Download Space-time description of squeezing

Z. Bialynicka-Birula and I. Bialynicki-Birula
Vol. 4, No. 10/October 19871J. Opt. Soc. Am. B
1621
Space-time description of squeezing
Zofia Bialynicka-Birula
Institute of Physics, Polish Academy of Sciences, Lotnikow 32/46, 02-668Warsaw, Poland
Iwo Bialynicki-Birula
Institute for Theoretical Physics, Polish Academy of Sciences, Lotnikow 32/46, Warsaw, Poland
Received February 5, 1987; accepted June 16, 1987
We present a field-theoretical description of the squeezed states of the electromagnetic field. A definition
of the
squeezed state is introduced that is a natural generalization of the definition for one or two modes. We
the squeezing produced by the medium with space- and time-dependent material coefficients e and show that
related to the photon-pair production. In Appendix A we describe the time evolution of the Gaussianis directly
squeezed
states in terms of the field variables.
1.
INTRODUCTION
In our opinion, the study of the squeezed states has reached
a stage at which it is worthwhile to develop a general frame-
work based on the field-theoretical formalism to encompass,
in a unified way, all possible situations. In this paper we
present such a description employing the vector field operators of the electric displacement D(r, t) and the magnetic
induction B(r, t), which fully describe the properties of the
quantized electromagnetic radiation.
We define the squeezing in terms of the two canonically
conjugate operators X and X2 built linearly from the field
operators D and B.
The Heisenberg equations of motion for the field operators are used to determine the dynamics of the system and,
in particular, to describe the processes that result in
squeezed states.
In the present paper, we restrict ourselves to the simplest
case when the back action of the quantized electromagnetic
field on the medium can be disregarded. The properties of
the medium, however, may vary in space and in time in an
arbitrary way, owing to the influence of some external driving forces. This medium will be described by two phenomenological real fields: the electric permittivity tensor e(r, t)
and the magnetic permeability tensor M(r,t). These tensors
will be treated as given functions of space and time.
We
show under what conditions such a medium generates radiation in a squeezed state.
We exhibit a close connection
between the squeezing and production of pairs of photons by
the medium. We express the squeezing parameters by the
probability amplitudes for the production of such pairs.
The study of the quantized electromagnetic field generated by the external medium is the next logical step after the
extensively studied case of the quantized radiation generated by the external currents. External currents produce the
coherent states of radiation, whereas the space- and timedependent external media produce squeezed states. In each
case the states belong to the class of Gaussons of the electromagnetic field, as defined by one of us' (called Gaussian pure
states by Schumaker 2 ).
0740-3224/87/101621-06$02.00
2. DEFINITION OF SQUEEZING
The standard definition of squeezing for one mode of radia-
tion (cf., for example, a review paper by Walls 3 ) is based on
the decomposition of the electric-field operator into the two
so-called quadrature components X1 and X2. These two
components can be viewedas the electric-field and the magnetic-field operators for this mode taken at t = 0. In our
field-theoretical description the corresponding operators
are defined as
X1[fl = d 3 rD(r, O) f(r),
X 2[g] =
J
d3 r B(r, 0) g(r),
(la)
(lb)
where the two real vector functions are introduced to extract
a particular mode from the complete field operators. In the
absence of the free charges, we can assume, without any loss
of generality, that both vector functions f and g are divergenceless,since their longitudinal parts do not contribute to
the integrals [Eqs. (1)].
We have selected D and B (and not, for example, E and H)
as our basic fields, because they form the pair of canonically
conjugate variables in the theory of the electromagnetic
field.4 The equal-time commutation relations for these operators are
[Bi(r, t), Dj(r', t)] = ihEijkk5(r
-r'),
[Bi(r, t), Bj(r', t)] = 0 = [D,(r, t), Dj(r', t)].
(2a)
(2b)
These commutators contain neither the dielectric permittivity nor the magnetic permeability, which would not be the
case for other pairs of the electric and magnetic fields.
The commutator of the X and X 2 operators is (h = 1)
[XI[f], X 2[g]] = i
d 3 rf(r)
[V X g(r)].
