Gravity induced Wave Function Collapse

0.28 MB·

by  Giulio Gasbarri

Gravity Induced Wave Function Collapse
G. Gasbarri,1,2,3 M. Toroˇs,2,3 S. Donadi,2,3 and A. Bassi2,3
1Abdus Salam ICTP, Strada Costiera 11, I-34151 Trieste, Italy
2Department of Physics, University of Trieste, Strada Costiera 11, 34151 Trieste, Italy
3Istituto Nazionale di Fisica Nucleare, Trieste Section, Via Valerio 2, 34127 Trieste, Italy
Starting fromanidea ofS.L. Adler[1], wedevelopa novelmodelofgravity-inducedspon-
taneous wave-functioncollapse. The collapse is drivenby complex stochastic fluctuations of
the spacetime metric. After having derived the fundamental equations, we prove the col-
7
1 lapseandamplificationmechanism,the twomostimportantfeaturesofaconsistentcollapse
0 model. Under reasonable simplifying assumptions, we constrain the strength ξ of the com-
2 plex metric fluctuations with available experimental data. We show that ξ 10−26 in order
n for the model to guarantee classicality of macro-objects, and at the same≥time ξ 10−20
≤
a in order not to contradict experimental evidence. As a comparison, in the recent discovery
J
of gravitational waves in the frequency range 35 to 250 Hz, the (real) metric fluctuations

0 reached a peak ξ 10−21.
1 ∼

]
h I. INTRODUCTION
p
-
t Thepossibility for quantummechanics to bethelimiting case of an underlyingnonlinear theory
n
a has been often considered in the literature [2–7]. A straightforward motivation is that linear
u
models typically are an approximation of nonlinear ones [6]. A stronger motivation is that they
q
[ open the way to solving the quantum measurement problem [8]. In this latter context, models

of spontaneous wave function collapse [9–12] provide a consistent phenomenology describing the
2
v collapseofthewavefunctionduringameasurement,viaextranonlinearandstochastictermsadded
6 to the dynamics. Due to their intrinsic nonlinearity, these models also offer a way out to some of
3
the puzzles in quantum gravity and cosmology [13–15].
2
2 The common feature of all collapse models is a classical noise, coupled nonlinearly to the quan-
0 tum wave function. The typical collapse equation, in the Itoˆ form, is:
.
1
0
i λ
7 dψ = Hˆ dt+√λ (Aˆ Aˆ )dW (Aˆ Aˆ )2dt ψ , (1)
1 t −~ 0 j −h jit j,t− 2 j −h jit  t
v: Xj Xj
 
i
X where Hˆ is the standard quantum Hamiltonian, Aˆ a set of self-adjoint commuting operators,
j j
{ }
r Aˆ = ψ Aˆ ψ and W a set of independent Wiener processes, which force the wave function
a h jit h t| j| ti j,t
to collapse towards one of the common eigenstates of the operators Aˆ [16]. The positive coupling
j
constant λ sets the strength of the collapse mechanism.
Eq. (1) should be considered as a phenomenological equation, raising the question why it takes
that form. A justification comes from the following argument firstproposed by Adler [7]. Consider
the Hamiltonian:
Hˆ = Hˆ +i~√λ Aˆ w , (2)
0 j j,t
j
X
where w = dW /dt is a set of independent white noises. It describes the coupling of a quan-
j,t j,t
tum system with a set of external classical noises, through the operators Aˆ . It is a reasonable
j
phenomenological ansatz, except for the fact the second term is anti-hermitian. As a consequence,
the norm of ψ is not conserved, jeopardizing the physical meaning of the wave function. The ob-
t
vious thing to do is to replace ψ with ψ / ψ , but this brings in a serious problem: the resulting
t t t
k k
2
equation is nonlinear and also the stochastic ensemble of states evolves nonlinearly, even in the
average. This leads to superluminal signaling [17]. The problem can be avoided if one adds extra
terms in Eq. (2), such that the master equation for density matrix ρ = E[ψ ψ ] associated to
t t t
| ih |
the ensemble becomes linear (and of the Lindblad type [18–20]). These new terms are precisely
those, which lead to Eq. (1). Appendix A contains the derivation of what outlined here.
