Download Electric and magnetic dipole allowed transitions of atoms for three

Prog. Theor. Exp. Phys. 2014, 033A01 (12 pages)
DOI: 10.1093/ptep/ptu018
Electric and magnetic dipole allowed transitions
of atoms for three-dimensionally isotropic left
handedness in a mixed atomic vapor
Jian Qi Shen1,2,∗
1
Centre for Optical and Electromagnetic Research, State Key Laboratory of Modern Optical
Instrumentations, East Building No. 5, Zijingang Campus, Zhejiang University, Hangzhou 310058, The
People’s Republic of China
2
Joint Research Centre of Photonics of the Royal Institute of Technology (Sweden) and Zhejiang University,
Zijingang Campus, Zhejiang University, Hangzhou 310058, The People’s Republic of China
∗
E-mail: [email protected], [email protected]
Received December 18, 2013; Accepted January 21, 2014; Published March 19, 2014
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Since previous negative-index atomic media based on quantum optical approaches are highly
lossy, a proposal for realizing a three-dimensionally isotropic left-handed atomic vapor medium
is suggested based on a mechanism of incoherent gain assisted atomic transitions. Two threelevel atomic systems are utilized for producing simultaneously negative permittivity and negative
permeability, respectively, in the same frequency band. We suggest that fine and hyperfine level
transitions of atoms (e.g., a hyperfine level transition in a hydrogen atomic system and a fine
level transition in an alkali-metal atomic system) would be applicable to realization of such
a negatively refracting atomic vapor. The attractive features of the present scenario include:
i) three-dimensionally isotropic negative indices; ii) incoherent gain wave amplification in the
negative-index atomic vapor; iii) tunable negative indices depending upon external fields. Such
a left-handed quantum optical medium can serve as a supporting substrate for lossy negativeindex materials for loss compensation. It can also be used in designing new quantum optical
and photonic devices (e.g., a subwavelength focusing system and a negative-index superlens for
perfect imaging) because of its attractive properties of three-dimensional isotropy and high-gain
wave amplification.
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Subject Index
1.
A01, A02, A64
Introduction
With the development of optical and photonic technologies, increasing attention has been paid to
new ways of manipulating electromagnetic wave propagation with artificial materials. Over the past
decade, a particularly flexible and promising approach to light propagation control has been artificial composite metamaterials [1–3]. One such group of metamaterials are the left-handed media
that simultaneously have negative permittivity and permeability [1–6]. In the literature, the lefthanded media that were fabricated successfully in earlier experiments were actually anisotropic in
nature [4–6]. Though there have been some new techniques for realizing isotropic metamaterials [7–
9], fabrication of ideally isotropic and homogeneous negatively refracting materials is believed to
remain a challenging issue [10]. Some scenarios have been suggested for achieving negative refractive indices of atomic vapor based on quantum optical approaches [11–16]. As it is an atomic vapor,
such a negatively refracting medium is three-dimensionally isotropic and homogeneous. However,
© The Author(s) 2014. Published by Oxford University Press on behalf of the Physical Society of Japan.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/),
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PTEP 2014, 033A01
J. Q. Shen
losses in all these materials are quite large, and they could not be employed in practical applications.
Recently, with the development of technology for artificial electromagnetic materials, loss reduction
in metamaterials becomes an increasingly important issue [17].
