Entanglement of scattered single photon with atom
Abstract
Single–photon which is initially uncorrelated with atom, will evolve to be entangled with the atom on their continuous kinetic variables in the process of resonant scattering. We find the relations between the entanglement and their physical control parameters, which indicates that high entanglement can be reached by broadening the scale of the atomic wave or squeezing the linewidth of the incident single–photon pulse.
pacs:
03.65.Ud, 42.50.Vk, 32.80.Lg.
I Introduction
Quantum entanglement is of fundamental importance in the theory of
quantum nonlocality1 nonlocality as well as in quantum
information2 QIT . Recently, photon–atom entanglement is
frequently discussed in their finite Hilbert spaces3 etagl fini , such as, the polarizations of photon or the internal states
of atom. With the progress of micro–cavity quantum
electrodynamics4 CQED and high coupling artificial
atom5 artificial atom , single photon raises its ability to
affect considerably not only the atom’s internal state but also its
external motion. As a result, it gives rise to some basic questions
related to the photon–atom entanglement on their infinite kinetic
degree of
freedom.
In recent studies6 Singlephoton 7 scattering ,
entanglement in the continuous kinetic variables between
single–photon and atom is mostly discussed in the process of
single–photon emission with atomic recoil, where the atom is
initially pumped to its excited level and the single–photon is
prepared “intrinsically” by the atomic spontaneous emission. In
our work, however, the resonant single–photon is initially injected
from a tuneable single–photon generator8 singlephoton generator , whereas an artificial atom is placed freely in vacuum on
its steady state (“artificial” indicates that the atomic coupling
to the single–photon is stronger than usual, which ensures the
interaction observablestatement ). We find that, after the
interaction, the scattered single–photon will be entangled to the
atom at a higher degree compared with the case of solely spontaneous
emission. We explain this phenomena as the coherent pumping of the
incident photon and evaluate it with a defined “entanglement
pumping coefficient”.
To describe the degree of entanglement, firstly, we use the ratio
() between the conditional and unconditional variance in momentum
to evaluate the two particles’ correlation in the probability
amplitude of their wave function, which is experimentally accessible
and can be seen as the “amplitude entanglement” in momentum
space9 photoionization 10 phase ent ; secondly, we use
the standard Schmidt decomposition11 Schmidt dec and treat
Schmidt number 12 Schmidt num as a criterion for the full
entanglement contained both in amplitude and phase. For both
criterions and , we revealed their dependencies on the
physical control parameters and , and compare them in
some region of interests, from which it is shown that: higher
entanglement can be achieved by either broadening the scale of the
atomic wave or squeezing the linewidth of the incident
single–photon. Transmitted photon is also considered, which is
different to the scattered photon, and exhibits little entanglement
with the atom due to its interference with the transparent wave (initially incident photon wave profile).
Ii theoretical analysis
As shown in Fig. 1 (a), the two–level atom with transition
frequency and mass
is placed freely in vacuum, the ground and excited states of which are denoted
by and , respectively. The incident
single–photon from some generator is resonant with the atom and
exhibits a superposed state of different fock states due to its
linewidth. For the realistic consideration in some
experiments13 exp1 , we fix the photon detector and atom
detector in opposite directions and make them both in the –
plane for simplicity as in Fig. 1 (b), the angle can be
chosen to observe the scattering in needed directions.
Under the rotating wave approximation (RWA) the Hamiltonian can be written in Schrödinger picture as:
(1) | |||||
where and denote atomic center–of–mass
momentum and position operators, denotes the
atomic operator (),
and are the
annihilation and creation operators for the light mode with photonic
wave vector and frequency ,
respectively. Note the summation is performed over all coupled modes
in the continuous Hilbert space. We also suppress the polarization
index in the summation as well as in photon state, since we can
always choose a particular
polarization to detect the photon. is the dipole coupling coefficient.
As there is only one photon in the interaction, the basis of the Hilbert space can be denoted as , where the arguments in the kets denote, respectively, the wave vector of the atom, and of the photon, and the atomic internal state. At time the state vector can therefore be expanded as:
(2) |
Substituting Eqs. (1) and (2) into Schrödinger equation yields:
(3) | |||
(4) |
where , are the slowly varying parts of and , i.e.:
(5) | |||||
(6) |
Suppose the atom is initially in the ground state and has zero average velocity, the initial condition can be set as:
(7) | |||
(8) |
where and
.
In this case, functions and
have zero center value and bandwidths and
separately. The coordinates are chosen as in Fig. 1
(b), where we make the incident direction as –axis.
is the normalized factor and
is
the resonant wave vector.