In the sense explained here above, the requirements of norm conservation and no-superluminal
signaling added to Eq. (2), give the desired collapse equation. Likely, a sensible nonlinear pre-
quantum theory, which leads to a dynamics for the wave function at the phenomenological level,
will naturally embody both requirements. The open issue now, is how to justify Hˆ in (2), in
particular why the coupling should be anti-hermitian, and what is the suitable choice for the
operators Aˆ , which select the basis along which the collapse occurs. While there is no answer to
j
the first question, more than the hope that the pre-quantum theory will provide a natural answer,
one can say more about the second question.
Quite often the literature suggest that the collapse is driven by gravity [21–28]. This is the
only hope one can have, to link the collapse to a known force, since all other forces as we know
them have been successfully quantized, therefore they cannot provide the anti-hermitian coupling
needed for the non-linear collapse. But there is a stronger motivation. The collapse scales with
the mass/size of the system [9, 10], and localizes the wave function in space. Then, the natural
candidate for the operators Aˆ is the local mass density mˆ(x) = m δ(3)(x xˆ ), coupled to a
j i i − i
noise w(x,t) spread through space [43]:
P
Hˆ = Hˆ +i~ ξ d3xmˆ(x)w(x,t). (3)
0 ˆ
p
A random gravitational field naturally provides such a coupling (see Appendix B), which would
contain an anti-hermitian part if the field has an imaginary component. In [1] arguments are
presented, as to why the metric could be classical and complex-valued. Following this idea we will
explore the consequences of assuming a complex non-white classical noise coupled to the local mass
density.
The paper is organized as follows. In Section II we derive, to the first meaningful perturbative
order,thegeneralcollapseequationforthewavefunction,aswellastheassociatedmasterequation,
in thecaseof N complex valued coloured randomnoises h (t), each coupled to an operator Aˆ . The
i i
literature so far considered only the case of real valued coloured noises [12]. In Section III we show
the collapse mechanism. In Section IV we consider specifically a noise field w(x,t) coupled to the
local mass density Mˆ(x) and discuss the amplification mechanism, one of the crucial properties of
any collapse model. In Section V we discuss the bounds on the spectrum of the noise, which are
set by current experiments. We conclude the paper with a discussion of the results (Section VI).
II. MASTER AND COLLAPSE EQUATIONS
We have seen how the idea of a complex gravitational stochastic background inducing the
collapse of the wave function leads to a collapse model where the noise is complex valued and
in general coloured. Since this has not been discussed in the literature so far, in this section we
derive the appropriate collapse equation and the master equation, following the same strategy as
in Appendix A for a real valued white noise.
The starting point is the following generalized Schro¨dinger equation
N
i~∂ φ = Hˆ +ξ Aˆ h (t)+Oˆ φ (4)
t t 0 i i t
| i | i
" #
i=1
X
3
where Aˆ are arbitrary self-adjoint operators and h (t) are N complex Gaussian noises, with zero
i i
average and correlation function:
E [h∗(t)h (τ)] = D (t,τ),
Q i j ij
E [h (t)h (τ)] = S (t,τ). (5)
Q i j ij
D (t,τ) and S (t,τ) are complex functions with magnitude of order 1 and Oˆ is an operator yet
ij ij
to be defined. The parameter ξ sets the strength of the noise, which is assumed to be small.
Following the scheme outlined in the introduction, we will determine Oˆ by the requirement of
no-faster-than-light signaling.
