Electric and magnetic dipole allowed transitions of atoms for three-dimensionally isotropic left handedness in a mixed atomic vapor

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 suggeste ...

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.

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][2][3]. One such group of metamaterials are the left-handed media that simultaneously have negative permittivity and permeability [1][2][3][4][5][6]. In the literature, the lefthanded media that were fabricated successfully in earlier experiments were actually anisotropic in nature [4][5][6]. Though there have been some new techniques for realizing isotropic metamaterials [7][8][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][12][13][14][15][16]. As it is an atomic vapor, such a negatively refracting medium is three-dimensionally isotropic and homogeneous. However, 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][12][13][14][15][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][21][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][21][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. 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.

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 transition in the "A" atom, the coupling coefficient is the electric Rabi frequency E = ℘ (A) 32 E p / , while for the pumped magnetic dipole allowed transition in the "B" atom, is the magnetic Rabi frequency 32 B p / . 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:ρ where the decay parameter 32 = γ 31 + γ 32 + γ 21 . The diagonal density matrix elements ρ 11 , ρ 22 , and ρ 33 agree with a constraint condition ρ 11 + ρ 22 + ρ 33 = 1. The parameters γ i j (i, j = 1, 2, 3) and γ ph 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 = ω 32 − ω with ω 32 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 ρ 21 , ρ 31 that characterize the quantum coherence: where 21 = γ 21 + λ and 31 = γ 31 + γ 32 + λ. It can be seen that both ρ 21 and ρ 31 will decay exponentially to zero, i.e., their steady values vanish. Thus, in what follows, we should consider only the density matrix elements ρ 11 , ρ 22 , ρ 33 , and ρ 32 .

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 assumeρ i j = 0 in Eq. (1). From the last two equations in (1), one can arrive at the off-diagonal density matrix elements ρ 32 and ρ 23 : where the decay parameter γ ≡ γ 31 +γ 32 +γ 21 +γ ph 2 . Substitution of Eq. (3) into the second formula in Eq. (1) yields Keeping the relation ρ 11 = 1 − ρ 22 − ρ 33 in mind, one can obtain by inserting Eq. (3) into the third formula in Eq. (1). The above two relations, i.e., (4) and (5), can be rewritten as where the parameters are defined as follows The solution to Eq. (6) is given by Then the explicit expression for ρ 33 − ρ 22 that can characterize the population inversion can be written as For a weak probe field (i.e., the square of the Rabi frequency, * , is negligibly small), the population inversion ρ 33 − ρ 22 can be reduced to the form It can be seen that the decay rate γ 21 should be larger than γ 32 (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 dipole resonant systems. The steady density matrix elements ρ Here, ε 0 and μ 0 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 ε r  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 (ε r and μ r ) and the microscopic electric/magnetic polarizabilities (β e and β m ) of the atoms.

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 transition frequency ω (A) 32 and the magnetic dipole transition frequency ω (B) 32 , 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) as ℘ 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 the electric dipole moment ℘ is m/(c℘) Aα = A 137 , where the coefficient A = 0.5 ∼ 2, α denotes 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 incident wave, there are two frequency detuning parameters: E = ω   Fig. 2 is Re ε r = −8.9 at the frequency detuning = −10.0γ and Re ε r = −3.0 at = 0.0γ . In this frequency detuning range [−10.0γ, 0.0γ ], the imaginary part Im ε r of the permittivity of the vapor medium is always negative (e.g., Im ε r = −1.5 at = −10.0γ and Im ε r = −0.24 at = 0.0γ ). In the range [−7.1γ, −2.5γ ] of the frequency detuning , the real part Re μ r of the relative permeability is negative (i.e., Re μ r = 0.0 at both = −7.1γ and = −2.5γ ), and the imaginary part Im μ r 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.

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 ( 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 8  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 × 10 25 atoms m −3 , the real part of ε r changes from −3.0 to 0.0 in the frequency detuning range [0.0, 50.0γ ], and the real part of μ r is Re μ r = −1.4 at = 0.0 as well as Re μ r = 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 μ r = 0.0. It should be noted that, in the frequency detuning range [0.0, 42.1γ ], the imaginary parts Im ε r , Im μ r , 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 9/12  level transitions and infrared/visible frequency transitions. In order to realize simultaneously negative permittivity ε r and permeability μ r , the electric dipole transition frequency ω (A) 32 and the magnetic dipole transition frequency ω (B) 32 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 systems, in which ω (A) 32 and ω (B) 32 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 frequencies), it can be relatively easy to find two atomic systems with ω (A) 32 and ω (B) 32 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][35][36]. The number density of atoms of hydrogen atomic vapor is 1 × 10 24 ∼ 2 × 10 25 atoms m −3 [34][35][36]. This