We proceed to solve the equations with Laplace transformation and single pole approximation14 single pole and yield:
(9) | |||
where the frequency shift and atomic linewidth are given as:
We can simplify Eq. (9), by replacing the term
with since the momentum bandwidth due to the recoil is normally much larger than the photon
linewidth ; also, we can replace with
in the term . With these approximations, the
first term in the curly bracket can be seen as the antifourier
transform of the product of the photonic shape and Lorentzian shape,
and will cause a decay at a time scale is due to the spontaneous emission. Then,
one can directly find that . In the further calculations, we ignore the
frequency shift since it can be treated as a modification of the
atomic transition frequency, and
regard the slowly varying function as a constant.
; the second decay
term
With the approximations mentioned above, from Eqs. (3) and (9), we obtain the steady solution of :
(10) | |||||
From Eq. (10), one sees that the final state is a superposition of
the transparent wave (initially incident photon wave profile,
depicted by the first term on the r.h.s.) and scattering wave
(second term on the r.h.s.). In the scattering part, the atom and
the photon are entangled due to the process of photon absorption and
emission with atomic recoil, which is sketched in Fig. 1 (c). One
may find that the Lorentzian–Gaussian factor in the scattering part
is very similar to that in the case of spontaneous emission with
recoil6 Singlephoton , where the Gaussian term is a reflection
of momentum conservation and
the Lorentzian term indicates the energy conservation.
The general formula (10) can be used to analyze the photon scattered in different directions. Without loss of physical generality, we choose the initial conditions for the atom as , and for the photon which is exactly the case if the incident single–photon is generated by spontaneous emission. As a remark, we point out that all the conclusions in the following keep available when the incident photon is chosen to be other shapes such as Gaussian or whatever.
Iii Amplitude Entanglement in Scattered photon
To make the physical results more evident and avoid unnecessary mathematical complexity, we focus our attention on the photon scattered perpendicular to the incident direction, i.e., . Then we project Eq. (10) into the subspace , with the same approximations used in Eq. (9), and yield:
(11) | |||||
where , ,
,
are all
defined dimensionless parameters. Note that and
contain all the physical parameters that determine the
nature of the atom–photon system, thus can be treated as physical
control parameters for the atom and the photon, respectively. We
neglect tiny terms in Eq. (11) due to
and in realistic conditions. is the
normalization factor where
.
From Eq. (11) and Fig. 2, one sees that, variables
and play the symmetric role in the two Lorentzian
functions. It makes the probability amplitude
localized along the diagonal of the
momentum space, which implies the nonfactorization of the
photon–atom wave function, and then will generate entanglement
between the two particles. In fact, we can treat the ratio () of
the conditional and unconditional variances for or
as an evaluation of entanglement9 photoionization . This ratio, compared to the Schmidt number ,
reveals more obvious analytic dependence for the entanglement on its
control parameters and , and is also
experimentally directly
accessible15 exp3 .
We proceed to calculate the ratio for variable , i.e., , where the unconditional variance is obtained from the single–particle observation as:
and coincidence measurement gives the conditional variance at some specified :
(13) | |||
Substituting Eqs. (11)–(13) into the definition of , we yield as a function of parameters and , the result of which is illustrated in Fig. 3 with fixed at the origin. From that, one can see that the entanglement increases monotonously when increases or decreases, which indicates that higher entanglement can be achieved by squeezing the linewidth of the incident photon or broadening the wave packet of the atom. In particular, when , we have:
(14) |
from which it is found that the entanglement increases linearly with
and will be abruptly enhanced when tends to
zero. As a remark, we emphasize that all the conclusions above
hold qualitatively the same either if is specified otherwise or one calculate the ratio
from the other variable .
The ratio , which can be obtained experimentally by comparing the
momentum dispersion variance, is an appropriate quantification for
the entanglement contained in the probability amplitude correlation
(thus can be seen as an evaluation of the “amplitude
entanglement”). Next, we can see that it reveals a correct varying
tendency for the entanglement with its control parameters. However,
the definition of is dependent on its representation space and
different choices for the basis of Hilbert space will cause distinct
values of . This is because we only use the amplitude of the
wavefunction to construct , and then all entanglements
included in phase10 phase ent is lost.
To obtain the “total entanglement”, we calculate the Schmidt
number12 Schmidt num and compare it with the entanglement
ratio in the following
section.
Iv Full entanglement in scattered photon
Mathematically, for a bipartite system in pure state, the entanglement of an unfactorable wavefunction can be completely characterized by the Schmidt number, which is denoted by , where are eigenvalues of the integral equation 11 Schmidt dec :
(15) |
the density matrix for photon is defined as:
(16) |
where, note that we have taken away the time–dependent phase in the density matrix since it does not contribute to entanglement. Although we do it with the photon, Schmidt number can be equally obtained through the atomic density matrix, and the eigenfunctions of atom can be related to those of photon through:
(17) |
where and form complete orthonormal sets for the photon and atom respectively. With these discrete modes, the unfactorable wavefunction can be expanded into a sum of factored products uniquely:
(18) |
Then, the Schmidt number , which is an estimation of the number
of modes that are “important” in making up the expansion of Eq.
