gðrÞ with density matrix are calculated in the coordinate representation as follows [29]: Z n 0 ^ rðr,r 0 ÞÞdðrr 0 Þd3 r 0 d3 r, ½g, ^ r ^ g^ ðr ÞÞrðr,r 0 Þ ^ ¼ ^ ¼ gðr g ¼ g^ r gðr ^ where the operator gðrÞ defined in the r-space acts only on the coordinate r of the function rðr,r 0 Þ. It is convenient to separate so called ‘‘longitudinal’’ and ‘‘trans^ ? , obeying ^ 99 and r verse’’ components of the density matrix7 r R 3 0 0 0 3 ^ ^ ^ ^ ^ conditions r r99 ¼ pr99 ¼ 0, r? ¼ r? ðr,r Þdðrr Þd rd r ¼ 0. Let this system in the absence of external monochromatic electric and constant magnetic fields be determined by Eq. (1), ^ ¼H ^ 0: where H ^2 ^ 0 ¼ p þU ðr Þ, H 2m 6
and the potential UðrÞ corresponds to the symmetric well: UðrÞ U x2 ,y2 ,z2 . The simplest relaxation operator obeying condition Eq. (5) in the coordinate representation can be chosen in the following way: ~^ G r ^ 990 , ^ ? r ^ 99 r ð33Þ R^ ¼ g r ?
d^ R^ ¼ 0,
o0 ¼ ig þ O21
153
We do not discuss here another frequency shift that is also induced by the interaction with the reservoir. It is defined by the imaginary part of the relaxation constant (ignored in the present analysis) which is defined by the Cauchy principal value of the integral over bath modes (see Refs. [4,24]). 7 The meaning of operators r99 ðr,r0 Þ and r? ðr,r0 Þ becomes clear if the coordinate representation is compared with matrix representation rmn [29]: rðr,r0 Þ ¼ Sm,n rmn cm ðrÞcnn ðr0 Þ. For ‘‘symmetric’’ basis, where r mm ¼ pmm ¼ 0, the components r99 ðr,r0 Þ and r? ðr,r0 Þ are defined by diagonal (m ¼ n) and offdiagonal (ma n) matrix elements, correspondingly.
where rUA ¼ 0, r AB ¼ B0 . In this case, firstly, it is necessary to add the interaction operator with an external field to the ^ 0: Hamiltonian H ^ ^ ¼H ^ 0 þ h, H
e e2 ^ þ Ap^ þ pA h^ ¼ A2 2mc 2mc2
ð34Þ
Secondly, as it was obtained in paper [13], the gauge invariance condition of initial equations of quantum mechanics requires that the relaxation operator be modified if supplementing the Hamiltonian by terms with vector potential9: ^ ¼ eiu R^ eiu r ^ , ^ ) R^ A r ð35Þ R^ r R e 0 where uðr,r 0 ,t Þ ¼ _c F Aðx,t Þdx. F Fðr,r Þ is a contour that connects the points r and r 0 , it is arbitrary if r A ¼ 0. If r A a 0, as it is shown in paper [13], the transformation (Eq. (35)) can also be used for construction of the relaxation operator in magnetic field to provide an invariance of the solution of von Neumann equations with respect to the electromagnetic potentials gauge transformation. At that the result does not depend on choosing the contour Fðr,r 0 Þ if the class of admissible contours is limited by condition of contour length vanishing for r-r 0 . Now we use the results of paper [13], where the properties of transformation (Eq. (35)) were analyzed for the ‘‘standard’’ ðstÞ ^ 990 . For an arbitrary ^ ? G r ^ 99 r relaxation operator R^ ¼ gr contour Fðr,r 0 Þ, belonging to the specified class, the following relations were proved: e e_ ^ ¼ i A, lim p^ 2 u ¼ div A, ðeiu Þr0 ¼ r ¼ 1, lim0 pu r-r r-r 0 c c As a consequence the following formulas were obtained for the ðstÞ ðstÞ ^ ¼ eiu R^ ^ . operator R^ A r eiu r ! eA ^ ðstÞ ^dR^ ðstÞ ¼ d^ R^ ðstÞ ¼ gd, p ^ RA A c ! eA ^ ^ ðstÞ ¼ g p , HRA ¼ G WW 0 , c ^r ^ . Repeating calculations described in paper [13] for where W ¼ H the operators R^ and R^ A defined by relations Eq. (33) and Eq.(35) we get somewhat different formulas: 8 In the coordinate representation the transposing operation is reduced to the ~^ ¼ r ~^ ¼ r ^ 99 , r ^ n? . interchange of arguments r and r 0 , i.e. r~ ðr,r0 Þ ¼ rðr 0 ,rÞ, at that r 99 ? 9 ^ can be used only if the interaction operator with electric The operator R^ r ^ and the vector potential is field is defined in the dipole approximation h^ E ¼ dE given by AB ¼ ½ð1Þ=ð2Þ½B0 ,r.
