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The spontaneous emission of a V-type Zeeman atom in a Fabry–Perot cavity ... or slower than that in free vacuum, while the spontaneous emission spectra are.
Zeng et al.

Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B

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Spontaneous emission spectrum of a V-type three-level atom in a Fabry–Perot cavity containing left-handed materials Xiaodong Zeng,2,* Mingzhang Yu,2 Dawei Wang,3 Jingping Xu,1,2 and Yaping Yang1,2,3 1

Key Laboratory of Advanced Micro-Structure Materials, Ministry of Education, Tongji University, Shanghai 200092, China 2 Department of Physics, Tongji University, Shanghai 200092, China 3 Beijing Computational Science Research Center, Beijing 100084, China *Corresponding author: [email protected] Received March 21, 2011; revised June 11, 2011; accepted July 22, 2011; posted July 25, 2011 (Doc. ID 144507); published August 26, 2011

The spontaneous emission of a V-type Zeeman atom in a Fabry–Perot cavity containing left-handed materials (LHMs) is investigated. Because of the strong indirect quantum interference induced by the refocusing and phase compensation of LHMs, the population evolution and the emission spectrum are much different from that in isotropic environments at different initial conditions. For the degenerate cases, by preparing different initial states, the population decays much faster or slower than that in free vacuum, while the spontaneous emission spectra are narrowed or broadened, respectively. For large detuning cases, the population exchange between two upper levels could be weakened, and the Fano minimum appears in the emission spectrum. In addition, the influence of the dipole orientation on the spectrum is discussed. © 2011 Optical Society of America OCIS codes: 160.3918, 270.1670.

1. INTRODUCTION Left-handed materials (LHMs) were first investigated by Veselago in 1948 [1]. Since both the effective permittivity and permeability of LHMs are negative in a certain frequency range, the refractive index is also negative according to Maxwell’s equations. Therefore, the electric field, magnetic field, and wave vector form a left-handed system, and the energy flux of the electromagnetic field is opposite to the wave vector. After the fabrication of LHMs in experiment [2], LHMs have attracted a great deal of attention during the past decade [3–8]. Potential applications have been widely proposed, such as in highly efficient low reflectance surfaces [9], superlenses [10,11], and broadband ground-plane cloaks [12]. Furthermore, atomic spontaneous decay in the presence of an LHM was recently investigated [13]. On the other hand, quantum interference among different decay channels of multilevel atoms is one of the basic topics in quantum optics, because it leads to many interesting effects, such as electromagnetically induced transparency [14], ultranarrow spectral lines [15], coherence trapping of populations [16], and lasing or gain without inversion [17]. Most of the previous studies about quantum interference assumed that the two degenerate transition dipole moments were nearly parallel or antiparallel. There are few systems that can satisfy the above condition. The Zeeman atom can have two nearly degenerate sublevels, but the two corresponding transition dipoles are orthogonal to each other. In 2000, Agarwal suggested that quantum interference between the two orthogonal dipoles can be achieved by placing the atom in an anisotropic vacuum [18]. Recently, Yang and co-workers pointed out that a cavity containing LHMs could provide a large anisotropic space and the Zeeman atom in it could have 0740-3224/11/092253-07$15.00/0

strong quantum interference [19,20]. Since such quantum interference originates from the coupling of one dipole with the reflected field emitted from the other dipole, they defined such interference as “indirect quantum interference.” Because of such indirect interference, the photon can exchange between the two upper levels, which induces the spontaneous evolution and the spectrum to be much different from those in free vacuum. By preparing different initial conditions, the decay could be canceled or accelerated, and the corresponding emission spectrum could be narrowed or broadened, respectively. In addition, we can change the atomic dipole directions to make the two degenerate dipoles have different decay rates and still have strong quantum interference. In this paper, we will investigate the population evolution and emission spectrum of a V-type Zeeman atom in such a cavity based on indirect quantum interference. This paper is organized as follows. In Section 2, we introduce the model and derive the atomic evolution equations and the formula of emission spectrum under proper approximations. In Section 3, we discuss atomic spontaneous emission in company with indirect quantum interference, and compare it with that in free vacuum. In last section, we draw a conclusion.

