ОПТИКА И СПЕКТРОСКОПИЯ, 2012, том 112, № 3, с. 399-404
= СЕДЬМОЙ СЕМИНАР Д.Н. КЛЫШКО
RELAXATION EQUATION FOR MUON SPIN TOMOGRAM IN PROBABILITY REPRESENTATION
© 2012 г. Yu. M. Belousov*, S. N. Filippov*, V. I. Man'ko**, and I. V. Traskunov*
*Moscow Institute of Physics and Technology (State University), 141700Dolgoprudnyi, Moscow Region, Russia **P.N. Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia
E-mail: firstname.lastname@example.org Received July 22, 2011
Abstract—The relaxation equations for spin density matrix has been widely used for the description of MuSR, NMR and in other areas. In this article the equation for spin tomogram of muon with taking into account relaxation phenomenon is written. The properties of tomographic probability distribution for the evolution tomogram are discussed.
Tomographic representation of quantum mechanics is currently being actively developed and great progress has been achieved in understanding the meaning and the application of this representation . However, all the results are obtained mostly for systems in equilibrium, whereas the density matrix formalism has been proved very useful in the description of the kinetics of nonequilibrium states and relaxation processes in open systems. In particular, the density matrix formalism has been successfully applied to the problems of magnetic relaxation, spectroscopy, quantum electronics, atomic and nuclear physics. The density matrix method has proved to be one of the most adequate to describe quantum nonequilibrium states and their evolution.
However, a universal equation, which could be applied to description of all relaxation processes in dynamical systems, hasn't been proposed yet, and it may be deemed improssible, since there is no universal interaction leading arbitrary dynamical system to statistical equilibrium. Instead of this, various models and approximations have been introduced, with their applicability essentially depending on the properties of the dynamical system and the environment, with which this system interacts. To describe the relaxation processes in dynamical systems the following approximations are most widely used: the short correlation time approximation, the random trajectory method, cumulants method and the others . In case of con-denced matter evolution the short correlation time approximation has been applied especially often. This approach allows to obtain the Generalized Master Equation for the subsystem density matrix [2—5]. In some particular cases this equation can be reinterpreted as the equation in terms of respective population of energetic levels. The equation for two level system (particularly for considered as qubits of spin 1/2 or qu-bit) which governs the evolution of levels populations
and polarization or magnetization can be derived from the Generalized Master Equation. The equations for magnetization for two level systems are exactly the same as the phenomenological Bloch equations which describe the magnetization of medium relaxation with the introduction of two parameters: longitudinal relaxation and transverse relaxation.
For spin systems, the Generalized Master Equation for density matrix can be simplified down to the more simple Wangsness—Bloch equation[6, 7], in which the relaxation time parameters are expressed in terms of parameters having the physical meaning of "spin flip" frequency v (see, for example ). The Wangsness—Bloch equation is widely used to describe the behavior of muon spin polarization in various media in the muon spin resonance experimental technique (mSR-method).
The Bloch equations as well as the equations for the levels population are widely used not only for qualitative interpretation of experimental data, but also to obtain quantitative results. In this regard it is reasonable to obtain and analyze the Wangsness—Bloch equation in the tomographic representation.
In Sec. 2 we will outline the Wangsness—Bloch equation and the area of its application. In Sec. 3 we will describe spin tomogram, going into details on the qubit case especially. In Sec. 4 the relaxation equation for qubit spin tomogram is derived, as well as the solution. The properties of the solution as a positive map are presented in the Sec. 5.
2. THE WANGSNESS-BLOCH EQUATION
We will not discuss here the general form of the Generalized Master Equation, due to its quite large and complex structure and because it is not the aim of this article. This equation and the related questions can be found in relevant sources [2, 3, 5]. However, in some cases, particularly, when dynamic subsystem has
nondegenerate spectrum, the equation in terms of density matrix elements in the basis of the Hamilto-nian eigenvectors can be simplified down to the form
Pmn + ^(m\[Ho + r, p]\n) =
^mn " ^ ^^mkPkk YmnPmn, k ^ m
where Ho is the unperturbated Hamiltonian of an isolated dynamical subsystem from the statistical ensemble, r — the Hamiltanian shift, the operator that can be interpreted as the effective Hamiltonian of the dynamical system in the environment. The physical meaning of the remaining parameters becomes clear when one looks separately at the equations only for the diagonal elements of the density matrix. Then the terms to the right are deprived of all the matrix elements but diagonal ones too:
" \ ^^mkPkk YmmPmm'
k ^ m
This is the Master Equation for the level populations. The parameters Wmk are the rates of the transitions from the level k to the level m and the parameters ymm are the intensities of the transitions from the level m to any other. To comply with the preservation of the sum of all the levels populations it's necessary that
k ^ m
Further, the requirement that density matrix be her-mitian, leads to Ymn = Y* .
