J.
Phys.
A:
Math. Gen.
19
(1986)
2525-2536.
Rinted in Great Britain
Generalised coherent states
and
Bogoliubov transformations
R
F
Bishop and
A
Vourdas
Theoretical Physics Group, Department of Mathematics, University of Manchester Institute
of Science and Technology,
Po Box 88, Manchester M60 lQD, UK
Received
8
July
1985,
in final form
16
December
1985
Abstract. We study the properties
of
the states
U&,
0,
h)lA) where
U,
is an operator
associated with the group
SU(1,
I),
and IA) is a standard (atomic or Glauber) coherent
state defined in terms of the usual boson creation and destruction operators
at
and
a.
We
show how these states may be viewed as ordinary coherent states in terms of the Bogoliubov
quasiparticles whose creation and destruction operators
bt
and
b
are associated with the
operators
at
and
a
by a Bogoliubov transformation. As an important example
of
the use
of these states, we show that they are the coherent states associated with a uniformly
accelerated (Rindler) observer moving through Minkowski space. Our previous results
then simply show how the Minkowski (inertial) vacuum appears to the Rindler observer
as a black-body radiator with a Planckian distribution corresponding to a temperature
proportional to the proper acceleration.
1.
Introduction
Since the introduction into quantum mechanics of the by now very well known standard
(atomic or Glauber (1963)) coherent states, more general coherent states which are
associated with particular irreducible representations of various Lie groups have also
been introduced by various authors (Barut and Girardello 1971, Radcliffe 1971,
Perelomov 1972, 1975, 1977, Feshbach and Tikochinsky 1977). Furthermore, other
coherent states not necessarily of this type have also been specially constructed with
reference to particular problems or applications (e.g. Nieto and Simmons 1979).
The physical motivation that leads
us
in the present paper to extend the range of
application of the standard coherent states is
our
desire to develop a rather broad
framework in which to embed the general phenomenon of clustering within a many-
body medium. In the first instance we restrict ourselves to pairing correlations, but
we have clearly in mind extensions to correlation phenomena concerning bound clusters
of more than two particles. Since much of condensed matter physics depends ultimately
on such correlations, any further attempt to motivate or to justify qualitatively our
results seems superfluous at this stage. Their ultimate justification must clearly rest
on their applicability and their power to suggest generalisations.
The standard coherent states are conventionally defined with respect to a set of
boson creation and destruction operators
at
and
a
respectively, as the eigenstates of
the destruction operator
a.
Continuing for the purposes of initial discussion with a
system of identical bosons, the introduction of the concept of correlated pairs leads
us
to consider the pairing operators
at*,
a'
and
uta.
In
0
2 we show how, since these
operators provide a simple realisation of the Lie algebra SU(1, l), they can be used
to construct a set
of
generalised coherent states associated with the corresponding Lie
0305-4470/86/132525
+
12$02.50
@
1986 The Institute of Physics 2525
2526
R
F
Bishop
and
A
Vourdas
group SU(1,l). We also show that these generalised coherent states may very usefully
be viewed as eigenstates, not of the annihilation operator
a,
but rather of some new
destruction operator
b
which can be associated with the operators
a
and
at
via a
Bogoliubov transformation. Furthermore we show that, although the new coherent
states are indeed
standard
coherent states with respect to the new operators
6
and
bt,
they have many very interesting and useful properties in connection with the operators
a
and
at
which have not previously been studied.
We should mention that other authors (Barut and Girardello 1971, Perelomov 1975,
1977, Feshbach and Tikochinsky 1977) have also previously introduced and studied
other coherent states of the group SU(1,
1).
However, we stress firstly that the states
discussed here are different from those previously considered; and secondly, and
perhaps more importantly, that whereas the previous works have provided hints as to
the relationship of their generalised coherent states with the Bogoliubov transformation,
in the present paper this relationship is demonstrated very clearly and explicitly. In
particular we show how merely by starting with the concept of pairing and hence from
the pairing operators given explicitly above, we are led inevitably via the general
concepts of coherent states to the Bogoliubov transformation itself.
A
further phil-
osophical contrast with previous work is that we regard this particular way of viewing
the Bogoliubov transformation as being very important and quite central to our stated
aims.
Thus,
by contrast with almost all other discussions of pairing phenomena in
which the Bogoliubov transformation is introduced in an extremely
ad
hoc
fashion,
the transformation is generated here by the formalism. With an eye towards building
on these foundations a broader formalism for higher clustering phenomena, the
importance to
us
of this particular aspect should be clear.