2. MODEL AND THE QUANTIZATION SCHEME We consider a V-type Zeeman atom placed in the middle of a Fabry–Perot cavity containing LHMs, as is shown in Fig. 1. The cavity wall is made of a LHM slab, with indices εA and μA , and thickness dA , mounted on a mirror. The distance between the two LHM slabs is 2dA . Interfaces are in the x–y plane, and the atom is located at the origin. The upper levels jai and jbi © 2011 Optical Society of America

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J. Opt. Soc. Am. B / Vol. 28, No. 9 / September 2011 A LHM

A A LHM

x atom 0

dA

dA

Perfect mirror

Perfect mirror

eΑ mΑ

Zeng et al.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^jN ðr; ωÞ ¼ ω ℏε0 Imεðr; ωÞ^f e ðr; ωÞ π sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏ þ∇× − Im½1=μðr; ωÞ^f m ðr; ωÞ; πμ0

a

b

z

where ε0 and μ0 are the permittivity and permeability in vacuum, respectively. Gðr; r0 ; ωÞ is the Green’s tensor in the environment of Fig. 1 satisfying the following equation:

c

dA

Fig. 1. Zeeman atom with two upper levels jai, jbi and a ground level jci is placed in the middle of two LHM slabs attached on two perfect mirrors. The thickness of the LHM slab is dA , and the distance between the slabs is 2dA .

are coupled by the same vacuum modes to the lower level jci with transition frequencies ωa and ωb , respectively. The corresponding transition dipoles of jai → jci and jbi → jci are orthogonal to each other, and the transition process jai → jbi is forbidden. The atomic dipole moment is given by d ¼ dðAac e1 þ Abc e2 Þ þ H:c:;

ð1Þ

 ∇×

 ↔ 1 ω2 ∇ × − 2 εðr; ωÞ Gðr; r0 ; ωÞ ¼ I δðr − r0 Þ: μðr; ωÞ c

jψðtÞi ¼ C a ðtÞe−iωa t jf0gijai þ C b ðtÞe−iωb t jf0gijbi þ Z ×

pffiffiffi ¼ ðez  iex Þ= 2;

ð2Þ

Aij ¼ jiihjjði; j ∈ fa; b; cgÞ are the atomic transition ði ≠ jÞ and population ði ¼ jÞ operators, ek ðk ¼ x; y; zÞ are the normalized Cartesian basis vectors, and d is the atomic dipole strength, chosen to be real. Under the rotating-wave approximation, the Hamiltonian of the total system can be derived as [20,21] Z ∞ X Z ^ ¼ ^ H d3 r dωℏω^f þ λ ðr; ωÞf λ ðr; ωÞ λ¼e;m

0

þ ℏωa jaihaj þ ℏωb jbihbj Z ∞ ^ ðþÞ ðrA ; ωÞ − ½jaihcjda · dωE 0 Z ∞ ^ ðþÞ ðrA ; ωÞ þ H:c:: dωE þ jbihcjdb ·

¼ δλλ0 δij δðr −

½^f λ;i ðr; ωÞ; ^f λ0 ;j ðr0 ; ω0 Þ ¼ 0;



ω0 Þ;



d3 r

λ¼e;m

dωe−iωt Cλc ðr; ω; tÞ · j1λ ðr; ωÞijci;

ð9Þ

where Cλc ðr; ω; 0Þ ¼ 0 and j1λ ðr; ωÞi ¼ ^f þ λ ðr; ωÞjf0gi. By sub_ stituting Eqs. (3) and (9) into Schrödinger equation iℏjψðtÞi ¼ ^ HjψðtÞi, we obtain the following differential equations: Z ∞ 1 dωe−iðω−ωa Þt C_ a ðtÞ ¼ − pffiffiffiffiffiffiffiffiffiffi ℏπε0 0  Z ω ω pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d3 r · da · Imεðr; ωÞ × c c

 ⃖ r  · Cmc ðr; ω; tÞ ; × ½GðrA ; r; ωÞ × ∇ ð3Þ

The first term on the right side of Eq. (3) represents the Hamiltonian of the LHM-assisted field, the second and the third terms are the Hamiltonian of the atom, and the last one is the interaction Hamiltonian. Here, ^f e ðr; ωÞ and ^f m ðr; ωÞ are basic bosonic vector operators, corresponding to the annihilation operators of electric and magnetic excitons [21]. They satisfy the following commutation relations: r0 Þδðω

0

X Z

× GðrA ; r; ωÞ · Cec ðr; ω; tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ −Im½1=μðr; ωÞ

