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Quantum Spin Dynamics in Time-Varying Magnetic Fields
Modelling Electron Spin Resonance
Harriet Walsh
Trinity College Dublin
at Miyashita Group
University of Tokyo
UTRIP 2015
Abstract
The mechanism of Electron Spin Resonance, widely used in
experiment o investigate free radicalsin materials, is examined
theoretically. The resonance phenomenon is discussed intuitively,
and thenin terms of the phenomenological Bloch equations and the
theory of linear response. Using a simplemodel of a single electron
weakly coupled to a boson bath at thermal equillibrium, the spin
dynamicsare found from first principles by a quantum master
equation. Hence the relaxation time parametersof the Bloch equation
are thus expressed in terms of microscopic parameters of the
system. Themaster equation dynamics were analysed by means of a
numerical simulation.
1 Introduction
The Electron Spin Resonance (ESR) phenomenon is widely applied
in the investigation of materialscontaining free radicals. Under a
static magnetic field H applied in the z-direction, Zeeman
splittingwith Hamiltonian
HZ = −gµBH · S = −gµBHSz (1)causes the two-fold degeneracy of
electron energy states to be lifted so that the state with
spinparallel to the static field has lower energy (a paramagnetic
interaction), as illustrated in figure 1.The system is then
‘probed’ by introducing a small sinusoidal magnetic field
transverse to the staticone, typically by electromagnetic waves in
the radio frequencies. We may express the field as
H(t) = (h cosωt,−h sinωt,H). (2)
The system then exhibits resonance behaviour when the frequency
of the probing field is tunedto the energy gap, that is,
~ω = gµBH. (3)At resonant frequencies, peaks of Lorentzian
curves are observed in the absorption spectrum (figure2). Thus ESR
can be employed to find values of g, Landé’s degeneracy factor,
present in the material.This encodes information about the angular
momentum available to free radicals.
1.1 Rabi oscillation
We consider the basic mechanism for ESR by looking at the spin
dynamics of a single isolated electronunder the Zeeman interaction
(1) and the magnetic field (2), where h is assumed to be much
smallerthan H, and ω is assumed to be small. The Schrödinger
equation can be expressed in terms of theraising and lowering
operators, as
i~gµB
∂
∂t|Ψ〉 = [−h (cosωtSx − sinωtSy)−HSz] |Ψ〉
=
[−h
2
(eiωtS+ + e−iωtS−
)−HSz
]|Ψ〉
1
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Figure 1: Zeeman split energy levels Figure 2: Peak in energy
absorption at resonance
The gyromagnetic ratio gµB~ will be denoted by γ. The static
field will cause |Ψ〉 to process aboutthe z-axis, so we transfrom
the equation to a rotating frame
|Ψ〉 = eiωtSz
|φ〉, (4)
yielding
i∂
∂t|Ψ〉 =
(−ωSzeiωtS
z
+ eiωtSz
i∂
∂t
)|Φ〉
= γ
[−h
2
(eiωtS+ + e−iωt
)−HSz
]eiωtS
z
|φ〉
⇒ i ∂∂t|φ〉 = e−iωtS
z[(ω − γH)Sz − γh
2
(eiωtS+ + e−iωtS−
)]eiωtS
z
|φ〉.
Since Sz commutes with itself, and demanding ω is very small,
such that eiωt ≈ 1, we may write
i∂
∂t|φ〉 =
[(ω − γH)Sz − γh
2
(S+ + S−
)]|φ〉 = [(ω − γH)Sz − γhSx] |φ〉
and at the resonant frequency ω = γH, that is, the frequency of
a photon with the energy of the gapdue to Zeeman splitting, the
solution in the rotating frame is
|φ(t)〉 = e−iγhSxt|φ(0)〉
andΨ(t) = eiγHS
zte−iγhSxt|Ψ(0)〉. (5)
To generalise to a statistical system, a mixed state is
described using a density matrix ρ(t) (definedfollowing Toda et.
al.[1] in appendix A). For a system under a time-varying magnetic
field H(t), thequantum Liouville equation yields
i∂
∂tρ = [−γH · S, ρ]
〈Ṡ〉 = TrṠρ= TrSρ̇
⇒ i ∂∂t〈S〉 = −γTrS ([HxSx, ρ] + [HySy, ρ] + [HzSz, ρ])
i∂
∂t〈Si〉 = −γTr
(Hx(SiSxρ− SiρSx
)+Hy
(SiSyρ− SiρSy
)+Hz
(SiSzρ− SiρSz
)).
Since the trace is additive and TrAB = TrBA, this gives
∂
∂t〈S〉 = γ〈S〉 ×H (6)
by the commutator relation [Sα, Sβ ] = �αβγiSγ . This is the
Euler equation.
The Liouville equation for the Hamiltonian of our system can be
expressed in the basis of Paulispin matrices
Sx =1
2
(0 11 0
)Sy =
1
2
(0 −ii 0
)Sz =
1
2
(1 00 −1
),
2
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Figure 3: System in a dissipative environment
by∂
∂tρ =
1
i~[H, ρ] = iγ
[(H he−iωt
heiωt −H
), ρ
](7)
1.2 Heuristic descriptions of resonance
The Rabi oscillation described by equation (5) is energy
conserving, so there should be no net powerabsorption or emmission
over all time. The absorption spectra obtained by ESR can be
explained bytwo different stories: energy dissipation due to the
environment, and approximation of the responseover a very short
time.