Since the norm of φ is not conserved, we consider the normalized state ψ = φ / φ ,
t t t t
| i | i | i || ||
which solves the equation
N
1
i~∂ ψ = Hˆ +ξ Aˆ h (t) i A hI(t) +Oˆ Oˆ Oˆ† ψ
t| ti 0 i i − h iit i − 2h − it | ti
" #
Xi=1(cid:16) (cid:17)
1
= Hˆ Hˆ Hˆ† ψ (6)
t− 2h t− tit | ti
(cid:18) (cid:19)
with
N
Hˆ = Hˆ +ξ Aˆ h (t)+Oˆ. (7)
t 0 i i
i=1
X
As expected, the normalized vector evolves according to a nonlinear stochastic dynamics. The
stochastic ensemble of pure states ρh = ψ ψ obeys the following dynamics:
t | tih t|
N
1
i~∂ ρh = Hˆ +ξ Aˆ h (t) i A hI(t) +Oˆ Oˆ Oˆ† ρh h.c. (8)
t t 0 i i − h iit i − 2h − it t −
" #
Xi=1(cid:16) (cid:17)
Taking the expectation value to compute the dynamics for the density matrix ρ = E[ρh], one
t t
obtains in general a non linear evolution for the ensemble, which implies the possibility of faster
than light signaling [17]. This can be avoided with a proper choice of the operator Oˆ. Contrary to
the white-noise case, identifying thecorrect form of Oˆ is very difficult(in general, impossible)since
the dependence of the right-hand-side of the above equation on the noise h is highly nontrivial.
This means that one is not able to compute the stochastic average and without such knowledge,
Oˆ cannot be determined. A way to circumvent the problem is to proceed perturbatively [29]. We
Taylor-expand ρh in terms of ξ:
t
ρh = ρh +ξρh +ξ2ρh + (ξ3) (9)
t 0,t 1,t 2,t O
where, for t = 0, all terms except the first one are zero. We expand also Oˆ in powers of ξ [44]:
Oˆ = ξOˆ +ξ2Oˆ + (ξ3). (10)
1 2
O
4
The perturbation expansion of Eq. (8) gives the following system of equations
i~∂ ρh =Hˆ ρh h.c.
t 0,t 0 0,t−
N
1
i~∂ ρh =Hˆ ρh + (Aˆ h (t) i A 0hI(t)+Oˆ + Oˆ Oˆ† 0 ρh h.c.
t 1,t 0 1,t i i − h iit i 1 2h 1 − 1it 0,t−
!
i=1
X
N
1
i~∂ ρh =Hˆ ρh + (Aˆ h (t) i A 0hI(t)+Oˆ Oˆ Oˆ† 0 ρh
t 2,t 0 2,t i i − h iit i 1− 2h 1 − 1it 1,t
!
i=1
X
N
1 1
i A 1hI(t) Oˆ + Oˆ Oˆ† 1+ Oˆ Oˆ† 0 ρh h.c. (11)
− h iit i − 2 2h 1− 1it 2h 2 − 2it 0,t−
!
i=1
X
where A n = Tr(Aˆρh ), and similarly for the other operators. We can formally solve the above
h it n,t
system of equations:
ρh =eiHˆ0tρ e−iHˆ0t
0,t 0
i N t 1
ρh = dτ Aˆ (τ t)h (τ) i A 0hI(τ)+Oˆ (τ t) O O† 0 ρh +h.c.
1,t − ~ ˆ i − i − h iiτ i 1 − − 2h 1− 1iτ 0,t
i=1 0 (cid:18) (cid:19)
X
i N t 1
ρh = dτ Aˆ (τ t)h (τ) i A 0hI(τ)+Oˆ (τ t) O O† 0 eiHˆ(t−τ)ρh e−iHˆ(t−τ)
2,t − ~ ˆ i − i − h iiτ i 1 − − 2h 1− 1iτ 1,τ
i=1 0 (cid:18) (cid:19)
X
i N t 1 1
dτ i A 1hI(τ) Oˆ (τ t)+ Oˆ Oˆ† 1 + O O† 0 ρh +h.c.