0

0 0 ½^f λ;i ðr; ωÞ; ^f þ λ0 ;j ðr ; ω Þ

ð8Þ

We assume the atom is in the superposition state between the upper levels initially, i.e., jC a ð0Þj2 þ jC b ð0Þj2 ¼ 1.The state function of the system at any time t can be written as

where e1;2

ð7Þ

ð4Þ ð5Þ

where λ; λ0 ¼ e; m and i, j run over all three spatial coordinates. The positive frequency part of the electric field in frequency space is Z ^ ðþÞ ðr; ωÞ ¼ iωμ0 d3 r0 Gðr; r0 ; ωÞ · ^jN ðr0 ; ωÞ: E ð6Þ The noise current operator ^jN ðr0 ; ωÞ is the function of bosonic vector operators [21]:

ð10aÞ

Z ∞ 1 dωe−iðω−ωb Þt C_ b ðtÞ ¼ − pffiffiffiffiffiffiffiffiffiffi ℏπε0 0  Z ω ω pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d3 r · db · Imεðr; ωÞ × c c × GðrA ; r; ωÞ · Cec ðr; ω; tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ −Im½1=μðr; ωÞ ⃖ r  · Cmc ðr; ω; tÞg; × ½GðrA ; r; ωÞ × ∇

ð10bÞ

1 ω2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C_ ec ðr; ω; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi 2 Imεðr; ωÞ ℏπε0 c × G ðr; rA ; ωÞ · ½da C a ðtÞe−iðωa −ωÞt þ db C b ðtÞe−iðωb −ωÞt ;

ð10cÞ

1 ω pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −Im½1=μðr; ωÞ C_ mc ðr; ω; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi ℏπε0 c ~ × G ðr; r ; ωÞ · ½d C ðtÞe−iðωa −ωÞt × ½∇ a a r A þ db C b ðtÞe−iðωb −ωÞt :

ð10dÞ

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Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B 0

Here, ½Gðr; r0 ; ωÞ × ∇⃖ r0 ij ≡ εjkl ∂rk Gil ðr; r0 ; ωÞ, and the indices i, j, k, and l run over all three spatial components. By integrating Eqs. (10c) and (10d) and substituting them into Eqs. (10a) and (10b), we get γ κ C_ a ðtÞ ¼ − a C a ðtÞ − a C b ðtÞeiωab t ; 2 2

ð11Þ

emission is canceled. For the dipole component normal to the interface, the reflected field has no phase difference as the emitted field, so the emission is accelerated. Because of the above reasons, p ≠ 0 and strong quantum interference appears [13,19,20]. Eqs. (11) and (12) have analytical solutions as C a ðtÞ ¼ C 1 es1 t þ C 2 es2 t ;

γ κ C_ b ðtÞ ¼ − b C b ðtÞ − b C a ðtÞeiωba t ; 2 2

ð12Þ

where ωij ¼ ωi − ωj ði; j ¼ a; b; i ≠ jÞ. Here we omit the frequency shifts induced by the environment. γ i ði ¼ a; bÞ are the decay rates from the corresponding upper jii level to jci, which are γi ¼

2ω2i  d · ImGðrA ; rA ; ωi Þ · di ; ℏε0 c2 i

ð13aÞ

and κ i are the quantum interference between two transitions: κi ¼

2ω2j ℏε0 c2

di · ImGðrA ; rA ; ωj Þ · dj :

ð13bÞ

In order to measure the degree of quantum interference, we pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi adopt the p ¼ κa κ b =ðγ a γ b Þ in Refs. [18–20]. When the frequency difference between two upper levels is small, it is reasonable to get γ a ≈ γ b ≈ ðΓ⊥ þ Γ∥ Þ=2, κa ¼ κ b ≈ ðΓ⊥ − Γj∥ Þ=2, and p ¼ ðΓ⊥ − Γ∥ Þ=ðΓ⊥ þ Γ∥ Þ, where Γ⊥ (Γ∥ ) are the spontaneous decay rates of the dipole moment d perpendicular (parallel) to the interface having the following expressions: 3 Γ∥ ¼ Γ0 Reμ0 n0 4   Z ∞ 2iβ0 dA 2iβ0 dA dk∥ k∥ 2r TE r TE e4iβ0 dA þ r TE þ r TE L e R e 1þ L R × TE 4iβ0 dA k0 β 0 1 − r TE 0 L rR e   2iβ0 dA − r TM e2iβ0 dA β2 2r TM r TM e4iβ0 dA − r TM L e R þ 20 1 þ L R ; ð14Þ TM 4iβ0 dA 1 − r TM k0 L rR e Z ∞ dk∥ k3∥ 3 Γ⊥ ¼ Γ0 Reμ0 n0 2 k30 β0 0   TM 4iβ0 dA þ r TM e2iβ0 dA þ r TM e2iβ0 dA 2r TM L rR e L R ; × 1þ TM 4iβ0 dA 1 − r TM L rR e

