transport properties for small quantum systems coupled to multiple baths with ... heat transport through short spin chains coupled to two thermal baths with.
Non-equilibrium evolution of quantum systems connected to multiple baths by Jinshan Wu
B . S c , The Beijing Normal University, 1999 M . S c , The Beijing Normal University, 2002
A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF Master of Science in The Faculty of Graduate Studies (Physics)
The University Of British Columbia October 2006 © Jinshan Wu, 2006
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In this thesis we develop a systematic formalism that allows the calculation of transport properties for small quantum systems coupled to multiple baths with different temperatures and/or chemical potentials. Our approach is based on a generalization of the projector operator technique, previously used to study the evolution towards equilibrium of systems weakly coupled to a single bath. Applying this technique to a system weakly coupled to multiple baths, we find a dynamical equation for the reduced density matrix of the system (the reduced density matrix is obtained by tracing out from the total density matrix the baths' degrees of freedom). Integration of this dynamical equation gives the time evolution of the biased system. After a time interval long enough compared to the characteristic time-scales of the problem, the system arrives to a non-equilibrium steady-state. The density matrix in this non-equilibrium steady state can also be calculated directly, avoiding the time-integration of the dynamical equation. In the second part of the thesis we study non-equilibrium steady states and heat transport through short spin chains coupled to two thermal baths with different temperature. Two new definitions for the local temperature in nonequilibrium are proposed, based on local spin and energy at different sites. The definition based on the local energy seems to be more reliable. We find that for small biases, the heat and spin currents increase proportionally to the bias, however non-linear behavior is observed for large biases. The temperature profile of the chain depends strongly on the parameters of the system.
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C o n t e n t s
Abstract
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Contents
iii
List of Figures
iv
Acknowledgements
v
1
Introduction
1
2
Dynamics of open systems 4 2.1 General formalism: system coupled to a single bath 5 2.1.1 Projector operator technique and the reduced Liouville equation 6 2.1.2 Alternative derivation: the non-correlation approximation 9 2.1.3 Lindblad form: general equation for the reduced density matrix 10 2.2 Applications: thermal equilibrium 11 2.2.1 One-state open systems 11 2.2.2 Multi-state open systems 15 2.3 The local coupling scheme is wrong 17
3
Spin chains coupled to thermal baths 3.1 Single spin coupled to a heat bath 3.2 Ising spin chain coupled to a heat bath 3.3 X Y chain coupled with one heat bath: thermal equilibrium . . . 3.3.1 Sample calculation for a 2-site X Y chain 3.4 X Y chain coupled to two baths 3.4.1 Sample calculation for a 2-site X Y chain 3.5 Definition of the local temperature and heat current 3.6 Current and local temperature profiles in non-equilibrium steadystates
20 20 22 25 25 28 29 32
Conclusion
50
4
Bibliography
38
52
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1.1
Temperature profile of a classical harmonic chain
2.1
A two-site chain coupled to a heat bath
3.1 3.2
3-spin X Y chain coupled with two heat baths Time evolution of off-diagonal elements of the density matrix for a biased system 3.3 ( t , i f > = b.
(2.51)
k
The only nonzero bath Green's functions are found to be: G l (±r) = tr (F^ (±r) i f > (0) p ) = (n ) e ± ^ l
k
B
G*;jt (±r) where in (l±n ), k
= tr [F1
B
2)
B
k
(±r) i f > (0) p ) = (1 ± n ) e^< )\ k
B
k
^
the — corresponds to a particle bath and the + to a phonon
13
Chapter 2. Dynamics of open systems bath. Then the reduced equation of motion for R (t) becomes: !
s
[XVkoe-^, WtaU^-^Rtg (t)] G * (r) +Wk^^\\V ae-^-^R {t)}Gll{r) _ [xv ae-^, R' (t) XV^e^-^] G^ (-r) 1
23kW dt
=
i r'dTJ W^kh S
I
T
k
aT
k
s
s
- [XV^aU^, R (t) XVkoe-W-r)] G ^ ( - T ) [a,a^R (t)] -^ e ^ + (l±n ) [a^aR's (t)] e^e'^ -(l±n ) [a,R' ( (or, we can assume that the system was coupled to the bath at t = —oo instead of t — 0). Then, in the interaction picture, we find the equation of evolution for the reduced density matrix of such an open system to be: ^R's (t) = n [ 2 a i ^ (t) at - t 0
+r
2
o i i
/ ( )_ ^ ( ) t ] t
[2a i?^ (t) a - a a i ? ^ (*) +
f
t
a
a
(t) aa ] , f
where
»"2
=
^A
2
pA
2
£ IVfcl (1 ± n )
£
/ 3 ( W - M ) zf
l
is the average number of phonons/particles with energy w in the bath (for phonons, p. = 0). Now we can solve the dynamical equation and check the stationary solution. Let us first consider the case of the fermionic two-level system. First, note that if the initial reduced density matrix is diagonal (as we assumed it to be, Rs(0) = |0)(0|), then this equation never gives rise to off-diagonal terms. In
14
Chapter 2. Dynamics of open systems
this case, Rs(t) = R s(t), and therefore we can work directly i n the Schrodinger picture. We set: J
Rs (t) =
- p$-
1
±
|o> (0| +
) 1 }( 1
,
t
( 2
.
5 7 )
so that the trace is preserved. The equation for p (t) is then straightforward to extract from E q . (2.54): j P (t) = 2 ( n - r ) - 2 ( n + r ) p (t), 2
t
(2.58)
2
p(0) = 0 . This has the stationary solution * = r 7 T 3 '
( 2
'
5 9 )
which for the fermionic bath is: p(oo) = This leads indeed to the expected equilibrium grandcanonical distribution R (oo) = —
i
s
-P(.u-v) 10> ). If b > 0, the latter type of terms are identically zero, whereas if b < 0, the former type of terms are identically zero. These are the terms that correspond to processes that violate conservation of energy, so their disappearance from the dynamical equation is expected. For specificity, let us assume that b > 0. In this case, after defining z
k
z
z
k
z
z
r =
^ J2\V \ S(u, -2b ) 2
2
h
2
k
k
k
z
and r =r[l+n(2b )},r 1
z
= rn(2b )
2
z
where
• is the average number of phonons of energy co = 2b , we finally find: k
—
=
r
i
[2a_R (t) a s
+
- v *_R'
+r [2o-+R (t) cr_ 2
s
+
0, then only the terms proportional to 5(u> —ft)contribute. The dynamical equation xu>kT
kk
k
iuJkT
k
k
27
Chapter 3. Spin chains coupled to thermal baths then reduces to: dR
(t)
s
dt
=
--[H ,R (t)] s
s
-r [cri mxRs(t) - miRs(t)cr± lX
- cr R (t)m2 + Rs(t)m cri ] (3.25)
iX
ltX
s
2
tX
where
mi
71
1 +n(fi) 0 0 l+n(Q)
0 n(fi) 1 +n(fi) 0
0 0 -n(fi)
0 n(ft) -1 -n(fi) 0
(3.26)
0
n(fi) 0 1 fn(Q) 0 0 1 + n(il) J_ (3.27) 1 + n(£2) m 0 0 -n(fi) 72 0 n(Q) - l - n ( f i ) 0 In other words, the matrices m i and m are similar to
