Although the problem of electromagnetic radiation by a quantum harmonic oscillator is considered in textbooks on quantum mechanics, some of its aspects have remained unclear until now. By this, we mean that usually the initial quantum states of both the oscillator and the field are assumed to be characterized by a definite energy level of the oscillator and definite occupation numbers of the field modes. In connection with growing interest in squeezed states, it would be interesting to analyze the general case when the initial states of both subsystems are arbitrary superpositions of energy eigenstates. This problem was considered in other work, where the power of the spontaneous emission was calculated in the case of an arbitrary oscillator's initial state, but the field was initially in a vacuum state. In the present article, we calculate the rate of the oscillator average energy, squeezing, and correlation parameter change under the influence of an arbitrary external radiation field. Some other problems relating to the interaction between quantum particles (atoms) or oscillators where the electromagnetic radiation is an arbitrary (in particular squeezed) state were investigated

Dodonov, V. V.

Nikonov, D. E.

English

NASA Technical Reports Server

N94-10571
HARMONIC OSCILLATOR INTERACTION
WITH SQUEEZED RADIATION
V.V.Dodonov, D.E.Nikonov
Moscow Institute of Physics and Technology
Zhukovsky, Moscow Region, 140160 Russia
Although the problem of the electromagnetic
quantum harmonic oscillator is considered in te×tbooks on
mechanics (see, e.g., [1]) some its aspects seem to
clarified until now. By this we mean that usually the
quantum states of both the oscillator and the field are
radiation by a
quantum
be not
initial
assumed
to be characterized by a definite energy level of the oscillator
and definite occupation numbers of the field modes. In connection _
with growing interest in squeezed states it would be interesting
to analize the general case when the initial states of both
subsystems are arbitrary superpositions of energy eigenstates.
This problem was considered partly in Refs. 2-4, where the power
of the spontaneous emission was calculated in the case of an
arbitrary oscillator's initial state (but the field was supposed
to be initially in a vacuum state). In the present article we
calculate the rate of the oscillator average energy and squeezing
and correlation parameter change under the influence of an
arbitrary e>::ternal radiation field. Some other problems relating
to the interaction between quantum particles (atoms) or
oscillators with the electromagnetic radiation being in arbitrary
( in particular, squeezed) state were investigated, e.g., in Refs
..=,-7•
Let us describe a charged harmonic
Hami Itonian
H = h_aa
0
and the field by a hamiltonian
oscillator by a
(i)
51
https://ntrs.nasa.gov/search.jsp?R=19940006116 2020-06-16T21:09:55+00:00Z
H m h _ _b b.J J J
J
here _ is the frequency of the oscillator, _ -
J
modes, a,b - corresponding destruction operators.
In a rather general case interaction can be
form
ones of
(2)
field
described in a
H = h _j I_J_a*b_+j X a_bj j + H.c. 1 (3)
(H.c. means hermitian conjugated part, _ and _. are constants).
J J
In Schrodinger picture an arbitrary initial state vector
I_(0) > evolves into a state vector l_=(t) > as predicted by
+ H + H.Schrodinger equation with Hamiltonian H = H ° I i
In interaction picture any Schrodinger operator Q changes
+ H
according to evolution operator U ° corresponding to H = H ° m
Q(t) = Uo(t)QUo(t). (4)
For example
a(t) = a exp(-i_t), b (t) = b, exp(-i_t). (5)
J 3 J
The interaction Hamiltonian in this picture
H
Z
J
generates evolution operator U(t) so that a state vector
picture defined as
I_(t) > = u+o(t) l_(t) >
will variate according to
= h _ IHja*b_e×p(i_t+i_t)+j Xa÷be×p(-iwt+i_t)j_ J + H.c.]
in
(&)
this
(7)
Iv(t)> = U(t) I_(0)>.
Expectation value in this picture
(8)
<Q>z = <_(t) iQl_,(t)> (9)
variates slowly, only due to interaction. On the other hand, it is
related to the conventional expectation value as follows
<O)"r : <'_slUoOUol_s >" (10)
After introducing designations we can pose several questions
to answer:
1. Can absorption and emission be distinguished iDa general case ?
2. Then how to calculate the rates of these processes ?
62
3. Is time ordering important in perturbation calculation for this
case ?
4. Does stimulated emission manifest itself ?
5. How does squeezing parameters of the oscillator and the field
vary ?
To calculate the rates of the processes we need to consider
infinitely long time intervals T _ _ in comparison with
oscillation period. But they must be much shorter than damping
time. Then the evolution operator has meaning of scattering matri>(