2255

C b ðtÞ ¼ −

2 γ ab

 s1 þ

ð16aÞ

    γa γ C 1 es1 t þ s2 þ a C 2 es2 t e−iωab t ; 2 2 ð16bÞ

where     γ γ s1;2 ¼ − a þ b þ iωab  D 2; 2 2 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 γa γb − þ iωab þ κa κb ; D¼ 2 2    γa κ D: C 1;2 ¼ ð∓Þ þ s2;1 C a ð0Þ þ a C b ð0Þ 2 2 The spontaneous emission spectrum Sðr; ωÞ observed at a position r is proportional to the Fourier transform of the field correlation function: ^ ð−Þ ðr; ωÞE ^ ðþÞ ðr; ωÞjψðtÞi Sðr; ωÞ ¼ limhψðtÞjE t→∞  ω4 μ2 d − ð2s1 þ γ a Þdb =κa C1 ¼ 2 0 ImGðr; rA ωÞ · a ω − ωa − is1 π  2 d − ð2s2 þ γ a Þdb =κ a þ a C 2 : ω − ω − is a

ð17Þ

2

Because of the indirect quantum interference of the Zeeman atom in the structure of Fig. 1, the properties of the spontaneous emission spectrum would be much different from those of two orthogonal dipoles in an isotropic environment, and also different from those of two parallel dipoles in previous studies [22].

3. SPONTANEOUS EMISSION ð15Þ

where Γ0 ¼ d2 ω30 =ð3πε0 ℏc3 Þ is the decay rate of dipole moment d with frequency ω0 in free space. r TE;TM (r TE;TM ) are L R the reflection coefficients of the left (right) slabs for the TE and TM polarization fields. Here, r TE;TM ¼ r TE;TM . L R If the quantum interference between the two transition channels vanishes, i.e., κi ¼ 0, p ¼ 0, then Eqs. (11) and (12) would be two independent differential equations, which means the population evolutions of the upper levels would not interfere with each other. In the present model (see Fig. 1), the phase shift for the cycle path is zero due to the negative phase shift in the LHM layer, and the LHM focuses the field to the atom and keeps the amplitude the same as that of the emitted field. For the dipole component parallel to the interface, the reflected field has a π phase shift at the mirror, so the

The spontaneous decay of the Zeeman atom is greatly influenced by the indirect quantum interference. In this section, we discuss this influence by examining the population in the upper levels and the emission spectrum. According to Eqs. (16) and (17), it is clear that the properties of the atomic population evolution and emission spectrum are dependent on the atomic initial states and the detuning between the two upper levels. In the following discussion, we adopt the parameters in Refs. [19,20] and set εA ¼ μA ¼ nA ¼ −0:999 þ i0:003 for the LHMs, r TE ¼ −0:99 and r TM ¼ 0:99 for the reflection coefficients of the mirrors for TE and TM waves, and dA ¼ λ0 ¼ 2πc=ωa for the width of the LHM slabs. A. Degenerate Case with Small ωab First, we choose a very small frequency difference between two upper levels ωab ≪ Γa0 , where Γao is the decay rate from jai to jci in free vacuum. In this case, it is a proper approximation that Gðωa Þ ≈ Gðωb Þ, γ a ≈ γ b , and κ a ¼ κb . Using Eqs. (14) and (15), we can get Γ⊥ ¼ 17:4Γa0 , Γ∥ ¼ 0:17Γa0 ,

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1.0

1.0

0.8

|Ca(t)|

2

|Ca(t)|

2

|Cb(t)|

2

0.8

2

0.6

2

0.6

P

P

|Ca(t)| +|Cb(t)|

0.4

0.4

0.2

0.2

0.0

0

1

2

3

4

5

6

Γaot Fig. 2. Evolution of jC a ðtÞj2 (dotted curve), jC b ðtÞj2 (dashed curve), and jC a ðtÞj2 þ jC b ðtÞj2 (dashed–dotted curve) of a Zeeman atom in the LHM cavity and jC a ðtÞj2 (solid curve) in free vacuum with C a ð0Þ ¼ 1, C b ð0Þ ¼ 0.