In order to achieve simultaneously negative permittivity and permeability through a quantum
optical mechanism (e.g., electric and magnetic dipole allowed transitions in atomic systems), simultaneous electric and magnetic dipole resonant transitions in the same narrow frequency band are
needed. Unfortunately, in almost all atoms, any electric and magnetic dipole transition frequencies
in one atomic system are in principle not equal. Thus, there is almost no chance for electric and magnetic dipole allowed transitions to occur simultaneously (i.e., with the same transition frequency) in
one kind of atomic system. Therefore, apart from the problem of high loss in the above left-handed
quantum optical materials [11–16], the most important bottleneck for such materials that should also
be resolved is the difficulty in finding a proper atomic level configuration in one atom, where simultaneous electric and magnetic dipole allowed transitions (in the same frequency band) can be driven by
a single electromagnetic wave. For this reason, we shall resort to some mixed atomic gaseous media,
in which there are two atomic systems provided by two kinds of atoms (one is for realizing negative
permittivity, and the other for negative permeability). For example, we find that there are two kinds
of atoms, i.e., neutral hydrogen and lithium atoms, in which the hyperfine and fine structure level
pairs can have almost the same transition frequencies [18,19]. (In general, the fine structure levels
are caused by atomic spectral line splitting due to relativistic corrections and electron spin contribution, and the hyperfine structure levels of atoms are formed because of nuclear spin and quantum
field effects, e.g., vacuum polarization.) Such hyperfine and fine structure level pairs can give rise
to electric and magnetic dipole allowed transitions, respectively. It is then possible for the electric
and magnetic dipole allowed transitions to be resonant with a single incident electromagnetic wave
(i.e., the electric and magnetic dipole allowed transitions are driven at the same frequency), and the
electric permittivity and the magnetic permeability can be negative simultaneously (in the same frequency band). As has been mentioned, losses in double-negative media should be overcome [17]. For
this purpose, we shall introduce a “pumping action” [20–22] into the present atomic systems in order
to compensate for losses. Then, the two bottlenecks (high loss and equal transition frequencies of
electric and magnetic dipole transitions) in the present left-handed quantum optical materials (mixed
atomic vapor) can be resolved.
In this paper we shall study a pumped three-level system {|1, |2, |3} (see Fig. 1) for both electric
and magnetic dipole allowed transitions. The schematic diagram of the atomic level configuration
in Fig. 1 is applicable to the systems for producing electric and magnetic polarizability. Levels |2
and |3 in the system shown in Fig. 1 are the fine or hyperfine structure levels. The population at
the ground level |1 can be pumped by a pump field to level |3 (at a pumping rate λ) [20–22]. For
the electric dipole allowed transition of an atomic system, levels |2 and |3 have opposite parity,
and then there is an atomic transition electric dipole moment ℘32 in the transition process from
level |2 to level |3, and the level pair |2–|3 can be coupled to the electric field of an incident
electromagnetic wave with mode frequency ω and electric Rabi frequency E (see Fig. 1). For the
magnetic dipole allowed transition of another atomic system, levels |2 and |3 have the same parity.
There is a transition magnetic dipole moment m 32 in the transition process from level |2 to level |3,
and the level pair |2–|3 can be coupled to the magnetic field of the same incident electromagnetic
wave of mode frequency ω and magnetic Rabi frequency B .
In the sections that follow, we shall address the optical behavior of the present double-negative
atomic vapor materials, including the characteristics of dispersion, tunability, and incoherent gain.
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Fig. 1. The schematic diagram of the pumped electric (or magnetic) dipole allowed transition for producing
negative permittivity (or negative permeability). Level |2 and level |3 are the hyperfine (or fine) structure
levels. The incident electromagnetic wave drives the |2–|3 transition. For the pumped electric dipole allowed
transition (|2–|3), the coupling coefficient is the electric Rabi frequency E , and for the pumped magnetic
dipole allowed transition (|2–|3), is the magnetic Rabi frequency B . The population at level |1 is pumped
to level |3 aiming at high-gain wave amplification in the negatively refracting atomic vapor.
For the first, we shall give the equation of motion of the density matrix governing the light-atom
interaction, and obtain the steady solution.
2.
