Download Quantum linear Boltzmann equation with finite intercollision time

PHYSICAL REVIEW A 80, 064104 共2009兲
Quantum linear Boltzmann equation with finite intercollision time
Lajos Diósi*
Research Institute for Particle and Nuclear Physics, P.O. Box 49, H-1525 Budapest 114, Hungary
共Received 1 June 2009; published 17 December 2009兲
Inconsistencies are pointed out in the usual quantum versions of the classical linear Boltzmann equation
constructed for a quantized test particle in a gas. These are related to the incorrect formal treatment of
momentum decoherence. We prove that ideal collisions with the molecules would result in complete momentum decoherence, the persistence of coherence is only due to the finite intercollision time. A corresponding
quantum linear Boltzmann equation is proposed.
DOI: 10.1103/PhysRevA.80.064104
PACS number共s兲: 03.65.Yz, 05.60.Gg, 03.65.Ta, 05.20.Dd
I. INTRODUCTION
Collisional theory of motion of a test particle through a
gas had been the oldest classical tool confirming atomic
kinetics—e.g., in Brownian motion 关1兴—or using atomic
kinetics—e.g., in Millikan’s experiment 关2兴. If the gas is dilute and hot, the interaction of the particle with the gas reduces to the repeated independent binary collisions between
the particle and the individual molecules. The dynamics of
the test particle is fully captured by the differential cross
section of the scattering: the linear Boltzmann equation
共LBE兲 describes the particle’s motion. For a quantum particle, the scattering matrix 1̂ + iT̂ is expected to capture the
dynamics which we describe by the quantum linear Boltzmann equation 共QLBE兲
d␳ˆ
= − i关Ĥ, ␳ˆ 兴 + L␳ˆ ,
dt
共1兲
where ␳ˆ is the density matrix, Ĥ is the free Hamiltonian
共including an energy shift due to the gas兲, and L is the Lindblad 关3兴 superoperator of the non-Hamiltonian evolution. The
microscopic derivation of L does not differ from the classical
derivation of the LBE as long as we assume a density matrix
具P兩␳ˆ 兩P⬘典 = ␳共P兲␦共P − P⬘兲 without superpositions in momentum P. The evolution of the diagonal density matrix is governed by the classical LBE applied to the distribution ␳共P兲.
This is why the classical kinetic theory has so far served as a
perfect effective theory for a quantum particle in those innumerable applications where complete momentum decoherence was a priori assumed. A stealthy request for a systematic theory emerged in the field of foundations where
quantum Brownian motion became the paradigm of how the
environment 共gas兲 “measures” the wave function 共of the particle兲 关4–7兴. For experiments, a proper quantum Brownian
particle was not available before, now it is, cf., e.g., 关8兴.
When we are interested in the local quantum dynamics of
the particle, we need the off-diagonal elements 具P兩␳ˆ 兩P⬘典 as
well, and then the microscopic derivation of L becomes
problematic. The classical derivation resists to a direct quantum extension. The first microscopic derivation was obtained
fifteen years ago 关9兴 by the present author. Subsequent works
*[email protected]; www.rmki.kfki.hu/~diosi
1050-2947/2009/80共6兲/064104共4兲
关10–12兴 by Vacchini and Hornberger—flavored by transient
debates 关13,14兴—led them to a refined version. Their up-todate review 关15兴 further elucidates the subject and can serve
as an exhaustive source of references for the reader. Microscopic derivations assume typically 共though not necessarily兲
the Maxwell-Boltzmann distribution for the molecular momentum k
␳g共k兲 =
冉 冊 冉 冊
␤
2␲m
3/2
exp −
␤k2
,
2m
共2兲
where m is the molecular mass and ␤ is the inverse temperature. To overcome the mentioned difficulties with the offdiagonal elements of ␳ˆ , Ref. 关9兴 suggested a formal resolution that became an irreducible part of all subsequent
derivations 关10–15兴. In all variants of QLBE, including a
most recent one 关16兴, L is not a linear functional of ␳g共k兲 as
it should have been, rather, L is a quadratic functional of the
square root of ␳g共k兲. In other words, L is the linear functional of
冑␳g共k兲␳g共k⬘兲.
共3兲
Therefore, no matter which QLBE is considered, we face the
following issue of consistency. Suppose the gas has two independent “components” consisting of the same molecules at
same densities while at two different temperatures. Then,
their contribution to the QLBE must be additive. But it is
not, since L in the QLBE 共1兲 is linear in the expression 共3兲,
not in ␳g共k兲 itself.
Before we construct a QLBE that is linear in ␳g共k兲, we
point out a surprising collisional decoherence effect apparently overlooked for a long time and discovered independently from the present work by Kamleitner and Cresser, too
关16兴. It may radically alter our understanding what QLBE
should be.
