We investigate the second-order non-linear interaction as a means to generate
entanglement between fields of differing wavelengths. And show that perfect
entanglement can, in principle, be produced between the fundamental and second
harmonic fields in these processes. Neither pure second harmonic generation,
nor parametric oscillation optimally produce entanglement, such optimal
entanglement is rather produced by an intermediate process. An experimental
demonstration of these predictions should be imminently feasible.Comment: 4 pages, 4 figure

Kirk McKenzie

Nicolai B. Grosse

P. D. Drummond

Ping Koy Lam

Warwick P. Bowen

English

arXiv

We investigate the second-order nonlinear interaction as a means to generate entanglement between fields of differing wavelengths and show that perfect entanglement can, in principle, be produced between the fundamental and second-harmonic fields in these processes. Neither pure second-harmonic generation nor parametric oscillation optimally produce entanglement; such optimal entanglement is rather produced by an intermediate process

Grosse, Nicolai B.

Bowen, Warwick P.

McKenzie, Kirk

Lam, Ping Koy

University of Queensland eSpace

PRL 96, 063601 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending17 FEBRUARY 2006Harmonic Entanglement with Second-Order Nonlinearity
Nicolai B. Grosse,1 Warwick P. Bowen,1,2 Kirk McKenzie,1 and Ping Koy Lam1
1Quantum Optics Group, Department of Physics, Faculty of Science, The Australian National University, ACT 0200, Australia
2Norman Bridge Laboratory of Physics, California Institute of Technology, Pasadena, California 91125, USA
(Received 6 July 2005; published 13 February 2006)0031-9007=We investigate the second-order nonlinear interaction as a means to generate entanglement between
fields of differing wavelengths and show that perfect entanglement can, in principle, be produced between
the fundamental and second-harmonic fields in these processes. Neither pure second-harmonic generation
nor parametric oscillation optimally produce entanglement; such optimal entanglement is rather produced
by an intermediate process.
DOI: 10.1103/PhysRevLett.96.063601 PACS numbers: 42.50.Dv, 03.67.Mn, 42.65.Yjκa1
ε
κb1
κa2
κb2
second-order 
nonlinearity 
input / output 
coupling mirror
mirror for
optical loss
a1,inˆ
b1,inˆ
a1,outˆ
b1,outˆ
b2,inˆ
a2,inˆ
FIG. 1 (color online). Schematic for the generation of optical
harmonic entanglement.Second-order nonlinear processes have found many ap-
plications in electronics, mechanics, optics, as well as
numerous other fields. More recently, in the field of quan-
tum optics it has been discovered that they offer an ideal
mechanism for the generation of entangled states of light.
As a result, many fundamental theories of quantum me-
chanics have been tested at an unprecedented level [1]. It is
surprising then to find that, to date, a comprehensive treat-
ment of entanglement generated through second-order
nonlinear processes has not been presented. Here, we
perform a broad analysis of entanglement in degenerate
processes of this kind. We demonstrate that, in principle,
perfect entanglement can be achieved between the funda-
mental and second-harmonic fields involved in the process.
This entanglement between harmonically related fields is
referred to as harmonic entanglement. It is well known that
quantum correlations and harmonic entanglement can be
achieved via second-harmonic generation (SHG) [2], and
furthermore, that highly squeezed states of light can be
obtained in the complementary process, optical parametric
oscillation (OPO) [3,4]. However, we find that optimal
harmonic entanglement is not generated by either of these
processes, but rather by an intermediate process, pump
depleted optical parametric amplification (OPA). In con-
trast, previous investigations of the quantum optical prop-
erties of OPA have almost exclusively been limited to the
small region of parameter space where pump depletion is
insignificant.
The study of entangled states began as a means to test
the counterintuitive predictions of the theory of quantum
mechanics. In recent years, however, focus has shifted as a
result of the realization that nonclassical states of light can
enhance many measurement, computation, and communi-
cation tasks [5,6]. That quantum mechanics, in principle,
allows such enhancement is significant in and of itself, and
has motivated a great deal of theoretical work. However,
emphasis has also been placed on experimental demon-
strations, which as yet have been primitive by comparison.
New tools are required for significant experimental
progress to be made. Both SHG and OPO have found
many applications in quantum optics experiments [7–9],06=96(6)=063601(4)$23.00 06360becoming standard tools in the field. The pump depleted
OPA studied here has the potential to become equally
significant. The generation of strong entanglement be-
tween optical fields at vastly different frequencies would
immediately facilitate many interspecies quantum infor-
mation protocols; for example, the frequency of states of
light could be drastically altered using interspecies quan-
tum teleportation protocols. Such a protocol would en-
hance the integrability of d**isparate nodes in a quantum
information network. Applications could be envisaged in
any situation where a nonclassical link is required between
two experiments at differing optical frequencies. Such a
situation might arise, for example, in experiments involv-
ing two different atomic species, or if a connection is
required to an atomic frequency standard. In addition, the
generation of harmonic entanglement will apply to other
systems that share the same second-order nonlinear inter-
action term in the Hamiltonian.
Analysis.—The system under analysis in this Letter
consists of a second-order nonlinear medium enclosed
within an optical resonator as shown in Fig. 1. The reso-
nator is coupled to the environment through two partially
reflective mirrors. One mirror represents an input/output
coupler, while the other represents uncontrollable coupling
loss. By assuming that the cavity only resonates a single
spatial eigenmode, the nonlinear medium induces an inter-
action between the two intracavity fields, giving [10]
_^a  aa^ a^yb^ A^in; _^b  bb^ 12a^2  B^in;
(1)1-1 © 2006 The American Physical Society
PRL 96, 063601 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending17 FEBRUARY 2006where a^ and b^ are Heisenberg picture annihilation opera-
tors describing the intracavity fundamental and second-
harmonic fields, respectively; a and b are the associated
total resonator decay rates;  is the nonlinear coup-
ling strength between the fields; and A^in and B^in repre-
sent the accumulated input fields to the system.
Throughout this Letter the partially reflective mirrors mod-
eling input/output coupling and loss are distinguished with
the subscripts ‘‘1’’ and ‘‘2,’’ respectively, while the input
and output fields are denoted by the subscripts ‘‘in’’ and
‘‘out.’’ Using this terminology a  a1  a2, b 
b1  b2, A^in 