The equation of motion of the density matrix
The present left-handed medium that can give rise to simultaneously negative permittivity and permeability is a mixture of two atomic vapors. We shall refer to the two atomic systems as the “A” and
“B” systems. The following model for treating the electromagnetic response and its dispersion characteristics is applicable to both the electric dipole transition in atomic system “A” and the magnetic
dipole transition in atomic system “B”. The electric and magnetic fields of the incident propagating wave drive the electric dipole allowed transition |2–|3 in an “A” atom and the magnetic dipole
allowed transition |2–|3 in a “B” atom, respectively. For the pumped electric dipole allowed transi(A)
tion in the “A” atom, the coupling coefficient is the electric Rabi frequency E = ℘32 Ep /, while
for the pumped magnetic dipole allowed transition in the “B” atom, is the magnetic Rabi frequency
(B)
B = m 32 Bp /. By using the semiclassical approaches for treating light-atom interaction [23,24],
one can obtain the equation of motion of the density matrix of an atomic system (“A” or “B”) as
depicted in Fig. 1:
ρ̇11 = −λρ11 + γ31 ρ33 + γ21 ρ22 ,
i
i
ρ̇22 = −γ21 ρ22 + γ32 ρ33 − ρ23 + ∗ ρ32 ,
2
2
i
i
ρ̇33 = λρ11 − (γ31 + γ32 ) ρ33 + ρ23 − ∗ ρ32 ,
2
2
32 + γph
i
+ i ρ32 + (ρ22 − ρ33 ) ,
ρ̇32 = −
2
2
32 + γph
i
ρ̇23 = −
− i ρ23 − ∗ (ρ22 − ρ33 ) ,
2
2
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(1)
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where the decay parameter 32 = γ31 + γ32 + γ21 . The diagonal density matrix elements ρ11 , ρ22 ,
and ρ33 agree with a constraint condition ρ11 + ρ22 + ρ33 = 1. The parameters γi j (i, j = 1, 2, 3)
and γph denote the spontaneous emission decay rates and the collisional dephasing rate, respectively.
The parameter λ represents the pumping rate of the population from the ground level |1 to the upper
level |3. The frequency detuning is defined as = ω32 − ω with ω32 the transition frequency
(between level |3 and level |2) and ω the mode frequency of the incident probe field exciting the
population at level |2 to level |3. In addition to the equations of motion in (1), there are two equations
of ρ21 , ρ31 that characterize the quantum coherence:
⎛
ρ̇21 = − ⎝
(21)
21 + γph
2
⎞
⎛
⎠ ρ21 ,
ρ̇31 = − ⎝
(31)
31 + γph
2
⎞
⎠ ρ31 ,
(2)
where 21 = γ21 + λ and 31 = γ31 + γ32 + λ. It can be seen that both ρ21 and ρ31 will decay exponentially to zero, i.e., their steady values vanish. Thus, in what follows, we should consider only the
density matrix elements ρ11 , ρ22 , ρ33 , and ρ32 .
3.
The exact steady solution of the equation of motion of the density matrix
We will obtain the exact expression for the steady solution of the density matrix elements in Eq. (1).
The steady solution can be obtained when we assume ρ̇i j = 0 in Eq. (1). From the last two equations
in (1), one can arrive at the off-diagonal density matrix elements ρ32 and ρ23 :
ρ32 =
i
2 (ρ22
− ρ33 )
,
γ + i
where the decay parameter γ ≡
Eq. (1) yields
ρ23
γ31 +γ32 +γ21 +γph
.
2
− 2i ∗ (ρ22 − ρ33 )
,
=
γ − i
(3)
Substitution of Eq. (3) into the second formula in
1
1
1 ∗
1
1 ∗
1
+
+ γ21 ρ22 =
+
+ γ32 ρ33 .
4
γ + i γ − i
4
γ + i γ − i
(4)
Keeping the relation ρ11 = 1 − ρ22 − ρ33 in mind, one can obtain
1
1 ∗
1
+
− λ ρ22
4
γ + i γ − i
1
1 ∗
1
+
+ (λ + γ31 + γ32 ) ρ33 + λ = 0
− 4
γ + i γ − i
(5)
by inserting Eq. (3) into the third formula in Eq. (1). The above two relations, i.e., (4) and (5), can
be rewritten as
Aρ22 = B ρ33 ,
C ρ22 + Dρ33 + λ = 0,
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(6)
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where the parameters are defined as follows
1
1
1
+
+ γ21 ,
A = ∗ 4
γ + i γ − i
1
1 ∗
1
+
+ γ32 ,
B= 4
γ + i γ − i
1
1 ∗
1
+
− λ,
C= 4
γ + i γ − i
1
1 ∗
1
+
+ (λ + γ31 + γ32 ) .