(3)
The uncertainty relation for these operators can be obtained in the standard fashion,5 and it reads as
© 1987 Optical Society of America
1622
Z.Bialynicka-Birula and L Bialynicki-Birula
J. Opt. Soc. Am. B/Vol. 4, No. 10/October 1987
AX,[f]AX2[g] >
d3 rf(r) [V X g(r)] -
'/2
(4)
In order to calculate the dispersions AX, we decompose the
field operators D and B at t = 0 into the creation and
annihilation operators of the photons with the right-handed
and the left-handed polarizations
-
2
3
d kw1/ 1[e(k)aR(k)
3]1/2
I
D(r) = [ 2
-h.c.,
e*(k)aL(k)]eikr
B(r) =
(5a)
3
I d kw1/21[e(k)aR(k)
1
+ e*(k)aL(k)eik.r + h.c.},
e*(k) = e(-k),
(6a)
k X e(k) = -iwe(k),
(6b)
e(k) = 0,
(6c)
2
w-2 kiki -iw-iEijkkk),
(6d)
= Iki.
The commutation relations for the creation and the annihilation operators are ( = R, L)
and
[ax(k), ax,t(k')] = 6,x,6(k
-
k').
atF + iVXF = -iV X (KF + KFt),
(5b)
lowing algebraic relations:
e(k)ej*(k) = /2(bij-
(12)
F(r, t) = 1/2 [D(r,t) + iB(r, t)],
which obeys the following field equations:
where the complex polarization vectors e(k) satisfy the fol-
e(k)
GENERATION OF THE SQUEEZED STATES
The generation of the squeezed state of the electromagnetic
radiation by the medium characterized by the electric permittivity tensor e(r, t) and the magnetic permeability tensor
/A(r,t) is described here in the Heisenberg picture. In this
picture the equations satisfied by the field operators D(r, t)
and B(r, t) have the form of the Maxwell equations. We
write these equations using the complex combination of the
6
two field operators first introduced by Kramers :
3.
(7)
In the vacuum state 10),with respect to these annhilation
operators, the dispersions AX, and AX2 are
(13a)
(13b)
V - F = 0,
where
(14a)
K = 1/2(-1 - 1 + Y-1 - 1),
K_ = /2 ('El-
'-1)
(14b)
The linear equations (13) for the field operators can, in
principle, be solved with the help of the appropriate Green
functions as if they were equations for ordinary functions.
In particular, the retarded Green function GR(r, t; r', t')
propagates the initial field operators Fin(r, t), according to
the formula
F(r, t) = Fin(r,t)
i
-
J
d'r'dt'GR(r, t; r', t') *V X [K+(r',t')
(15)
*F(r', t') + K-(r',t') F(r', t')]1.
The retarded Green function for the left-hand side of Eq.
(13a) can be written as the following 3 X 3 matrix:
X [f] =
d3kwl(k)12]
[/2
AX21g] = ['/2
J
d3kwlI(k)2].
(8a)
(8b)
We choose the normalization of the vector functions f and g
in such a way as to obtain, in the vacuum state, the standard
values3 (AX, = 1/2= AX2 ) for each dispersion:
Jd
3
k2wlf(k)1
2
= 1=
2
d3k2wAg(k)J .
(9)
The restriction imposed by the uncertainty relation [Eq. (4)]
on the product AX1AX2 is the strongest when the following
equality holds:
w9(k) = +ik X f(k).
(10)
This condition simply means that the vector functions f and
g describe the electric and magnetic fields of the same mode.
Under conditions (9) and (10),the uncertainty relation takes
the form
AX1[f]AX 2 [g] > 1/4.
GRij(r,t; r',t') =
of the two dispersions falls bolow the vacuum value 1/2,in
3
accordance with the definition used in the one-mode case.
iEiJkV)DR(r- r', t - t'),
(16)
where DR(r, t) denotes the retarded Green function for the
scalar wave equation.
Instead of solving Eq. (15) in full generality, we restrict
ourselves here to the lowest-order perturbation with respect
to K4+. This is justified if both Eand ,u do not differ substantially from unity. Also,we are interested in the properties of
the radiation propagating in the region of space and time far
in the future where it is no longer influenced by the medium.