2a1
p
A^1;in 

2a2
p
A^2;in, and B^in 
2b1
p
B^1;in 

2b2
p
B^2;in.
The solution to Eqs. (1) is obtained through the tech-
nique of linearization, where operators are expanded in
terms of their coherent amplitude and quantum noise op-
erator, so that a mode a^i   a^i with ha^ii  0, and
second-order terms in the quantum noise operators are
neglected. First, the steady-state coherent amplitudes of
the fundamental and second-harmonic intracavity fields
are obtained. After a stability analysis has been made on
the solutions, the system shows a range of interesting
behavior such as monostability, bistability, out-of-phase
monostability, and self-pulsation [10]. For clarity, we nor-
malize the driving fields to their respective critical transi-
tion values, i.e., the critical amplitude, 1;in;c, for self-
pulsation in SHG, and the threshold amplitude 1;in;c for
OPO, such that
1;in;c  2a  b2ba  b
1=2


2a1
p (2)
1;in;c  ab 2b1p : (3)
The normalized driving fields are then denoted d 
1;in=1;in;c and d  1;in=1;in;c. The quantum fluctua-
tions of the intracavity fields can be obtained from the
fluctuating part of Eq. (1)
 _^a  aa^ b^ a^y  A^in
 _^b  bb^ a^ B^in:
(4)
These equations can be easily solved by taking the Fourier
transform into the frequency domain. We want to inves-
tigate harmonic entanglement between the amplitude X^
and phase X^ quadratures of the two output fields. These
quadratures are related to the annihilation operators via
X^  a^ a^y and X^  ia^y  a^. Writing the solution to
Eqs. (4) in terms of field quadratures, we find
XA;in
XA;in
XB;in
XB;in
2
6664
3
7775 
A B C D
B A D C
C D E 0
D C 0 E
2
6664
3
7775
Xa
Xa
Xb
Xb
2
6664
3
7775; (5)
where fX^	a ; X^	b g and fX^	A;in; X^	B;ing are the intracavity and
06360accumulated input field quadratures, respectively, and
A	  a  i!
 jj cos, B  jj sin, C 
jj cos, D  jj sin, E  b  i!, with  
Arg ,   Arg , and ! is the side-band detection
frequency. The output fields of the resonator can then
be directly obtained using the input-output formal-
ism, X^	A1;out 