and p ¼ 0:98, which is different from the case of two (anti-) parallel dipoles (p ¼ 1) and the case of two orthogonal dipoles in an isotropic environment (p ¼ 0). By preparing the atom initially in the state C a ð0Þ ¼ 1, C b ð0Þ ¼ 0, we plot the population evolution of levels jai and jbi in Fig. 2. We can see that the population jC a ðtÞj2 in the upper level jai can decay to the lower level jci and then jump in part to the other upper level jbi. For the case in free vacuum, there is no population in level jbi because the quantum interference between the two transitions disappears. During the starting period, the population jC b ðtÞj2 in level jbi increases with time, and goes to a maximum value (about 0.22) at time t ≈ 0:5=Γa0 . Then jC a ðtÞj2 and jC b ðtÞj2 decay to zero with the same decay rate. It is obvious that, at the beginning, jC a ðtÞj2 or the total population in upper levels (jC a ðtÞj2 þ jC b ðtÞj2 ) decreases faster than that in free vacuum (the solid curve). The appearance of such a phenomenon is because the decay rate γ a of the atom in our structure is much larger than that in free vacuum. But after jC b ðtÞj2 reaches the maximum value, the indirect quantum interference between the two transition channels leads to the population exchanging between the two upper levels. The jC a ðtÞj2 is the same as jC b ðtÞj2 . Because the phase difference between C a ðtÞ and C b ðtÞ approximates to π, the decay is inhibited and is much slower than that in free vacuum. Compared with the complete quantum interference, the population cannot be trapped on the upper levels forever, because the two decay channels cannot cancel each other absolutely for p ≠ 1. In Fig. 3, we plot theptotal ffiffiffi population evolution for pffiffiffiinitial states C a ð0Þ ¼ C b ð0Þ ¼ 2=2 and C a ð0Þ ¼ −C b ð0Þ ¼ 2=2. It shows that the total population of the two upper levels pffiffiffi pffiffiffi with C a ð0Þ ¼ C b ð0Þ ¼ 2=2 (C a ð0Þ ¼ −C b ð0Þ ¼ 2=2) decays much faster (slower) than that in free vacuum. If the atomic pffiffiffi initial state is C a ð0Þ ¼ C b ð0Þ ¼ 2=2, as the indirect quantum interference inhibits the emission of x components of the dipoles and enhances the emission of z components of the dipoles, jC a ðtÞj2 and jC b ðtÞj2 decrease in exponential form expð−Γ⊥ tÞ. Here, Γ⊥ is equal to 17:4Γa0 , which is much larger than Γa0 , so the atom decays much faster than that in free vacuum. In contrast, if the atomic initial state is

0.0

0

1

2

3

4

5

6

Γa0t

pffiffiffi 2 Fig. 3. Evolution of jC a ðtÞj2 þ jC bpðtÞj ffiffiffi with C a ð0Þ ¼ C b ð0Þ ¼ 2=2 (dashed curve), C a ð0Þ ¼ −C b ð0Þ ¼ 2=2 (dottedpcurve) of a Zeeman ffiffiffi atom in the LHM cavity and C a ð0Þ ¼ C b ð0Þ ¼ 2=2 (solid curve) in free vacuum.

pffiffiffi C a ð0Þ ¼ −C b ð0Þ ¼ 2=2, the π phase difference between C a ð0Þ and C b ð0Þ induces the indirect quantum interference to cancel the emission of z components of the dipoles and accelerates the emission of x components of the dipoles. jC a ðtÞj2 and jC b ðtÞj2 decrease in exponential form expð−Γ∥ tÞ, while Γ∥ is equal to 0:17Γa0 and much smaller than Γa0 , so the atom decays much slower than that in free vacuum. Because of the indirect quantum interference, the population evolution of the upper levels displays different behavior compared with that in free vacuum. The decay time is prolonged or reduced, which is determined by the atomic initial state. The indirect quantum interference and the initial atomic state also influence the spontaneous emission spectrum. In the following discussion, the spectra at the position r ¼ ð−λ0 ; 0; 0Þ are considered. By preparing the atom initially in state C a ð0Þ ¼ 1, C b ð0Þ ¼ 0, we get s1 ¼ −Γ∥ =2, s2 ¼ −Γ⊥ =2, C 1 ¼ 1=2, and C 2 ¼ 1=2. From Eq. (17), the spontaneous emission spectrum of Zeeman atom can be rewritten as Sðr; ωÞ ¼

