Negative refraction with tunable absorption in an active dense gas of atoms

We consider a gas of five-level atoms with an energy level structure that is shown in figure 1. In the following, we first motivate the chosen model system, and discuss the significance of different coupling fields. A complete definition of the model in terms of the master equation will be given in section 2.3. A specific feature of the scheme is that it contains an electric and a magnetic dipole transition at roughly the same energy ₄₃ ≈ ₂₁, where _ij = _i − _j, and _i is the energy of state i. An incoming probe field with frequency ω_b ≈ ₂₁ therefore couples near-resonantly to the transition |3〉–|4〉 with its electric field component E_b and to the transition |1〉–|2〉 with its magnetic field component B_b.

Close to the resonance ω_b ≈ ₂₁ one thus finds that both the electric permittivity Re() < 0 and the magnetic permeability Re(μ) < 0 become negative. As a result, the refractive index n, which is given by $n = \eta \sqrt {\epsilon \,\mu }$ if magneto-electric cross-couplings can be neglected, exhibits a negative real part. Here, η∈{1,−1} determines the branch of the square root function, which is determined by the condition that the imaginary part Im(n) is positive in a passive system, as both Im(,μ) > 0.

The strength of the magnetic response is weaker than the electric response by a factor of α², where α = 1/137 is the fine-structure constant. For this reason one requires rather large densities of N ∼ 10¹⁷ cm⁻³ to achieve negative refraction in such an atomic gas. To enhance the magnetic response we apply two (strong) coupling laser fields E_a and E_c which together with the electric probe field component induce the coherence ρ₂₁ = 〈2|ρ|1〉 which drives the magnetic dipole moment of the atom. Here, ρ denotes the density matrix of a single atom. The two fields E_a,c drive transitions between states |1〉–|3〉 and |2〉–|4〉, respectively. We choose the frequencies of both fields to be equal (ω_a = ω_c) and almost resonant with the transition |1〉–|3〉 [29].

The third coupling laser field E_d, which resonantly couples states |4〉–|5〉, serves a different purpose. It allows shifting electric and magnetic probe field resonances with respect to each other, because it causes an Autler–Townes splitting of the fourth level into two dressed states at ₄ → ₄ ± ℏ|Ω₅₄|/2, where Ω₅₄ is the Rabi frequency associated with this coupling laser to be defined later. This is useful for two reasons. First, it allows one to bring electric and magnetic resonances closer to each other, if they are separated in energy in the bare atom. This is a way of circumventing the problem of non-degenerate electric and magnetic probe field transitions [16]. In metastable neon, for example, the two resonances are separated in energy by the bare gap

$$\begin{eqnarray} &&\Delta = (\epsilon_{43} - \epsilon_{21})/h = 100 \, {\rm GHz} = 2 \pi \times \; 10^4 \gamma, \end{eqnarray} \tag{ 1 }$$

where we have used that γ = 10⁷/(2π)Hz is a typical value for the spontaneous decay rate in neon. In order to close this gap via an induced Stark shift, we consider applying a strong laser field with Ω₅₄ ≈ 10⁴γ. Since neighboring transitions are still detuned at the least by 15Ω₅₄, such large light shifts are feasible without inducing unwanted transitions. We define the resulting effective gap as

$$\begin{eqnarray} &&\Delta'=-\Delta + \Omega_{54}/2 . \end{eqnarray} \tag{ 2 }$$

It turns out, however, that it is not mandatory to close this gap completely. Due to the large density, the electric permittivity exhibits a negative real part Re() ≈ −2 already relatively far away from the resonance [30]. More importantly, the states |2〉 and |3〉 are connected by a two-photon transition induced by the fields E_c and E_b, which becomes important only for non-zero effective gap frequencies Δ'.