More specifically, what we actually show is that, just as the standard coherent states
may be viewed as eigenstates of the original single-boson destruction operator
a,
so
our generalised SU(1,l) coherent states may be viewed as eigenstates of a new
single-boson destruction operator
6,
which is itself generated from the operators
a
and
at
by the usual Bogoliubov canonical transformation. In other words, we demonstrate
that the generalised coherent states appropriate to paired bosons that we construct
may also be viewed as standard coherent states of the Bogoliubov quasiparticles.
From the discussion and motivation above, it is clear that a class
of
physical systems
to which our results may be applied is those described by Hamiltonians at least
approximately bilinear in the underlying boson fields. Apart from such obvious
examples as superfluidity and the parametric excitation of a quantum oscillator, there
are perhaps less obvious applications in both quantum optics and relativistic field
theory and general relativity.
Within quantum optics and quantum electronics, similar states to those that we
describe in
0
2
have recently been described by several authors (Yuen 1976, Caves
1982, Walls 1983). Just as the standard coherent states were introduced into quantum
optics by Glauber (1963) as one-photon coherent states appropriate to the radiation
field from a conventional single-photon laser,
so
our generalised SU(1,l) paired
coherent states may be viewed as two-photon coherent states in connection with the
possibility of a two-photon laser.
In
this context our generalised coherent states have
become known as ‘squeezed’ states for reasons that we discuss when we take this point
up
again in our concluding remarks set out in
8
4.
As
a concrete example of how our results may be applied we discuss in
Q
3
an
example drawn from relativity and quantum field theory, namely the relative nature
of the vacuum (and other) states for an inertial observer and for
a
uniformly accelerated
Generalised coherent states and
Bogoliubov
transformations
2527
observer. From a mathematical point of view the fundamental operators
U2(u,
A),
defined in
9
2,
which generate the particular representation of the group
SU(1,
1)
of
relevance to us here, provide a unitary isomorphism of a Hilbert space onto itself by
mapping each ket
Is)
of the space into another ket
1s;
uA)=
U2(u,
A)ls)
belonging to
the space. We explain in
§
3
that there is an interesting physical interpretation of this
mapping in the relativistic case of Minkowski space. If the operators
a
and
a’
are
now associated with quanta appropriate, say, to solutions of the massless Klein-Gordon
equation in the Minkowski metric appropriate to an inertial observer, then it turns out
that the operators
b
and
bt
are associated with the corresponding solutions in the
so-called Rindler metric appropriate to an observer undergoing uniform acceleration.
Some previously known results for this rather important example (which has a close
bearing on the phenomenon of Hawking radiation from a black hole) can then be
rather simply demonstrated and extended, using our general results.
More generally we believe that the generalised paired coherent states that we
introduce in
9
2,
their properties and the results that we discuss there will be very
useful for practical calculations in the many other fundamental problems in quantum
field theory and many-body theory where the Bogoliubov transformation continues to
play an important role (e.g. Hsue
et
al1985). After the discussion in
§
3
of the particular
application mentioned above, we conclude in
§
4
with some remarks concerning
possible extensions and generalisations of this work.
2. The generalised coherent states
We begin our discussion by considering the unitary operators
exp(Aa’
-
A*a) exp(i4) AEC
+ER
which form the so-called Weyl (or Heisenberg-Weyl) group, where
a
and
a’
are the
usual boson destruction and creation operators. Together with the identity operator
Z
they satisfy the Heisenberg commutation relations
[a,
U+]
=
z
(1)
and
so
generate a Lie algebra-the so-called Weyl (or Heisenberg-Weyl) algebra. The
phase factor exp(i+) plays no further role in our arguments and will henceforth be
omitted. Ordinary (or standard) coherent states are defined as usual by
IAP U,(A)IO) U,(A)=exp(Aa’-A*a) (2)
a’aln)
=
nln) In>= (n!)-”2(at)nlo>
(3)
where
10)
is the vacuum,
a10)
=
0.
In terms of the states
In)
of definite boson number
the standard coherent state has the form
m
IA)
=
exp(-i)A12) (n!)-”2Anln>
n=O
and is easily seen to be an eigenstate of the destruction operator
alA)
=
AIA).
2528
R
F
Bishop
and
A
Vourdas
We next consider a representation of the group
SU(1,
1)
realised with the unitary
operators
U2(p,
8,
A)=
exp(-$p e-leat2++p eLea2) exp(iAata)
P,
6,
A
E
U: U,
=
I.
(6)
K-
G
:a2
Ko=$ata+i
(7)
[KO,
K*I=*K+
[K-, K,]
=
2Ko.
(8)
The three operators
K,,
K-
and
KO
defined as
K
=1
t2
+-,a
satisfy the Lie algebra of
SU(
1,
l),
namely
In
terms of the operator
U,
of equation
(6)
we finally introduce the states [A;
p8A)
defined as
1-4;
POA)
=
U2(P,
8,
A
)I4
=
U2(P,
6,
A
1
Ul(A)IO)
(9)
which are the prime objects of study in this paper. We point out immediately that the
states [A), given in equation
(4),
are eigenstates of the operator
K-
KIA)
=
;A,IA)
(10)
but
not
of the operator
KO.
