Table Of ContentThe Generalized Schr¨odinger-Langevin equation
Pedro Barguen˜o1 and Salvador Miret–Art´es2
1
Departamento de F´ısica, Universidad de los Andes, Apartado A´ereo 4976,
Bogot´a, Distrito Capital, Colombia ([email protected])
2
Instituto de F´ısica Fundamental, CSIC, Serrano 123, 28006, Madrid, Spain (s.miret@iff.csic.es)
Inthiswork,wederiveageneralization oftheso-called Schr¨odinger-LangevinorKostinequation
for a Brownian particle interacting with a heat bath. This generalization is based on a nonlinear
interaction model providing a state-dependent dissipation process exhibiting multiplicative noise.
Two straightforward applications to the measurement process are then analyzed, continuous and
weakmeasurementsintermsofthequantumBohmiantrajectoryformalism. Finally,itisalsoshown
that the generalized uncertainty principle, which appears in some approaches to quantum gravity,
4 can be expressed in terms of thisgeneralized equation.
1
0
2
Quantum stochasticity constitutes a very broad and quasi-particle tunneling in Josephson systems [9], in the
n active field of research within quantum mechanics. Real Langevin canonical formulation of chiral two level sys-
a physical systems do not exist in complete isolation and tems [10], in atom surface scattering [11] and so on.
J
one then speaks about open quantum systems [1–3]. Within the Markovian regime, the standard Langevin
7
Thus, the interaction of a quantum system with its en- equation with multiplicative noise is reached. The main
1
vironment can not be totally neglected leading to an en- purpose of this work is to derive a generalization of the
] tanglement between them. The corresponding theory SL or Kostin equation for nonlinear dissipation which is
h
encompasses a series of formalisms and approaches de- termed the generalized SL equation (GSLE).
p
veloped to deal with this complex but fundamental is- Once this equation is established, three straightfor-
-
t sue. Three main approaches are usually considered in ward applications dealing with two measurement pro-
n
a this context: (i) effective time-dependent Hamiltonians, cesses and quantum gravity are analyzed. The first one
u (ii) nonlinear (logarithmic) Schr¨odinger equations and considers continuous measurement. As is known, the
q (iii) the system-plus-bath model within a conservative mere presence of an observing apparatus should con-
[
scenario. Obviously, links among them can be found. siderable affect the behavior of the measuring system.
1 For example, in the last approach, and for one dimen- These frequent measurements are at the origin of the
v sional systems, the so–called Caldeira-Leggett Hamilto- so-called Zeno [12–14] and anti-Zeno effects. [15, 16]
5
nian [4] is the starting point leading to the generalized Veryrecently,Nassar[17]hasproposedanonlinearloga-
0
Langevin equation (GLE). One of the key issues is the rithmicSchr¨odingerequationundercontinuousmeasure-
4
4 interaction term which by construction is linear in the ment as a generalization originally due to Mensky [18]
. bath coordinates. The dependence on the system vari- and Bialynicki-Birula and Mycielski [19]. The establish-
1
ableisthroughafunctionf(x)whichusuallyisseparable mentofadividinglinebetweentheclassicalandquantum
0
4 and linear (linear dissipation). This scenario is known regimes is one of the main aspects of the measurement
1 as a state–independent dissipation and can be seen as process. [20]
v: a measurement of particle’s position by a reservoir in Thesecondapplicationfocusesonweakmeasurements.
i von Neumann’s sense. [1] This function also appears in For this goal,it is pertinent to analyze the consequences
X
an additional term in the total Hamiltonian in order to of this GSLE from a hydrodynamical point of view or
r avoidtherenormalizationoftheinteractionpotential. In following Bohmian mechanics [3]. This can be carried
a
the MarkovianregimewithOhmicfriction,thisGLEbe- outbyreplacingthewavefunctionψ (andψ∗)byitsreal
comes the standard Langevin equation for a Brownian amplitude and real quantum phase. As it is well known,
particle where noise is additive. Kostin [5] established whenconservativesystemsareconsidered,thegradientof
the link between this standard Langevin equation with this quantum phase gives the momentum of the particle
the Schr¨odinger equation leading to a nonlinear, loga- from the so-called guiding condition (see, for example,
rithmic equation termed the Schr¨odinger-Langevin (SL) Ref. [3]). Weakvalues wereproposedbyAharonovet al.
or Kostin equation. This type of equations are quite dif- [21]andareconsideredto playakeyroleinfundamental
ferent from others existing in the literature such as the problems of quantum mechanics (see, for example, Refs.