This two-photon transition is the motivation to apply an additional incoherent light field which is resonant with the transition |3〉–|4〉. It transfers population between the two levels with rates r₃₄ = r₄₃ = r. In combination with spontaneous emission from |4〉 to |2〉, it effectively pumps population from state |3〉 into state |2〉. As soon as the population of state |2〉 exceeds the one of state |3〉, i.e. for ρ₂₂ > ρ₃₃, the probe field E_b is amplified by means of this two-photon process. It is then more likely for the transition to occur in the direction |2〉 → |3〉 than in the reverse direction. Since this direction involves the emission of a probe field photon, we observe gain in the electric probe field component for ρ₂₂ > ρ₃₃. Choosing equal coupling laser frequencies ω_a = ω_c ensures that this two-photon resonance is always located at the position of a magnetic probe field resonance close to δ₂₁ = 0. The two-photon virtual intermediate level is separated from state |4〉 by the tunable effective gap Δ' defined in equation (2). For sufficiently large Δ', the two-photon transition will thus be the dominant process around δ₂₁ ≃ 0. This is important, as it allows to obtain gain in the electric probe field component exactly at those frequencies where the magnetic response is strong.

We now define the electromagnetic linear response functions and the index of refraction n. As stated above, negative refraction Re(n) < 0 requires that both the electric and the magnetic component of an electromagnetic probe wave couple near-resonantly to the system. We are thus interested in the linear response of the medium to a weak probe field of frequency ω_b with electric component E_b and magnetic field component H_b. The electric polarization P and magnetization M induced in the medium at the frequency ω_b are given by [20]

$$\begin{eqnarray} {\boldsymbol P}(\omega_b) &= \chi_{EE}(\omega_b) {\boldsymbol E}_b + \xi_{EH}(\omega_b) {\boldsymbol H}_b/4 \pi, \end{eqnarray}\noindent \tag{ 3a }$$

$$\begin{eqnarray} {\boldsymbol M}(\omega_b) &= \chi_{HH}(\omega_b) {\boldsymbol H}_b + \xi_{HE}(\omega_b) {\boldsymbol E}_b/4 \pi. \end{eqnarray} \tag{ 3b }$$

We now bring the vector relations in equations (3) into a simpler scalar form, where the response functions reduce to complex scalars. To this end, we focus on a circularly polarized probe beam that propagates in the z-direction in the following. Its wavevector reads k = nk₀e_z with k₀ = 2π/λ_b. The electric and magnetic field components are given by

$$\begin{eqnarray} &&{\bf E}_b = \frac{E_b}{2} {\bf e}_{\pm 1} \,\mathrm{e}^{-\mathrm{i} \omega_b t} + {\rm c.c.} \end{eqnarray} \tag{ 4 }$$

and H_b = ∓ie_±1 e^−iω_btH_b/2 + c.c. with polarization unit vectors that are defined as ${\bf e}_{\pm 1} {=} \mp ({\bf e}_x \pm \mathrm {i} {\bf e}_y)/\sqrt {2}$. The two signs indicate the two circular polarizations σ^±. The electric polarization then becomes

$$\begin{eqnarray} &&{\bf P}(\omega_b) = \frac{P}{2} {\bf e}_{\pm 1} \,\mathrm{e}^{-\mathrm{i} \omega_b t} + {\rm c.c.} \end{eqnarray} \tag{ 5 }$$

with amplitude

$$\begin{eqnarray} P &= \chi_{EE} E_b + \xi_{EH} \frac{\mathrm{i} H_b}{4 \pi} . \end{eqnarray} \tag{ 6 }$$

The real (imaginary) part of the susceptibility χ_EE describes the electric response in phase (out of phase) with the incoming probe field. The response from the cross-coupling adds to the electric polarization via ξ_EH. Accordingly, we derive for the amplitude M of the induced magnetization ${\boldsymbol M}(\omega _b) {=} (\mp \mathrm {i} \frac {M}{2} {\bf e}_{\pm 1}\, \mathrm {e}^{- \mathrm {i} \omega _b t} + {\mathrm { c.c}})$ the response relation

$$\begin{eqnarray} &&M = \chi_{HH} H_b - \mathrm{i} \xi_{HE} \frac{E_b}{4 \pi} . \end{eqnarray} \tag{ 7 }$$

It is useful to define the electric permittivity ε and the magnetic permeability μ as usual as

$$\begin{eqnarray} &&\varepsilon = 1 + 4 \pi \chi_{EE}, \end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray} &&\mu = 1 + 4 \pi \chi_{HH}. \end{eqnarray} \tag{ 8b }$$

We can now calculate the refractive index n, which depends on the probe wave polarization. For circular σ^±-polarization, we obtain [19, 20]

$$\begin{equation} n_{\pm} = \eta \sqrt{\varepsilon \mu - \frac{\left( \xi_{EH} + \xi_{HE} \right)^2}{4}} \; \pm \; \frac{\mathrm{i}}{2} \left( \xi_{HE} - \xi_{EH} \right). \end{equation} \tag{ 9 }$$

As before, η∈{1,−1} determines the branch of the square root function, which we determine as discussed in section 2.1. We note that in our system the chiralities turn out to be negligible compared to the direct response coefficients and μ. The refractive index is therefore approximately given by $n =\eta \sqrt {\epsilon \mu }$, and thus is independent of polarization.