D=− 4
γ + i γ − i
(7)
The solution to Eq. (6) is given by
ρ33 = −
λA
,
CB + AD
ρ22 = −
λB
.
CB + AD
(8)
Then the explicit expression for ρ33 − ρ22 that can characterize the population inversion can be
written as
ρ33 − ρ22 =
1 ∗
4 1
γ +i
+
1
γ −i
λ (γ21 − γ32 )
(2λ + γ31 + γ21 ) + λ (γ32 + γ21 ) + γ21 (γ31 + γ32 )
.
(9)
For a weak probe field (i.e., the square of the Rabi frequency, ∗ , is negligibly small), the population
inversion ρ33 − ρ22 can be reduced to the form
ρ33 − ρ22 λ (γ21 − γ32 )
.
λ (γ32 + γ21 ) + γ21 (γ31 + γ32 )
(10)
It can be seen that the decay rate γ21 should be larger than γ32 (i.e., the lifetime of state |3 should
be longer than that of state |2) in order to realize the population inversion.
The aforementioned prescription of atomic transitions is applicable to both electric and magnetic
(A)
dipole resonant systems. The steady density matrix elements ρ32 (in the electric dipole resonant
(B)
system) and ρ32 (in the magneticdipole resonant system) can be used to characterize the electric and
magnetic dipole allowed transitions, respectively. The expressions for the microscopic electric and
(A) (A)
(B) (B)
magnetic polarizabilities of the atoms are given by βe = 2℘23 ρ32 /(ε0 Ep ), βm = 2μ0 m 23 ρ32 /Bp ,
which can also be rewritten as
(A)
βe =
2|℘32 |2 (A)
ρ ,
ε0 E 32
(B)
βm =
2μ0 |m 32 |2 (B)
ρ32 .
B
(11)
Here, ε0 and μ0 denote the permittivity and the permeability, respectively, of free vacuum.
In order to achieve the negative permittivity and the negative permeability, the chosen vapor should
be dense, so that one should consider the local field effect, i.e., one must distinguish between the
applied macroscopic fields and the microscopic local fields that act upon the atoms in the vapor
when addressing how the atomic transitions are related to the electric and magnetic susceptibilities
[25]. The relative permittivity in atomic vapor A and the relative permeability in atomic vapor B are
given by εr = 1 + N (A) βe /(1 − N (A) βe /3) and μr = 1 + N (B) βm /(1 − N (B) βm /3), respectively,
where N (A) and N (B) denote the atomic concentrations (total numbers of atoms per unit volume)
of these two atomic vapors. These two formulae are the electric and magnetic Clausius–Mossotti
relations [25,26] that can reveal the connection between the macroscopic quantities (εr and μr ) and
the microscopic electric/magnetic polarizabilities (βe and βm ) of the atoms.
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4.
J. Q. Shen
A numerical example of simultaneously negative permittivity and permeability
We shall in this numerical example present the characteristics of dispersion of the simultaneously
negative permittivity and permeability in the mixed atomic vapor. Although the theoretical mechanism presented here can be employed in both fine and hyperfine level transitions and infrared/visible
frequency transitions, we will concentrate our attention on the fine/hyperfine level transitions occurring at microwave/terahertz frequencies, because it is relatively easy to find the electric dipole
(A)
(B)
transition frequency ω32 and the magnetic dipole transition frequency ω32 , which are equal or
very close, and hence the fine/hyperfine level transitions are more realistic than the infrared/visible
frequency transitions for simultaneously negative permittivity and permeability.