so–called stochastic Schr¨odinger equation and the Lin- [22, 23]). Even more, as has recently pointed out by
blad equation for the density matrix. [2] Hiley [24] in the context of Bohmian mechanics, these
For a nonlinear function f(x), the open quantum weakvaluesaremerelytransitionprobabilityamplitudes
system displays a state dependent dissipation process which,ingeneral,arecomplexmagnitudes. Inparticular,
andthecorrespondingGLEexhibitsmultiplicativenoise. fortwosequentialmeasurements(oneweakandtheother
[6, 7] Typical examples of nonlinear functions take one strong) of complementary observables such as, for
place, for example, in rotational tunneling systems [8], example,momentumandposition,therealpartprovides
2
usthevelocityoftheBohmianparticleandtheimaginary where the time–dependent friction (memory kernel) is
parttheso-calledosmoticvelocityduetothe presenceof given by
a gradient of the quantum probability. On the contrary,
when we are dealing with open quantum systems, the α(t)= 1 d2i cos(ω t) (6)
new guiding condition is issued from our the previous m i miωi2 i
derived GSLE. X
Finally, the third application deals with the general- and the external force (noise term) is expressed as
ized uncertainty principle (GUP), which appears in the
d
contextoftheunificationofquantummechanicsandgen- ξ(t)=− d x (0)+ i f(0) cos(ω t)
i i m ω2 i
eralrelativityinseveralproposals(see[25]andreferences i (cid:20)(cid:18) i i (cid:19) (cid:21)
X
therein). This principle gives place to deformed commu- p (0)
i
−d sin(w t) . (7)
tation relations which are linear or quadratic in particle i i
m ω
momenta. Inthelinearcase,whichcorrespondstodouble Xi (cid:20) i i (cid:21)
special relativity theories [26], this fact leads to express In the Markovian regime, the memory kernel is a δ–
the corresponding dynamics in terms of a gravitational function in time, giving place to an Ohmic dissipation
frictionwithinthe GLEframeworkwithagivenfunction with a time–independent friction. Within this regime
f(x). [25] Therefore, the corresponding GSLE can then andforanonlinearcoupling,thecorrespondingstandard
be easily derived in terms of this particular function. Langevin equation reads as
Without loss of generality, a one dimensional problem
is considered. For open systems, it is usual to split the mx¨(t)+mα[f′(x)]2x˙(t)+V′(x)=f′[x(t)]ξ(t). (8)
totalHamiltonianintothreepartsincludingsystem,bath
and mutual coupling, in such a way that Noticethattherandomforceismultipliedbythe deriva-
tiveofthe functionf(x)givingplacetoastochasticpro-
H =Hs+Hb+Hsb, (1) cess with multiplicative noise. When the system–bath
coupling is linear, that is, for f(x) = x, the standard
where
Langevin equation for additive noise (that is, when the
p2 noisetermisnotmultipliedbyanysystemfunction)with
H = +V(x) (2)
s 2m Ohmic friction is recovered
standsfortheHamiltonianoftheisolatedsysteminpres- mx¨(t)+mαx˙(t)+V′(x)=ξ(t). (9)
ence of a force field given by the potential V(x);
Followingthe Kostinprocedure[5],the generalizedSL
1 p2
H = i +m ω2x2 (3) equation(that is, for anyf(x)) can be obtainedby writ-
b 2 i (cid:18)mi i i i(cid:19) ing first the Schr¨odinger equation as
X
istheHamiltonianforthebath,whichactsasareservoir, 2
∂ψ 1 ∂
and can be represented as an infinite set of harmonic i~ = −i~ +V(x)+Vd+Vr ψ, (10)
∂t 2m ∂x
oscillators; and " (cid:18) (cid:19) #
f2(x)d2 where Vd and Vr are the dissipative and random poten-
Hsb = m ω2i −2dif(x)xi (4) tials to be specified later on. The quantum mechanical
i (cid:20) i i (cid:21) current is defined as
X
expressingthe interactiontermbetweenthe isolatedsys-
1 ∂ ∂
tem and the bath, d being appropriate coupling con- J = ψ∗ −i~ ψ+ψ −i~ ψ∗ (11)
i 2m ∂x ∂x
stants. Thefunctionf(x)is,ingeneral,anonlinearfunc- (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
tion of the system variable x. The term with the square
∗
Then,from∂ψ/∂tand∂ψ /∂tandEq. (10)wefindthat
of f(x) gives the so–called counter term introduced to
compensate the renormalizatonof the potential. Follow- d
ing the standard procedure where the bath degrees of hxi= Jdx
dt
freedom are eliminated, the equation of motion for the Z
d2 ∂J
corresponding system dynamics in the Heisenberg pic- hxi= dx. (12)
dt2 ∂t
ture of quantum mechanics is given by the GLE [4] Z
′ ′ where<.> is the expectationvalue ofa givenoperator.