2a1
p
X^	a  X^	A1;in, X^	B1;out 

2b1
p
X^	b 
X^	B1;in [11]. We can now proceed to investigate the presence
of entanglement between the output fundamental and
second-harmonic fields. A bipartite Gaussian entangled
state is completely described by its correlation matrix
[12], which has the following elements
Cklmn  12hX^kmX^ln  X^lnX^kmi  hX^kmihX^lni; (6)
where fk; lg 2 f;g and fm; ng 2 fA1;out; B1;outg.
The standard characterization of continuous variable
entanglement is to measure the quantum correlations be-
tween two fields via the EPR criterion [13] and to apply the
inseparability criterion [12]. Before the inseparability cri-
terion can be applied, however, the correlation matrix is
required to be in standard form II, which can be achieved
by application of the appropriate local-linear-unitary-
Bogoliubov operations (local rotation and squeezing op-
erations) [12]. The product form of the degree of insepa-
rability [14] is given by
I 

CI CI
q
k 1=k ; (7)
C	I  kC		aa  1=kC		bb  2jC		ab j (8)
k 

C		bb  1
C		aa  1

1=2 (9)
I < 1 is a necessary and sufficient condition of insepara-
bility and therefore entanglement. We will use I as a
measure of entanglement. The degree of EPR paradox,
which measures the level of apparent violation of the
Heisenberg uncertainty principle achieved by the state,
can also be found from elements of the correlation matrix.
" 

Caa  jC

ab j2
Cbb

Caa  jC

ab j2
Cbb

: (10)
The state is entangled when " < 1. Note that " is mini-
mized when the correlation matrix is in standard form. We
therefore restrict our analysis to this case, denoting the
optimized degree of EPR paradox as "o.
In this work, the optimization of the correlation matrix
was achieved numerically. The parameters used for making
our calculations were a1  1, a2  0:01, b1  10,
b2  0:1,   1, !  0 which were chosen to resemble
the squeezing/entanglement sources of recent experiments
[14].
Results.—Using these analytical results, we mapped out
the degree of optimized EPR paradox ("o) of harmonic1-2
PRL 96, 063601 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending17 FEBRUARY 2006
entanglement across the parameter space of fundamental
and second-harmonic driving field amplitudes. This has
been visualized in Fig. 2 where a gray-scale-compatible
color map has been assigned to a range of EPR values on a
dB log (base 10) scale. Areas that are printed in dark ink
signify strong entanglement. White areas are not en-
tangled, and are either at the standard quantum limit or
greater (classical thermal states). The vertical d and
horizontal d axes of the plot coincide with the special
cases of OPO and SHG, respectively. Regions of interest
have been individually labeled. Note that in order to rep-
resent bistability in the map, the left- and right-hand sides
of the map must be mentally folded along the OPO axis.
The map of driving fields is clearly divided into the three
distinct areas that correspond to the stable branches of the
steady-state solutions. These are: (i) the bistable region
above OPO threshold, (ii)–(iii) the monostable solution
showing parametric amplification at (ii) and deamplifica-
tion at (iii), and finally, the so-called out-of-phase solution,
which has a complex-valued fundamental field solution, is
labeled with (iv). We find that the system produces a
maximum entanglement of up to "o  13:8 dB at the
intersection of (i)–(ii) and also when crossing (iii)–(iv).
These regions correspond exactly to the critical boundaries
where a branch change in the classical solution occurs.
A more detailed analysis of entanglement in SHG is
shown in Fig. 3(a) where the EPR measure is plotted
concomitant with the inseparability criterion as the ampli-
tude of the driving field is varied. The strength of entan-
glement finds a maximum of "o  1:9 dB and I  3:0 dB
for the system driven with a fundamental field of d 
	0:1. In SHG, entanglement is produced for all nonzero
driving field amplitudes. However, the strength of the
entanglement does not become arbitrarily high if loss in
the system is made arbitrarily small. This is a bound set by
the SHG process itself. Note that for a traveling-wave SHG-0.03 -0.02 -0.01 0 0.01 0.02 0.03
-2
-1
0
1
2
-14
-12
-10
-8
-6
-4
-2
≥ 0
β d 
 
