- Email: [email protected]
Microscopic derivation of a class of non-linear dissipative schrödinger-like equations
Physicu
Publishing
IllA (1982) 364-370 North-Hollnnd
MICROSCOPIC NON-LINEAR
DERIVATION
DISSIPATIVE
Co.
OF A CLASS
SCHRGDINGER-LIKE
OF EQUATIONS
N. GISIN* Dipartemrnf
de Physique
Theoriqrre.
Unicrrsite
Received
The general
formalism
of master equations
the pure states to derive a class of non-linear spin-i coupled
to a bath of two-level
make a connection
with Sz-Nagy’s
t/e GenPre.
6 August
I?!
I GvnPw 4.
1981
is used together Schrodinger-like
with a projection equations.
systems and the Rloch equations theorem
Swifrerland
on the dilations
which preserves
The result is applied to a are recovered.
of contracting
Finally
we
semi-groups.
1. Introduction
The master equation formalism has been widely studies and applied with success to numerous problems involving an irreversible dynamical evolution of an open quantum system C, coupled to its surrounding (bath) X2. The idea is to project the unitary evolution of the whole, system + bath, on the state space of the system alone. The bath, on which the thermodynamic limit has been taken, is assumed to be in a fixed state, not influenced by the interaction master equation thus obtained, which is Hamiltonian Hlz. The generalized formally exact, limits’,‘,3).
is then
simplified
by
some
Markovian
approximations
or
The Pauli master equation is deduced from the partial-trace projection. Note that it does not preserve the pure states. We present another example of the above scheme, in which the pure states are preserved. Such an example is interesting on several accounts. From a fundamental point of view we believe it is important to see by an explicit example that a dissipative quantum evolution is not necessarily stochastic. but may be deterministic in the sense of preserving the pure states4). Moreover, although we use exactly the same approximations and limits that lead to the Pauli master equation, we expect that far from equilibrium large differences exist between the different approaches (section 2). Next our proposal puts in a new perspective Sz-Nagy’s well-known theorem on dilations of contracting semi-groups (section 3). Last, but not least, our approach is a simple example of how a non-linear evolution can be deduced from the linear Schrodinger equation. *Partially
0378437
supported
by the Swiss National
I /82/0000-0000/$02.7S
@
Science
Foundation.
1982 North-Holland
MASTER
2. Formal deduction
EQUATION
FOR STATE VECTORS
365
and exact results
Let %‘, (resp. 2,) be the Hilbert space corresponding to the system (resp. to the bath). We suppose that to the whole, system and bath, is associated the tensor usual
product
X1 @ X2’). The Hamiltonian
of the whole
is assumed
of the
form: H = H,@1++@H~z+hH,z.
(I)
Let XOE 2’2 be the state of the bath which, according of master equations, we assume to be time-independent, E2,yo. As projection
to the general in particular
theory H2x0 =
we choose
P = I @ PO, where
(2)
PO is the projection
The theory of general equation of
onto the state x0. We identify master
equations
gives
X, with P - Xl @I X2.
the formally
exact
evolution
(pt = P e-‘“‘cPo@ x0.
(3)
It reads
+t = - t(H, + APH,,P)cp,
- A*
I
PHn(I
- P)
0 x
exp[ - i(H, + HZ+ h(PHnP
x (I - P)H,zPq,--s
- P)))s]
ds.
(4)
Going over to the interaction functions of the bath decrease approximation Cp,= - i(H, + APHr2P)~, where
+ (I - P)Hn(l
picture, and assuming sufficiently fast, one
that gets
the the
correlation Markovian
- A’(B + iA)cp,,
A and B are two linear
operators,
formally
(5) self-adjoint,
defined
by
D
B + iA =
I
Pei(H1+Hz’sH12(Z- P)e-i(H~+H~s(l - P)HIZP
ds.