We choose the transition electric dipole moment of the electrically resonant system (i.e., system
(A)
A) as ℘32 = 3.0 × 10−29 C·m [27,28]. The typical values for the decay rates due to spontaneous
(A)
(A)
(A)
emission are γ31 = 3.0 × 107 s−1 , γ32 = 1.0 × 107 s−1 , γ21 = 2.0 × 107 s−1 [29,30], and the
(A)
collisional dephasing rate γph = 4.0 × 107 s−1 [31]. Then the decay rate, γ (A) , from level |3
(A)
(A)
(A)
(A)
to level |2, which is defined as γ (A) ≡ (γ31 + γ32 + γ21 + γph )/2, is 5.0 × 107 s−1 . This
parameter (rewritten as γ = 5.0 × 107 s−1 ) will be used to normalize the frequency detuning in the following figures. The number density of atoms of the electrically resonant atomic vapor is
N (A) = 1.1 × 1022 atoms m−3 .
For the magnetically resonant system (system B), we choose the atomic transition magnetic dipole
(B)
moment as m 32 = 1.1 × 10−22 C·m2 s−1 (the typical atomic magnetic dipole moment has the order
of magnitude of cα℘ with ℘ = 3.0 × 10−29 C·m. Here, α and c denote the electromagnetic fine
structure constant and the speed of light in vacuum, respectively). The typical values for the decay
(B)
(B)
(B)
rates due to spontaneous emission are γ31 = 2.0 × 107 s−1 , γ32 = 0.0 × 107 s−1 , γ21 = 1.6 ×
(B)
107 s−1 , and the collisional dephasing rate is γph = 3.0 × 107 s−1 . Then the decay rate from level |3
(B)
(B)
(B)
(B)
to level |2, γ (B) (defined as γ (B) ≡ (γ31 + γ32 + γ21 + γph )/2), is 3.3 × 107 s−1 . Since there
is incoherence-assisted wave amplification in the atomic vapor, the chosen electric and magnetic
Rabi frequencies (E and B ) of the incident weak probe field should be very small (e.g., E =
1.0 × 10−6 γ and B = E
(B)
m 32
(A)
℘32 c
= 1.2 × 10−8 γ ). All the aforementioned parameters (atomic and
optical) will be adopted throughout this paper. The number density of atoms of this magnetically
resonant atomic vapor as shown in Figs. 2 to 4 is N (B) = 2.0 × 1024 atoms m−3 , and in Fig. 5 we will
choose another two cases (N (B) = 8.0 × 1024 and 3.0 × 1025 atoms m−3 ) for additional discussions.
It can be found that the atom concentration of the magnetically resonant vapor (vapor B) should be
two or three orders of magnitude larger than that of the electrically resonant atomic vapor (vapor A).
This can be interpreted as follows: In atomic systems, the magnetic polarizability is only about one
percent of the electric polarizability, because the ratio of the atomic magnetic dipole moment m to
A
, where the coefficient A = 0.5 ∼ 2, α denotes
the electric dipole moment ℘ is m/(c℘) Aα = 137
the electromagnetic fine structure constant, and c is the speed of light in vacuum (c℘ has the same
dimension as the magnetic dipole moment m, and hence the ratio m/(c℘) is dimensionless). In order
for the required negative permittivity and permeability to have comparable values, the magnetically
resonant atomic vapor (vapor B) should be more dense than the electrically resonant atomic vapor
(vapor A).
Since we have two atomic systems interacting with the electric and the magnetic fields of a single
(A)
(B)
incident wave, there are two frequency detuning parameters: E = ω32 − ω and B = ω32 − ω.
Obviously, these two frequency detuning parameters are quite close, since we have chosen proper
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Fig. 2. The characteristics of dispersion of both the real and the imaginary parts of the relative permittivity εr ,
the relative permeability μr , and the refractive index n of the incoherent gain atomic vapor when the incident
propagating field is very weak, e.g., the electric and magnetic Rabi frequencies are E = 1.0 × 10−6 γ and
B = 1.2 × 10−8 γ , respectively.