A. Single collision decoheres momentum
Let us consider the kinematics of a single collision between the particle and a molecule. As a Gedanken experiment, suppose we prepared the molecule with momentum k
before the collision and we detected it with momentum k
− Q after the collision, i.e., we know the momentum transfer
Q. Note that the interactions of these preparation and detection devices with the molecule are irrelevant for the reduced
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PHYSICAL REVIEW A 80, 064104 共2009兲
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dynamics of the particle, yet such preparation and detection
devices are relevant for our arguments. We get dramatic conclusions for the reduced state ␳ˆ of the particle since, for
elementary kinematic reasons, the observation of Q is
equivalent with the observation of one component of the
particle’s momentum P. Let P stand for the postcollision
momentum, then from energy conservation
k2 共P − Q兲2 共k − Q兲2 P2
+
=
+
,
2m
2M
2m
2M
冉 冊
M
Q
M
k储 −
−1
,
m
m
2
共5兲
where the subscripts 储 refer to the components parallel to Q.
It is obvious that after our single collision the particle’s density matrix ␳ˆ , whatever it was before the collision, becomes
perfect diagonal in P储. Gradually, after many collisions, the
state ␳ˆ becomes a mixture of plane waves, no off-diagonal
mechanism will be left at all. This result contradicts to what
we have so far learned as QLBE. The art of deriving a plausible QLBE has always been about the nontrivial treatment
of the off-diagonal elements. The above decoherence mechanism, however, shows that standard scattering theory and
independent collisions do not yield any QLBE for the offdiagonals. The apparent off-diagonal dynamics may well
have been artifacts of the square-root fenomenology 共3兲.
In reality, the state of the particle is not necessarily the
delocalized one advocated by our decoherence argument. In
fact, our argument is relevant for the case of very dilute gas
or for very small collision cross sections. Then, the density
matrix has no off-diagonal elements at all. Such perfect momentum decoherence assumes perfect energy conservation of
the collision process. In quantum scattering, it requires infinite time and this is the loophole where we concentrate. If
the intercollision time is moderate long, it still leaves ample
room for energy balance uncertainty and momentum coherence.
T̂ =
1
2␲mⴱ
冕
ˆ
dkdQeiQX f共k̂ⴱf ,k̂ⴱi 兲␦共Êⴱfi兲兩k − Q典具k兩,
We outline the derivation of QLBE 共1兲 where L remains a
linear functional of ␳g共k兲 and momentum decoherence is described correctly in function of the intercollision time ␶. The
full quantum derivation introduces the box-normalized
single-molecule density matrix
␳ˆ g =
共2␲兲3
␳g共k̂兲.
V
共6兲
We follow Ref. 关9兴. Starting from the precollision initial state
␳ˆ 丢 ␳ˆ g, we calculate the collisional change of the particle’s
reduced state. The parts responsible for in- and out-scattering
are respectively given by the following expressions:
⌬␳ˆ 兩in = Vng trg共T̂␳ˆ 丢 ␳ˆ gT̂†兲,
共7兲
共9兲
where the ket and bra stand for the momentum eigenstates of
the molecule. The function f is the scattering amplitude. The
quantities marked by ⴱ are the initial and final 共pre- and
postcollision兲 c.o.m. momenta, while Êⴱfi is the c.o.m. energy
balance, respectively,
k̂ⴱi =
M
m
P̂,
ⴱk −
M
Mⴱ
k̂ⴱf = k̂ⴱi − Q,
Êⴱfi =
1
1
ⴱ 2
共k̂ⴱ兲2 .
ⴱ 共k̂ f 兲 −
2m
2mⴱ i
共10兲
Here, M ⴱ = M + m and mⴱ = Mm / M ⴱ. We substitute the Eqs.
共6兲 and 共9兲 into Eq. 共7兲, yielding
⌬␳ˆ 兩in =
2␲ng
mⴱ2
冕冕
ˆ
dk␳g共k兲dQeiQX
ˆ
⫻f共k̂ⴱf ,k̂ⴱi 兲␦共Êⴱfi兲␳ˆ f̄共k̂ⴱf ,k̂ⴱi 兲␦共Êⴱfi兲e−iQX .