(6)
0
The relationship between the eqs. (4) and (5) can be studied by the same techniques as for the usual Pauli master equation. In particular the singular coupling and the van Hove limit can be applied to the eq. (4). In the latter one lets the coupling constant A go to zero, and simultaneously one lets the time t go to infinity in such a way that tA* = T remains fixed; r is called the resealed time. A general theorem due to E.B. Davie8) gives the exact conditions under
N. GISIN
366
which the solution of the equation (4) tends, solution of (5) with the resealed time. With our choice decreasing function
of P, defined of time. But,
when
A goes
to zero,
to the
by (2), the norm of the vector cp, is a as long as cp,f 0, C,C+ defines a ray of the
Hilbert space X,, i.e. ‘p, defines a pure state of the system. In order to avoid possible difficulties with the limit t + 3~ (cpf could tend to zero), and to be consistent with the usual at each time:
The evolution
equation
+,t = -iH&
choice
of normalization,
of $, corresponding
+ AI((
we normalize
to (5) is easily
the vector
cp,
deduced:
- B)&,
(7)
where Ho = H, + APHlzP + A’A and (B),, = (C/JIB/J/~). The equation (7) defines a class of non-linear dissipative Schrodinger-like equations. We shall call them the Master equations for state vectors, since our formal deduction shows a close connection with the usual Pauli master equation. operators,
Using’) one easily proves that if Ho and then the eq. (7) has a unique solution:
B are
both
bounded
-,H”f-~h~RI rlrr =
(‘eiH,t-AtBr
(8)
. e-iHc,t?~Bf),4Jl/2’
Let us analyze this solution in the simple case discrete spectra. First, if Ho and B commute, then i(B),,
Ho and
this projection
is non-vanishing.
B have
(9)
= -2A2((B2)+,-(B);,)~0
and $cIItends as t + 130 to the (normalized) an eigensubspace of B. The corresponding which
where
projection of the initial state & on eigenvalue is the lowest one for Next,
if
Ho and B do not commute,
but if A is small enough, then the asymptotic state I,!J~ is close to an eigenvector C#Iof Ho for which (B), is minimal. As an example let us consider the evolution of one spin-i coupled to a bath of spins-$, i.e. a bath of two-level systems: X,=C?, The whole matrices:
X*=&C’. 1-I is supposed
to be in a magnetic
field,
so that,
with
u the Pauli
MASTER
EQUATION
FOR STATE
VECTORS
where al acts on the ith spin. The coupling Hamiltonian HI2 =
9 Ai(UlU++ da_),
367
is assumed to be
where (++= ux ‘2 ic+y
i=l
and the state of the bath is chosen as
Consequently Pe
PHIZP = 0 and
i(H,+Hz)sH,2,-i(HI+Hz)SH12p
In the thermodynamic limit (N + 00, n + 00, n/N = 8 = constant) the sums become integrals. If the distributions of the A: as functions of the oi are Lorentzian, then, as is well known, the correlation functions of the bath decrease exponentially and one gets m B+iA=h*
II
Oe- i(oo-0)s
e-s/r
c 2l -
(+z +
(1
_
f)),+i(w-~)s,-s/T
++,.p
0
-2 =
(coo-
:2+
l/7: rl
1 i + (i-e);+$wo-o) [
&+ (i- e)i(w,c I
0)
I
.I
err P, I
(10)
where W (resp. ,i) is the mean Larmor frequency (resp. mean coupling constant), and T, is the correlation time. According to (7) and (10) the Markovian master equation for the state vector of our one spin-f system reads:
(11) This equation has been studied in detail in*). The right-hand side of (11) introduces in the corresponding m = (u)$, a term, known as the Landau-Lifshitz friction used by several authors to describe the damping of a
second term on the evolution equation of term, which has been classical spin (see for
N. GISIN
368
example’)).
Note
that
(11) is stable
which are macroscopically
under
variations
undistinguishable.
From
of the state
of the bath
(11) one deduces:
(u;)ti, = th where
(12)
Co is an integration
ingly (a,)$, tends
constant
asymptotically
depending
on the initial
state
$0. Accord-
to 1 (resp.
- l), whenever 0 > 4 (resp. 0
eq. (11) gives a microscopic model of the wellBloch equationn). that if one uses an equation of the same type as (11) of a spin toward the instantaneous direction of a field B(t) = B. + B,(t), where B,(t) is a rotating field, is the so-called quasiperiodic state (or permanent the phenomenological theory of Redfield’<).
3. Inverse problem and dilations In this conditions
section we briefly study the following question: under can an evolution equation of the form of our eq. (7). i.e.