154
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
d^ R^ A ¼ d^ R^ ¼ 0,
^ p
! eA ^ eA RA ¼ 2g p , c c
^ R^ A ¼ G WW 0 , H ð36Þ
(only the last equality remained the same). Note that the gauge-invariant relaxation operator is constructed by transformation of the relaxation operator given in the absence of magnetic field. This method cannot take into account possible dependence of relaxation constants g and G on magnetic field. This task is beyond the capabilities of the phenomenological approach. As for ‘‘equilibrium’’ energy W 0 , it is defined only by reservoir temperature, so its preservation under transformation (Eq. (35)) is additional check of this approach robustness. Consider the equation for density matrix i h^ i ^ r_^ ¼ H, r^ þ RA r^ _ ^ and R^ A r ^ defined by Eqs. (34),(35) and (33). with operators H ^ (Eq. (36)), we get equations Using relations for the operator R^ A r for
averaged
values
of
dipole r and energy W ¼ H^ r^ :
^, d ¼ d^ r
current
^ ¼ j ¼ ^jr
e ^ ^ m ½p½ðeAÞ=ðcÞ
e e _ € ^ þ oH b d_ ¼ d þ2gd þ rU r r^ EðtÞ, m m
ð37Þ
_ j ¼ d,
ð38Þ
_ þ G WW ¼ EðtÞd_ W 0
ð39Þ
azz z2 Þ the term me rU r^ in expression Eq. (37) is reduced to e ^ ^ ^ m rU r ¼ Yd, where Y is a diagonal tensor: Yxx ¼ axx =m, Yyy ¼ ayy =m, Yzz ¼ azz =m. So the system becomes closed. So, we obtained equation, which are absolutely equivalent to the equations for classical anisotropic oscillator with a friction placed in constant magnetic field and HF field. It is interesting to compare Eqs. ((37)–(39)) with corresponding formulas obtained in paper [13] for the standard relaxation ðstÞ model R^ A :
_ þ GWW ¼ EðtÞ d_ þ gd W 0
e2 1 m Y? o2 þ ooH 2igo
ð42Þ
^ are supposed to be Here the transversal components of tensor Y equal Y? for simplicity. At the same time from Eq. (40) we get:
wðstÞ þ ¼
e2 1 m Y? o2 þ ooH þ g2 2ig o oH =2 Þ
ð43Þ
The frequency profiles of real and imaginary parts of susceptibilities defined by Eqs. (42) (43) are depicted in Fig. 1. It is obvious that for different relaxation operators the antiHermitian components of susceptibilities differ by the factor ½1 oH =2o . In the resonant region where 9O þ oH =2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi o9, g 5 oH (here O ¼ Y? þ o2H =4 ) there is only quantitative difference:
wþ
e2 1 2mO O þ oH =2 oi 1 þ oH =2O Þg,
wðstÞ þ
e2 1 2mO O þ oH =2 oig
wðstÞ þ wþ
where B0 ¼ bB0 , b is a unit vector along magnetic field, R oH ¼ eB0 =mc is a gyrofrequency, r^ ¼ rðr,r0 Þdðrr0 Þd3 r0 d3 r ¼ 1. For a three-dimensional oscillator U ¼ ½1=2 axx x2 þ ayy y2 þ
e2 e _ € ^ þ oH b d_ þ gd ¼ d þ2gd þ g2 d þ rU r r^ Eðt Þ m m
wþ ¼
There is no difference at all in the region of high frequency, where o b O, oH , g:
2
from Eq. (37) d ¼ Rew þ E þ eiot , where
ð40Þ
ð41Þ