Spin-lattice relaxation, where a two state system interacts with
a bath at equillibrium, can beregarded in terms of the the
populations N− of the lower energy state and N+ of the higher
energystate (figure 3). Following an argument presented by
Slichter[2], we assume there is a process allowingtransition
between each state, with probabilities W↑ of a transition N− → N+
and W↓ of a transitionN+ → N−. Then
∂
∂tN+ = N−W↑ −N+W↓
and since at thermal equillibrium the populations will not
change,
0 = [N−W↑ −N+W↓]eqN−N+
=W↓W↑
= eβγH
by the Boltzmann distribution, where the states are split by a
field H. The unequal upward anddownward transition probabilities
are allowed because of the thermal bath, which can provide
energyto, or absord energy from, the system. In terms of the total
population and the difference betweenstate populations, N = N− +N+,
n = N+ −N−,
∂
∂tn = N (W↑ −W↓)− n (W↑ +W↓) .
By putting
n0 = NW↑ −W↓W↓ +W↑
;1
T= W↑ +W↓
the rate equation is∂
∂tn =
n0 − nT
so the difference in state populations decays to the
equillibrium difference as
n = n0 +Ae−t/T .
If we additionally consider transitions resulting from the
application of an oscillating magneticfield to the system, this
should be an energy conserving process with equal probability W for
upwardsand downwards transition (figure 4). Just considering the
isolated system,
∂
∂tN+ = W (N+ −N−) ⇒
∂
∂tn = −2Wn.
With both effects considered,∂
∂tn = −2Wn+ n0 − n
T
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Figure 4: Balancing of level populations by an energy conserving
process
Figure 5: Energy absorbing time interval
and the steady state solution (such that ∂∂tn = 0) gives a rate
of energy absorption
∂
∂tE = n∆EW = n0∆E
W
1 + 2WT. (8)
The absorption of energy when the field is at the resonant
frequency can be predicted consideringonly the isolated system.
Considering the time evolution of the spin state in Rabi
oscillation at theresonant frequency (3), determined by (5), in the
limit as h, the oscillating field amplitude, tends to0, the period
of rotation about the x axis tends to infinity. Hence the part of
the motion that weobserve in a finite time will only be energy
absorbing (figure 5).
2 Two models of resonance
2.1 Bloch Equation
Where the single electron two-state system is within a larger
material, the interaction of spins causestransverse relaxation, and
tunnelling between Zeeman levels causes longitudinal relaxation.
This ismodelled phenomenologically by
∂
∂t〈S〉 = γ〈S〉 ×H− τ−1 (〈S〉 − χH) (9)
where τ is the relaxation time tensor, which has xx and yy
components equal to the transverserelaxation time T2 and zz
component equal to the longitudinal relaxation time T1, which are
notpredicted by this model. χ is the isothermal susceptibility.
This is the energy conserving equation of motion (6) with
dissipative terms that simply modelthe systems tendency to reach
the equillibrium state 〈S(t)〉 = H(t). If the interaction of local
spins ismodelled as random frequency modulation, in the motional
narrowing limit the relaxation becomessimple exponential decay,
consistent with this equation[3].
Writing in terms of the macroscopic parameter, the net
magnetisation M =≡ 〈s〉, for the field(2) the equations are
∂∂tMx = γ (MyH +Mzh sinwt)− M
x−χh coswtT2
∂∂tMy = γ (−MxH +Mzh coswt)− M
y+χh sinwtT2
∂∂tMz = γ (−Mxh sinwt−Myh coswt)− M
z−χHT1
.
Neglecting transient effects, the system is expected to be
stationary in a frame rotating with the
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field (4), so there is the following special solution (expressed
in terms of ω0 = γH and ω1 = γh):
Mx =1+ω0(ω0−ω)T22 +ω
21T1T2
1+(ω0−ω)2T22 +ω21T1T2
χh cosωt
+ ωT21+(ω0−ω)2T22 +ω
21T1T2
χh sinωt
My = ωT21+(ω0−ω)2T22 +ω
21T1T2
χh cosωt
− 1+ω0(ω0−ω)T22 +ω
21T1T2
1+(ω0−ω)2T22 +ω21T1T2
χh sinωt
Mz =1+(ω0−ω)2T22 +(ω0−ω)ω
−10 ω
21T1T2
1+(ω0−ω)2T22 +ω21T1T2
χH
and for a weak transverse field h, the ω21 terms may be
neglected, leaving no dependence on T1, thelongitudinal relaxation
time.
This model assumes that the rotation of the field is
sufficiently slow that the spin may rotatealong with the field. If
the rotating field is weak and fast, the spin may instead tend to
go intoequillibrium with the static field only. The
phenomenological equations then become
∂∂tMx = γ (MyH +Mzh sinwt)− M
x
T2∂∂tMy = γ (−MxH +Mzh coswt)− M
y
T2∂∂tMz = γ (−Mxh sinwt−Myh coswt)− M
z−χHT1
,
(10)known as the Bloch equations.