4, = -i HO+, + (WI),,
- HI)&.
with Ho and H, two self-adjoint operators, be associated evolution on a larger space in the sense of section 2?
which
(13) to an
unitary
A simple necessary and sufficient condition is that HI is bounded below. Indeed this condition is necessary since, as mentioned in the previous section, the norm of the solution of eq. (5) decreases with time. Consequently the operator B defined by (6) is necessarily positive. But it may contain a multiple of the identity, and this term cancels in eq. (7) due to the normalization. Thus the operator HI of (13) must be of the form HI = B - M - I, where B is a positive operator, and M a real number. This condition is also sufficient. Indeed if HI is bounded below, then the operator B defined by B = HI + M - I, where M is the lower bound of HI, is positive, and ((B)* - B)$ =
MASTER
EQUATION
FOR STATE VECTORS
369
((HI), - HI)+ for all +. Now exp{( -iHo- B)t} is a contraction semi-group and, according to Sz-Nagy’s theorem”), it can be dilated to a unitary evolution. Accordingly the dissipative evolution equation (13) with H, = k . Ho (k > 0) can be considered as describing the reduced dynamics of a system coupled to a heat bath if and only if Ho is bounded below. This is interesting since with this choice the equation (13) implies d(Ho)/dt = -2k(AHo)*, i.e. the mean value of Ho is a decreasing function of time for all states except the eigenstates. The eigenstates turn out to be asymptotic states, all semi-stable except for the ground-state which is stable (if it exist)“). One can apply this last result to the Hamiltonian of the harmonic oscillator and we then find an evolution which is analogous to the classical damped oscillator: If the initial state is the coherent state defined by the point (~0of the classical phase space, then the solution is at each time a coherent state and the corresponding function (Y,is the solution of the classical equations of the damped oscillator”).
4. Conclusion
We have shown that, by using the same approximations which are necessary to deduce the Pauli master equation, it is possible to construct a master equation for state vectors. The theorems which give the exact relationship between the dynamics of the reduced density matrix and the dynamics of the whole2v6), also apply to our approach. The non-linear term in our eq. (7) comes from the fact that one associates a ray (pure state) of the whole to a ray (pure state) of the system and that representating rays by normalized vectors (defined up to a global phase), this correspondence is not linear. The results of the simple examples of sections 2 and 3 are physically sound. Finally let us make two remarks. First we wish to emphasize that the master equation (4) is reversible, in spite of the appearance of an integral for which one often gives the antropomorphic interpretation of a memory2). Only the Markovian approximation describes an irreversible evolution. The latter approximation is usually applied to the Pauli master equation which governs the evolution of the density matrix of the system. But, and this leads to our second remark, one could a priori also think of making some approximation on the Schrodinger equation which governs the evolution of the state vector of the system. This could be done as presented in this paper, or, at least in a first step, by simply adding non-linear phenomenological terms to the Schrddinger equation, in the same way as one adds friction terms to the the different classical canonical equations. Due to these non-linearities
N. GISIN
370
microscopic towards
distributions
distributions
accordingly, equilibrium suggested
corresponding
which
no evolution however,
one
by the example
to one given
may correspond equation
for
expects
these
of section
2.
density
to different the
density
matrix
density matrix
non-linearities
evolve
matrices exist.
Close
to be irrelevant,
and, to as
Acknowledgement I wish to thank Profs E.B. Davies, for several enlightening discussions.
G.G. Emch,
Ph.A.