To find the electric polarization P(ω_b) and magnetization M(ω_b) induced in the atomic gas, we have to calculate the electric and magnetic dipole moments of the atoms at the probe field frequency ω_b. The total response of the medium is given by a superposition of the individual responses of the atoms. These are determined by the steady-state density matrix ρ of an atom in the driven laser field configuration of figure 1. Specifically, the steady-state coherences ρ₄₃ and ρ₂₁ govern the induced polarization and magnetization at the probe field frequency as

$$\begin{eqnarray} &&{\boldsymbol P} = N \left(\rho_{43} {\boldsymbol d}_{34} + {\rm c.c.}\right), \end{eqnarray}\noindent \tag{ 10a }$$

$$\begin{eqnarray} &&{\boldsymbol M} = N \left(\rho_{21} \boldsymbol{\mu}_{12} + {\rm c.c.}\right), \end{eqnarray} \tag{ 10b }$$

We now set up a master equation for the five-level atoms in the laser configuration of figure 1, which allows us to calculate the steady-state density matrix and thus the response of the medium. Since the density of the gas is rather large, we have to take into account nonlinearities arising from a resonant atom–atom interaction.

In the setup of figure 1, the probe field couples to the transition |3〉–|4〉 with its electric field component and to the transition |1〉–|2〉 with its magnetic field component. Coupling fields drive the transitions |1〉–|3〉, |2〉–|4〉 and |4〉–|5〉. Without a probe field, the atom is in a superposition of the states |1〉 and |3〉. In the presence of the probe field, the atom also evolves into the other states. The zero-order subspace {|1〉,|3〉} is also connected to the other states by the incoherent pump field r₃₄.

The time evolution of a single such five-level atom with density matrix ρ is governed by the master equation [31]

$$\begin{eqnarray} \dot{\rho} &=& \frac{1}{\mathrm{i} \hbar} \left[ H, \rho \right] - \sum_{j,k} \, \frac{\gamma_{jk}}{2} (\, |j\rangle \langle j| \rho + \rho |j\rangle \langle j| -2 |k\rangle\langle j|\rho|j\rangle\langle k| \, ) \end{eqnarray} \tag{ 11 }$$

with the system Hamiltonian given by

$$\begin{eqnarray} &&\fl H = \sum_{j=1}^5 \epsilon_j |j\rangle\langle j| - \Bigg\{ \boldsymbol{\mu}_{21} \cdot {\boldsymbol B}_{\rm L} \,\mathrm{e}^{-\mathrm{i} \omega_b t} |2\rangle \langle 1| + {\boldsymbol d}_{43} \cdot {\boldsymbol E}_{\rm L} \,\mathrm{e}^{-\mathrm{i} \omega_b t} |4\rangle \langle 3| + \frac{\hbar \Omega_{31}}{2}\, \mathrm{e}^{-\mathrm{i} \omega_a t} |3\rangle\langle 1| \nonumber \\ &&+ \frac{\hbar \Omega_{42}}{2} \mathrm{e}^{-\mathrm{i} \omega_c t} |4\rangle \langle 2| + \frac{\hbar \Omega_{54}}{2} \mathrm{e}^{-\mathrm{i} \omega_d t} |5\rangle \langle 4| + {\rm h.c.} \Bigg\} . \end{eqnarray} \tag{ 12 }$$