(18), serves as a quantitive measurement of entanglement7 scattering 12 Schmidt num . Note is independent from
representation since all keep the same in different
representations, thus can be seen as a quantity of the full
entanglement information (both amplitude
and phase entanglement) kept in the collective wavefunction.
Since Eq. (15) is not analytically solvable, we use a discrete
eigenvalue equation to approximate the integral equation. Up to a
reliable precision, we use matrices to carry out
the diagonalization, and collect some of the results in Fig. 4,
where we also compare Schmidt number
with the amplitude entanglement ratio .
From the numerical results, we find that, similar to the ratio ,
rises linearly with parameter and will increase
rapidly when the linewidth of incident photon is squeezed narrower
to the atomic linewidth , i.e, ; secondly, when
is fixed, the slope of is always larger
than
that of , which means that more entanglement information will transfer to phase when becomes larger,
and this phenomena will become more evident when
is reduced, e.g., when ,
whereas , which indicates that more than half
of the entanglement information will be unavailable by momentum dispersion observation when
goes large on this condition.
Another phenomena is notable, when , i.e., the linewidth of the incident photon is not squeezed and can be prepared directly by spontaneous emission from the same atom, we find whereas in the case of spontaneous emission6 Singlephoton . This difference indicates that, although in both cases, entanglement is generated from momentum conservation in the process of photon emission with atomic recoil, the absorption of the incident photon will add some entanglement due to its coherent pumping effect. As is linear with (or ), we define the “entanglement pumping coefficient” as:
since the constant term in plays a minor
role when entanglement is large. The defined coefficient
shows the times that entanglement is increased by the coherent
pumping of an incident photon. As it is independent on the atomic
parameter, it reflects the ability of entanglement of the photon
separately. We collect some numerical results in Fig. 5 and fit it
with within ,
from which, one sees that increases rapidly when
diminishes, which also implies that, if the incident
photon is prepared monochromatically on its limit condition, i.e.,
, the scattered photon will be highly
entangled to the recoiled atom.
We plot the amplitude of the first three Schmidt modes for the
photon with and in Fig. 6. We find that
their number of peaks in momentum space is proportional to the
Schmidt mode index, but the separations of different peaks are more
distinct than in the case of spontaneous
emission7 scattering .
V transmitted photon
To consider the transmitted photon, we make the observation angle , and yield the collective wavefunction from Eq. (10):
One can see that, in Eq. (19), the first term describes that the two
particles are free of interaction and keep their initial factorable
wave form; the second term reflects the entanglement. Usually, the
second term is much smaller than the first one since , but one can enlarge it by choosing some
special physical system, such as the artificial atom with low
excited level and high coupling to its resonant modes. However, this
improvement can add few entanglement between the transmitted photon
and recoiled atom, because interference between the two terms in Eq.
(19) will weaken the correlation of the two particles at a great
deal. To make it clear, we show the contour and density plots for
the probability amplitude of in Fig. 2 on an artificial
condition in this situation.
, and
yield
The eigenfunctions of transmitted photon for the first three modes
with and are collected in Fig. 6, from
which one can see that, due to the interference, the corresponding
modes of the transmitted photon exhibit one peak less than that of
the
scattered photon.
Vi conclusion
We analyze the physically fundamental interaction between a single
photon and a free artificial atom in vacuum. With a few physical
approximations, the general solution of the photon–atom wave
function is obtained, from which, it is found that the initially
uncorrelated particles will evolve to be entangled due to momentum
conservation in scattering. To evaluate the entanglement in the
scattering, firstly, we use an experimentally accessible parameter
, which denotes the ratio between momentum variance in
single–particle and in coincidence observations, and yield its
simple dependences on the two physical control parameters
and ; secondly, we use
standard Schmidt decomposition to reveal the full entanglement
information and find out its varying tendency similar to that of
, which indicates that high entanglement can be achieved by
either squeezing the linewidth of the incident photon or broadening
the scale of atomic wave packet. Furthermore, compared with
spontaneous emission, we defined a parameter to evaluate
the entanglement enhancement due to the coherent pumping effect of
the resonant incident photon. In the end, we found out that, for the
transmitted photon, one
can expect little entanglement due to the interference between the transparent and scattered wave.
Acknowledgments
One of the authors (HG) acknowledges J. H. Eberly for his
discussions when drafting this manuscript. This work is supported by
the National Natural Science Foundation of China (Grant No.
10474004), and DAAD exchange program: D/05/06972 Projektbezogener
Personenaustausch mit
China (Germany/China Joint Research Program).
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