(B)
(A)
atomic systems for ω32 → ω32 . We shall define a new frequency detuning parameter: =
(A)
(B)
(E + B )/2, and the two frequency detunings can be expressed as E = + (ω32 − ω32 )/2
(A)
(B)
and B = − (ω32 − ω32 )/2 in terms of . In the present illustrative example, we choose
(A)
(B)
ω32 − ω32 = −2π × 9.0 × 107 s−1 . The two pumping rates in Fig. 2 are given by λ(A) = 4.0γ
and λ(B) = 2.0γ . The real part of the relative permittivity in Fig. 2 is Re εr = −8.9 at the frequency detuning = −10.0γ and Re εr = −3.0 at = 0.0γ . In this frequency detuning range
[−10.0γ , 0.0γ ], the imaginary part Im εr of the permittivity of the vapor medium is always negative
(e.g., Im εr = −1.5 at = −10.0γ and Im εr = −0.24 at = 0.0γ ). In the range [−7.1γ , −2.5γ ]
of the frequency detuning , the real part Re μr of the relative permeability is negative (i.e.,
Re μr = 0.0 at both = −7.1γ and = −2.5γ ), and the imaginary part Im μr is always negative in this frequency detuning range. Besides, it can be found that the real part of the refractive
index is Re n = −4.4 at = −7.1γ and Re n = −0.47 at = −2.5γ , and the minimum of Re n is
−4.6 at = −6.9γ . In this range [−7.1γ , −2.5γ ], the imaginary part Im n of the refractive index
is negative with its maximum −0.46 at = −3.3γ . Thus, in the range of [−7.1γ , −2.5γ ] of the
frequency detuning , both the permittivity and the permeability have simultaneously negative real
parts, and the resulting refractive index has a negative real part.
The tunable dispersion characteristics of the permittivity of the electrically resonant atomic vapor
medium and the permeability of the magnetically resonant atomic vapor are presented in Figs. 3
and 4. It can be seen that in a quite broad range of the pumping rates λ(A) , λ(B) , the present mixed
atomic vapor can exhibit simultaneously negative permittivity and permeability, and the wave propagating inside can be amplified because the pumped population transfer in the atomic systems leads
to the incoherent gain of the atomic vapor.
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Fig. 3. The tunable dispersion characteristics of the permittivity (of the electrically resonant vapor) and the
permeability (of the magnetically resonant vapor) depending upon their respective pumping rates λ(A) , λ(B) .
5.
Discussions
In the preceding section, we have shown that such a mixed atomic vapor can have negative permittivity and permeability in the same frequency band. As the negative indices are sensitive to the atom
concentration of the vapor, we will address its dispersion characteristics in more detail. Two cases of
atomic concentration N (B) = 8.0 × 1024 atoms m−3 and N (B) = 3.0 × 1025 atoms m−3 are shown
in Fig. 5. In the case of N (B) = 8.0 × 1024 atoms m−3 , the real part of εr changes from −3.0 to
−1.5 in the frequency detuning range [0.0, 10.0γ ], and the real part of μr is Re μr = −0.58 at
= 0.0 and Re μr = 0.0 at = 6.8γ . Then the real part of the refractive index is Re n = −1.3
at = 0.0 and Re n = −0.029 at = 10.0γ , and the real part of the refractive index is −0.20 at
= 6.8γ , where Re μr = 0.0. It should be noted that, in the frequency detuning range [0.0, 6.8γ ],
the imaginary parts Im εr , Im μr , and Im n are negative and their magnitudes are small, e.g., they
are in the range [−0.24, 0.0]. The imaginary part Im n = −0.15 at = 0.0 and Im n = −0.16 at
= 6.8γ . The imaginary part of the refractive index, which has a minimum of magnitude |Im n|, is
Im n = −0.11 at = 4.7γ . Thus, the amplification factor across one wavelength at this frequency is
exp (2π |Im n|) = 1.9, and such a weak probe field will propagate through a path of 24 wavelengths
before its electric Rabi frequency is amplified to 10γ (once the electric Rabi frequency E has the
order of magnitude of 10γ , the negative refractive index of the atomic vapor is unstable, because
the large negative imaginary part of the refractive index will lead to a large amount of gain, and the
optical nonlinearity will spoil the negative-index effect. For example, one can see that the population
inversion term (9) is a function of the square of the Rabi frequency, ∗ , and hence the refractive
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Fig. 4. The tunable dispersion characteristics of the refractive index and the relative impedance depending
upon the pumping rate λ(B) . The other pumping rate λ(A) remains constant, λ(A) = 4.0γ .