Martin
and C. Piron
References I) L. Prigogine and P. Resibois, Physica 27 (1961) 629. F. Haake, Springer Tracts in Modern Physics, Vol. 66, Springer. (Berlin, Heidelberg. New York, 1973). 2) V. Gorini et al., Rep. Math. Phys. 13(1978) 149. 3) Ph.A. Martin, Lecture Notes in Physics 103 (Springer, Berlin, 1979). 4) C. Piron, to be published in Essays in Honor of Wolfgang Yourgrau, Ed. Alwyn van der Merwe (Plenum, New York). 5) D. Aerts, The one and the many, these de doctorat. Universite Libre de Bruxelles. 1981. 6) E.B. Davies, Math. Ann. 219 (1976) 147. 7) M.S. Berger, Nonlinearity and Functional Analysis, Acad. Press. 1977 (Theorem 3.1.23). 8) N. &sin, Spin relaxation and dissipative Schrddinger-like evolution equations, to appear in Helv. Phys. Acta. 9) R. Kubo and N. Hashitsume, Prog. Th. Phys. Suppl. 46 (1970) 210. IO) B. Sz-Nagy and C. Foias. Harmonic Analysis of Operators on Hilbert Space (North-Holland. Amsterdam, 1970). 11) N. Gisin, J. Phys. Al4 (1981) 2259. 12) E.B. Davies, Quantum theory of an open systems (Acad. Press, Oxford, 1976) p. 154. 13) A. Abragam, The Principles of Nuclear Magnetism (Clarendon, Oxford, 1979). 14) R.H. Young and W.J. Deal, J. Math. Phys. ll(l970) 3298. See also G. Lochak and A. Alaoui. Ann. Foundation Louis de Broglie 2 (1977) 87. 15) A.G. Redfield, Phys. Rev. 96 (1955) 1787.
Publishing
IllA (1982) 364-370 North-Hollnnd
MICROSCOPIC NON-LINEAR
DERIVATION
DISSIPATIVE
Co.
OF A CLASS
SCHRGDINGER-LIKE
OF EQUATIONS
N. GISIN* Dipartemrnf
de Physique
Theoriqrre.
Unicrrsite
Received
The general
formalism
of master equations
the pure states to derive a class of non-linear spin-i coupled
to a bath of two-level
make a connection
with Sz-Nagy’s
t/e GenPre.
6 August
I?!
I GvnPw 4.
1981
is used together Schrodinger-like
with a projection equations.
systems and the Rloch equations theorem
Swifrerland
on the dilations
which preserves
The result is applied to a are recovered.
of contracting
Finally
we
semi-groups.
1. Introduction
The master equation formalism has been widely studies and applied with success to numerous problems involving an irreversible dynamical evolution of an open quantum system C, coupled to its surrounding (bath) X2. The idea is to project the unitary evolution of the whole, system + bath, on the state space of the system alone. The bath, on which the thermodynamic limit has been taken, is assumed to be in a fixed state, not influenced by the interaction master equation thus obtained, which is Hamiltonian Hlz. The generalized formally exact, limits’,‘,3).
is then
simplified
by
some
Markovian
approximations
or
The Pauli master equation is deduced from the partial-trace projection. Note that it does not preserve the pure states. We present another example of the above scheme, in which the pure states are preserved. Such an example is interesting on several accounts. From a fundamental point of view we believe it is important to see by an explicit example that a dissipative quantum evolution is not necessarily stochastic. but may be deterministic in the sense of preserving the pure states4). Moreover, although we use exactly the same approximations and limits that lead to the Pauli master equation, we expect that far from equilibrium large differences exist between the different approaches (section 2). Next our proposal puts in a new perspective Sz-Nagy’s well-known theorem on dilations of contracting semi-groups (section 3). Last, but not least, our approach is a simple example of how a non-linear evolution can be deduced from the linear Schrodinger equation. *Partially
0378437
supported
by the Swiss National
I /82/0000-0000/$02.7S
@
Science
Foundation.
1982 North-Holland
MASTER
2. Formal deduction
EQUATION
FOR STATE VECTORS
365
and exact results
Let %‘, (resp. 2,) be the Hilbert space corresponding to the system (resp. to the bath). We suppose that to the whole, system and bath, is associated the tensor usual
product
X1 @ X2’). The Hamiltonian
of the whole
is assumed
of the
form: H = H,@1++@H~z+hH,z.
(I)
Let XOE 2’2 be the state of the bath which, according of master equations, we assume to be time-independent, E2,yo. As projection
to the general in particular
theory H2x0 =
we choose
P = I @ PO, where
(2)
PO is the projection
The theory of general equation of
onto the state x0. We identify master
equations
gives
X, with P - Xl @I X2.
the formally
exact
evolution
(pt = P e-‘“‘cPo@ x0.
(3)
It reads
+t = - t(H, + APH,,P)cp,
- A*
I
PHn(I
- P)
0 x
exp[ - i(H, + HZ+ h(PHnP
x (I - P)H,zPq,--s
- P)))s]
ds.