In the limit where Mz is in its equillibrium state, say M0, an
analytic solution may be found, aspresented by Slichter[4]. Using
the rotating frame (4), in which the field (2) is
H̃ = (h̃, 0, H̃),
the equations transform as
∂
∂tM̃x =e−iωt
∂
∂tMx − iωM̃x
=γe−iωt[(MyHz −MzHy)−
Mx
T2
]− iωM̃x
=γM̃yH̃ −(
1
T2− iω
)M̃x
∂
∂tM̃y =γ
(M̃zh̃−MxH̃
)−(
1
T2+ iω
)M̃y.
In terms of M̃+ ≡ 〈S+〉 ≡ M̃x + iM̃y the equation of motion
is
∂
∂tM̃+ = −M̃+
(1
T2+ iγH̃ − iω
)+ iγh̃M̃z
and allowing M̃z = M0, and noting that H̃ = H, h̃ = h,
M̃+ = Ae−t/T2e−i(γH−ω)t +iT2γhM0
1 + iT2(γH − ω).
In the laboratory frame, the magnetisation is
Mx = M̃x cosωt+ M̃y sinωt
and so at a time long after the rotating field is first applied
when the first term is sufficiently decayedto be negligeable,
putting ω0 ≡ γH, the x magnetisation is
M̃x = γM0T2(ω0 − ω)T2
1 + (ω − ω0)2T 22h.
The absorption of energy is typically measured in terms of the
magnetic susceptibility χ, ameasure of the degree to which the
field magnetises the system, by the relation
M(t) = χH(t); χ = χ′ + iχ′′, (11)
where the admittive and absportive parts of the susceptibility
are labelled χ′ and χ′′ respectively.Expressing the rotating field
as hr(t) = he
iωt, there is a relation
Mx(t) = χhr(t) ⇒ Mx(t) = (χ′ cosωt+ χ′′ sinωt)h (12)
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so the absorptive part of the susceptibility is
χ′′ =
(γM0T2
2
)1
1 + (ω − ω0)2T 22. (13)
This is the familiar Lorentzian curve centred on the resonant
frequency, and can be compared toequation (8) derived by simple
intuition. However this accuracy is only phenomenological: it
remainsto relate the relaxation time parameters T1, T2 to some
physical proprties of the system.
2.2 Kubo formula
In the small probing field limit, we can approximate the
perturbation to the system at equillibriumcaused by the oscillating
field to first order, and hence employ the theory of linear
response, inparticular the Kubo formula.
Following Kubo[5], we consider a system in equillibrium with
HamiltonianH, to which an externalgeneralised force K(t) is
applied. In order to neglect transient effects, we let the force be
appiedfrom t = −∞. Let A be the dynamic quantity conjugate to K, so
that the total Hamiltonian is
Ht = H−AK(t). (14)
Treating this force as a perturbation, the Liouville equation of
motion
∂
∂tρ =
1
i~[Ht, ρ] = iLtρ
is solved to first order of the external force (see appendix B).
The system starts in equillibrium,determined by the Boltzmann
distribution:
ρ(−∞) = ρeq = Ce−βH.
Putting ρ(t) = ρ(−∞) + ∆ρ(t) + . . ., we have
∆ρ(t) =
∫ t−∞
dt′ei(t−t′)LiLext(t′)ρeq =
∫ t−∞
dt′e(t−t′)[H,ρ]/i~ 1
i~[AK(t′), ρeq].
The system’s response is observed in the change in an observable
B,
∆〈B(t)〉 = TrB∆ρ(t) = 1i~
∫ t−∞
dt′K(t′)Trρeq[A(0), B(t− t′)]
where B(t) = e−iLtB = eitH/~Be−itH/~ is the time evolution of B
in the unperturbed system. Theresponse function φBA(t) is defined
as
φBA(t) =1
i~Trρeq[A,B(t)] =
1
i~〈[A,B(t)]〉eq =
1
i~Tr[ρeq, A]B(t)
so that
∆〈B(t)〉 =∫ t−∞
dt′K(t′)φBA(t− t′).
So the expression for the response ∆〈B(t)〉 is linear in K and is
the superposition of delayed effects.For a periodic force
K = K0eiωt,
the response can be written as∆〈B(t)〉 = χBA(ω)K0eiωt
with the admittance defined by the Laplace transformed response
function,
χBA(ω) =
∫ ∞0
φBA(t)e−iωtdt =
∫ ∞0
1
i~〈[A,B(t)]〉eq e
−iωtdt.