The free-field operator in the far future will be denoted by
Fout(r, t). Similarly, Fin(r*t) describes the free radiation in
the region of space-time before the interaction with the
medium started. The relation between the field operators
in these two asymptotic regions in the lowest order of perturbation theory, obtained from Eq. (15), reads
Fout(r, t) = Fin(r, t) - iV X (Ot- iV X)
X (r
+
(11)
The squeezed state is defined as the state for which either
(aijt -
-r',
t -t')[K+(r',
J
d'rdtDR
t') Fin(r', t')
K-(r',0 -Fin (r', t')].
(17)
For the free-field operators Finand Foutthe decomposition
into creation and annihilation operators can be made for all
times:
Z. Bialynicka-Birula and I. Bialynicki-Birula
in
Fout(r, t) = i[2(27r) 3 ]-/
2
Vol. 4 No. IO/October 1987/J. Opi. Soc Am.
in
Jd3kcw/e(k)[aRou (k)
(vX20ut[g] )2 -(AX
t
(18)
Substituting these formulas into Eq. (17), we obtain a set
of linear relations between the out and the in operators:
ut(k,) = a'N(kl) + i(2ir)
'1
1 2
d3 k2 (W@cL
2) /
(19)
where
+(k -k 2 ) e(k2),
MRL(kl, k2) = e*(kl) - R(k,
-
(20a)
k2 ) e*(k2 )
= MLLR*(k
2 , kj),
(20b)
MLL(kj,k2) = -e(kl)
k+(k, - k2) e*(k),
(20c)
NRR(kl, k2 ) = e*(kl)
k.(k, + k2 ) e*(k2),
(20d)
NRL(kl,
e*(kl) - R+(k, + k2 ) e(k2 ),
k2 )
(20e)
NLL(kl, k2) - e(kj) - R-(k, + k2 ) e(k2).
(20f)
The arguments of the four-dimensional Fourier trans-
forms
tion:
R4
are evaluated according to the following prescrip-
R,(k + k2 ) = kR(k, + k2 , W1+ ( 2 .
(21)
The relations between the out and the in operators will
now be used to compare the properties of the radiation in the
past and in the future. We assume that there was no radiation in the distant past so that the radiation observed in the
future was generated by the medium. That means that the
state of the system can be characterized as the vacuum state
join)for the in operators. In order to find the conditions for
squeezing, we calculate the dispersions of Xout[f] and
X2 out[g] in the state oin). Omitting straightforward but
lengthy calculations, we quote only the final result below:
(,Ax outm)2- (Xin[f])
x [-2+(kl)
=
2
=
i(4r)-
J J
d3 klwl
f+(kl) k_(k + k2) * +(k2)
-
f(kl) k_(k + k2) *f_(k2) - C.C]
i(8ir-l'J d3k,
d
k2 W2
J
d 3'k2,(w2
+(k)
+ 9-(kl) L(kl + k2) 9-(k2 ) - C.C.}
= -i(8r-1
J d3 k,
Jd 3 k2
(ki))
Fl(k,
+ k 2 ) [(k 2 X (k 2 )]
+ WlW29(kl)
*A-1(k, + k2) j(k 2) - c.c.j, (22b)
where Z-1(k)and Ai1(k) are four-dimensional Fourier transforms of cl(r, t) and Mp'(r, t), whose arguments are understood according to Eq. (21). In Eqs. (22) we used the decomposition of the Fourier transforms of the mode functions f
and g into their circularly polarized components,
f+(k) = ((k)
e*(k))e(k) = /2 [i(k) + i-lk
f(k) = [(k)
e(k)]e*(k)
=
/2[?(k
-
X f(k)],
(23a)
i-lk X 1(k)]. (23b)
We consider now the functions f and g normalized according to Eq. (9) and related to each other by Eq. (10). Then,
the values of the dispersions in the past, describing the
fluctuations in the vacuum, are
AX,in[f]
=
/2
AXin[g].