(4)
Going over to the interaction functions of the bath decrease approximation Cp,= - i(H, + APHr2P)~, where
+ (I - P)Hn(l
picture, and assuming sufficiently fast, one
that gets
the the
correlation Markovian
- A’(B + iA)cp,,
A and B are two linear
operators,
formally
(5) self-adjoint,
defined
by
D
B + iA =
I
Pei(H1+Hz’sH12(Z- P)e-i(H~+H~s(l - P)HIZP
ds.
(6)
0
The relationship between the eqs. (4) and (5) can be studied by the same techniques as for the usual Pauli master equation. In particular the singular coupling and the van Hove limit can be applied to the eq. (4). In the latter one lets the coupling constant A go to zero, and simultaneously one lets the time t go to infinity in such a way that tA* = T remains fixed; r is called the resealed time. A general theorem due to E.B. Davie8) gives the exact conditions under
N. GISIN
366
which the solution of the equation (4) tends, solution of (5) with the resealed time. With our choice decreasing function
of P, defined of time. But,
when
A goes
to zero,
to the
by (2), the norm of the vector cp, is a as long as cp,f 0, C,C+ defines a ray of the
Hilbert space X,, i.e. ‘p, defines a pure state of the system. In order to avoid possible difficulties with the limit t + 3~ (cpf could tend to zero), and to be consistent with the usual at each time:
The evolution
equation
+,t = -iH&
choice
of normalization,
of $, corresponding
+ AI((
we normalize
to (5) is easily
the vector
cp,
deduced:
- B)&,
(7)
where Ho = H, + APHlzP + A’A and (B),, = (C/JIB/J/~). The equation (7) defines a class of non-linear dissipative Schrodinger-like equations. We shall call them the Master equations for state vectors, since our formal deduction shows a close connection with the usual Pauli master equation. operators,
Using’) one easily proves that if Ho and then the eq. (7) has a unique solution:
B are
both
bounded
-,H”f-~h~RI rlrr =
(‘eiH,t-AtBr
(8)
. e-iHc,t?~Bf),4Jl/2’
Let us analyze this solution in the simple case discrete spectra. First, if Ho and B commute, then i(B),,
Ho and
this projection
is non-vanishing.
B have
(9)
= -2A2((B2)+,-(B);,)~0
and $cIItends as t + 130 to the (normalized) an eigensubspace of B. The corresponding which
where
projection of the initial state & on eigenvalue is the lowest one for Next,
if
Ho and B do not commute,
but if A is small enough, then the asymptotic state I,!J~ is close to an eigenvector C#Iof Ho for which (B), is minimal. As an example let us consider the evolution of one spin-i coupled to a bath of spins-$, i.e. a bath of two-level systems: X,=C?, The whole matrices:
X*=&C’. 1-I is supposed
to be in a magnetic
field,
so that,
with
u the Pauli
MASTER
EQUATION
FOR STATE
VECTORS
where al acts on the ith spin. The coupling Hamiltonian HI2 =
9 Ai(UlU++ da_),
367
is assumed to be
where (++= ux ‘2 ic+y
i=l
and the state of the bath is chosen as
Consequently Pe
PHIZP = 0 and
i(H,+Hz)sH,2,-i(HI+Hz)SH12p
In the thermodynamic limit (N + 00, n + 00, n/N = 8 = constant) the sums become integrals. If the distributions of the A: as functions of the oi are Lorentzian, then, as is well known, the correlation functions of the bath decrease exponentially and one gets m B+iA=h*
II
Oe- i(oo-0)s
e-s/r
c 2l -
(+z +
(1
_
f)),+i(w-~)s,-s/T
++,.p
0
-2 =
(coo-
:2+
l/7: rl
1 i + (i-e);+$wo-o) [
&+ (i- e)i(w,c I
0)
I
.I
err P, I
(10)
where W (resp. ,i) is the mean Larmor frequency (resp. mean coupling constant), and T, is the correlation time. According to (7) and (10) the Markovian master equation for the state vector of our one spin-f system reads:
(11) This equation has been studied in detail in*). The right-hand side of (11) introduces in the corresponding m = (u)$, a term, known as the Landau-Lifshitz friction used by several authors to describe the damping of a
second term on the evolution equation of term, which has been classical spin (see for
N. GISIN
368
example’)).