The Fourier transform of the correlations of two quantities X,Y
are related by the Kubo-Martin-Schwinger relation (derived in
section 3.1, at equation (18)):∫ ∞
−∞〈X(0)Y (t)〉e−iωtdt = eβ~ω
∫ ∞−∞〈Y (t)X(0)〉e−iωtdt. (15)
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So, ∫ ∞−∞〈[X(0), Y (t)]〉e−iωtdt = (1− e−β~ω)
∫ ∞−∞〈X(0)Y (t)〉e−iωtdt
By (11), the magnetic susceptibility χ measures the
magnetisation of a system responding to anexternal magnetic field
(2), to which M(0) is conjugate. For the imaginary, absorptive
part,
χ′′(ω) =1
~
∫ ∞0
〈[M(0),M(t)]〉eq cos(ωt)dt
and as both cosine and the commutator
[M(0),M(t)] = [M(−t),M(0)] = [eitHM(0)e−itH,M(0)] =
[M(0),M(−t)]
are even in time, by (15) we can write
χ′′BA(ω) =1
2~(1− e−β~ω)
∫ ∞−∞〈M(0)M(t)〉eq e
−iωtdt. (16)
This formula can be explicitly evaluated using the set of
eigenvectors and corresponding eigen-values, {|n〉, En}Dn=1 of the
Hamiltonian[6],
H|n〉 = En|n〉
. Then, using Mx(t) = e−iHtMxe−iHt,
〈Mx(0)Mx(t)〉eq =∑n
〈n|Mxe−iHtMxe−iHt|n〉/Z =∑m,n
|〈m|Mx|n〉|2ei(Em−En)t−βEn/Z.
Fourier transforming,∫ ∞−∞〈Mx(0)Mx(t)〉eq e
−iωtdt =∑m,n
|〈m|Mx|n〉|2e−βEn∫ ∞−∞
e−i(ω−(Em−En))tdt/Z
=∑m,n
|〈m|Mx|n〉|2e−βEn2πδ(ω − (Em − En))/Z
and the spectrum is given by an ensemble of delta functions:
χ′′(ω) =1− e−βω
2
∑m,n
|〈m|Mx|n〉|2e−βEn2πδ(ω − (Em − En))/Z
For our two state system where the splitting is due to a field
H, and only considering ω > 0, thisis
χ′′(ω) = π1− e−β~ω
1 + e−β~ωδ(ω − γH) = π tanh β~ω
2δ(ω − γH). (17)
3 An approach from first principles
To derive a model for the process of ESR from first principles,
we look at a single electron, Zeemansplit and probed by a magnetic
field (2) that is coupled to a bath of bosons. Assuming the
coupling isweak, the evolution of the system can be determined by
treating the bath interaction as a perturbationto second order,
that is, by a quantum master equation.
3.1 Derivation of a quantum master equation
The total Hamiltonian of the system (S) and the bath (B) has the
form
HT = HS +HB + λHI
where λ is the coupling coefficient, and the equation of motion
is
∂
∂tρ =
1
i~[HT , ρ] ≡ iLρ.
The behaviour of the electron will be described by the reduced
density matrix ρS , which is the pro-jection of the information in
the NS×NB dimensional Hilbert space of total system and
environment
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onto the NS dimensional Hilbert space of the system. Since the
electron has two spin state, NS = 2,while the boson bath has an
infinite dimensional Hilbert space, NB =∞. We define
ρS = Pρ = TrBρ
where TrBρ contracts the density matrix ρ to an operator on the
NS states of the system, by ‘tracingout’ using ρB , the density
matrix operator on the NB states of the bath at thermal
equillibrium.The dissipative equation of motion will have the
form
∂
∂tρS =
1
i~[HS , ρs] + Γ(ρ) = iLS + Γ(ρ).
Putting P ′ = 1− P, the equation of motion is seperated as
∂
∂tPρ = PiLPρ+ PiLP ′ρ
∂
∂tP ′ρ = P ′iLPρ+ P ′iLP ′ρ
and solving for P ′iLPρ by ∂∂tx = Ax+B ⇒ x(t) =
∫ tt0e(t−τ)AB(τ)dτ + e(t−τ)Ax(0),
Γ(ρ) = PiL∫ Tt0
e(t−τ)P′(iL)P ′iLPρ(τ)dτ + PiLe(t−t0)P
′(iL)P ′ρ(0).
Up to second order in the coupling parameter λ, where 1i~ [λHI ,
ρ] ≡ iλLIρ and
1i~ [HS +HB , ρ] ≡
iL0ρ
∂
∂tρ(2)S =iLSρ
(2)S
+ λ2TrBiLI∫ tt0
e(t−τ)iL0 iLIρBρ(2)S (τ)dτ
+ λ2TrBiLIe(t−t0)iL0∫ 10
dxP ′e−x(t−t0)iL0(t− t0)iLIP ′ex(t−t0)iL0P ′ρ(t0)
and the third term may be disregarded, as it should be
negligeable after integration.We can write explicitly
Γ(ρ(2)S ) = TrB
(λ
i~
)2 [HI ,
∫ tt0
e−i(t−τ)(HS+HB)[HI , ρBρ(2)S (τ)
]ei(t−τ)(Hs+HB)
]dτ
The interaction Hamiltonian shall be adopted as
HI =∑i
XiYi
where {Xi} are the operators of the system, corresponding to
operators {Yi} on the bath. Thecommutator can be untangled and,
since bath operators commute with system operators, written
Γ =− λ2
~2TrB
∑i,j
∫ tt0
(Yie−i(t−τ)HBYjρBe
i(t−τ)HBXie−i(t−τ)HSXjρ
(2)S (τ)e
i(t−τ)HS
− Yie−i(t−τ)HBρBYjei(t−τ)HBXie−i(t−τ)HSρ(2)S (τ)Xjei(t−τ)HS
− e−i(t−τ)HBYjρBei(t−τ)HBYie−i(t−τ)HSXjρ(2)S (τ)ei(t−τ)HSXi
+ e−i(t−τ)HBρBYjei(t−τ)HBYie
−i(t−τ)HSρ(2)S (τ)Xje
i(t−τ)HSXi)dτ
It is convenient to express the bath operators in terms of their
correlations, using the interactionrepresentation,
〈Yi(t− τ)Yj〉 = TrBe−i(t−τ)HBYie−i(t−τ)HBYjρB .