Here, _j is the energy of state |j〉 and the other terms describe the interaction with the laser fields in the long-wavelength and dipole approximations. The fields have frequencies ω_n with n∈{a,b,c,d}. The laser detuning on transition |i〉–|j〉 reads

$$\begin{eqnarray} &&\hbar \delta_{ij} = \hbar \bar{\omega}_{ij} + \epsilon_{ij} \end{eqnarray} \tag{ 13 }$$

with $i,j \in \{1,\ldots , 5\}$, where $\bar {\omega }_{ij} {=} \bar {\omega }_i {-} \bar {\omega }_j$ and _ij = _i − _j, and we have introduced the angular frequencies

$$\begin{eqnarray} &&\bar{\omega}_1 = \omega_c + \omega_d + \omega_b, \end{eqnarray}\noindent \tag{ 14a }$$

$$\begin{eqnarray} &&\bar{\omega}_2 = \omega_c + \omega_d, \end{eqnarray}\noindent \tag{ 14b }$$

$$\begin{eqnarray} &&\bar{\omega}_3 = \omega_b + \omega_c + \omega_d - \omega_a, \end{eqnarray}\noindent \tag{ 14c }$$

$$\begin{eqnarray} &&\bar{\omega}_4 = \omega_d, \end{eqnarray}\noindent \tag{ 14d }$$

$$\begin{eqnarray} &&\bar{\omega}_5 = 0. \end{eqnarray}\noindent \tag{ 14e }$$

$$\begin{eqnarray} &&{\boldsymbol E}_m=\frac{E_m}{2} \boldsymbol{\varepsilon}_m \,{\rm e}^{- \mathrm{i} \omega_m t} + {\rm c.c.} \end{eqnarray} \tag{ 15 }$$

with polarization vector ε_m (m∈{a,c,d}) give rise to the complex Rabi frequencies

$$\begin{eqnarray} \Omega_{31} &= ({\boldsymbol d}_{31} \cdot \boldsymbol{\varepsilon}_a) E_a /\hbar, \end{eqnarray}\noindent \tag{ 16a }$$

$$\begin{eqnarray} \Omega_{42} &= ({\boldsymbol d}_{42} \cdot \boldsymbol{\varepsilon}_c) E_c /\hbar, \end{eqnarray}\noindent \tag{ 16b }$$

$$\begin{eqnarray} \Omega_{54} &= ({\boldsymbol d}_{54} \cdot \boldsymbol{\varepsilon}_d) E_d /\hbar, \end{eqnarray} \tag{ 16c }$$

It is important to note that the Hamiltonian equation (12) contains instead of the external probe fields E_b, H_b the actual local fields E_L, B_L inside the medium. These contain contributions from the surrounding atoms in the medium as described by the polarization P and magnetization M. The corresponding Rabi frequencies are thus defined as

$$\begin{eqnarray} &&\Omega_{43} = (\boldsymbol{d}_{43} \cdot \boldsymbol{\varepsilon}_b) E_{\rm L}/\hbar, \end{eqnarray}\noindent \tag{ 17a }$$

$$\begin{eqnarray} &&\Omega_{21} = (\boldsymbol{\mu}_{21} \cdot \boldsymbol{\varepsilon}_b) B_{\rm L}/\hbar. \end{eqnarray} \noindent \tag{ 17b }$$

The second part of equation (11) describes spontaneous decay and decoherence arising due to elastic collisions. The decay rate on transition |i〉 → |j〉 is denoted by γ_ij and set to γ_ij = γ if |i〉 → |j〉 is electric dipole allowed (E1) and set to γ_ij = α²γ for metastable E2, M1 transitions, where α = 1/137 is the fine-structure constant. To model elastic collisions, we add an effective dephasing constant γ_C to the decay rates of the off-diagonal density matrix elements ρ_jk in equation (11), such that they read

$$\begin{eqnarray} &&\tilde{\gamma}_{jk} = \sum_l (\gamma_{jl} + \gamma_{kl})/2 + \gamma_{\mathrm{C}} \end{eqnarray} \tag{ 18 }$$

for $j \neq k\in \{1,\ldots ,5\}$. In our numerical calculations, we set γ_C = γ.

As already mentioned, the Hamiltonian equation (12) contains the actual local fields inside the medium, whereas the medium response equation (3) is formulated in terms of the external probe fields E_b, H_b. It turns out that atom densities exceeding N ∼ 10¹⁶ cm⁻³ are required to obtain negative refraction, meaning that many particles are found in a cubic wavelength volume Nλ³_b ≫ 1. Then, the local probe fields E_L, H_L experienced by the atoms may considerably deviate from the externally applied probe fields E_b, H_b in free space, since they contain contributions from neighboring atoms. Therefore, single-atom results for ρ₄₃ and ρ₂₁ cannot describe the system and thus are not shown here.