− ~ ˆ h iiτ i − 2 − 2h 1 − 1iτ 2h 2 − 2iτ 0,t
i=1 0 (cid:18) (cid:19)
X
(12)
where Aˆ (t) is the operator Aˆ in the interaction picture at time t:
i i
Aˆ (t) = e~iHˆ0tAˆ e−~iHˆ0t, (13)
i i
and similarly for the operator Oˆ.
Now we are in the position to compute a closed equation for the averaged density matrix
E[ρh]. We plug the solutions in Eq. (12) into Eq. (9); in this way the stochasticity is entirely
t
contained in polynomials of h, whose correlations are known. We can then explicitly compute
the stochastic average of each term. Collecting all pieces together, we arrive at the following
perturbative equations for the ensemble, which is valid up to order ξ2.
i~∂ ρ =Hˆ ρ h.c.
t 0,t 0 0,t
−
1
i~∂ ρ =Oˆ ρ + O O† 0ρ h.c.
t 1,t 1 0,t 2h 1− 1it 0,t−
i N t
i~∂ ρ = dτ S (t,τ)(Aˆ A 0)(Aˆ (τ t) A (τ t) 0)ρ
t 2,t − ~ ˆ ij i −h iit j − −h j − it 0,t
0
i,j=1
X
i N t
+ dτ D (t,τ)(Aˆ (τ t) A (τ t) 0)ρ (t,τ)(Aˆ A 0)
~ ˆ ij j − −h j − it 0,t i −h iit
0
i,j=1
X
i N t
+ dτ (S (t,τ) D (t,τ)) A (A (τ t) A (τ t) 0) 0ρ
~ ˆ ij − ij h i j − −h j − it it 0,t
0
i,j=1
X
1
+Oˆ + O O† 0ρ h.c. (14)
2 2h 2 − 2it 0,t−
5
The above equations are again non-linear. The non-linear terms can be removed by choosing
Oˆ =0
1
i N t
Oˆ =+ dτ(S (t,τ) D (t,τ))Aˆ (Aˆ (t τ) A (t τ) 0)
2 ~ ˆ ij − ij i j − −h j − it
0
i,j=1
X
i N t
dτ(S (t,τ) D∗(t,τ)) A 0Aˆ (τ t) (15)
− ~ ˆ ij − ij h iit j −
0
i,j=1
X
Substituting this expression in Eq. (14) and re-summing the Taylor series one arrives at:
i ξ2 t N
∂ ρ = Hˆ ρ dτ D (t,τ) Aˆ Aˆ (τ t)ρ Aˆ (τ t)ρ Aˆ + (ξ3)+h.c.
t t − ~ 0 t− ~2 ˆ ij i j − t− j − t i O
0
iX,j=1 (cid:16) (cid:17)
(16)
or equivalently:
i ξ2 N t
∂ ρ = [Hˆ ,ρ ] dτ DR(t,τ)[Aˆ ,[Aˆ (τ t),ρ ]]+
t t −~ 0 t − ~2 ˆ ij i j − t
0
i,j=1
X (17)
iξ2 N t
dτ DI (t,τ)[Aˆ , Aˆ (τ t),ρ ]+ (ξ3),
− ~2 ˆ ij i j − t O
0
iX,j=1 n o
where the superscript R/I stands for the real/imaginary part. Note that the correlator S (t,τ)
ij
does not appear. The collapse equation for the wave function turns out to be, up to second order
in ξ2:
N
i~∂ ψ = Hˆ +ξ (Aˆ h (t) i A hI(t)
t| ti 0 i i − h iit i
" !