f [x(t)]ξ(t)=mx¨(t)+V (x)
Now, by performing the same type of averaging into
t
′ ′ ′ ′ ′ ′ Eq. (8)
+mf [x(t)] dtα(t−t)f [x(t)]x˙(t)
Z0
′ 2 ′ ′
(5) mhx¨(t)i+mαh[f (x)] x˙(t)i+hV (x)i=hf (x)ξ(t)i, (13)
3
and comparing it with Eq. (10) and using Eq. (12), we where the new quantum mechanical current is now de-
can identify terms leading to fined as
J˜≡f′(x)2J (16)
∂V
ψ∗ − d ψdx=αm f′(x)2Jdx. (14)
∂x which is coupling–dependent. On the other hand, the
Z (cid:18) (cid:19) Z
generalized random potential due to the heat bath cor-
responding to the random force ξ(t) can be written as
Thus, the damping potential is a functional of the wave
function and can be expressed as
V =−f(x)ξ(t). (17)
r
J˜ Finally, the corresponding GSLE can be then expressed
∗
V [ψ,ψ ,f]=−mα dx, (15) by
d ψψ∗
Z
∂ψ 1 ∂ 2 J˜
i~ = −i~ +V(x)−mα dx−f(x)ξ(t)−W(t) ψ. (18)
∂t 2m ∂x ψψ∗
" (cid:18) (cid:19) Z #
where W(t)=hV i arises from the requirement that the pling function f is assumed to be linear in the system
d
integration of Eq. (15) with respect to x must be equal variable, Eq. (18) reduces to the standard SL or Kostin
to the expectation values of the kinetic and potential equation [5].
energies through the total Hamiltonian. As mentioned A new and straightforward generalization of Eq. (18)
by Kostin [5], this term can be removed from Eq. (18) is when continuous measurement is considered. As has
by introducing the transformation of the wave function been recently shown, this process can also be described
ψ(x,t) = eiθ(t)φ(x,t). We also note that when the cou- byanonlinearlogarithmicSchr¨odingerequation[17,20].
Thus, we have
∂ψ 1 ∂ 2 J˜
i~ = −i~ +V(x)−mα dx−f(x)ξ(t)−W(t)+W (x,t) ψ. (19)
∂t 2m ∂x ψψ∗ κ
" (cid:18) (cid:19) Z #
where W (x,t) = −i~κ[ln|ψ(x,t)|2 −hln|ψ(x,t)|2i] and coupled equations,
κ
κ gives the resolution of the continuous measurement.
2
These two basic decoherence mechanisms are thus put ∂S 1 ∂S
=− −(V +V +V +Q) (21)
d r
on equal footing. This more general equation should be ∂t 2m ∂x
(cid:18) (cid:19)
applicable to the Zeno and anti-Zeno effects in presence ∂A2 1 ∂S∂A2 A2∂2S
=− − , (22)
of an environment displaying nonlinear dissipation and, ∂t m∂x ∂x m ∂x2
ingeneral,totheso-calledenvironmentinduceddecoher-
where
ence. [27]
Our next goalis to propose an equationgoverningthe ~2 ∂2A
Q(x,t)=− (23)
weak measurement process in presence of a heat bath. 2mA ∂x2
Forthis purpose,insteadofdealingwiththe twofieldsψ
∗ is the quantum potential. Note that the first and second
andψ ,Eq. (18)canbewrittenintermsofthehydrody-
equationsarethequantumHamilton-Jacobiandcontinu-
namicalorquantumtrajectoryformulationbyexpressing
ityequations,respectively. Moreover,thecurrentdensity
the wave function in polar form with a real amplitude
is expressed in this formalism as
A(x,t) and real phase S(x,t) as
∗
ψψ ∂S
J = . (24)
ψ(x,t)=A(x,t)eiS(x,t)/~. (20) m ∂x
As is known, the gradient of the wave function phase
Then, Eq. (10) can now be split into a system of two isassociatedwiththetrajectorymomentumbymeansof
4
p(x,t) = ∂S(x,t)/∂x (the guiding condition). By differ- Moreover,the partial integration in Eq. (27) leads to
entiating Eq. (21), the time evolution of p(x,t) is given
by df 2 df 2 df d2f
pdx= S−2 S dx, (28)
dx dx dxdx2
Z (cid:18) (cid:19) (cid:18) (cid:19) Z
∂p p ∂p ∂
=− − (V +Vd+Vr +Q). (25) which gives place naturally to the coupling–dependent
∂t m∂x ∂x
phase
According to Eq. (8), which can be interpreted in
df 2 df d2f
terms of the Lagrangian framework of hydrodynamics, S˜≡ S−2 S dx. (29)
dx dxdx2
the corresponding quantum Newton-Langevin equation (cid:18) (cid:19) Z
including the dissipative and random sources can be ex-
which can be straightforwardlyexpressed as
pressed as
J˜
∂p =− p ∂p − ∂ (V +Q) S˜=m ψψ∗dx (30)
∂t m∂x ∂x Z
−αf′(x)2p−f′(x)ξ(t). (26) with
V [ψ,ψ∗,f]=−αS˜. (31)
Then, by integrating Eq. (26) with respect to x, we ob- d
tain
Importantly, Eq. (31) includes, as a special case, the
Bohmian version of the Kostin equation when the cou-
− ∂S = p2 +V +Q+α f′(x)2pdx pling is linear. In this case, J˜→ J and Vd[ψ,ψ∗,f] →
∂t 2m ∗
V [ψ,ψ ,x] = −αS, which is the dissipative potential
Z d
+f(x)ξ(t)+C(t), (27) expressed within the Bohmian formalism [20, 28, 29].