As is easily seen there are the same differences in equations with standard and modified relaxation operator as for two-level system discussed in part IV. In particular, the stationary solution of Eq. (37) for the dipole moment in a constant electric field ^ 1 E does not depend on relaxation constants. The d ¼ ½ e2 =ðmÞY energy exchange between HF field and the medium (Eq. (39)) is _ described by usual term Ed (without any additional terms), as it should be for the correct relaxation description. It is essential that not only terms quadratic in relaxation constant but also additional terms of the order pgoH d b g2 d appear in the system with magnetic field and standard relaxation operator. The last terms affect the anti-Hermitian component of the electric susceptibility in the same order p½oH =o as magnetic field affects the Hermitian component. Indeed, let as consider the circularly polarized monochromatic field EðtÞ ¼ ReE þ eiot , orthogonal with respect to the constant magnetic field, and rotating with frequency o in the direction of dipole rotation in magnetic field. In that case we get
e2 1 m o2 2igo
However, the difference becomes significant as the frequency decreases. The expression (Eq. (42)) defines conductivity of constant sign in all frequency range, whereas expression Eq. (43) predicts paradoxical sign change of conductivity when frequency goes through the value o ¼ oH =2.
7. The relaxation operator in the Heisenberg–Langevin equation The Heisenberg–Langevin approach is popular for different applications on a par with density matrix approach. In frame of secondary quantization representation this approach leads to the following equation for the system dynamics: d þ i h^ þ i ^ þ þ þ a^ a^ m ¼ H, a^ n a^ m Rmn a^ 1 a^ 1 , a^ 1 a^ 2 ,. . .a^ n a^ m . . . þ F^ mn , dt n _ ð44Þ þ
where a^ n n a^ m are the creation and annihilation operators of the quantum–mechanical states 9n 4 and 9m 4 correspondingly, R^ mn is the relaxation operator, F^ mn is the d-correlated Langevin source describing the system fluctuations (see, for example Refs. [1,35]). As is easily seen, the standard commutation relations for the þ operators a^ n and a^ m lead to the following commutation relations þ for the products a^ n a^ m : þ þ þ þ ð45Þ ½a^ p a^ q , a^ n a^ m ¼ a^ p a^ m dnq dmp a^ n a^ q , and this result does not depend on either bosons or fermions are considered. The presence of the Langevin source should provide the fulfillment of commutation relation (Eq. (45)) even if the relaxation processes are taken into account (see, for example Ref. [1]). Although the evolution equation for the operator (Eq. (44)) formally differs from equation for the density matrix (Eq. (1)) by the sign before the commutator term (see Refs. [29,19]), this equation can be reduced to the von Neumann equation form if we ^ in the secondary quantization represenwrite the Hamiltonian H ^ ¼ PH ^ pq a^ þ a^ q into tation. Indeed, substituting the expression10 H p pq
equation (Eq. (44)) and taking into account commutation
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
m 2 Im e2
155
m 2 Re e2
Fig. 1. The anti-Hermitian (a) and Hermitian (b) parts of susceptibilities as dependence of the field frequency. The solid lines correspond to w þ defined by Eq. (42), the dashed lines correspond to wðstÞ þ defined by Eq. (43). The parameters: oH =O ¼ 0:5, g=O ¼ 0:1.