 
 
se
co
n
d-
ha
rm
o
n
ic
 d
riv
in
g 
fie
ld
 
 
 
 
[th
res
ho
ld=
1]
αd  fundamental driving field  [threshold=1]
increasing entanglem
ent
[dB]
SHG
OPO
(i)
(iii)
(iv)
(ii)
FIG. 2 (color online). A plot of the EPR entanglement strength
as a function of the fundamental, , and second harmonic, ,
driving field amplitudes. Bistable OPA (i) above and (ii) below
threshold. Parametric deamplification regime (iii) above and
(iv) below threshold.
06360process the strength of entanglement has been found to be
"o  12 dB [2].
Contrast this behavior with the OPO process, which for
below threshold, is not entangled, and only becomes en-
tangled once the system is pushed beyond threshold. This
is shown in Fig. 3(b). The maximum entanglement pro-
duced is "o  9:1 dB and I  7:5 dB both for d 
	1:1. Note that as loss in the model is made arbitrarily
small and for a driving field approaching OPO threshold,
the entanglement becomes arbitrarily strong.
If we view the total input power to the system as a
resource, then it is interesting to see how the strength of
entanglement changes, as the power splitting ratio between
the fundamental and second-harmonic field is varied, while
keeping the total input power constant. We define the
splitting fraction to be R  j1;inj2=12 j1;inj2  j1;inj2,
and follow a path, parametrized by angle R, as defined by
d  1;in;c=1;in;c

2
p
sinR, and d 


p
cosR.
In Fig. 4 the total input power  has been set to 90% in
(a) and 400% in (b) of the power required to reach OPO
threshold. A given angle in the plot describes not only the
splitting fraction for fundamental and second-harmonic
power, but also the relative phase between them (0 or ).
The radial distance in the plot corresponds to the EPR
measure of entanglement in a dB scale. A circle is drawn
at 0 dB to mark the classical limit. The shaded areas
highlight the entanglement.
We find that for a total input power required to reach
90% OPO threshold, there is a choice of two splittings of
R  99:9% and R  77%, that both give a maximum
amount of entanglement of "o  6:5 dB. These parameters
correspond to neither a pure OPO nor a pure SHG process,
but correspond rather to parametric amplification/deampli-
fication in a moderately pump depleted/enhanced regime.
In Fig. 4(b) we increase the resource available by setting
the total input power to 400% of OPO threshold (for which-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-10
0
10
20
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-3
-2
-1
0
1
EP
R
  a
nd
  I
ns
ep
.
 