TheconstantofintegrationC(t)canbedefinedinsuch
where C(t) is an arbitrary time function resulting from a way that the overallphase of the wavefunction should
the space integrationto be specifiedlater on. This equa- not affect its evolution. This requirement is satisfied
tion gives the evolution of the wave function phase in when C(t) = −αhS˜i. Therefore, C(t) ≡ W(t) and for
presence of damping which corresponds to a generalized ageneralnon–linearcoupling,theGSLEorKostinequa-
Caldeira–Leggett coupling. Clearly, the random poten- tion can be rewritten in terms of the modified quantum
tial is also expressed in this case by Eq. (17). action, S˜, as
2
∂ψ 1 ∂
i~ = −i~ +V(x)−α S˜−hS˜i −f(x)ξ(t) ψ. (32)
∂t 2m ∂x
" (cid:18) (cid:19) #
(cid:16) (cid:17)
Thus, the corresponding generalized Hamilton-Jacobi the so-called osmotic velocity due to the presence of the
equationforthenonlineardissipationisgivenbyEq. (21) gradientofthe quantumprobability. Inthis context,the
with V given by Eq. (31) and V by Eq. (17). Notice, same weak value is obtained in presence of an environ-
d r
however, that the continuity equation is the same ex- mentbutnowψisgovernedbyEq. (18)forlinearand/or
pressed by Eq. (22). As before, the term W could also nonlinear dissipation.
κ
beaddedfordescribingthecontinuousmeasurementpro-
cess.
After Hiley [24], the sequential measurement of the
momentum and position of a particle of mass m is given Finally, let us apply this formalism to the GUP. As
by the weak value or transition probability amplitude shownrecently[25],thedeformedcommutationrelations
hx|p|ψi = ∂S − i ∂A2 (33) [x,p]=i~ 1− mγpcp ,whereγisthedimensionlessGUP
hx|ψi ∂x 2A2 ∂x parameter(cid:16)and mp is(cid:17)the Planck mass, can be expressed
as [x,p] = i~ 1− αp where α ≡ γ/m c is a friction
2 p
where ψ obeys the standard (conservative and linear) coefficient. Moreover, this commutation relation leads
(cid:0) (cid:1)
time-dependentSchr¨odingerequationandisexpressedin to aGLE witha position–dependentcouplingthatturns
polar form according to Eq. (20). The real part is the out to be f(x) = x V′(y)dy. In this case, working
0
Bohmian velocity of the particle and the imaginary part withinthe Bohmianformalismwearriveatthe following
R p
5
damping potential for the GUP,
2γ
V =−αS˜=− pV(x). (34)
d
m c
p
Therefore, the corresponding GSLE can be written as
2
∂ψ 1 ∂ 2γ 2γ
i~ = −i~ +V(x) 1− p − hpV(x)i ψ. (35)
∂t 2m ∂x m c m c
" (cid:18) (cid:19) (cid:18) p (cid:19) p #
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(1987).
eralSL equationvalidfor quantumprocessesinpresence
[10] A.Dorta-Urra,H.C.Pen˜ate-Rodriguez,P.Barguen˜o,G.
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[20] A. Nassar and S. Miret-Art´es, Phys. Rev. Lett. 111,
through Grant No. FIS2011-29596-C02-C01. S. M.-A.
150401 (2013).
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