Different approaches have been proposed in the literature to solve this problem. One approach, which is expected to be valid for moderate densities, consists of expanding the steady-state density matrix elements ρ₂₁ and ρ₄₃ to linear order in the local fields E_L, H_L to obtain the medium polarization and magnetization. Then, the local and external fields are related by the Lorentz–Lorenz (LL) formulae [30, 32]

$$\begin{eqnarray} &&{\boldsymbol E}_{\rm L}={\boldsymbol E}_b + \frac{4 \pi}{3} {\boldsymbol P}, \end{eqnarray} \tag{ 19a }$$

$$\begin{eqnarray} &&{\boldsymbol H}_{\rm L} = {\boldsymbol H}_b + \frac{4 \pi}{3} {\boldsymbol M}, \end{eqnarray} \tag{ 19b }$$

Here, we use an alternative method. We relate the local fields via the LL-relations to the external fields already in the Hamiltonian in equation (11). The corrections by the induced polarization P and magnetization M render the master equation nonlinear, since P and M given in equation (10a) itself depend on ρ [33]. Specifically, we obtain the complex Rabi frequencies

$$\begin{eqnarray} \Omega_{43} &= ({\boldsymbol d}_{43} \cdot \boldsymbol{\varepsilon}_b) E_{\rm L}/\hbar = w_E \gamma + \frac{8 \pi N d_{43}^2}{3 \hbar} \tilde{\rho}_{43}, \end{eqnarray}\noindent \tag{ 20a }$$

$$\begin{eqnarray} \Omega_{21} &= ({\boldsymbol \mu}_{21} \cdot \boldsymbol{\varepsilon}_b) H_{\rm L}/\hbar = w_H \gamma + \frac{8 \pi N \mu_{21}^2}{3 \hbar} \tilde{\rho}_{21} , \end{eqnarray} \tag{ 20b }$$

$P {=} 2 N \tilde {\rho }_{43} d_{34}$

$\tilde {\rho }_{43} {=} \rho _{43} \,{\mathrm { e}}^{\mathrm {i} \omega _b t}$

$M {=} 2 N \tilde {\rho }_{21} \mu _{12}$

$\tilde {\rho }_{21} {=} \rho _{21} \,{\mathrm { e}}^{{\rm i} \omega _b t}$

$$\begin{eqnarray} &&w_E = \frac{d_{43} E_b}{\hbar \gamma} \ll 1, \end{eqnarray}\noindent \tag{ 21a }$$

$$\begin{eqnarray} &&w_H = \frac{\mu_{21} H_b}{\hbar \gamma} \ll 1 . \end{eqnarray} \tag{ 21b }$$

To find the linear response coefficients χ_αα and ξ_αβ (α,β∈{E,H}) introduced in equations (3), we numerically integrate the nonlinear differential equations of motion (11) using a standard Runge–Kutta-like algorithm [35] until the system has reached its steady state. This is done for a number of different electric field amplitudes E_b, holding the magnetic field amplitude H_b fixed. Linear regression of the relevant coherences ρ₂₁,ρ₄₃ as a function of E_b allows to extract the response coefficients χ_αα and ξ_αβ as slope m_α and y-axis intercept b_α from equations (3). We note again that in the case of a circularly polarized probe field, these equations describe scalar relations. In the following, we choose E_b ∼ e₋₁ and thus H_b ∼ ie₋₁.