index will depend critically on the electric and magnetic Rabi frequencies, i.e., the negative refractive
index of the atomic vapor is unstable). In the case of N (B) = 3.0 × 1025 atoms m−3 , the real part of
εr changes from −3.0 to 0.0 in the frequency detuning range [0.0, 50.0γ ], and the real part of μr
is Re μr = −1.4 at = 0.0 as well as Re μr = 0.0 at = 42.1γ . Then the real part of the refractive index is −2.1 at = 0.0 and 0.026 at = 50.0γ . The real part of the refractive index will be
−0.021 at = 42.1γ , where Re μr = 0.0. It should be noted that, in the frequency detuning range
[0.0, 42.1γ ], the imaginary parts Im εr , Im μr , and Im n are negative and their magnitudes are small.
For example, the imaginary part Im n = −0.12 at = 0.0 and Im n = −0.029 at = 42.1γ . The
imaginary part Im n (which has the minimum of |Im n|) is −0.016 at = 38.7γ . The amplification
factor across one wavelength at this frequency is exp (2π |Im n|) = 1.1, and such a weak probe field
will propagate through 163 wavelengths before its electric Rabi frequency is amplified to 10γ . Now
in this case, the negative refractive index can be said to be sufficiently stable, i.e., it is tolerant toward
the change in the field intensity of the probe field. It should be pointed out that the wavelength of
the incident propagating wave in the present illustrative example is about 0.01 m, which is comparable to practical sizes of objects, and hence the diffraction effect would be significant. But by taking
full advantage of the negative indices, such a mixed vapor medium can be used in the technique of
super-resolution imaging (e.g., for testing such a novel effect of enhancement of imaging resolution),
where a low-loss three-dimensional metamaterial lens is required [32,33].
The numerical example in the preceding section is presented for a general mixed atomic vapor
with typical atomic and optical parameters. The theoretical model is suitable for both fine/hyperfine
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Fig. 5. The behavior of dispersion of both the real and the imaginary parts of the relative permittivity εr , the
relative permeability μr , and the refractive index n of the incoherent gain atomic vapor when the electric Rabi
frequency E = 1.0 × 10−6 γ and the magnetic Rabi frequency B = 1.2 × 10−8 γ . The two pumping rates
are λ(A) = 4.0γ and λ(B) = 2.0γ , the same as those used in Fig. 2.
level transitions and infrared/visible frequency transitions. In order to realize simultaneously negative
(A)
permittivity εr and permeability μr , the electric dipole transition frequency ω32 and the magnetic
(B)
dipole transition frequency ω32 should be equal or very close (and hence these two dipole-allowed
transitions can be driven by the same electromagnetic wave). But we find that two such atomic sys(A)
(B)
tems, in which ω32 and ω32 of the infrared/visible frequency transitions are very close, are in
fact rarely seen. However, for the fine/hyperfine level transitions (at microwave and terahertz fre(A)
(B)
quencies), it can be relatively easy to find two atomic systems with ω32 and ω32 very close. In
other words, atomic systems with transition frequencies in the infrared/visible band would not be
realistic for realizing the negative refractive index, while atomic systems which can give rise to
fine/hyperfine level transitions would be potential candidates for achieving such a negative refractive
index with simultaneously negative permittivity and permeability. One such candidate is a mixture of neutral hydrogen atomic vapor and neutral lithium atomic vapor. As we know, hydrogen
molecular vapor is commonly seen in nature, while hydrogen atomic vapor is rarely seen. Under
certain proper conditions, however, hydrogen atomic vapor can also be stable against recombination. In the literature, there has been some research on preparing hydrogen atomic vapor samples
by using thermal compression methods and ultraslow deposition techniques [34–36]. The number density of atoms of hydrogen atomic vapor is 1 × 1024 ∼ 2 × 1025 atoms m−3 [34–36]. This
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J. Q. Shen