(24)
The departures from the vacuum values in the future are
given by the right-hand sides of Eqs. (22a) and (22b). These
expressions differ only in sign, so that the dispersions of
X1outand X2 0utstill saturate the uncertainty relations. This
result, however, is valid only in the lower order of perturba-
tion theory.
The form of the expressions in formulas (22) clearly set
the conditions on the medium and on the mode function f [g
is given by Eq. (10)] under which the squeezing occurs. In
particular, the medium must be described by either or
varying in time; also their Fourier transforms must be different from zero in the proper region of the momentum space.
4. PRODUCTION OF PHOTON PAIRS AND
SQUEEZING
aA°ut(k) = exp(i4)aXif(k)exp(-iP).
Xf(k,) *F'(k + k2) * (k2 )
+ [k X (kl)] *A7'(k + k2 ) [k2 X (k 2)]
-(k2)
The connection between the squeezing and the photon-pair
production is most easily seen in the S-matrix language. In
order to derive this relation, we express the linear relations
[Eq. (19)] connecting the out and the in operators as a unitary transformation:
k+(k + k2) *f(k 2 )
-
d 3klw, f d 3 k2 W2
= i(4ir)'J
+ 9+(kl) *R_(lk + k2 )
X (k, X
X [Mx,(kl, k2)a , (k2 ) + NAx,(kl,k2 )a ,int(k2)],
MRR(kl, k2 ) =-e*(kl)
2
X {-29+(kl) * R+(k, + k2 )
in
X exp(-icot + ik r) + aL t(k)exp(iwt - ik r)].
a
2 in[g])
1623
-
c.c.1, (22a)
(25)
Such unitary transformations are the standard tool in the
description of the squeezed states.2 In the first order of
perturbation theory the Hermitian operator 4' has the form
1624
Z. Bialynicka-Birula and I. Bialynicki-Birula
J. Opt. Soc. Am. B/Vol. 4, No. 10/October 1987
4b= -(47r)-l
J
d 3 klwlll 2
J
d3 k21/2 [aRt(kl)MRR(kl, k 2 )aR(k2 )
photon pair from the vacuum, each photon created by the
operator aloutt[f] is
+ 2aRt(kl)MRL(kl, k2)aL(k2) + aLt(kl)MLL(kl, k2)aL(k2 )
All[f, f]
All[f,
X (kl, k2 )aLt(k2 ) + aLt(kl)NLL(kl, k 2 )aLt(k 2 ) + h.c.],
(26)
which is obtained by comparing Eq. (25) with Eq. (19). In
this formula all the creation and annihilation operators are
the in operators. It is instructive to rewrite formula (26) as
an integral over space and time:
JJ
dt
d3 rl/2tD'(r, t) [(r,
+ Bin(r, t)
[
l(r, t)
(27)
(28)
We can even write a closed expression for the S operator as
the time-ordered exponential valid to all orders of perturbation theory:
J
dtH1 (t)],
The photon observed in the future in an arbitrary state is
created by a linear combination of the creation operators
with right and left polarizations. The combinations that
appear in the operators X1 and X2 play the distinguished
They will be denoted by alt[f] and a 2 t[g],
aloutt[f] = if d3k(2c,,)l/2{[i*(k) e(k)]aLft(k)
a2 outt[g] =
[*(k) . e*(k)]aRoutt(k),
J
d 3 k(2w)l/ 2 J[g*(k)
A 2 2 [g, g]
.
(30a)
e(k)]aLoutt(k)
+ [*(k) e*(k)]aRoUtt(k)},
-i(2)-/
2
(Oi.l(alin[f])2,l'Oin)
(Xlin[f])
-
2
= (2)-1/2 Re All[f, f].
=
-i(2)-1/
2
(0inli(a2 in[g]) 2 4'l0in),
(AX2 out[g])2 - (AX2in[g])2 =
(32)
(33)
(2)71/2
Re A 2 2 [g, g]-
(34)
(35)
Thus the photon-pair production is a necessary condition for
squeezing.
The relation between the dispersions AX1,2 and the photon-pair production amplitudes were obtained here in the
lowest order of perturbation theory, but, with a certain
amount of field-theoretical trickery, one can generalize these
results to arbitrary order.