S transforming initial state Iv(O)> w li> to resulting one Ir>.
From Heisenberg equation one gets
S = expT(-iT), (11)
where all products are believed time-ordered (designated with
subscript T), and T - matrix is given by
T = $ H (t)/h dt. (12)
I
For our particular case
T = 2_h _ [ X a_b 6(_-_) + H.c.l, (13)
j J J J
here the terms with _ vanish because of a factor 6(_+w). Further
J
6
J
E 6(_-_). Delta function originates as a limit of an integral
J
T/2
Int = (14)$ exp (i or) dt
-T/2
(here the initial instant
Limits of this integral are
Int _ _,
in time re-designated as -TI2).
Int _ 2n _(13), if T _
if _ _ O, (14')
(14")
Conventional techique in quantum
follows [8]. T - matrix is splitted in!o two
part
T- = 2n E _a_b 6(w-_)
j J _ J
and hermitian _onjugated emission part T _.
electrodynamics is as
parts - absorption
Then
(15)
probability for
63
time T of absorption ( and similarly emission ) is declared as
P = 12, (16)
f
where summation is performed over a complete set of possible final
states. If rewritten in a form
T- = 2_M-6(Ef-E_) ,
(17)
where Ef and E_ are energies of final and initial states, it shows
employing (14) that (16) expresses the well-known Fermi's rule
2_
P = T E l<flM-li>I 2 6(et-ei)" (18)
f
But is it always valid and why probability is defined in this
manner ?
The expansion of S - matrix (11) is as follows
S = 1 - i (T*+T-) - (T 2) /2 + ...
T
The identity of normalization must be valid
perturbation, i.e. for all powers of T as it
the first power of coupling constant :
I = <rlr> " + + 
+ + - + ..-
in all orders
is proportional
(19)
of
to
(20)
Then terms from second to fifth can be interpreted as a
probability of transitions in the second order, since the first
and the sixth will be probability to stay in the initial state. So
conventional procedure ignores the second and the fifth terms. It
is possible only if T-li> is orthogonal to T*li>. It can happen
when either field or the oscillator is in energy eigenstate. Then
actually only _wo levels are involved in any sort of transitions.
In this case emission and absorption can be distinguished- That is
on obtaining after measurement one of If> states we can tell a
result of absorption from a result of emission.
For arbitrary initial state they cannot be distinguished
experimentally. But the total probability of emission and
absorption together in (20) does not have physical meaning.
Therefore we have to revise our approach. More well-grounded
64
procedure is to calculate not probabilities but observable
variations :
_<Q>z = - . (21)
Besides, we do not need to introduce the Fock basis If>, but deal
only with the initial state.
Since the observable variation is expected to grow with time,
to calculate the rates of the processes we need to consider only
terms proportional to long time T. We will see later that
expressions like (21) contain terms with factor 6(_-_) and terms
J
with 62(_-w) under a sign of summation. One power of delta
J
function disappear because of summation over the continuum of
modes. The rest one power will transform to factor T. So terms
with delta function of infinitely little difference to the first
and zeroth powers will give non-growing with time observable
variation. Consequently, these terms represent dressing bare
states by virtual quanta. Terms with the second powers of delta
function will give time-Proportional variations of observables.
Just these terms correspond to transitions with creation of real
quanta.
For our case we need S - matrix up to the second order of
perturbation. In this order a time-ordered product
(TZ)T =__ dt• _£ dt 2 _x(t•lHx(t2)IT/h2 , (22)
where
I H (t)H it ),[H (t)H (t)]- z 2 x .
x • x 2 T Hx (t•) Hx(t2) ,
is different from non-ordered product
(T 2) = TxT ÷ T 2
T di.f
by a term
if t > t
if t > t
i 2'
(23)
(24)
t
00 2
T 2 = $ dt $ dt [H (t)H (t)] /h 2 , (25)di.f --00 2 -_0 • x 2 Z •
The latter expression depends on time like
t
2 exp(i (A-_)Z] - I
/" dt $ dtlexp(iAt +ii_t ) = 4_ 6(A+_) lim • (26)
-_0 2 --_ 2 •
z_ i (;_-_)
65
Such terms do If A = w + w then theJ
last factor in (26") is not singular. Terms with A = _ - _ orJ
opposite give a contribution to T2dLf
exp (2i AZ) -l+exp (-2i AZ)-I
12 [ab*,a*b ] , (27)E IXj j j 2i A
which is not singular either. So T 2 contains first powers ofdif
delta functions and can be neglected compared to T (the former is
coupling constant )_ times less).
J
not vanish only if A = -O.
We arrive to an assumption
S = 1 -iT - TxT/2,
that leads to ,
_<Q >
I
1 [T,Q]] IS>
= x - 1_. iX.2rr6.<b.> - _ £J J J . Jj
• I •
A<bk > = -iXk2n6k<a> - _ Xk2n6 k _. X2n6,
x j J J
A<a*a> = i _ ()_<bea> - X<a*b >)2n6- _ l×kl2(2n6k)2<a*a",'
I .j J J J J J Ic
1 X* * > * * _ • (30c)
+ ._ _: ()'j k<bkbj • + XjXk<bjbk_) (2n):636 k
k.j