 ez ω4 μ20 d2 ImGðr; r ; ωÞ · A 2 ω − ωa þ i Γ2⊥ 2π  iex 2 þ Γ∥ : ω − ωa þ i 2

ð18Þ

This means the spectrum can be divided into two parts: the spectrum of the x-polarized field with width Γ∥ and the z-polarized field with width Γ⊥ . In Fig. 4, we plot the spontaneous emission spectrum as function of ω − δ ¼ ω − ðωa þ ωb Þ=2. Because of Γ∥ (Γ⊥ ) much smaller (larger) than the spontaneous decay rate in free vacuum, the emission spectrum of the x (z)-polarized field is narrower (flatter) compared with that in free vacuum. The total spectrum is sharp on the top part near ω − δ ¼ 0 and broad in other regions. Compared with the population decay process, the faster population decay during the starting time period contributes to the flat part of the spontaneous emission spectrum, and the slower decay after a certain time leads to one peak in the top spectrum. The behavior of the spontaneous emission spectrum is quite different from that in free space. If the atom

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Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B

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18 16

10 14 12

S(r,ω)

S(r,ω)

8 6

10 8 6

4

4

2

2

-10

-5

0

5

10

ω -δ

Fig. 4. Spontaneous emission spectrum with ωab ¼ 0 and C a ð0Þ ¼ 1, C b ð0Þ ¼ 0.

is in an isotropic environment, there is no quantum interference, the spontaneous emission from the Zeeman atom (initially in one upper level) is the same as that from a two-level atom, and the corresponding spectrum is Lorentzian, so the ultranarrow spectrum comes from the indirect quantum interference. pffiffiffi If the atom is in the state C a ð0Þ ¼ C b ð0Þ ¼ 2=2 and pffiffiffi C a ð0Þ ¼ −C b ð0Þ ¼ 2=2 initially, we can get s1 ¼ −Γ∥ =2, pffiffiffi s2 ¼ −Γ⊥ =2 and C 1 ¼ 0, C 2 ¼ 2=2 for C a ð0Þ ¼ C b ð0Þ ¼ pffiffiffi pffiffiffi pffiffiffi 2=2 (or C 1 ¼ 2=2, C 2 ¼ 0 for C a ð0Þ ¼ −C b ð0Þ ¼ 2=2). The spontaneous emission spectrum corresponding to different atomic initial states can be rewritten as 2 ez ω4 μ20 d2 ImGðr; r ; ωÞ · A Γ ⊥ 2 ω − ω þ i π a 2 pffiffiffi ¼ C b ð0Þ ¼ 2=2;

Sðr; ωÞ ¼

2 ex ω4 μ20 d2 ImGðr; r ; ωÞ · A Γj∥ 2 ω − ωa þ i 2 π pffiffiffi ¼ −C b ð0Þ ¼ 2=2:

Sðr; ωÞ ¼

0 -15

15

for C a ð0Þ ð19aÞ for C a ð0Þ ð19bÞ

This shows that the emission spectrum stems frompffiffithe ffi z-polarized field with width Γ⊥ for C a ð0Þ ¼ C b ð0Þ ¼ 2=2 (or pffiffiffi the x-polarized field with width Γ∥ for C a ð0Þ ¼ −C b ð0Þ ¼ 2=2). The two transition dipoles from upper p levels jai and ffiffiffi jbi to lower level jci are d ¼ dðe þ ie Þ= 2 and a z x pffiffiffi pffiffiffi db ¼ dðez − iex Þ= 2. In the case of C a ð0Þ ¼ C b ð0Þ ¼ 2=2, the phase difference between two x-polarized components of the field emitted from two transition dipoles is π and the amplitude of the x-polarized field is canceled to zero. Corresponding to the z-polarized component of the field, the phase difference is zero and the amplitude of the z-polarized field is doubled. As a result, only the z-polarized component remains in the emitted field. if the atomic initial state is pffiffiConversely, ffi C a ð0Þ ¼ −C b ð0Þ ¼ 2=2, there exists an additional phase difference π between the two levels. The phases of two x-polarized components of the field are the same, and the amplitude of the x-polarized field is enhanced. The phase difference between the two z-polarized components of the field is π and the amplitude of the z-polarized field is reduced to