i=1
X
iξ2 N t
+ dτ (S (t,τ) D (t,τ))Aˆ (Aˆ (t τ) A (t τ) )
~ ˆ ij − ij i j − −h j − it
0
i,j=1
X
iξ2 N t
dτ(S (t,τ) D∗(t,τ)) A Aˆ (τ t)
− ~ ˆ ij − ij h iit j −
0
i,j=1
X
iξ2 N
dτ(S (t,τ) D (t,τ))( A A (τ t) 2 A A (τ t) )
− 2~ ˆ ij − ij h i j − it− h iith j − it
i,j=1
X
iξ2 N
dτ(S∗(t,τ) D∗(t,τ))( A (τ t)A 2 A A (τ t) ) ψ . (18)
−2~ ˆ ij − ij h j − iit− h iith j − it | ti
i,j=1
X

Performing the Markovian limit by imposing D (t,s) = δ(t s)D˜ (t) and S (t,s) = δ(t s)S˜ (t)
ij ij ij ij
− −
6
one ends up with the following stochastic Schro¨dinger equation in the Stratonovich form:
N
i~∂ ψ = Hˆ +ξ (Aˆ h (t) i A hI(t)
t| ti 0 i i − h iit i
" !
i=1
X
iξ2 n 1
+ (S˜ (t) D˜ (t))[(Aˆ A )(Aˆ A )+ ( A A + A A 2 A A )]
~ ij − ij i −h iit j −h jit 2 h i jit h j iit− h iih ij
i,j=1
X
iξ2 N i2ξ n
(S˜I (t) D˜I (t))( A A 2 A A )+ D˜I (t) A Aˆ ψ . (19)
− ~ ij − ij h i jit− h j iit ~ ij h iit j| ti
i,j=1 i,j=1
X X

This equation is a generalization of Eq. (7.43) in [11]. The first two lines correspond to Eq. (7.43),
with the replacement γ S˜ (t) D˜ (t), taking also into account that in our case the operators
ij ij
→ −
A are not assumed to commute; the third line is associated to the complex part of the noise, while
i
in [11] the noise was assumed to be real.
In the next sections we will discuss the main consequences of Eqs. (17) and (18): the collapse
of the wave function, the presence, under certain conditions, of an amplification mechanism and
the experimental predictions.
III. COLLAPSE OF THE WAVE FUNCTION
Wenowestablishunderwhichconditionsthedynamicsgiven byEq.(18), whenH = 0, induces
0
the collapse of the state vector ψ into one of the eigenstate of Aˆ , assuming that these operators
| it i
commute with each other and therefore have a common set of eigenstates. We will follow the
procedure outlined in Sec. IIa of [29]. We neglect the standard evolution since we are focusing on
the collapse process. This approximation, in general not true, is good for macroscopic objects. In
fact, given theamplification mechanism, whichwewilldescribein thenextsection, theeffect ofthe
collapse increases with the mass of the system, making it dominant with respect to the standard
evolution for large objects.
We consider the stochastic average of the variance V (t) = Aˆ2 Aˆ 2 of an operator Aˆ which
A h it−h it
commutes with each Aˆ . One may prove that, for any n:
i
E[ Aˆn ] = Tr[ρ Aˆn] = Tr[ρ Aˆn] = E[ Aˆn ]. (20)
t t 0 0
h i h i
Then, exploiting the perturbation series in Eq. (12) and performing the stochastic average one can
obtain:
E[ A 2] = E[ A ]
h it h i0
2ξ2 N t τ
dτ ds SR(t,τ) DR(t,τ) A (A A A (A A + (ξ3). (21)
− ~2 ˆ ˆ ij − ij hh i0 i −h ii0i0hh i0 j −h ji0i0 O
0 0
iX,j=1 (cid:16) (cid:17)
Given the above result, one can now compute the stochastic average of the variance V (t), arriving
A
at:
2ξ2 N t
E[V (t)] =V (0) dτ F (τ) AA A A AA ) A A + (ξ3)
A A − ~2 ˆ ij h ii0−h i0h ii0 h j i0−h i0h ji0 O
0
iX,j=1 (cid:16) (cid:17)(cid:16) (cid:17)
(22)
7
where:
τ
Fij(τ) = ˆ dS DiRj(τ,s)−SiRj(τ,s) . (23)
0
(cid:0) (cid:1)
Accordingto[45]thepositivityofF(x,y,τ)inthelimitt isasufficientconditiontoguarantee
→ ∞
the reduction properties of Eq. (18). In fact, whenever F is non negative, Eq. (22) implies that,
for large times AA A A converges to 0 for any realization of the noise, with the only
i τ τ i τ
h i −h i h i
possible exception of a subset of measure 0. In particular, when Aˆ is equal to A we have
i
lim A A A A = lim V (t)= 0. (24)
t→∞h i iiτ −h iiτh iiτ t→∞ Ai
This means that any initial state converges asymptotically, with probability 1, to one of the eigen-
states of the operator Aˆ .