5.
form [Eq. 26)] but with the coefficients containing now all
powers of K±.
-
=
(29)
which is the lowest order that reproduces the previous result.
Since the interaction Hamiltonian is quadratic only in the
field operators, the complete S operator can also be written
as exp(-i4'), with the new operator 4' again of the quadratic
role in our study.
respectively:
(31)
Similarly, the dispersion AX2 is related to the photon-pair
production amplitude A 22 [g, g]:
We can easily understand this result once we observe that
S = T exp[-i
f]
(AXOut[f]) 2
4' is simply the time integral of the interaction Hamiltonian
for the electromagnetic field interacting with the medium.
It must be so because, after all, the unitary relation between
the in and the out operators is given by the S operator
Fout(r, t) = StFin(r, t)S.
10in).
where the operator 4' is given by Eq. (26). This matrix
element can be easily calculated by using the commutation
relations [Eq. (7)] for the creation and the annihilation operators. The resulting expression has the same form as the
integral that appears in the formula (22a) for the dispersion
AX,. Hence we expressed this dispersion by the photonpair creation amplitude:
t) - 1] Din(r,t)
1] Bin(r, t)I.
-
2
In the lowest order of perturbation theory, we obtain
+ aRt(kl)NRR(kl,kl)aRt(k2 ) + 2aRt(kl)NRL
4' =
= (2)-1/2 (0outl(alout[f])
(30b)
where i and g are the Fourier transforms of the mode functions introduced earlier. When the projections of the mode
functions f and g on the polarization vector e (or e*) vanish,
then the operators [Eq. (30)] create photons with the lefthanded (or right-handed) polarization. For linearly polarized hotons the two projections on e and on e* have the
same absolute value.
The probability amplitude Ali[f, f] for the creation of the
CONCLUSION
Our description of squeezed states may certainly be criticized as an exercise in elementary quantum field theory that
does not lead to any progress in our understanding of the
physics of these phenomena. To defend ourselves against
such criticism, we would like to point out that the description of squeezing based on the field-theoretical foundations
can be more easily extended to situations in which the standard approaches may fail. In particular, in this description,
the modes of the quantized electromagnetic field in which
the squeezing is observed do not have to coincide with the
modes that play the decisiverole in the dynamics. Also, the
observed modes are completely general; they do not have to
be monochromatic. In the description that uses the in and
out operators, we do not need the Hamiltonian. This may
make the analysis of squeezing by dispersive media much
easier.
In our description of squeezing we have used the language
of the creation and annihilation operators to facilitate the
comparison with the standard description of this effect.
However, certain interesting features of the squeezed states
of the electromagnetic field are not seen in this representation; even the term Gausson or Gaussian pure state cannot
be truly justified. Fortunately, quantum electrodynamics
offers an alternative representation of the field operators
and the state vectors, which enables one to see explicitly the
Gaussian form of the squeezed states produced by the medium and to study their time evolution. This representation
may be called the Schr6dinger representation because it is
completely analogous to the standard representation used in
Z. Bialynicka-Birida and I. Bialynicki-Birula
Vol. 4, No. 10/October 1987/J. Opt. Soc. Am. B
wavemechanics, in which the position operator is represented by the multiplication by x, the momentum operator is
represented by -id,9 and the state is described by a function
of x. In Appendix A we give an outline of the field-theoreti-
cal description of the Gaussons in the Schrodinger representation and show, in particular, what it is that is actually
The simplest Gausson of them all is the vacuum state described by the wave functional To[A],
X [V A(r)]Ir
Gaussons are a natural extension of the coherent states.
Coherent states are simply the displaced vacuum states in
which the real and the imaginary parts of the displacement
are interpreted as the classical (average) magnetic and electric fields. The Gaussons, however, in addition to being
displaced, also have their shape modified.
In this appendix we briefly describe the Gaussons and
determine their time evolution in the absence of the interactions with the medium, followingthe approach developed in
Ref. 1.