-10

-5

0

10

5

15

ω -δ

Fig. 5. p Spontaneous emission spectrum with ωab ¼ 0 and ffiffiffi pffiffiffi C a ð0Þ ¼ C b ð0Þ ¼ 2=2 (dashed curve) and C a ð0Þ ¼ −C b ð0Þ ¼ 2=2 (solid curve).

zero, so only the x-polarized component occurs in the emitted field (see Fig. 5). From Figs. 4 and 5, it is seen that the indirect quantum interference could affect the spontaneous emission spectrum, and the spectral narrowing or broadening is determined by the atomic initial state. B. Case of Large ωab Let us consider the case of a large detuning between two upper levels, i.e., ωab ≥ Γa0 . In Fig. 6, we plot the population evolution for different detuning ωab . By preparing the atom in upper level jai initially, the decay rate of the population in level jai increases as the detuning ωab increases. Because of indirect quantum interference, the population in the lower level jci can jump in part to the other upper level jbi. As the detuning between the reflected field and the resonant frequency ωb increases, the population coming back to the upper level jbi decreases. If the parameters of the system, γ a , γ b , κa , κb , and ωab , satisfy the relations γ a − γ b ≠ 0 and ωab ≠ 0 or γ a − γ b ¼ 0 and ω2ab ≥ κ a κb , the indirect quantum interference leads to the population exchanging between the two upper levels, and the population evolution displays an oscillatory behavior during the decay process (see Fig. 7). 1.0 2

|Ca(t)| (ωab=2Γa0 )

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Γ0 t Fig. 6. Evolution of jC a ðtÞj2 (solid curve), jC b ðtÞj2 (dotted curve) with ωab ¼ 2Γa0 , and jC a ðtÞj2 (dashed curve), jC b ðtÞj2 (dashed–dotted curve) with ωab ¼ 8Γa0 .

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Γa0t Fig. 7. Population evolution of jC b ðtÞj2 with C a ð0Þ ¼ 1, C b ð0Þ ¼ 0, and ωab ¼ 20Γa0 .

In Fig. 8, we plot the spontaneous emission spectrum with initial state C a ð0Þ ¼ 1, C b ð0Þ ¼ 0 and different ωab . The emission spectrum takes a Fano-type form [23] characterized by a Fano minimum corresponding to the transition frequency ωb . It is different from the case of quantum interference between two parallel dipoles where the Fano minimum is replaced by a completely dark point [22,24]. The Fano minimum cannot be completely dark, mainly because the indirect quantum interference here is incomplete for the dissipation of LHMs. This Fano minimum is lower as the frequency difference ωab increases. The spectrum at the peak close to ωa is narrow, whereas the spectrum at the other peak is flat. In Fig. 9, we p plot ffiffiffi the spectrum of the atom initially in C a ð0Þ ¼ C b ð0Þ ¼ 2=2 with different ωab . If the atom is in isotropic surroundings, there will be no indirect quantum interference between the two orthogonal dipoles, and the spectrum will be a combination of emission from two twolevel atoms. Consequently, the two peaks corresponding to the two transition dipoles in the emission spectrum could not separate until the detuning of the two upper levels is larger than maxðγ a =2; γ b =2Þ. In the present structure, there is indirect quantum interference between the two transitions with orthogonal dipoles, and the emission spectrum is not a simple

0 -15

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pffiffiffi Fig. 9. Spontaneous emission spectrum with C a ð0Þ ¼ C b ð0Þ ¼ 2=2 and ωab ¼ 1Γa0 (solid curve), 3Γa0 (dashed–dotted curve), and 6Γa0 (dotted curve).

combination of p emission from two two-level atoms. For ffiffiffi C a ð0Þ ¼ C b ð0Þ ¼ 2=2, the spectra has two peaks and there is a darklike line at the transition frequency from the middle point of two upper levels to the lower level, which is due to a destructive interference (See Fig. 9). Even if ωab (such as Γa0 , 3Γa0 ) is much smaller than γ a;b =2 (γ a ≈ γ b ≈ 8:7Γa0 ), the two peaks in the emission spectrum will also separate, and the spontaneous spectrum near two pffiffiffi peaks could be extremely narrow. For C a ð0Þ ¼ −C b ð0Þ ¼ 2=2, the destructive interference will merge the two peaks to one. In the above discussion, the angle θ between the dipole da and the z axis is π=4. If we change the angle θ from 0 to π=2, the strength of quantum interference increases from zero to a maximal value in the region 0 ≤ θ ≤ π=4, and then decreases to zero in the region π=4 ≤ θ ≤ π=2 (See Fig. 10). In Fig. 11, we plot the spontaneous emission spectrum with different directions of dipole da . The ratio of the decay rate γ a =γ b decreases with θ. For example, if θ ¼ π=6; π=4; π=3, we can obtain γ a =γ b ¼ 2:92; 1:0; 0:34, respectively, and the strengths of the quantum interference are nearly the same (See Fig. 10). A darklike line also occurs at the frequency ωb . The Fano minimum becomes smaller as γ a =γ b increases, and the frequency of the right peak is farther away from ωa for bigger θ.