i
A related question is how fast the wave function collapses. The decoherence rate of the asso-
ciated master Eq. (17) provides a good measure. If we set Hˆ = 0, we immediately obtain the
0
decoherence rate in the basis of the common eigenstates of the operators Aˆ :
i
ξ2 N t τ
ρ (α,β) =exp dτ dsDR(τ,s)(α α α β α β +β β )
t −~2 ˆ ˆ ij i j − i j − j i j i
0 0
i,j=1
X

+iDI (τ,s)(α α +α β +α β +β β ) ρ (α,β)
ij i j i j j i j i 0
!
ξ2 N t τ
=exp dτ dsD (τ,s)(α α β β ) D∗ (τ,s)(α β +α β ) ρ (α,β)
−~2 ˆ ˆ ij i j − i j − i,j i j j i  0
0 0
i,j=1
X
  (25)
where ρ (α,β) = α ρ β and α (β ) is one element of the basis, i.e. Aˆ α = α α . It is worth
t t i i
h | | i | i | i | i | i
studying the case where there is only one collapse operator and the correlation is real and delta
correlated in time, i.e.
D(τ,s) = τ δ(τ s) (26)
0
−
with τ a real parameter with the dimensions of a time. Then Eq. (25) reduces to:
0
2
ρt(α,β) = e−ξ~τ20t(α−β)2ρ0(α,β) (27)
where the decoherence rate is constant in time and is determined by τ ξ2.
0
IV. MASTER EQUATION FOR THE CENTER OF MASS AND THE AMPLIFICATION
MECHANISM
After the collapse of the wave function, the next fundamental requirement for a good collapse
model is the amplification mechanism: the center of mass wave function of a composite system
should collapse with a rate which increases with the size of the system. This is necessary in order
for the equation to preserve the quantum properties of microscopic systems and, at the same time,
to guarantee the classical properties of macroscopic objects.
8
Instead of considering the problem in full generality as done in the previous two sections, we
focus our analysis to the case of interest here: the collapse noise coupled to the mass density
operator mˆ(x). In this case Eq. (17) takes the form:
i
∂ ρ = [Hˆ ,ρ ]+
t t −~ 0 t
ξ2c4 t
− ~2 ˆ dxˆ dyˆ dτ DR(x−y,t−τ)[mˆ(x),[mˆ(y,τ −t),ρt]]+ (28)
0
iξ2c4 t
dx dy dτ DI(x y,t τ)[mˆ(x), mˆ(y,τ t),ρ ],
− ~2 ˆ ˆ ˆ − − { − t}
0
where DR and DI are the real and the imaginary parts of the correlation function of the noise
field [46] D(x,y;t,τ) = E[h∗(x,t)h(y,τ)]. In writing the above equation, we assumed that the
noise is statistically homogeneous over space and time: DR,I(x,y,t,τ) = DR,I(x y,t τ). We
− −
consider a system of N point-like particles. The the mass density function is:
N N
m
mˆ(x) = m δ(x xˆ ) = i dQe~iQ·(x−xˆi). (29)
i − i (2π~)3 ˆ
i=1 i=1
X X
Substituting Eq. (29) into Eq. (28) and performing the integration over x and y we arrive at the