By e−i(t−τ)HSρ(2)S (τ)e
i(t−τ)HS = ρ(2)S (t) we can write Γ in terms of u = t− τ as
Γ = −λ2
~2∑i,j
∫ 0t−t0
(〈Yi(u)Yj〉XiXj(−u)ρ(2)S (t)− 〈Yj(−u)Yi〉Xiρ
(2)S (t)Xj(−u)
− 〈Yi(u)Yj〉Xj(−u)ρ(2)S (t)Xi + 〈Yj(−u)Yi〉ρ(2)S (t)Xj(−u)Xi
)du.
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Figure 6: Exchange between system and bath
The correlations of the bath can be expressed in terms of its
frequency spectrum,
Φij(u) ≡ 〈Yi(u)Yj〉 =∫ ∞−∞
dωeiωuΦij [ω],
and noting
〈Yj(u)Yi〉 = TrBeiuHBYje−iuHBYiρB= TrBe
−i(u+iβ~)HBYiei(u+iβ~)HBYie
i(u+iβ~)HBYjρB = Φij(−u− iβ~)
⇒ Φij [ω] = e−β~ωΦji[−ω]. (18)This is the Kubo-Martin-Schwinger
condition, and is a fundamental characteristic of the
quantummechanical regime, as e−β~ω tends to unity at the classical
limit.
We now employ a Markov approximation: assuming that the
system-bath interactions occur on atime scale much faster than the
evolution of the system, we suppose they have been in contact
sincethe infinite past, so that the range for the integration over
u become [0,∞) (using the stationarityproperty). This type of
time-coarsing is common to the derivation of all Langevin-type
equations[3].The dissipative term becomes
Γ = −λ2
~2
∫ ∞0
du
∫ ∞−∞
dωeiωu∑ij
[XiXj(−u)ρ(2)S (t)Φij [ω]−Xiρ
(2)S (t)Xj(−u)Φji[−ω]
−Xj(−u)ρ(2)S (t)XiΦij [ω] + ρ(2)S Xj(−u)XiΦji[−ω]
].
Where the bath is made up of bosons, the spectrum is that of
independent harmonic oscillators,
HB =∑α
~ωαb†αbα, (19)
where b†α, bα are the creation and annihilation operators on the
bath, respectively. As we consider thedynamics for a small probing
field, only the system operators due to the static field are
consideredin the interaction with the bath. In particular this
allows a Markov approximation to be made,as it is still valid if
the field oscillates on a timescale similar to the system-bath
interaction. TheHamiltonian of the system with only the static
field is
HS = −~γH(Sz +
1
2
)= −~γHS+S−
and the system’s ladder operators S+, S− correspond to those of
the bath, bα and b†α. So theinteraction Hamiltonian may be
written
HI =∑α
(καS+bα + κ
†αS−b†α) = S
+B + S−B†, (20)
encoding boson creation and annihilation in the bath when the
system goes between its two states(figure 6).
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The average population of an excited state at energy ~ωα, 〈n〉 =
TrB e−βHBZB
b†αbα is found by
recalling the eigenvalue relations for the ladder operators[8],
so that the probability of boson absorp-tion, say |n〉 → |n+ 1〉, is
n|bα|2 and the probability of emission |n+ 1〉 → |n〉 is (n+ 1)|bα|2.
As thebath is in equillibrium, the absorption and emission rates
for an level must be equal for any level, soletting Nn+1 be the
number of bosons in state |n+ 1〉, Nn be the number of bosons in
state |n〉,
Nn〈n〉|bα|2 = Nn+1(〈n〉+ 1)|bα|2
⇒ 〈n〉〈n〉+ 1 =Nn+1Nn
= e−~ωα
TrBe−βHB
ZBb†αbα =
1
eβ~ωα − 1
and by similar argument for a grand state, 〈n〉 = TrB e−βHBZB
bαb†α , finds
〈n〉 − 1〈n〉 =
NnNn−1
= e−~ωα
TrBe−βHB
ZBbαb†α =
1
1− e−β~ωα .
The equation of motion may be written explicitly in the terms of
the reduced density matrix,
ρS =
(ρ11 ρ12ρ21 ρ22
).
Where h and ω are small, and Rabi oscillation occurs, from (7)
the term for the isolated system is
∂
∂tρS = iγ
[−(
H he−iωt
heiωt −H
),
(ρ11 ρ12ρ21 ρ22
)]= −iγ
(−ρ12heiωt + ρ21he−iωt 2ρ12H − he−iωt(ρ11 − ρ22)−2ρ21H +
heiωt(ρ11 − ρ22) ρ12heiωt − ρ21he−iωt
)To find explicitly the dissipative term, we note that for each
bath operator correlation
∑i,j〈Yi(u)Yj〉,
only 〈B†B〉 and 〈BB†〉 contribute, and they have corresponding
system operators S+S− and S−S+.So writing Γ as
Γ =− λ2
~2(Γ1 + Γ2 + Γ3 + Γ4)
=− λ2
~2∑i,j
∫ ∞0
(〈Yi(u)Yj〉XiXj(−u)ρ(2)S (t)− 〈Yj(−u)Yi〉Xiρ
(2)S (t)Xj(−u)
− 〈Yi(u)Yj〉Xj(−u)ρ(2)S (t)Xi + 〈Yj(−u)Yi〉ρ(2)S (t)Xj(−u)Xi
)du
the four matrices of the sum are found taking the continuous
limit of the spectrum of the thermalbath, and are expressed in
terms of the Pauli spin matrices.