80 70 60

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Fig. 8. Spontaneous emission spectrum with C a ð0Þ ¼ 1, C b ð0Þ ¼ 0, and ωab ¼ 2Γa0 (dotted curve), 4Γa0 (dashed–dotted curve), 8Γa0 (solid curve).

0.0 0.0

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Fig. 10. Strength of quantum interference of a Zeeman atom in the LHM cavity varying with θ.

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Fig. 11. Spontaneous emission spectrum with C a ð0Þ ¼ 1, C b ð0Þ ¼ 0, ωab ¼ 2Γa0 , and θ ¼ π=6 (dotted curve), π=4 (solid curve), and π=3 (dashed–dotted curve).

4. SUMMARY In this paper, we consider a V-type Zeeman atom with two orthogonal transition dipoles that is located in the focal plane of LHM lenses. The population evolution and spontaneous emission spectrum are investigated in detail. Because of the properties of refocusing and phase compensation of LHMs, there exists strong indirect quantum interference between the two orthogonal dipoles. If the atom is in state C a ð0Þ ¼ 1, C b ð0Þ ¼ 0 initially, the total population decay could be faster during the starting period and slower after a certain period. The population exchange between the two upper levels occurs due to existence of the indirect quantum interference and can display an oscillatory behavior for large frequency difference ωab . There is a sharp peak close to ωa or the Fano minimum at ωb in the spontaneous spectrum. When the atom is initially in the coherent pffiffiffi superposition of the two pupper ffiffiffi levels C a ð0Þ ¼ C b ð0Þ ¼ 2=2 (or C a ð0Þ ¼ −C b ð0Þ ¼ 2=2), the population decay becomes faster (or slower) than that in free vacuum, the spontaneous emission spectra are narrowed (or broadened), and there could be a darklike line. The properties of the spontaneous emission spectrum are different from the cases of two parallel dipoles or two orthogonal dipoles in an isotropic environment.

ACKNOWLEDGMENTS This work is supported in part by the National Natural Science Foundation of China (NSFC) (Nos. 91021012, 10904113), the Foundation of the Ministry of Science and Technology (Nos. 2007CB13201 and 2011CB922203).

REFERENCES 1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Usp. 10, 509–514 (1968).

7. 8.

9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

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D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handedmedia simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett. 89, 183901–183904 (2002). A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002). C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401–107405 (2003). L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys. Rev. B 66, 155411–155427 (2002). M. L. Povinelli, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Toward photonic-crystal metamaterials: creating magnetic emitters in photonic crystals,” Appl. Phys. Lett. 82, 1069–1071 (2003). D. R. Smith and N. Kroll, “Negative refractive index in lefthanded materials,” Phys. Rev. Lett. 85, 2933–2936 (2000). J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405–077408 (2003). R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009). J. Kästel and M. Fleischhauer, “Suppression of spontaneous emission and superradiance over macroscopic distance in media with negative refraction,” Phys. Rev. A 71, 011804–011807 (2005). S. E. Harris, J. E. Field, and A. Imamoglu, “Nolinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990). P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77, 3995–3998 (1996). P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a three-level system,” J. Phys. B 15, 561–573 (1982). M. O. Scully and S. Y. Zhu, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989). G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84, 5500–5503 (2000). Y. P. Yang, J. P. Xu, H. Chen, and S. Y. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100, 043601–043604 (2008). J. P. Xu and Y. P. Yang, “Quantum interference of V-type threelevel atom in structures made of left-handed materials and mirrors,” Phys. Rev. A 81, 013816–013823 (2010). H. T. Dung, S. Y. Buhmann, L. Knöll, and D. G. Welsch, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68, 043816–043830 (2003). S. Y. Zhu, R. F. Chan, and C. P. Lee, “Spontaneous emission from a three-level atom,” Phys. Rev. A 52, 710–716 (1995). U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76, 388–391 (1996).