The new set of variables can be introduced which consists of Cartesian projections of the spin polarization (or magnetization) vector. For spin 1/2 this set gives a complete description of density matrix:
PZ = Pl1 - P22 > Px = P12 + P2I ' Py = '(P12 - P21 )'
The equation for these variables can be put down as
PZ = -(P, - P.6q))/T1, Px = fflPy - PJ T1,
Py = - ffl Px - Py/T2 '
where the new denotations are introduced with T- =
= W12 + W21 and T21 = y12. T1 and T2 are usually called longitudinal and transversal relaxation times respectively. The frequency ® defines the precession of mag-
netization vector and depends on the eigenvalues of the effective Hamiltonian H = Ho + 1" :
® = ( E2 - E1 ) / h '
The equations (5) are the same as the fenomenological Bloch equations for magnetization when no oscillating magnetic field applied.
The remaining definitions on the equation coefficients can be made by addition af some physical matters. If the relaxation appears through the interaction with a surrounding system in thermal equilibrium with temperature T than the parameters Wmk relate one to another like
Wmk/ Wkm = exp [-( Em - Ek )/T] .
In case of significantly high temperature the parame ters become equal to each other Wmk ~ Wkm and botl longitudinal and transversal relaxation time are de fined only by one parameter v:
W21 = W12 = 1 /27\ = 1/2 T2 = Y12 = v, (8)
which can be interpreted as the frequency of the spin flip. Under these conditions the relaxation equations can be easily put down in a quantum representation-independent operator form
P + h[H, H] = v(°P° - 3P)/2'
The operator-vector o is composed of the Pauli matrices, the energy shift is incorporated in the Hamiltonian term. In relation to mSR the equation (8) is called Wangsness—Bloch equation.
3. SPIN TOMOGRAM
The tomographic approach provides description of quantum states in terms of standard probability distributions as alternative to wave function or density matrix. Such tomographic probability distribution were introduced both for continous variables  and discrete variables (spin, qubits) [8—10]. The tomographic description of quantum states is called probability representation . Some evolution equations for conti-nous variables were written in probability representation [1, 8, 11]. The kinetic equations for discrete variables are still needed to be rewritten in the form of equations for the tomographic probabilities distributions (called tomograms). One of the aims of this article is to consider the simplest kinetic equation for particle with spin 1/2 in the form of the equation for the spin tomogram. The evolution of density matrix is given by map of initial density matrix onto density matrix at the time t. This map is so called positive map . In the set of positive maps there exists so called completely positive maps (called also channels). Also the aim of this work is to consider the solution of the ki-
netic equation for qubit with isotropic relaxation as depolarizing channel .
The spin tomogram is introduced as an alternative to the description of spin quantum state by density operator p . For spin with spin quantum numberj the tomogram w( j)(m, U) is defined as follows:
wJ)(m, tf) = <jm\lf ptf\jm),
The operator U can be parameterized by three parameters which define the rotation, for example by Euler angles. But spin tomogram can instead be treated as the function of only two of them since any unitary operator that commutes with the spin projection on the direction n doesn't change the tomogram. The Euler angle which is associated with that rotation can be excluded from the tomogram variables set. Due to this observation, from now on we use the notation for spin tomogram w(j)(m, n), where the unit vector is completely identified by two angles:
n = (cos9sin0, sin9sin0, cos0).
To be an operator U for that tomogram, any can be taken, for which it's true that (Jn) U\ jm) = m jm) .
Tomogram of spin 1/2 state can be presented in the form of probability vector with two components:
w ( tf) =
f w(+1/2, CO N w (-1/2, CO
The vector read
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