In the Heisenberg picture, which we have used in the main
body of this paper, the time evolution is introduced through
the equations for the field operators. In the Heisenberg
picture the squeezing appears as the mixing of the creation
and the annihilation operators or as the photon-pair creation. A different picture of squeezing is obtained in the
Schr6dinger picture, in which the time evolution is introduced through the changes in the state vector,
idt(t) = HIl(t).
(Al)
In the quantum theory of the electromagnetic field the state
vector 41(t),or the wave function, describes the state of the
electromagnetic field, and the operator H stands for the
Hamiltonian of the electromagnetic field.
In the standard approach, the Hamiltonian is expressed in
terms of the photon creation and annihilation operators, and
the state vector is expanded into the photon-number states.
This (Fock) representation
is not so natural, however, when
it comes to the study of the Gaussons. Their Gaussian form
T[Alt] = N(t)exp{-'/
2
f d r f d3r'[V X A(r)
3
X Wij(r, r'; t)[V X A(r')
B(r)P[A
= V X A(r)T [A],
(A3a)
D(r)T[A]
= -ib/bA(r)'If[A].
(A3b)
The Hamiltonian in this representation takes the form
H =
1/2
f d rt-[/6/A(r)]
3
2
+ [V X A(r)] 2 1.
B(r, t)]i
B(r', t)1
(A6)
a.Ki1(r, r'; t)
=f-if
V *B = 0,
V-E= 0,
(A7a)
(A7b)
d3rlKik(r,rj; t)Kkj(rl; r'; t)
+ i(ViVj - 5jA)6(r -r'),
(A8)
where Kij is related to Wijby the two curl operations acting
on both vector indices i and j:
Kij(r, r';t) =
-ikakWIm(r,
r'; n
(A9)
Thus the average fields obey the classical Maxwell equations, as do the ordinary coherent states. These equations
determine the motion of the center of mass of the Gausson,
and they are decoupled from the equations for the W that
determine the internal motion of the Gausson.
The squeezing phenomena are described by the 3 X 3
matrix Wij(r, r'; t). As was shown in Ref. 1, the nonlinear
(A4)
In this representation the Gaussons show their identity
best:
(A5)
This function is a solution of the Schrodinger equation (Al),
with the Hamiltonian .of the electromagnetic field taken in
the representation [Eq. (A4)], provided that the two vector
functions E(r, t) and B(r, t) and the matrix kernel Wij(r, r';
t) obey the time-evolution equations listed below7:
(A2)
wave function, which depends on the particle coordinates.
The magnetic induction operator B(r) and the electric displacement operator D(r) are represented as the multiplication by the function V X A(r) and as the differentation with
respect to A(r), respectively:
-
-
+ if d3 rE(r, t) A(r)}-
dtE= V X B,
i.e., on the field variable, in full analogy with the Schr6dinger
r- 2 [V X A(r')]}-
electromagnetic field, for which the quadratic part in the
exponent differs from that given in formula (A5), describe
squeezed states of the electromagnetic field.
The most general Gausson of the electromagnetic field is
described by the wave function of the form
atB = -V X E,
[Alt],
-
d 3 r'
as in the vacuum function [Eq. (A5)]. All Gaussons of the
representation in quantum mechanics. In such a representation the wave function (t) depends on the values of the
vector potential A(r):
=
d 3r
This is the only Gausson that does not change during the
time evolution. All other Gaussons undergo complicated
squeezing and tumbling motions, but they always retain
their Gaussian form, i.e., their wave functions remain exponential functions of the quadratic expressions built from the
functions A(r).
The Gaussons of the electromagnetic field represent a
natural generalization of the states of the quantum-mechanical oscillator described by Gaussian wave functions. All
coherent states are obtained by adding a term linear in A(r)
into the exponent while keeping the quadratic part the same
is seen only when one uses the analog of the Schr6dinger
Tt
J J
To[A] = N exp{(27r)-2
being squeezed.