共11兲
A similar elaboration applies to ⌬␳ˆ 兩out as well. And now, we
alter 共and simplify兲 the old derivation 关9兴. In standard quantum scattering theory, both the preparation of precollision
and the emergence of postcollision states take infinite long
time and the transition operator 共9兲 contains the ␦ function
for perfect energy conservation. In our case, however, the
intercollision time ␶ is finite
␶=
B. QLBE with finite intercollision time
共8兲
where the single-collision contributions has been enhanced
by the total number Vng of molecules at density ng. The
standard form of the transition operator for the 共spinindependent兲 scattering with initial laboratory momenta k
and P, respectively, reads
共4兲
we have the following relationship:
P储 =
⌬␳ˆ 兩out = − 21 Vng trg兵T̂†T̂, ␳ˆ 丢 ␳ˆ g其,
冑␲ ␤ m
␴ng
共12兲
,
␴ is the total scattering cross section. We take this feature
into the account if in T̂, we use the “smoothened” ␦-function
␦␶共E兲 instead of ␦共E兲
␦␶共E兲 =
sin共␶E/2兲
,
␲E
共13兲
and we extend the transition amplitude f共kⴱf , kⴱi 兲 off-shell.
We shall approximate d␳ˆ / dt 兩in by ⌬␳ˆ 兩in / ␶, when we substitute Eq. 共11兲, it yields
冏 冏
d␳ˆ
dt
=
in
2␲ng
␶mⴱ2
冕冕
ˆ
dk␳g共k兲dQeiQX
ˆ
⫻f共k̂ⴱf ,k̂ⴱi 兲␦␶共Êⴱfi兲␳ˆ f̄共k̂ⴱf ,k̂ⴱi 兲␦␶共Êⴱfi兲e−iQX . 共14兲
This form, and a similar form for d␳ˆ / dt 兩out, give d␳ˆ / dt
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= d␳ˆ / dt 兩in + d␳ˆ / dt 兩out and explicitly indicate the following
Lindblad structure 共1兲 of our QLBE:
2␲ng
d␳ˆ
= − i关Ĥ, ␳ˆ 兴 +
dt
␶mⴱ2
冕冕
冉
冊
2DxxP̂␳ˆ P̂ =
共15兲
V̂共k,Q兲 = e
f共k̂ⴱf ,k̂ⴱi 兲␦␶共Êⴱfi兲.
共16兲
The cautious reader might notice that the average intercollision time ␶ depends on ␳g共k兲, so that L’s exact linear dependence on ␳g共k兲 is arguable. Whether a refined means of including the random intercollision time exists, we do not
know yet. We expect the following features of the new
QLBE. In very dilute gas, we can take ␶ → ⬁ and ␦␶ → ␦, the
off-diagonal elements of ␳ˆ become completely suppressed
and our QLBE recovers the standard LBE for the diagonal
elements of the density matrix. If the gas is less dilute, the
LBE remains valid for the diagonal part of ␳ˆ while the offdiagonal part reappears and then ␶ becomes the relevant
quantity for it. We are going to illustrate these features.
d␳ˆ
= − i关Ĥ, ␳ˆ 兴 − D pp关X̂,关X̂, ␳ˆ 兴兴 − Dxx关P̂,关P̂, ␳ˆ 兴兴
dt
2M
冕
关␦␶⬘共E fi兲兴2dk储 =
m
Q
关X̂,兵P̂, ␳ˆ 其兴.
1 ng
6m
冕冕
冏冉
冕
关␦␶⬘共E fi兲兴2dE fi =
The resulting equation reads
2DxxP̂␳ˆ P̂ =
␶3 1 2␲ng
24␲ M 2 ␶m
冏冉
共17兲
冉
1
dQQdk⬜␳g k⬜ + Q
2
1
1
⫻ f k⬜ − Q,k⬜ + Q
2
2
冊冏
冊
2
.
共18兲
Different QLBEs do not agree upon the value of the position
diffusion 共momentum decoherence兲 constant Dxx. Our QLBE
共15兲 provides a ␶-dependent Dxx. We are going to derive it.
Since the Dekker equation 共17兲 corresponds to the second
order Taylor expansion of the QLBE 共15兲 in the operators
X̂ , P̂, we shall compare the respective coefficients of the term
P̂␳ˆ P̂ in the Dekker equation and in the QLBE. In the latter,
the leading term comes from the expansion of the ␦␶共Êⴱfi兲
factors around P̂ = 0. To calculate their argument for m / M
→ ⬁, recall Eq. 共10兲
共20兲
冕冕
冉
m ␶3
.
Q 24␲
1
dQdk⬜␳g k⬜ + Q
2
冊冏
2
共21兲
冊
1
QP̂␳ˆ P̂Q,
Q
共22兲
where rotational symmetry allows for substituting QP̂␳ˆ P̂Q
by 共Q2 / 3兲P̂␳ˆ P̂. We get finally that the quantum position diffusion 共momentum decoherence兲 is proportional to the classical momentum diffusion 共18兲
Dxx =
For a heavy Brownian particle 共M Ⰷ m兲, in the diffusion limit
共cf., e.g., 关9,15兴兲, this equation is recovered by all QLBEs.