relations (Eq. (45)), we get an equation of the same form as usual equation for the density matrix: i ih r_^ mn ¼ H^ mp r^ pn r^ mp H^ pn þ R^ mn r^ 11 , r^ 21 ,. . .r^ mn . . . þ F^ mn , ð46Þ _ ^ mn corresponds ^ mn ¼ a^ nþ a^ m . We emphasize that symbol r where r to matrix where each element is an operator (unlike standard density matrix). In such an approach the averaged value of physical quantity, which is given in the secondary quantization P þ representation g^ ¼ g pq a^ p a^ q , is calculated by formula g¼
pq
oC 9 ð0Þ
P
^ nm 9Cð0Þ 4 , where 9Cð0Þ 4 is the initial quang mn r
m,n
tum state of the system, symbol /. . .S denotes averaging over Langevin source statistics. The presence of Langevin source does not affect the equation for the averaged values of dynamic variables (see, for example, Ref. [1]). So it follows from identity of forms of equations (Eqs. (1) (46)) that the proof procedure of relaxation operator properties in von Neumann equation can be carried over to the properties of ^ 21 ,. . .r ^ mn . . . . In particular, for the basis ^ 11 , r the operator R^ mn r with real elements 9m 4 the standard expression ^ mn (see Refs. [1,35]) should be ‘‘antisymmetrized’’: R^ m a n ¼ gmn r ^ nm . ^ mn r R^ m a n ¼ gmn r It is worth to note that such a modification of relaxation term in Eq. (46) does not affect commutation properties of Langevin source, which provide conservation of commutation properties (Eq. (45)) with an allowance for relaxation processes: j k ^ mm dðtt 0 Þ: ^ nn r F^ mn tÞF^ nm ðt 0 Þ ¼ 2gmn r
8. Conclusion In this paper we propose a new and simple method for improving accuracy of description of relaxation processes in quantum systems. We have investigated the problem of violation of two fundamental properties of relaxation operators in master equation that follow from initial relation for the unreduced quantum system, but can be lost due to various approximations. The first condition provides the correct relation between macroscopic current and polarization of the quantum medium; the other one is defined by thermodynamic arguments. We have shown that widely used 10 ^ pq , if It makes sense to preserve the operator sign over matrix elements H ^ takes into account interaction of the atom with any nonthe Hamiltonian H classical system.
conventional constant-rate relaxation model does not meet these requirements and the use of this model under some conditions can lead even to ‘‘unphysical’’ instability. The proposed correction lies in the antisymmetrization of standard relaxation term for the off-diagonal density matrix elements. This phenomenological method is described for any quantum systems with discreet number of basis function, as well as for a quantum system described by density matrix in coordinate representation. The dissipative three-dimensional oscillator in arbitrary oriented magnetic field is considered in detail. The proposed modification of relaxation operator leads eventually to more simple and noncontradictory equations for dynamics of averaged values of such physical quantities as current, dipole moment and energy of the system. This method can be applied for both Von Neumann and Heisenberg–Langevin equations. Properly introduced relaxation operator does not significantly modify relaxation time scales but can modify the detailed properties of quantum states. The proposed method can be used for the accurate description of quantum system evolution, including estimation of asymptotical entanglement dynamics (see, for example Refs. [25,36,37]), investigation of the systems with strong qubits–environment interaction [33,15,38], development of optical frequency standards [39,40] and many other actual problems of quantum optics and atomic physics.
Acknowledgments This work was supported by Russian Foundation for Basic Research Grant no. 11-02-97079, Federal Target Program ‘‘Research and Development in Priority Directions of Development of Russia Scientific-Technological Complex for 2007-2013’’ (Grant contract no. 07.514.11.4162). The authors are grateful to I.D. Tokman, Vl.V. Kocharovsky and V.Ye. Semenov for the helpful discussions.
References [1] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, England, 1997. [2] R.H. Pantell, H.E. Puthoff, Fundamentals of Quantum Electronics, Wiley, New York, 1969. [3] V.M. Fain, Y.I. Khanin, Quantum Electronics. Basic Theory, 1, MIT, Cambridge, 1969. [4] K. Blum, Density Matrix Theory and Applications, Plenum, New York, 1981. [5] H. Haken, in: L. Genzel (Ed.), Laser Theory in Encyclopedia of Physics, Vol. XXV/2c, Springer, Heidelberg, 1970.