APPENDIX: TIME EVOLUTION OF GAUSSONS
IN FREE SPACE
1625
they are described by a Gaussian functional of A.
matrix equation (A8) can be reduced to a linear one and
solved explicitly in simple cases.7 It is the matrix Wj (or its
curl Kij) that describes the shape of the Gaussian wave function and its intricate time evolution. The squeezing of this
1626
Z.Bialynicka-Birula and 1.Bialynicki-Birula
J. Opt. Soc. Am. B/Vol. 4, No. 10/October 1987
Gaussian shape in just one direction is exactly what corresponds to a normal squeezing of the quantum state of the
electromagnetic radiation, but there are many other ways in
which this (infinite-dimensional) Gaussian "monster" can
change its shape.
These internal motions have no classical
interpretation, although they contribute' to all physical
variables such as the energy, the momentum, and the angular momentum of the field.
ACKNOWLEDGMENT
This research was supported by grant CPBP 01.07.administered by the Polish Academy of Sciences.
tuations, F. Haake, L. M. Narducci, and D. Walls, eds. (Cambridge U. Press, Cambridge, 1986), pp. 159-170.
2. B. L. Schumaker, "Quantum mechanical pure states with Gaussian wave functions," Phys. Rep. 135, 317-408 (1986).
3. D. F. Walls, "Squeezed states of light," Nature 306, 141-146
(1983).
4. The fact that the D field and not the E field is the canonically
conjugate momentum was previously recognized by M. Born and
L. Infeld in their study of nonlinear electrodynamics [M. Born
and L. Infeld, "On the quantization of the new field equations I,"
Proc. R. Soc. London Ser. A 147, 522-546 (1934); "On the quantization of the new field equations II," Proc. R. Soc. London Ser. A
150,141-166 (1935)]. Cf. also I. Bialynicki-Birula and Z. Bialyn-
icka-Birula, Quantum Electrodynamics (Pergamon, Oxford,
1975).
5. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970),
pp. 158-161.
6. H. A. Kramers, Quantum Mechanics (North-Holland, Amster-
REFERENCES AND NOTES
1. I. Bialynicki-Birula,
"Beyond the coherent states:
Gaussons of
the electromagnetic field," in Coherence, Cooperation and Fluc-
Iwo Bialynicki-Birula
Zofia Bialynicka-Birula
f/
t
em
dam, 1957), pp. 418-441.
7. There is a systematic error in Ref. 1 where these equations were
derived; viz., the imaginary unit i should be replaced by-i everywhere.
received the
Zofia Bialynicka-Birula
M.S. degree in physics (1956) and the
Ph.D. degree in theoretical physics
(1963) from Warsaw University, Poland.
She spent a year in 1970 as a research
<
associate at the University of Pittsburgh
Iwo Bialynicki-Birula was born in 1933
in Warsaw, Poland. He received the
M.S. degree in physics in 1956 and the
Ph.D. degree in theoretical physics in
After
1959 from Warsaw University.
two years at the University of
, Aspending
and one semester in 1981 as a visiting
Rochester, he returned to Warsaw Uni-
professor at the University of Southern
versity.
at the University of Pittsburgh.
s23H jjrently,
Cur-
my of Sciences in Warsaw and heads the
research group on quantum optics. Her main interests are quantum electrodynamics,quantum optics, and, in particular, the theory
of atomic processes in strong laser fields.
She co-authored (with I.
professor of
Polish Academy of Sciences. He has
spent extensive periods of time as visit-
she is professor of physics in the
Institute of Physics of the Polish Acade-
He is currently
physics at Warsaw University and at the
Institute for Theoretical Physics of the
California, Los Angeles, and one in 1985
ing professor at the University of Pittsburgh, the University of Rochester, and the University of Southern
California. His main interests are in quantum field theory and its
applications.
His work in quantum electrodynamics
[he co-auth-
Bialynicki-Birula) the monograph Quantum Electrodynamics
ored a monograph Quantum Electrodynamics (Pergamon, New
(Pergamon, New York, 1975). She is a member of the Polish Physical Society.
Siegert shift, on the interaction of charges with intense photon
York, 1975)] led him to research in quantum optics on the Blochbeams, and on collective effects in such interactions.
He is a mem-
ber of the Polish Physical Society, the American Physical Society,
and the European Physical Society. He has also been elected a
member of the Polish Academy of Sciences and of the Norwegian
Royal Society of Sciences and Letters.