Γ1 =
∫ ∞0
[S+S−(−u)ρ(2)S 〈B(u)B
†〉+ S−S+(−u)ρ(2)S 〈B†B(u)〉
]du
=
∫ ∞0
du
[(1 00 0
)eiuγH
(ρ11 ρ12ρ21 ρ22
)∑α
|κα|2e−iuωα
1− e−β~ωα
+
(0 00 1
)e−iuγH
(ρ11 ρ12ρ21 ρ22
)∑α
|κα|2eiuωα
eβ~ωα − 1
]=
(ρ11 ρ120 0
)∫ ∞0
dωD(ω)|κ(ω)|2 11− e−β~ω
∫ ∞0
eiu(γH−ω)du
+
(0 0ρ21 ρ22
)∫ ∞0
dωD(ω)|κ(ω)|2 1eβ~ω − 1
∫ ∞0
e−iu(γH−ω)du
and the real part of Γ1 is
Re[Γ1] = πD(γH)|κ(γH)|2[
1
1− e−β~γH
(ρ11 ρ120 0
)+
1
eβ~γH − 1
(0 0ρ21 ρ22
)].
10
-
(The contribution from the Cauchy principle part has been shown
to represent the Lamb shift dueto dynamic renormalisation[7], and
shall be neglected here).
Next,
Γ2 =
(ρ22 00 0
)∫ ∞0
dωD(ω)|κ(ω)|2 1eβ~ω − 1
∫ ∞0
eiu(γH−ω)du
+
(0 00 ρ11
)∫ ∞0
dωD(ω)|κ(ω)|2 11− e−β~ω
∫ ∞0
e−iu(γH−ω)du
The real part is then
Re[Γ2] = πD(γH)|κ(γH)|2[
1
1− e−β~γH
(0 00 ρ11
)+
1
eβ~γH − 1
(ρ22 00 0
)].
With
Γ3 =
(ρ22 00 0
)∫ ∞0
dωD(ω)|κ(ω)|2 1eβ~ω − 1
∫ ∞0
eiu(γH−ω)du
+
(0 00 ρ11
)∫ ∞0
dωD(ω)|κ(ω)|2 11− e−β~ω
∫ ∞0
e−iu(γH−ω)du
Γ4 =
(ρ11 0ρ21 0
)∫ ∞0
dωD(ω)|κ(ω)|2 11− e−β~ω
∫ ∞0
e−iu(γH−ω)du
+
(0 ρ120 ρ22
)∫ ∞0
dωD(ω)|κ(ω)|2 1eβω − 1
∫ ∞0
eiu(γH−ω)du
The dissipative term then has a real part
Re[Γ] = −2λ2πD(γH)|κ(γH)|2
~2(eβ~γH − 1)
(ρ11e
β~γH − ρ22 ρ12 eβ~γH+1
2
ρ21eβ~γH+1
2ρ22 − ρ11eβ~γH
).
3.2 Analysis and simulation
In analogy with the Bloch equations (10), the time evolution of
each component of the electron’sspin in the Bloch sphere is found,
from the equation of motion
∂
∂tρ(2)S =− iγ
(−ρ12heiωt + ρ21he−iωt 2ρ12H − he−iωt(ρ11 − ρ22)−2ρ21H +
heiωt(ρ11 − ρ22) ρ12heiωt − ρ21he−iωt
)− 2λ
2πD(γH)|κ(γH)|2
~2(eβ~γH − 1)
(ρ11e
β~γH − ρ22 ρ12 eβ~γH+1
2
ρ21eβ~γH+1
2ρ22 − ρ11eβ~γH
)by the relation
〈Ȧ〉 = TrρȦ = Trρ̇A.Denoting the prefactor of the dissipative
term by
C(γH) ≡ λ2πD(γH)|κ(γH)|2
~2(eβ~γH − 1) ,
then (using normalisation ρ11 + ρ22 = 1)
〈Ṡx〉 =12
(ρ̇12 + ρ̇21)
=− iγH(ρ12 − ρ21) + γh(ρ11 − ρ22) sinωt− C(γH)eβ~γH + 1
2(ρ12 + ρ21)
=γ〈Sy〉H + γ〈Sz〉h sinωt− C(γH)eβ~γH + 1
2〈Sx〉
〈Ṡy〉 =12
(−iρ̇12 + iρ̇21)
=− γH(ρ12 + ρ21) + γh(ρ11 − ρ22) cosωt− C(γH)eβ~γH + 1
2(−iρ12 + iρ21)
=− γ〈Sx〉H + γ〈Sz〉h cosωt− C(γH)eβ~γH + 1
2〈Sy〉
11
-
〈Ṡz〉 =12
( ˙ρ11 − ˙ρ22)
=− iγh(−ρ12eiωt + ρ21e−iωt
)− C(γH)
(ρ11e
β~γH − ρ22)
=− γ〈Sy〉h cosωt− γ〈Sx〉h sinωt− C(γH)(〈Sz〉(eβ~γH + 1)− (eβ~γH −
1)
).