156
M. Tokman, M. Erukhimova / Journal of Luminescence 137 (2013) 148–156
[6] G.S. Agarwal, Quantum Statistics Theories of Spontaneous Emission and Their Relation to Other Approaches, in: G. Hohler (Ed.), Springer Tracts in Modern Physics, Vol. 70, Springer-Verlag, Heidelberg, 1974. [7] R.P. Feynman, F.L. Vernon Jr., Ann. Phys. 24 (1963) 118. [8] S. Khademi, S. Nasiri, Int. J. Mod. Phys. B 18 (1993) 7. [9] H. Dekker, Phys. Rep. 80 (1981) 1. [10] Y. Subouti, S. Nasiri, e-print arXiv:quant-ph/051124. [11] M.S. El Naschie, Chaos Solitons Fractals 32 (2007) 271. [12] M. Blasone, E. Gelegheni, P. Jizba, G. Vitello, Phys. Lett. A 310 (2003) 393. [13] M.D. Tokman, Phys. Rev. A 79 (2009) 053415. [14] A. Kurcz, A. Capolupo, A. Beige, E. Del Giudice, G. Vitiello, Phys. Rev. A 81 (2010) 063821. [15] F. Beaudoin, J.M. Gambetta, A. Blais, Phys. Rev. A 84 (2011) 043832. [16] A. Stokes, A. Kurcz, T.P. Spiller, A. Beige, Phys. Rev. A 85 (2012) 053805. ¨ onen, ¨ [17] J. Salmilahto, P. Solinas, M. Mott Phys. Rev. A 85 (2012) 032110. [18] L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Course of Theoretical Physics. Electrodynamics of Continous Madia, Vol. 8, Butterworth-Heinemann, Oxford, 1984. [19] L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics. Statistical Physics, Vol. 5, Pergamon, London, 1980. [20] R. Ravinder, Puri, Mathematical Methods of Quantum Optics, Springer Series in Optical Sciences (2001). Vol. 79; Springer-Verlag Berlin and Heidelberg GmbH and Co. K; W.T. Rhodes. [21] A.G. Litvak, V.Yu Trakhtengerts, Sov. Phys. JETP 33 (1971) 921. [22] M.D. Tokman, Sov. J. Plasma Phys. 10 (1984) 331–335. [23] H.-P. Breuer, F. Petruccione, The theory of Open Quantum Systems, Oxford University Press, New York, 2002. [24] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom–Photon Interactions, Wiley, New York, 1992.
[25] Chris Fleming, N.I. Cummings, Charis Anastopoulos, B.L. Hu, J. Phys. A Math. Theor. 43 (2010) 405304. [26] W.J. Munro, C.W. Gardiner, Phys. Rev. A 53 (1996) 2633. [27] G. Lindblad, Commun. Math. Phys. 40 (1975) 147. [28] V. Gorini, A. Kossakowski, E.C.G. Sudarsham, J. Math. Phys. 17 (1976) 821. [29] L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Nonrelativistic Theory, Oxford, Vol. 3, Pergamon, 1977. [30] R.K. Wangness, Phys. Rev. 98 (1955) 927. [31] A. Abragam, The Principles of Nuclear Magnetism, Clarendon, Oxford, 1961. [32] J.J. Sakurai, Modern. Quantum Mechanics., Revised Edition (1994), AddisonWesley, San Fu Tuan. [33] P. Forn-Diaz, J. Lisenfeld, D. Marcos, J.J. Garcia-Ripoll, E. Solano, C.J.P.M. Harmans, J.E. Mooij, Phys. Rev. Lett. 105 (2010) 237001. [34] T. Chakraborty, Comment. Condens. Matter Phys. 16 (1992) 35. [35] M.D. Lukin, Rev. Mod. Phys. 75 (2003) 457–472. [36] R.C. Drumond, M.O. Terra Cunha, J. Phys. A Math. Theor. 42 (2009) 285308. [37] J. Novotny´, G. Alber, I. Jex, Phys. Rev. Lett. 107 (2011) 090501. [38] J. Casanova, G. Romero, I. Lizuain, J.J. Garcia-Ripoll, E. Solano, Phys. Rev. Lett. 105 (2010) 263603. [39] C.W. Chou, D.B. Hume, J.C.J. Koelemeij, D.J. Wineland, T. Rosenband, Phys. Rev. Lett. 104 (2010) 070802. [40] A.D. Ludlow, et al., Science 319 (2008) 1805.