The Hermitian phase formalism of Pegg and Barnett allows for direct calculations of the phase variance and, consequently, the number-phase uncertainty product. This gives us a unique opportunity, inaccessible before, to study the number-phase uncertainty relations for optical fields in a direct way within a consistent quantum formalism. A few examples of fields generated in nonlinear optical processes are studied from the point of view of their number-phase uncertainty relations

Tanas, Ryszard

English

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NUMBER-PHASE UNCERTAINTY RELATIONS
FOR OPTICAL FIELDS
Ryszard Tana_
Nonlinear Optics Division, Institute of Physics, Adam Mickiewicz University,
Grunwaldzka 6, 60-780 Poznad, Poland,
Abstract
The Hermitian phase formalism of Pegg and Barnett allows for direct calculations of the
phase variance and, consequently, the number-phase uncertainty product. This gives us a
unique opportunity, inaccessible before, to study the number-phase uncertainty relations for
optical fields in a direct way within a consistent quantum formalism. A few examples of
fields generated in nonlinear optical processes are studied from the point of view of their
number-phase uncertainty relations.
1 Number-phase uncertainty relations
Pegg and Barnett [1] introduced the Hermitian phase formalism, which is based on the observation
that in a finite-dimensional state space the states with well-defined phase exist. Thus they restrict
the state space to a finite (s + 1)-dimensional Hilbert space k_ spanned by the number states ]0),
I1),..., Is). In this space they define a complete orthonormal set of phase states by
IO_)- _l _exp(inO_)ln), m-O, 1,...,s, (1)
n=O
where the values of 8_ are given by
2?rm
e. = eo+ --. (2)
s+l
The value of O0 is arbitrary and defines a particular basis set of (s + 1) mutually orthogonal phase
states.
The Pegg-Barnett Hermitian phase operator is defined as
$
&e - E e=le._><O._l. (3)
rn----O
Of course, phase states (1) are eigenstates of the phase operator (3) with the eigenvalues 8=
restricted to lie within a phase window between 80 and 00 + 2rs/(s + 1). The Pegg-Barnett
prescription is to evaluate any observable of interest in the finite basis (1) and only after that to
take the limit s --+ oo.
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Since the phase states (1) are orthonormal, (0,_10,_,) = 3,_m,, the kth power of the Pegg-Barnett
phase operator (3) can be written as
$
= E le ><e l. (4)
rrt _-- 0
Substituting eqs. (1) and (2) into eq. (3) and performing summation over m yields the phase
operator explicitly in the Fock basis
^ s_" 27r exp[i (n - n')O0] In)(n' I (5)
(be - 0o + --s+l + s+l E exp[i(n-n')2rt/(s+l)]-i
It is well apparent that the Hermitian phase operator _0 has well defined matrix elements in the
number state basis and does not suffer from such problems as the original Dirac phase operator.
A detailed analysis of the properties of the Hermitian phase operator was given by Pegg and
Barnett [1]. As the Hermitian phase operator is defined, one can calculate the expectation value
and variance of this operator for a given state of the field If}.
The Pegg-Barnett phase operator (5), expressed in the Fock basis, readily gives the phase-
number commutator [1]:
2rr (n - n')exp[i(n - n')O0]
[_0,r_] = s + 1 E exp-_(-n 7 _s-7+-]-_)] - 1In)<n'l' (6)
Equation (6) looks very different from the faanous Dirac postulate of the phase-number commu-
tator.
Having defined the Hermitian operators for the number and phase variables and knowing their
commutator, we can easily test the number-phase Heisenberg uncertainty relation for any given
field with known number state decomposition.
> (7)Ac}eAh
For physical states the number-phase commutator can be considerably simpified [1], and its
expectation value in the physical state IP) can be expressed in terms of the phase distribution
function P(S0), which makes calculations of this quantity pretty simple.
(Pl [_0,¢_11P) = -i[1 - (s + 1)l(P110o>1=] (s)
-+ -i[1- 2_P(Oo)] (9)
In the next Sections we give a few exasnples of the number-phase uncertainty relations calcu-
lated using the above formulas.
2 Examples
2.1 Anharmonic oscillator model
The anharmonic oscillator model is described by the Hazniltonian
1. t2 2 (lo)
268
where g and &t are the annihilation and creation operators of the field mode, and _ is the coupling
constant, which is real and can be related to the nonlinear susceptibility X (a) of the medium if
the anharmonic oscillator is used to describe propagation of laser light (with right or left circular
polaxization) in a nonlinear Kerr medium. If the state of the incoming beam is a coherent state
]a0), the resulting state of the outgoing beam is given by
I_(r)> = 0(,)]c,o) = exp(-Ic_ol2/2) _n.lexp i n(n-1)In>, (11)
rl.m0
where r = -nt.
4.5
3:/--
!2/
1.5_10.5
0
0 1 2 3 4 5 6
tau
FIG. 1. Evolution of the uncertainty product (lhs of eq. (7) - upper curve) and
its lower bound (rhs of eq. (7) - lower curve) for the anharmonic oscillator state with
I 01 = 4,
The appearance of the nonlinear phase factor in the state (11) modifies essentially the properties
of the field represented by such a state with respect to the initial coherent state la0}. It was
shown by Tanad [2] that a high degree of squeezing can be obtained in the anharmonic oscillator
model. Squeezing in the same process was later considered by Kitagawa and Yamamoto [3] who
used the name crescent squeezing because of the crescent shape of the quasiprobability distribution
contours obtained in the process.