Thus, the arbitrary relaxation times introduced in the Bloch
equations may be related to thecoupling between the system and its
surroundings and the density of states at resonance. Thetransverse
relaxation time T2 is found as
1
T2=λ2π
~2D(γH)|κ(γH)|2 coth β~γH
2. (21)
The longitudinal relaxation brings the system into the system
equillibrium state determined by theFermi distribution,
〈Sz〉eq = Tr(
1 00 −1
)(e−β~γH
1+e−β~γH0
0 11+e−β~γH
)= tanh
β~γH2
, (22)
and1
T1=
2λ2π
~2D(γH)|κ(γH)|2 coth β~γH
2=
2
T2. (23)
This relates the rates absorption due to spin-spin interaction
(transverse) and due to tunnelingbetween Zeeman levels
(longitudal).
The special solution of the Bloch equation (13) in the t → ∞
limit as 〈Sz〉 reaches equillibriummay be applied. Using the values
in (21) and (22), this is
χ′′(ω) =
(γ~2 tanh β~γH
2T2
2λ2πD(γH)|κ(γH)|2 coth β~γH2
)1 + (ω − γH)2
(~2
λ2πD(γH)|κ(γH)|2 coth β~γH2
)2 .In the λ→ 0 limit as the system becomes uncoupled from the
bath, the behaviour predicted by theKubo formula is recovered, and
the absorptive susceptibility is given by (17).
The equations of spin motion were numerically integrated, from
an initial state spin-up along thez axis. By (11) and (12), the
absorption of energy by the system and environment is observed as
thephase lag between the oscillating field and transverse
magnetisation. Hence, the absorption curvewas found by numerical
computation of the first Fourier sine coefficient:
χ′′(ω) =ωπ
h
∫ t0+2π/ωt0
〈Sx(t)〉 sin(ωt)dt. (24)
The numerical integration was carried out at large t0 to avoid
transient effects (figure 8).The Kubo formula behaviour (17) was
also found numerically, where the coupling parameter λ
was set to zero, and where the probing field magnitude h was
small. The time evolution of 〈Sz〉 inthe small h regime shows the
accuracy of a linear response approximation, and it can be
understoodthat the evaluation of (24) was carried out in a finite
time interval in which power was absorbed.Beyond this regime, there
was no absorption observed. This is again explained by the plot of
〈Sz(t)〉,although it should be noted that such a plot is somewhat
spurious the equations of motion assumeRabi oscillation occurs and
so are only valid for small h (figure 9).
4 Conclusions
The quantum master equation method outlined here cannot
universally accurately determine thedynamics of systems in
dissipative environments. The perturbative approach is only valid
for weakcoupling, and the Markov approximation makes large
assumptions about the nature of the spin-bathinteraction.
Nonetheless, the electron spin resonance phenomenon can be quite
successfully modelled by amaster equation, and in this way the
dynamics predicted by Langevin type phenomenological equa-tions and
by linear response theory can be related to the microscopic
behaviour of the system.
12
-
Figure 8: Time evolution of 〈Sx(t)〉, simulated for oscillating
fields near and at the resonant frequency, with the absorptioncurve
found by the (24)
Figure 9: Plot of 〈Sz(t)〉 for small h, left, where Kubo formula
behaviour is observed in the isolated system, and for largeh,
right, where energy absorption is not observed
13
-
Acknowledgments
I wish to very sincerely thank Professor Seiji Miyashita for his
extensive help and guidance, as wellas all the members of Miyashita
group for their generosity and help. It was a privilege to learn
fromthis group.Thank you also to the Graduate School of Science, in
particular the International LiaisonOffice, for making UTRIP the
wonderful thing it was.
Appendix A Density matrix formalism
The density matrix formalism succinctly expresses expectation
values from both quantum mechanicalstate vectors and statistical
ensemble averages. Let {|φn〉} be an orthonormal basis, so that
thenormalised state vector of the system may be expressed as |ψ(t)〉
=
∑n cn(t)|φn〉. An observable A
of the system will have expectation value
〈A〉 = 〈ψ(t)|A|ψ(t)〉 =∑n,m
cnc†m〈φn|A|φm〉,
and matrix representationAmn = 〈φn|A|φn〉
so that〈A〉 =
∑n,m
Amncnc†m.
We define the density operator,
ρ̂(t) = |ψ(t)〉〈ψ(t)| =∑n,m
cnc†m|φn〉〈φm| ⇒ 〈A〉 = TrAρ̂.
Consider the matrix representation of the Hamiltonian, Hnl =
〈φn|H|φl〉. The Schrödingerequation,
i~ ∂∂t|ψ〉 = H|ψ〉 ⇒ ~ ∂
∂tcn =
∑l
Hn,lcl,
so as Hij = H†ji (Hermitian),
i~ ∂∂tcnc†m =
∑l
(Hnlclc
†m − cmc†lHlm
).