Note
that
(11) is stable
which are macroscopically
under
variations
undistinguishable.
From
of the state
of the bath
(11) one deduces:
(u;)ti, = th where
(12)
Co is an integration
ingly (a,)$, tends
constant
asymptotically
depending
on the initial
state
$0. Accord-
to 1 (resp.
- l), whenever 0 > 4 (resp. 0
eq. (11) gives a microscopic model of the wellBloch equationn). that if one uses an equation of the same type as (11) of a spin toward the instantaneous direction of a field B(t) = B. + B,(t), where B,(t) is a rotating field, is the so-called quasiperiodic state (or permanent the phenomenological theory of Redfield’<).
3. Inverse problem and dilations In this conditions
section we briefly study the following question: under can an evolution equation of the form of our eq. (7). i.e.
4, = -i HO+, + (WI),,
- HI)&.
with Ho and H, two self-adjoint operators, be associated evolution on a larger space in the sense of section 2?
which
(13) to an
unitary
A simple necessary and sufficient condition is that HI is bounded below. Indeed this condition is necessary since, as mentioned in the previous section, the norm of the solution of eq. (5) decreases with time. Consequently the operator B defined by (6) is necessarily positive. But it may contain a multiple of the identity, and this term cancels in eq. (7) due to the normalization. Thus the operator HI of (13) must be of the form HI = B - M - I, where B is a positive operator, and M a real number. This condition is also sufficient. Indeed if HI is bounded below, then the operator B defined by B = HI + M - I, where M is the lower bound of HI, is positive, and ((B)* - B)$ =
MASTER
EQUATION
FOR STATE VECTORS
369
((HI), - HI)+ for all +. Now exp{( -iHo- B)t} is a contraction semi-group and, according to Sz-Nagy’s theorem”), it can be dilated to a unitary evolution. Accordingly the dissipative evolution equation (13) with H, = k . Ho (k > 0) can be considered as describing the reduced dynamics of a system coupled to a heat bath if and only if Ho is bounded below. This is interesting since with this choice the equation (13) implies d(Ho)/dt = -2k(AHo)*, i.e. the mean value of Ho is a decreasing function of time for all states except the eigenstates. The eigenstates turn out to be asymptotic states, all semi-stable except for the ground-state which is stable (if it exist)“). One can apply this last result to the Hamiltonian of the harmonic oscillator and we then find an evolution which is analogous to the classical damped oscillator: If the initial state is the coherent state defined by the point (~0of the classical phase space, then the solution is at each time a coherent state and the corresponding function (Y,is the solution of the classical equations of the damped oscillator”).
4. Conclusion
We have shown that, by using the same approximations which are necessary to deduce the Pauli master equation, it is possible to construct a master equation for state vectors. The theorems which give the exact relationship between the dynamics of the reduced density matrix and the dynamics of the whole2v6), also apply to our approach. The non-linear term in our eq. (7) comes from the fact that one associates a ray (pure state) of the whole to a ray (pure state) of the system and that representating rays by normalized vectors (defined up to a global phase), this correspondence is not linear. The results of the simple examples of sections 2 and 3 are physically sound. Finally let us make two remarks. First we wish to emphasize that the master equation (4) is reversible, in spite of the appearance of an integral for which one often gives the antropomorphic interpretation of a memory2). Only the Markovian approximation describes an irreversible evolution. The latter approximation is usually applied to the Pauli master equation which governs the evolution of the density matrix of the system. But, and this leads to our second remark, one could a priori also think of making some approximation on the Schrodinger equation which governs the evolution of the state vector of the system. This could be done as presented in this paper, or, at least in a first step, by simply adding non-linear phenomenological terms to the Schrddinger equation, in the same way as one adds friction terms to the the different classical canonical equations. Due to these non-linearities
N. GISIN
370
microscopic towards
distributions
distributions
accordingly, equilibrium suggested
corresponding
which
no evolution however,
one
by the example
to one given
may correspond equation
for
expects
these
of section
2.
density
to different the
density
matrix
density matrix
non-linearities
evolve
matrices exist.
Close
to be irrelevant,
and, to as
Acknowledgement I wish to thank Profs E.B. Davies, for several enlightening discussions.
G.G. Emch,
Ph.A.
Martin
and C. Piron
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