The response functions χ_EE and ξ_EH are obtained from the electric polarization

$$\begin{equation} {\boldsymbol P} = \frac{P}{2}\boldsymbol{\varepsilon}_b \,\mathrm{e}^{- \mathrm{i} \omega_b t} + {\rm c.c.} = N \left(\tilde{\rho}_{43} \,\mathrm{e}^{- \mathrm{i} \omega_b t} {\boldsymbol d}_{34} + {\rm c.c.}\right), \end{equation} \tag{ 22 }$$

where $\tilde {\rho }_{43}{=} \rho _{43} \,\mathrm {e}^{\mathrm {i} \omega _b t}$ is the slowly varying part of the coherence. We find that

$$\begin{equation} \frac{P}{2} = N \tilde{\rho}_{43} d_{34} = \chi_{EE} \frac{E_b}{2} + \xi_{EH} \frac{\mathrm{i} H_b}{8 \pi} \end{equation} \tag{ 23 }$$

and thus

$$\begin{eqnarray} \tilde{\rho}_{43} &= \chi_{EE} \frac{\hbar \gamma}{2 N d_{34}^2} w_E + \xi_{EH} \frac{\mathrm{i} \hbar \gamma}{8 \pi N d_{34} \mu_{21}} w_H \end{eqnarray} \tag{ 24 }$$

with the small expansion parameters defined in equations (21). Solving the nonlinear master equation (11) for different values of w_E keeping w_H fixed, we calculate $\tilde {\rho }_{43} {=} m_E w_E + b_E$ and thus

$$\begin{eqnarray} &&\chi_{EE} = \frac{ 2 N d_{34}^2}{\hbar \gamma} m_E = \frac{ 3 N \lambda^{3}_{b}}{16 \pi^3} m_E, \end{eqnarray}\noindent \tag{ 25a }$$

$$\begin{eqnarray} &&\xi_{EH} = -\mathrm{ i} \frac{8 \pi N d_{34} \mu_{21}}{\hbar \gamma w_H} b_E =-\mathrm{ i} \frac{3 N \alpha \lambda^{3}_{b}}{4 \pi^2 w_H} b_E . \end{eqnarray} \tag{ 25b }$$

$\sqrt {3\gamma _{ij}\hbar c^3/(4 \omega _{ij}^3)}$

$$\begin{eqnarray} &&\frac{\mathrm{i} M}{2} = \mathrm{i} N \tilde{\rho}_{21} \mu_{12} = \chi_{HH} \frac{\mathrm{i} H_b}{2} + \xi_{HE} \frac{E_b}{8 \pi} , \end{eqnarray} \tag{ 26 }$$

one obtains the response coefficients for the magnetization from $\tilde {\rho }_{21} {=}m_H w_E + b_H$ as

$$\begin{eqnarray} &&\chi_{HH} = \frac{3 N \alpha^2 \lambda^{3}_{b}}{16 \pi^3 w_H} b_H, \end{eqnarray}\noindent \tag{ 27a }$$

$$\begin{eqnarray} &&\xi_{HE} = \mathrm{i} \frac{3 N \alpha \lambda^{3}_{b}}{4 \pi^2} m_H . \end{eqnarray} \tag{ 27b }$$

In figure 2, the permittivity and the permeability μ of the medium are shown as a function of the probe field detuning from the magnetic resonance δ₂₁. The resulting refractive index n can be seen in figure 3. Without incoherent pumping the system is passive and thus ε, μ and n have a positive imaginary part, i.e. the medium absorbs. Whereas absorption is small for the electric response due to local field effects [30], the losses are significant in the magnetic component. The fact that Re() ≈ −2 and the electric losses are small Im() ∼ 1/N follows directly from the Clausius–Mossotti relation applied to a simple oscillator model [30]. The local-field-induced shift of the magnetic resonance away from δ₂₁ = 0 to positive δ₂₁ can also be qualitatively understood within this approach. We observe power broadening in all response functions due to the nonlinearities that appear in the master equation because of the local fields that include near-dipole–dipole effects (see equations (20)). With gradually increasing the incoherent pumping rate between levels |3〉 and |4〉, we find that the imaginary part of ε around zero detuning turns negative, indicating that the two-photon transition |2〉–|3〉 amplifies the probe field. The magnetic permeability is mostly unaffected by the incoherent field. The real part of n is negative for all the shown detunings. It is worth pointing out that this even includes regions with Re(μ) > 0, where negative refraction can occur since Re() < 0 and absorption is dominant, Im(μ) ≫ Re(μ) [36].