With this formalism, an observable may be averaged over a
statistical ensemble, that is, manysystems of the same structure
under the same microscopic conditions. A mixed state is
representedby weighting each state with its probability. If the
state |ψi(t)〉 has probability pi, the density matrixshall be
generalised as
ρ(t) ≡∑i
pi∑n,m
(ci)n(ci)†m|φn〉〈φm|
The quantum Liouville equation is
i~ ∂∂tρ = [H, ρ].
This preserves probability, analogous to the preservation of
phase volume by the classical Liouvilletheorem[1].
Looking at an observable A (Hermitian) in the Heisenberg
picture, we put
〈Ȧ(t)〉 = TrρȦ(t) = Trρ i~
[H, A]
and since TrAB = TrBA, this is equivalent to
TrA1
i~[H, ρ] = Trρ̇(t)A
in the Schrödinger picture.
14
-
Appendix B Time-dependent perturbation theory
For a system with total Hamiltonian
H(t) = H0 + αV (t),
whereH0 has no time dependence and α is a constant parameter, we
seek to expand the wavefunction|ψ(t)〉 as a power series in α. We
employ the interaction representation
|ψ(t)〉int = eiH0t/~|ψ(t)〉; |ψ(0)〉int = |ψ(0)〉
to express the Schrödinger equation as
i~ ∂∂t|ψ(t)〉int = eiH0t/~αV (t)e−iH0t/~|ψ(t)〉int =
αVint(t)|ψ(t)〉int.
Introducing a time evolution operator Uint(t, t0), such that
|ψ(t)〉int = Uint(t, t0)|ψ(t0)〉int ⇒ i~∂
∂tUint(t, t0)|ψ(t0)〉int = αVint(t)Uint(t, t0)|ψ(t0)〉int.
Then, by the boundary condition Uint(t0, t0) = I there is a
self-consistent equation
Uint(t, t0) = I−i
~
∫ tt0
dt′αVint(t′)Uint(t
′, t0),
which may be substituted into itself giving
Uint(t, t0) = I−i
~
∫ tt0
dt′αVint(t′)
[I− i
~
∫ t′t0
dt′′αVint(t′′)Uint(t
′′, t0)
],
and repeating this substitution we have an expression
Uint(t, t0) =
∞∑n=0
(−iα~
)n ∫ tt0
dt1 · · ·∫ tn−1t0
dtnVint(t1)Vint(t2) · · ·Vint(tn).
Expanding in basis functions {|n〉}, among which |i〉 is the
initial state, the coefficients cn(t) suchthat
|ψ(t)〉int =∑n
cn(t)|n〉
are expanded as
Uint(t, t0)|i〉 =∑n
|n〉〈n|Uint(t, t0)|i〉 =∑n
cn(t)|n〉
⇒ cn(t) = δni − αi
~
∫ tt0
〈n|Vint(t′)|i〉 − α1
~2
∫ tt0
dt′∫ t′t0
dt′′〈n|Vint(t′)Vint(t′′)|i〉 . . .
and the c(0)n terms with no α dependence are constant in time,
representing the initial state[9].
The same approach may be taken in the density matrix formalism,
where the Liouville equationis
∂
∂tρ =
1
i~[H0 + αV (t), ρ] = i (L0 + αL(t)) .
As the density matrix is composed of the cn(t) components, the
expansion is, to first order,
ρ(t) = ρ(t0) + αi
∫ tt0
dt′Lint(t′)ρ(t0) +O(α2)
= ρ(t0) + αi
∫ tt0
dt′ei(t−t′)L0L(t′)ρ(t0) +O(α2).
15
-
References
[1] M. Toda, R. Kubo and N.Saito, Statistical Physics I:
Equillibrium Statistical Mechanics, 2ndEd. (Springer, 1992), p.
17.
[2] C.P. Slichter, Principles of Magnetic Resonance, 3rd Ed.
(Springer, 1996), p. 16.
[3] R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II:
Nonequillibrium Statistical Me-chanics, 2nd Ed. (Springer, 1991),
p. 128.
[4] C.P. Slichter, Principles of Magnetic Resonance, 3rd Ed.
(Springer, 1996), p. 32.
[5] R. Kubo, Rep. Prog. Phys. 29 (255), (1966).
[6] H. Ikeuchi, Miyashita Laboratory Handout (Unpublished).
[7] T. Mori and S. Miyashita, Miyashita Laboratory Handout
(Unpublished).
[8] R. Feynman, R. Leighton and M. Sands, The Feynman Lectures
on Physics, Volume III (Cali-fornia Institute of Technology, 1965),
p. 4-4.
[9] B. Simons, Time-dependent perturbation theory, Cavendish
Laboratory Handout (WWW Doc-ument,
http://www.tcm.phy.cam.ac.uk/~bds10/aqp/handout_dep.pdf).
16
http://www.tcm.phy.cam.ac.uk/~bds10/aqp/handout_dep.pdf
IntroductionRabi oscillationHeuristic descriptions of
resonance
Two models of resonanceBloch EquationKubo formula
An approach from first principlesDerivation of a quantum master
equationAnalysis and simulation
ConclusionsDensity matrix formalismTime-dependent perturbation
theory