The Pegg-Barnett Hermitian phase formalism has been applied for studying the phase prop-
erties of the states (11) by Gerry [4], who discussed the limiting cases of very low and very high
light intensities, and by Gantsog and Tana.4 [5], who gave a more systematic discussion of the
exact results.
In Fig. 1 we show the evolution of the number-phase uncertainty product as given by the lhs
of ineq. (7) (upper curve) and its lower bound as given by the rhs of ineq. (7) (lower curve) for
269
the state (11) of the anharmonic oscillator assuming that the mean number of photons ]ao[ 2 = 4.
It is seen that the number-phase uncertainty product rapidly increases at the early stage of the
evolution, which is due to the rapid randomization of the phase, since the photon statistics remain
all the time Poissonian with the number of photons variance equal to the mean number of photons
laol 2. This is a typical behavior for mean numbers of photons greater than unity. It is also seen
that the states generated in the anharmonic oscillator model are never the minimum uncertainty
or intelligent states. The level of noise is much bigger than its lower bound allowed by quantum
mechanics. Since the dynamics is periodic, after time r = 2_r the system returns back to its initial
state. It can be shown [5] that for [a0[ 2 >> 1 the number and phase uncertainty product takes the
approximate analytical form
1
<(A)2>(_0((A)2>6 = (_ + [ao[2r2), (12)
explicitly showing rapid increase of the uncertainty product from the value 1/4 known for the
coherent state.
2.2 Jaynes-Cummings model
The model is described by the Hamiltonian (at exact resonance)
/3/ : hw(ata +/_) + hg(/_ta + ]_at), (13)
where a t and _ are the creation and annihilation operators for the field mode; the two-level atom
is described by the raising, /_t, and lowering, /_, operators and the inversion operator 1_, and g
is the coupling constant.
To study the phase properties of the field mode we have to know the state evolution of the
system. After dropping the free evolution terms, which change the phase in a trivial way, and
assuming that the atom is initially in its ground state and the field is in a coherent state [C_o), the
state of the system is found to be
oo
[¢(t)> = __b. exp(inOo)[COS(x/ngt)[n,g>-isin(v/-ngt)[n-l,e>], (14)
_----0
where [g) and [e) denote the ground and excited states of the atom, the coefficients b, are the
Poissonian weighting factors of the coherent state a0) and O0 is the coherent state phase (phase of
So). The main oscillations of the uncertainty product reflect the oscillations of the phase variance,
which has its extrema for the revival times (in the figure time T = gt/(2_r[ao[) is scaled in the
revival times). Small oscillations seen on the figure stem from the oscillations of the photon number
variance and have only minor effect on the overall behavior. They are associated with the revivals
of the rapid l_bi oscillations in the model. However, this is the phase variance that smoothly
oscillates in the time scale of the subsequent revivals. In this way, the well known phenomenon of
collapses and revivals has obtained clear interpretation in terms of the cavity mode phase [6].
270
14 v
12,
10
_8
2-
00 1 2 3 4 5 6 7 8 9 10
T
FIG. 2. Same as fig. 1 but for the Jaynes-Cummings model with lao[ 2 = 20.
8 i i i i
7
o
o 0.5 1 1.5 2 2.5 3 3.5 4 4,5 5
g!
FIG. 3. Same as fig. 1 but for the down conversion with quantum pump with the
initial mean number of photons equal to 4.
271
2.3 Down conversion with quantum pump
The parametric down conversion with quantum pump is governed by the Hamiltonian
= //0 + HI : hw_ + 2hw _'t_' + _g(b*a 2 + bat2), (15)
where a (a t) and b (/_t) are the annihilation (creation) operators of the signal mode of frequency
w and the pump mode at frequency 2w, respectively. The coupling constant g, which is real,
describes the coupling between the two modes.
Phase properties of this system have been described by Gantsog et al. [7] and Tana_ and
Gantsog [8] and the details of the calculations can be found there. Here, in Fig. 3, we show,
as in previous examples, the evolution of the number phase uncertainty product and its lower
bound for this process. For finite initial mean number of photons the number-phase uncertainty
product remains finite during the evolution contrary to the parametric approximation under which
it rapidly explodes to infinity.
3 Conclusions
All above examples, are typical examples of the fields generated in nonlinear optical processes,
and they show clearly that nonlinear processes typically evolve to quantum states which are far
from being the minimum uncertainty or intelligent states.
Acknowledgments
I thank Drs. Ts. Gantsog and A. Miranowicz for many discussions and cooperation. This work
was supported by the Polish Committee for Scientific Research (KBN) under the grant No. 2
P03B 128 8.
References
[1] D. T. Pegg and S. M. Barnett, Phys. Key. A 39,1665 (1989).
[2] 1%. Tana_, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, Eds., (Plenum, New
York, 1984), p. 645.
[3] H. Kitagawa and Y. Yamamoto, Phys. 1%ev. A 34, 3974 (1986).
[4] C. C. Gerry, Opt. Commun. 75,168 (1990).
[5] Ts. Gantsog and R. Wana_, J. Mod. Opt. 38, 1021 (1991).
[6] H. T. Dung, 1%. Tana_, and A. S.Shumovsky, Opt. Commun. 79,462 (1990)
[7] Ts. Gantsog, 1%. Tana_, and R. Zawodny, Opt. Commun. 82,345 (1991)
[8] R. Tana_ and Ts. Gantsog, Quantum Opt. 4, 245 (1992); Phys. 1%ev. A 45, 5031 (1992)
272


Number-Phase Uncertainty Relations for Optical Fields

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