They yield the same 共classical兲 momentum diffusion constant D pp and friction constant ␩
␤␩ = D pp =
dQdk␳g共k兲
1
1
⫻ f k⬜ − Q,k⬜ + Q
2
2
A common test of QLBE is Dekker’s equation 关17兴 containing the fenomenological constants ␩, D pp, and Dxx of
friction, momentum and position diffusion 共i.e., position and
momentum decoherence兲, respectively,
−i
冕冕
where we set P̂ = 0 in the arguments of f, assuming a slowly
varying f. Now, we change the integration variable according
to dk储 = 共m / Q兲dE fi and perform the integral over E fi. We can
ignore the variation of ␳共k兲 and f共k − Q , k兲 with the variation
of k储, which can only take values from a narrow vicinity of
Q / 2 because of the narrow support of the ␦␶⬘. Hence, we
substitute k储 ⬇ Q / 2 in ␳g and f, then we perform the integral
C. Diffusion limit
␩
1 2␲ng
M 2 ␶m2
共19兲
⫻兩f共k − Q,k兲兩2关␦␶⬘共E fi兲兴2QP̂␳ˆ P̂Q,
where the Lindblad operators read
ˆ
iQX
冉 冊
1
1
Q
Q
k−
− QP̂ ⬅ E fi − QP̂.
m
2
M
M
Therefore, we get
dk␳g共k兲dQ V̂共k,Q兲␳ˆ V̂†共k,Q兲
1
− 兵V̂†共k,Q兲V̂共k,Q兲, ␳ˆ 其 ,
2
Êⴱfi =
冉冊
1 ␶
12 M
2
D pp .
共23兲
This result is in agreement with the conclusion of our Gedanken experiment: the longer the intercollision time, the stronger the momentum decoherence is. According to a universal
constraint 关3,17兴 of complete positivity of the Dekker equation 共17兲, the inequality Dxx ⱖ 共␤ / 4M兲2D pp must be satisfied,
which puts a lower bound on position diffusion at fixed friction ␩. The QLBE introduced in Ref. 关11兴 predicts this minimum position diffusion. All other previous QLBEs 关9–16兴
predict similar position diffusion suggesting that it is irrelevant at high temperatures. Our result is different: position
diffusion 共momentum decoherence兲 goes with ␶2, not with
␤2. The above complete positivity condition gives the constraint ␶kBT ⱖ 冑3ប / 2 for ␶, where T is the temperature; we
restored the standard physical dimensions. Substituting Eq.
共12兲 and introducing the thermal de Broglie wavelength ␭T
of the molecules, we get
␭T␴ng ⱕ 冑2␲/3,
共24兲
suggesting that our QLBE works until a volume ␭T␴ will
contain more than cca. 1 molecule on average. This may
happen when the density ng is too high, or the temperature is
too low, or the cross section 共e.g., the size of the particle兲 is
too big. But otherwise, there is an ample regime for our
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PHYSICAL REVIEW A 80, 064104 共2009兲
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particle to develop a localized wave function subject to our
QLBE.
II. SUMMARY AND OUTLOOK
Quantum dynamics of a particle in interaction with a dilute gas is definitely tractable by the collision model. We
pointed out—in agreement with Ref. 关16兴—that the coherence between different momentum eigenstates is heavily
suppressed 共cf. related remarks in 关18兴兲 and the very persistence of any localized coherent dynamics is only due to the
uncompleted quantum collisions, i.e., to the finite intercollision time ␶. This brings a dramatic change in our understanding the quantum behavior of the test particle. Previously,
momentum decoherence 共i.e., position diffusion兲 was considered a weak quantum effect irrelevant at high temperatures.
Now, we must claim that such weak momentum decoherence
predicted by the previous QLBEs was an artifact. Momentum decoherence is a dominating effect. In very dilute gas, at
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very small scattering rates, the true dynamics is just the classical diffusion 共cf. LBE兲 through the momentum eigenstates.
This means, in particular, that a gradual momentum decoherence under controlled conditions may become testable, e.g.,
in fullerene 关8兴 or in nano-object interference experiments
关19兴.
There is no guarantee that momentum decoherence has a
universal Markovian description. The proposed QLBE captures the influence of finite intercollision time ␶ within the
concept of independent collisions while the mechanism is
beyond it: the second collision literally interferes with the
first one. Might then we not think of a Lindblad structure L
that is linear functional of ␳共k兲␳共k⬘兲? In any case, our QLBE
with the finite collision time represents a progressive fenomenology compared to the earlier QLBEs.
ACKNOWLEDGMENTS
This work was supported by the Hungarian OTKA under
Grants No. 49384 and No. 75129.
关9兴
关10兴
关11兴
关12兴
关13兴
关14兴
关15兴
关16兴
关17兴
关18兴
关19兴
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