expression:
i
∂ ρ = Hˆ ,ρ +
t t −~ 0 t
ξ2c4h N mi m t
− ~2 (2πi~)j3 ˆ dτ ˆ dQD˜R(Q,t−τ) e−~iQ·xˆi, e~iHˆ0(τ−t)e~iQ·xˆje−~iHˆ0(τ−t),ρt +
0
iX,j=1 h h ii
iξ2c4 N m m t
− ~2 (2πi~)j3 ˆ dτ ˆ dQD˜I(Q,t−τ) e−~iQ·xˆi, e~iHˆ0(τ−t)e~iQ·xˆje−~iHˆ0(τ−t),ρt
0
iX,j=1 h n oi
(30)
where we defined:
D˜β(Q,t τ):= drDβ(r,t τ)e~iQ·r with β = R, I. (31)
− ˆ −
We are interested in describing the dynamics of the center of mass of the composite system. In
particular, we have in mind the case of a rigid body. We introduce the center of mass coordinates:
N N
m
Xˆ = ixˆ , Pˆ = qˆ , (32)
i i
M
i=1 i=1
X X
(33)
and the relative coordinates
m
rˆ = xˆ Xˆ i (1,...,N 1), pˆ = qˆ iPˆ i (1,...,N 1),
i i − ∈ − i i− M ∈ −
N−1 m N−1
 rˆN = − mNi rˆi, pˆN = − pˆi,
Xi=1 Xi=1
 (34)

9
whereM = N m isthetotalmassofthesystem. Theoperatorsrˆ andpˆ arenotindependent
i=1 i N N
(they are definedin terms of the other relative positions and momenta) butit is convenient to keep
P
them to make the notation simpler. These new variables obey to the following commutation
relations:
m
[Xˆ ,Pˆ]= i~ [rˆ ,pˆ ] = i~ δ i
i j ij
− M
(cid:16) (cid:17)
[Xˆ ,rˆ]= [Xˆ ,pˆ ] = [rˆ ,rˆ ] = [rˆ ,Pˆ] = 0. i,j (1,...,N 1) (35)
i i i j i
∈ −
We introduce the center of mass density matrix as
ρCM := Tr (ρ )
t REL t
where Tr () denotes the partial trace over the relative coordinates. We study the effect of the
REL
·
partial trace on the operators in the three lines of Eq. (30). Assuming that Hˆ = HˆCM+HˆREL, the
0 0 0
term in the first line simplifies as
Tr ([HˆCM +HˆREL,ρ ]) = [HˆCM,ρCM].
REL 0 0 t 0 t
The double commutator in the second line can be expanded as the sum of four terms:
TrREL e−~iQ·xˆi, e~iHˆ0(τ−t)e~iQ·xˆje−~iHˆ0(τ−t),ρt = TrREL e−~iQ·xˆie~iHˆ0(τ−t)e~iQ·xˆje−~iHˆ0(τ−t)ρt
Tr (cid:16)h e~iHˆ0(τh−t)e~iQ·xˆje−~iHˆ0(τ−t)ρ e−~iQ·xˆi ii(cid:17)Tr e−(cid:16)~iQ·xˆiρ e~iHˆ0(τ−t)e~iQ·xˆje−~iHˆ0(τ−t) (cid:17)
REL t REL t
− −
+Tr (cid:16)ρ e~iHˆ0(τ−t)e~iQ·xˆje−~iHˆ0(τ−t)e−~iQ·xˆi (cid:17) (cid:16) (cid:17)(36)
REL t
(cid:16) (cid:17)
We consider the first term, as the calculations for the remaining terms are similar. Exploiting the
commutativity of the relative and center of mass degree of freedoms, we rewrite the exponential
operators in Eq. (36) as
e−~iQ·xˆi = e−~iQ·Xˆe−~iQ·rˆi, e~iHˆ0(τ−t) = e~iHˆ0CM(τ−t)e~iHˆ0REL(τ−t), (37)
so that
Tr e−~iQ·xˆie~iHˆ0(τ−t)e~iQ·xˆje−~iHˆ0(τ−t)ρ = (38)
REL t
(cid:16) (cid:17)
Tr e−~iQ·rˆie~iHˆ0REL(τ−t)e~iQ·rˆje−~iHˆ0REL(τ−t) e−~iQ·Xˆe~iHˆ0CM(τ−t)e~iQ·Xˆe−~iHˆ0CM(τ−t) ρ .