We stress this point because in the literature the
SU(1,l)
coherent states are usually generated by letting
U,
or a similar operator act
on
an
eigenstate of KO.
Next we note the important relation
exp(-$p e-lsK+
+fp
e"K-)
=
euK+ e'% e-'*K-
U
-e-'@
(11)
tanh(
$p
)
T
=
In(
1
-
/ai2)
(12)
where
which has been given by Perelomov
(1977).
Equation
(11)
is valid for any operators
K,,
K-
and KO which satisfy the algebra of equation
(8),
not just the particular
representation of
it
given in equation
(7).
(We note that it is simplest, and sufficient,
to prove equation
(11)
for the representation Ko-,+u3, K++$~(U~*~U,) in terms of
the usual Pauli spin matrices
U#,
i
=
1,2,3.)
Using the result of equations
(1 1)
and
(12),
we may write the operator
U2(p,
8,
A)
of equation
(6)
in the equivalent form
(13)
where
A
is real and U, given by equation
(12),
is complex and with modulus
IuI
<
1.
Equation
(13)
is particularly useful in simplifying the derivation of later results.
u,(u,
A)
=
exp(faat2)(1
-
1~/~)~+~/~+~/~
exp(-$U*a2) exp(iAata)
For example, we can easily prove the very important relations
U2(q A)aU:(v,
A)
=
e-'*(
1
-
1~1~)-~'~(a
-aut)
=
b
U2(cr,
A)utUi(u,
A)
=
e'*(l
-
1~1~)-~'~(a~
-
U*U)
=
bt
(14)
either by making use of equation
(13)
and the unitarity
of
U,,
or directly from equation
(6).
We readily see from equation
(14)
that the operators
b
and
bt
again obey the
boson commutation relations
[b, bt]
=
I
(15)
and that the transformation from the operators
a
and
at
to the operators
b
and
b'
is
just the usual Bogoliubov transformation. From equation
(14)
and the fact that
U,
is
unitary we can trivially prove for any function
f(a,
at)
the relation
(16)
UJ(
U,
a
')
U:
=
f(
b, b
t,
e
U2f(
U,
(I
')
=
f(
b, b
t,
U,.
Generalised coherent states and Bogoliubov transformations
2529
By
making use of equation (13) and the trivial relation
exp(4ata)lA)
=
exp[flAI2(le’l2- 1)]IA e’) (17)
which is readily proved from equations
(3)
and
(4),
we can now write the states of
equation (9) in the equivalent form
/A;
CA)=
U2(a,
A)(A)
=
U2(G,
A)
UMIO)
=
(1
-
/o/~)’/~
exp(-fa*A2 eZiA -flaA12)
x
exp($aat2)1A eiA(l
-
(a)2)1’2).
In
the special case
A
=
0
=
a,
we simply get the standard coherent states, lA;
00)
=
\A),
defined in equation (2). Equation (16) immediately implies that
U2a
=
bU2,
and hence
that the states \A;
ah)
are eigenstates of the destruction operator
b,
blA;
ah)
=
bU2(a, A)\A)
=
U2(W,
A)+)
=
AIA;
UA)
(19)
by making use of equation
(5).
The special case A
=
0
is of particular interest, since
equation (19) implies the relation
b(0;
a)
=
0
(20)
where we have written
10;
CA)
=
10;
a),
as equation (18) shows that this state is indepen-
dent of
A.
Just as the state
10)
was defined by the relation
a(O)=O
to be the ground
state with respect to the operators
a
and
at
(i.e. the vacuum for a-type bosons),
so
the state
10;
a)
=
U2(a,
A)(O)
obeys equation (20) and
is
therefore the vacuum for b-type
bosons. By making further use of equation (16) we get the relation
U,(
a,
A
)
exp( Aa
-
A*
a
)
=
exp(
Ab
’
-
A* b
)
U,(
a,
A
)
\A;
uA)
=
exp(Abt
-
A*b)(O;
a).
(21)
(22)
and hence from equations
(2),
(9) and (21):
Thus we see very clearly that the states /A;
aA)
may be viewed as ordinary coherent
states with respect to the operators
b
and
bt,
i.e. they are the standard coherent states
of the Bogoliubov quasiparticles
(or
6-type bosons).
For
fixed values of
a
and
A,
they
therefore obey the well known properties of ordinary coherent states, e.g.,
(D;
aA/A;
aA)=exp(D*A-iID(2-f)A12)
(23)
I(D;
(TAIA;
CA)(’
=
exp(-ID
-
A(’)
where