[dB
]
EP
R
  a
nd
 In
se
p. 
[dB
]
αd
βd
(a)
(b)
FIG. 3 (color online). Harmonic entanglement observed in
(a) SHG and (b) OPO. Both the EPR (solid lines) and the
inseparability (dotted lines) entanglement measures are plotted
as function of the fundamental and second-harmonic field am-
plitudes, respectively.
1-3
R=75% R=99.9%
R=77%
R=25%
R=100%
R=0%
-10-15 0 +10+5-5 [dB]
classical lim
it
SHG
OPO(a)
R=75%
R=25%
R=100%
R=0%
SHG
OPO
-10-15 0-5 [dB]
classical limit
(ii)
(iii)
(iv)
(i)
(b)
R=43%
R=97%
FIG. 4 (color online). Polar plot of harmonic entanglement
strength with a total input power of (a) 90% of threshold and
(b) 400% of threshold. Shaded areas denote the presence of
harmonic entanglement.
PRL 96, 063601 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending17 FEBRUARY 2006stable operation of OPO has been demonstrated). The
bistability displayed by the system in region (i) is now
clearly visible. As we drive the system from (i) through to
region (ii), the entanglement approaches a strength of "o 
13:8 dB at R  97%. The same strength of entanglement is
also found at R  43%, which is exactly at the boundary
between the regions (iii) and (iv). Arbitrarily strong entan-
glement can be produced in these boundary regions, pro-
vided that the system has sufficiently low loss, and is
driven above threshold. In practice, the maximum entan-
glement in region (iii)–(iv) will be easier to control than
(i)–(ii), due to the narrowness of the latter.
From these two examples, we can see that stronger
entanglement is made available when the system is driven
by higher total input powers. This supports the view, that
the total input power to the system is a resource for the
generation of entanglement. To relate this result to a po-
tential experimental demonstration, we stress that the
amount of total input power needed to access at least "o 
6 dB of entanglement is near to that required to reach
threshold in OPO, where both the detection of such entan-
glement, and the total input power required, are experi-
mentally accessible [4].06360So far, we have only considered the case where the
driving fields are coherent states. Although an in-depth
discussion is contained in a forthcoming paper, we can
qualitatively describe the effect that squeezed driving fields
have on the harmonic entanglement generation for this
system. In short, squeezing of the fundamental and/or the
second-harmonic field in the correct combination of am-
plitude/phase quadratures results not only in stronger en-
tanglement, but also extends the regions of strong
entanglement closer in toward the origin, thereby reducing
the total input power needed to observe a given strength of
entanglement.
Summary.—We have shown that arbitrarily strong en-
tanglement between a fundamental field and its second
harmonic can, in principle, be generated by the second-
order nonlinear interaction. The maximal strength of this
harmonic entanglement, as measured by the EPR and
inseparability criteria, is produced neither by pure SHG
nor parametric oscillation, but rather by an intermediate
process. We considered the total input power that drives the
nonlinear interaction as a resource for the strength of
entanglement, and found that an experimental demonstra-
tion of harmonic entanglement using optical techniques
should be attainable.
This research is supported by the Australian Research
Council Discovery Grant scheme. W. P. B. would like to
acknowledge financial support from the Center for the
Physics of Information, California Institute of Technology.1-4[1] A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49,
91 (1982); Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C.
Peng, Phys. Rev. Lett. 68, 3663 (1992).
[2] H. M. Wiseman, M. S. Taubman, and H. A. Bachor, Phys.
Rev. A 51, 3227 (1995); M. K. Olsen, Phys. Rev. A 70,
035801 (2004).
[3] L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev.
Lett. 57, 2520 (1986).
[4] P. K. Lam et al., J. Opt. B 1, 469 (1999).
[5] T. C. Ralph and E. H. Huntington, Phys. Rev. A 66,
042321 (2002).
[6] C. H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993).
[7] A. M. Lance et al., Phys. Rev. Lett. 92, 177903
(2004).
[8] X. Li et al., Phys. Rev. Lett. 88, 047904 (2002).
[9] W. P. Bowen et al., Phys. Rev. A 67, 032302 (2003).
[10] P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta
27, 321 (1980).
[11] M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386
(1984).
[12] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys.
Rev. Lett. 84, 2722 (2000).
[13] M. D. Reid and P. D. Drummond, Phys. Rev. Lett. 60,
2731 (1988); M. D. Reid, Phys. Rev. A 40, 913 (1989).
[14] W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph,
Phys. Rev. Lett. 90, 043601 (2003); Phys. Rev. A 69,
012304 (2004).


Harmonic entanglement with second-order nonlinearity

Caltech Authors - Main

http://authors.library.caltech.edu/3652/1/GROprl06.pdf

Harmonic entanglement with second-order non-linearity

Abstract

Similar works

Full text

Available Versions

University of Queensland eSpace

Caltech Authors - Main