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Increasing the pump rate r decreases the absorption over the whole range of displayed probe detunings. Since we can continuously change the pump rate, we can identify the physically correct square root branch of n if we follow the path in the complex plane starting from the root with a positive imaginary part for the passive system. Increasing r thereby continuously decreases Im(n) in a region around δ₂₁ = 0 with a width of a few γ. Finally, for suitable incoherent pump rates, negative refraction occurs without absorption or even with amplification, see figure 4. In this frequency range, the FOM becomes very large due to the vanishing imaginary part. The physical mechanism for the crossover from passive to active can easily be understood by following the path of ε, μ, n² and n in the complex plane as a function of the pump rate r, see figure 5. Without pumping (step 1), the system is passive and shows absorption. With increasing the pump rate, Im(ε) becomes negative, which compensates for the losses via the magnetic component. Thus, Im(n) decreases and finally vanishes, while Re(n) remains negative. In figure 4, one finds FOM > 10 for a frequency range of more than 30γ. FOM > 50 is achieved over a range of about γ.

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We now turn to the second set of parameters. The second set is based on a lower density, which, however, is still large compared to the typical densities applied in light propagation through coherently prepared atomic media. Another main difference lies in the much lower effective gap Δ' for the case with lower density. Nevertheless, in both cases the effective gap is large enough to render two-photon processes important.

It can be seen from figure 6 that the permittivity and permeability are qualitatively similar to the ones in figure 2 for larger density. This, in particular, applies also to the dependence on incoherent pumping. Generally, the permittivity shows a stronger dependence on detuning, with a tilted base line, which is due to the lower value of Δ'. Consequently, the index of refraction, shown in figure 7, and the evolution of ,μ,n² and n in the complex plane are qualitatively similar to the case with higher density (see figures 3 and 5).

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Nevertheless, quantitatively, the lower density leads to a smaller range of negative refraction with a low absorption and thus a high FOM. Results are shown in figure 8. The detuning range with the imaginary part of n close to zero is about one decay rate γ; the range with the absolute value of the real part exceeding that of the imaginary part (FOM > 1), however, is about 10γ. This is a significantly smaller region than in the high density case. Therefore, as expected, the facilitated implementation due to the lower density comes at the price of a reduced performance in terms of lossless negative refraction.

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Finally, we estimate the effect of Doppler broadening on our results. For this, we use Antoine's equation, logp = A − B/(C + T), where p is the vapor pressure in units of bar, T is the temperature in kelvin, and A = 3.75641, B = 95.599 and C = −1.503 are parameters taken from NIST [37] for neon. We further relate the density N to the pressure and temperature via N = p/(k_BT) with k_B the Boltzmann constant. From these two relations, we obtain for the larger density N₁ = 2.55 × 10¹⁷ cm⁻³ a temperature of T₁ = 15.1 K, and for the smaller density N₂ = 5 × 10¹⁶ cm⁻³ a temperature of T₂ = 13.8 K. Note that these temperatures are slightly below the temperature range (15.9–27 K) given in [37] for the parameters A,B and C.

We assume a Maxwell–Boltzmann velocity distribution of the atoms in the laser propagation direction with the most probable velocity $v_m {=} \sqrt {2\, k_{\mathrm { B}}\, T/m}$ with m the mass of a single atom. The Doppler shift thus leads to an additional detuning Δ_D with a Gaussian distribution [38]

$$\begin{eqnarray} &&f(\Delta_{{\rm D}}) d\Delta_{{\rm D}} = \frac{1}{\sqrt{\pi} \, k_{\rm B}\, v_m }\, \mathrm{e}^{- [\Delta_{{\rm D}}/(k_{\rm B} v_m)]^2}\, d\Delta_{{\rm D}} \end{eqnarray} \tag{ 28 }$$

over which we average our medium response coefficients. The full-width at half-maximum of this distribution evaluates to a Doppler width of $\delta \omega {=} \sqrt {8 \ln (2) k_{\mathrm { B}} T/m}$. It should be noted that it is not obvious whether the averaging over the microscopic medium response provides the full picture. For example, modifications to the nonlinear density-dependent local field corrections could arise due to the motion of the atoms. If the probe field transitions for atoms with different velocities are shifted out of resonance relative to each other, the local field at one of the atoms due to the presence of the second atom will be different from the effect of a resonant atom. This could, for example, effectively reduce the density entering the local field corrections for a particular atom to that of mutually resonant atoms moving with similar velocities. Such effects, however, are beyond the scope of this paper.