REL t
(cid:16)h ih i (cid:17)
We assume the motion of the relative coordinates to be a small fluctuation around the equilibrium
positions r within the solid (e.g. in a crystalline structure), i.e. rˆ(t) = r +∆rˆ(t), where the
i0 i i0 i
fluctuations ∆rˆ(t) are negligible with respect to the spatial correlation length of the noise [47]
i
within the time t τ . Under this approximation, the first square bracket in the second line of
Eq. (38) becomes e−−~iQ·(ri0−rj0) and we obtain:
TrREL e−~iQ·xˆie~iHˆ0(τ−t)e~iQ·xˆje−~iHˆ0(τ−t)ρt
(cid:16) (cid:17)
e−~iQ·(ri0−rj0)e−~iQ·Xˆe~iHˆ0CM(τ−t)e~iQ·Xˆe−~iHˆ0CM(τ−t)ρCM,
≃ t
10
which depends on center of mass operators only. The other three terms can be computed in the
same way and therefore we get:
Tr e−~iQ·xˆi, e~iHˆ0(τ−t)e~iQ·xˆje−~iHˆ0(τ−t),ρ
REL t
(cid:16)h h ii(cid:17)
= e−~iQ·(ri0−rj0) e−~iQ·Xˆ, e~iHˆ0CM(τ−t)e~iQ·Xˆe−~iHˆ0CM(τ−t),ρCM .
t
h h ii
Similarly, for the operators in the third line of Eq. (30) we obtain:
Tr e−~iQ·xˆi, e~iHˆ0(τ−t)e~iQ·xˆje−~iHˆ0(τ−t),ρ
REL t
(cid:16)h n oi(cid:17)
= e−~iQ·(ri0−rj0) e−~iQ·Xˆ, e~iHˆ0CM(τ−t)e~iQ·Xˆe−~iHˆ0CM(τ−t),ρCM .
t
h n oi
Combining the previous results, we arrive at the following master equation for the center of mass
i
∂ ρCM = HˆCM,ρCM +
t t −~ 0 t
ξ2c4h 1 tidτ dQD˜R(Q,t τ)A(Q) e−~iQ·Xˆ, e~iHˆ0CM(τ−t)e~iQ·Xˆe−~iHˆ0CM(τ−t),ρCM +
− ~2 (2π~)3 ˆ ˆ − t
0
h h ii
iξ2c4 1 tdτ dQD˜I(Q,t τ)A(Q) e−~iQ·Xˆ, e~iHˆ0CM(τ−t)e~iQ·Xˆe−~iHˆ0CM(τ−t),ρCM
− ~2 (2π~)3 ˆ ˆ − t
0
h n oi
(39)
with:
N
A(Q):= mimje−~iQ·(ri0−rj0) = ρ(Q/~)2 (40)
| |
i,j=1
X
where
ρ(k) := dxρ(x)e−ikx (41)
ˆ
is the Fourier transform of the classical mass density distribution ρ(x) := N m δ(x rcl).
i=1 i − i
The master equation (39) for the center of mass wave function has the same structure as the
P
single particle master equation, with the addition of the amplifying factor A(Q), which keeps track
of the fact that we are dealing with a composite object, not a point-like particle.
Typically, the noise correlatiors DR(r,t τ) and DI(r,t τ) are expected to have spatial cutoffs
− −
(the noise correlation length), respectively rR and rI . As for the case of the CSL model, it is
C C
interesting to study the behavior of the amplification factor in two limiting cases (for a more
detailed proof of what follows, see [30]):
1. When the particles are at distances smaller than the noise correlation lengths rR, rI , they
C C
contribute coherently, giving a factor ( m )2;
∝ i i
2. When the particles are at distance largePr than the noise correlation lengths rR, rI , they
C C
contribute incoherently giving a factor m2.
∝ i i
P