• := * @ * *a "::" 2
_<bkbk> i (×x<a bk> - _k<bka>)2m5 k + IX)_! 2<a , (2n6 k)
( -o -, : • ?
- n6 k X k E X_n6j<bjbk/ + )_ E X2_6.<bkb '> " (30d)
• j J ,} J J
These variations are expressed in terms of expectation in
the initial state (designated with triangle brackets), can
define quadrature component variances by
D(P,Q) = _ <pQ> + <Qp> - <Q>. (31)
Their variations can be e:...'pressed similar to
= _<aa> - 2<a> A<a> - (_<a>)z (32)
_D(a'a) x x x x "
values
One
(30a)
(30b)
This kind of variance is important because in canonical
6_
coordinate-momentum space ImD(a,a) corresponds to correlation and
ReD(a,a) - to squeezing. In Schrodinger picture they rapidly
transfer from each to other.
Retaining in (30) and (32) only terms proportional to T and
dividing by T we obtain time derivative equations. From them we
clearly see that radiation damping
" I: ..Ix,i122rt6j (_3)
determines variation of amplitudes
d
dt <a'..x = 2_ <a>, (34a)
d
1
d-_ <bk>z = - 2 IXki22rt6k <bk>" (34b)
These equations coincide with those obtained usually in the
frame of Wigner - Weisskopf approximation. Field modes and the
oscillator exchange their energies. As a result there is no effect
of stimulated emission but only two independent fluxes of energy:
d
d-_ <a*a>x = - _,<a*a> + _ l_k122n6k<bkbk>, (34c)
d k
_-_ <b_bk> x - IXk122_6k<a*a> + l_k122n6k<bkbk>. (34d)
Squeezing-correlation parameter behaves in a similar way :
d
__ = 2 2n6kD (bk, bk)dt D(a'a)z - _,D(a,a) - _ )_k
d k
d--_ D(bk'bk)x = - _22n6k D(a'a) - l>_k122n6kD(bk'bk )"
(34e)
(._'-,4f )
Further development can be made for the specific
of coefficients in Hamiltonian (3). For the continuum
summation is substituted by integration over phase
summation over polarization inde>_es r
-, _ $ Up do> d_'}
j r.
with volume V, solid angle element di% mode frequency density
2
b_
8nsc s
expressions
of modes
space and
(35)
(36)
6?
Decomposition of vector potential A(r,t) over mode variables is
h
A(r,t) = E _ coV e (b (t)e×p(ikr) + H.c.). (37)J J J _ J
where e is a polarization vector and k- a wave vector of j-th
j J
mode.
Gauge invariance substitution of oscillator momentum p * p -
ea leads to the interaction Hamiltonian (3)
-epX eZX z
H = + -- • (38)
m 2m
Here e,m are the charge and the mass of the oscillator. The second
term in this case proves to be a unity operator in state-space of
the oscillator. Hence it results in an infinitely little
renormalization of field energy because of a factor I/V (for
infinitely large volume V). The coupling constant will be
(39)
where 0 is the angle between a polarization vector and the
J
oscillation direction.
On the other hand, from the Hamiltonian in another gauge form
H = - eqE, (40)
where q is a coordinate of the oscillator and E is the electric
field vector, it follows that the coupling constant
t
X' = _-J • (41)
J J
But as all expressions contain delta functions 6(w-_), constants
o
(39) and (41) coincide. We see that it is one of the cases when
gauge transform, performed over state vectors in the absence of
vector potential and corresponding to a change from gauge form
(40) to (38), does not make any difference. These transforms were
considered in detail in Ref. 10,
Einstein's stimulated coefficient can be also introduced.
However it is different from a common one - it depends on the
angle and expresses radiati'on power instead of probability :
68
2 _e
_e c os
B --2.u Ix I"= 2m_
o
The spontaneous emission coefficient is obtained
Integration should be performed over solid angles of
vectors (they are also isotropical ly distributed) ,
vectors of modes :
2 2 2
(d • b_
= I B dO = 6_m_ c s "
4_sc 5 o
(42)
from (33) •
pol arisati on
not wave
(43)
A light beam containing several close modes has an energy density
W = _ p<b*b>h_ (44)
r
or W = 8 W dO. (45)
It will allow us to express eqs.(34) through physically
meaningfull values.
d[h_K.a_a>] - - _h_Ka*a> + $ BW dO,
d ,
[WV] " B [_a*a> - W 1 ,
(4&a)
(46b)
d D(b,b)
d-[ D(a,a) - - zD(a,a) + f BW
i_b+b>
dO, (46c)
d
[WVD(b,b)I - B [pf_,b'b>D(a,a)- W D(b,b)]. (46d)
All above discussed enables us to answer posed questions :
1. In general absorption and emission can not be distinguished.
2. So not Fermi's rule but e×pectation values should be used to
calculate the rates of these processes.
3. Time ordering in this case is not important up to the second
order of perturbation.
4. Stimulated emission does not manifest itself in the final
result.
5. Energy and squeezing-correlation parameters behave in a similar
way : there are independent interchange fl_×es of them
proportional to their current values.
69
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7O

Harmonic oscillator interaction with squeezed radiation

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