atomic vapor density will be sufficient for us to achieve the negative permittivity of the hydrogen atomic vapor. Now the atomic system “A” (i.e., the neutral hydrogen atomic system) and the
atomic system “B” (i.e., the neutral lithium atomic system) can lead to electric and magnetic
dipole responses. The energy levels of the neutral hydrogen atomic system {|1, |2, |3}, e.g.,
{12 s 1 , 22 s 1 , 22 p 3 }, are {0.0000, 82258.9544, 82259.2850} cm−1 . Thus, the electric dipole |2–|3
2
2
2
transition (i.e., the transition between the hyperfine levels 22 s 1 , 22 p 3 ) is resonant with the electro2
2
magnetic wave at wave number 0.3306 cm−1 [18]. The energy level of the neutral lithium atomic
system {|1, |2, |3}, e.g., {22 s 1 , 22 p 1 , 22 p 3 }, is {0.000, 14903.622, 14903.957} cm−1 . Then the
2
2
2
magnetic dipole |2–|3 transition (i.e., the transition between the fine levels 22 p 1 , 22 p 3 ) is res2
2
onant with the electromagnetic wave at wave number 0.335 cm−1 [19]. Therefore, the transition
frequency of the electric dipole allowed transition (|2–|3) in the neutral hydrogen atomic system is
(elec)
ω32 = 2π × 0.992 × 1010 s−1 , and the transition frequency of the magnetic dipole allowed tran(magn)
= 2π × 1.005 × 1010 s−1 . Then
sition (also |2–|3) in the neutral lithium atomic system is ω32
(magn)
(elec)
− ω32 , between these two kinds of atomic systems is
the transition frequency difference, ω32
quite small, namely, the electromagnetic responses of the present electric and magnetic dipole transitions can occur simultaneously (i.e., in the same narrow frequency ranges). In other words, they
can be driven by a single incident electromagnetic wave at frequency 2π × 1010 s−1 . Therefore, we
expect that hydrogen atomic vapor and alkali-metal atomic vapor (neutral lithium atomic vapor) can
be utilized to produce simultaneously negative permittivity and negative permeability.
6.
Concluding remarks
The electric and magnetic dipole allowed transitions in atoms can give rise to intriguing electromagnetic responses, e.g., simultaneously negative permittivity and permeability under certain proper
conditions (as has been shown in the preceding sections). We have demonstrated the existence of
atomic-vapor negative indices in an illustrative example. The present scenario of negative indices
can have some novel characteristics:
(1) The negative indices of the medium are three-dimensionally isotropic. Since it is an atomic
vapor, the present negatively refracting medium is three-dimensionally isotropic and homogeneous. This is particularly essential for designing devices such as a subwavelength focusing
system or negative-index superlens for perfect imaging [32,33].
(2) The pumped population transfer can compensate for losses in the present left-handed atomic
medium, or give rise to high-gain wave amplification.
(3) The negative indices are tunable through external fields, including pump fields.
Quite recently, some new schemes for compensating for losses in metamaterials have been suggested,
e.g., Soukoulis et al. have studied both theoretically and experimentally the loss compensation (or
light amplification) in some metamaterial structure units (e.g., split ring resonator and its array) with
gain systems (i.e., four-level layer underneath gain and InGaAs single quantum well gain [37–40]).
We point out that the negative-index materials suggested in the present paper can also be utilized as
a supporting substrate for lossy negative-index materials for loss compensation. We expect that such
a new mechanism of incoherent gain assisted transitions (particularly the fine and hyperfine level
transitions) of atoms could offer a possible route to high-gain three-dimensionally isotropic negative
index materials.
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PTEP 2014, 033A01
J. Q. Shen
Acknowledgements
This work is supported in part by the National Natural Science Foundation of China under Grants Nos.
11174250, 91233119, 60990320, and the Program of Zhejiang Leading Team of Science and Technology
Innovation.
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