In our calculations, we scale all frequencies to the decay rate γ on transition $|2\rangle {\leftrightarrow } |4\rangle$, which is approximately γ = 10⁷/(2π) s⁻¹. We thus take this transition as a reference, and obtain Doppler widths of δω₁ ≈ 333γ and δω₂ ≈ 318γ for the wavelength λ_c at the two densities N₁ and N₂. The Doppler broadenings for laser fields with wavelengths λ other than that of the reference transition are different from δω_i by a factor of λ_c/λ. In particular, the Doppler broadening on the probe transitions is about 15 times smaller, since λ_b ≈ 15λ_c. The probe field Doppler width is thus about 21γ.

As a first step, we Doppler averaged our results in figures 4 and 8. We found that while the Doppler broadening has a detrimental effect on the results, nevertheless even with full Doppler broadening taken into account negative refraction with significant FOM = |Re(n)/Im(n)| is achieved over a broad spectral range. Overall, the modifications of the results due to Doppler broadening are consistent with the estimated probe transition Doppler width of about 21γ. The Doppler broadening has a stronger effect on the results of figure 8 compared with figure 4, since the range of probe field detunings over which negative refraction is observed is lower in this case.

But there are important differences compared with the results without Doppler averaging. Due to the different wavelength on the various transitions, the two-photon resonance between states |2〉 and |3〉 via state |4〉 occurs at different probe field detunings for different atom velocities, such that its effect is reduced in the Doppler averaging. Also, we found that for the parameters of figures 4 and 8, only atoms in a certain velocity range exhibit negative refraction. Thus, the Doppler averaged result contains contributions, both with negative and positive index of refraction. The optimum incoherent pump rates to eliminate absorption depend on the atom velocity as well. Finally, since the linewidth of the dipole-forbidden transition between |1〉 and |3〉 is suppressed by α², already small Doppler shifts detune the pump field Ω₃₁ strong enough to significantly change the medium response to the electric probe field component. In effect, the Doppler averaged results have a more involved dependence on the various system parameters compared with the non-averaged results, and a straightforward enhancement of the results by incoherent pump fields becomes more challenging with increasing Doppler width.

Nevertheless, we found that the concept of using active media to improve the performance of atomic negative refractive index media can also be applied in Doppler broadened vapors. For this, we replaced the coherent pumping Ω₃₁ by a broadband incoherent pump field between |1〉 and |3〉, such that the electric response becomes less dependent on the Doppler shift. Results are shown in figure 9 for slightly adjusted control field parameters, but with the same density as in figure 4. The two curves (i) show the real part of the index of refraction n (the lower half of the figure) and the FOM divided by 10 (FOM/10, upper half) in the passive medium without Doppler broadening. It can be seen that already in this passive case negative refraction with FOM of more than 20 can be achieved over a spectral range of several γ. The curves (ii) show the corresponding results with full Doppler broadening. Whereas the system still exhibits negative refraction, the maximum FOM is reduced to about eight due to the averaging. But, as shown in curves (iii), rendering the Doppler broadened system active by applying an incoherent pump field between states |3〉 and |4〉 leads to significant enhancement of the FOM, which in this case approaches 100. Interestingly, for the case with Doppler broadening, the FOM and the overall performance are not monotonically improved with increasing the incoherent pump rate. Rather, increasing the pump rate first worsens the results, and only toward slightly higher pump rates it leads to a strong increase in the FOM as shown in figure 9. This more complicated dependence again arises from the averaging of the different results for the various atom velocities.

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Our estimates above apply to a thermal gas vapor, which leads to rather large Doppler widths. Alternative implementations such as in ultracold gases, solid-state quantum optics or with Doppler-free laser configurations in related level structures could lead to significantly lower Doppler widths even at high atom densities. Interestingly, we found that for some parameters, moderate Doppler broadening can even lead to an enhancement of the FOM compared with the case without Doppler broadening. This again points to the rich interplay between incoherent pump, Doppler broadening and medium response, such that independent control over density and Doppler broadening would certainly be desirable. In any case, we can conclude from our analysis that the general concept of significantly improving the performance of a dense gas of atoms as a negative refractive index medium by rendering it active via the application of suitable pump fields proves beneficial also for the Doppler broadened case in thermal vapors.