Quantum Light on Demand

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May 22, 2018 - intensities. The presence of the coupling Ωij between the ... lated by a photon emitted at Rj and detected at r. .... transition frequency can be seen for both concurrence ... and observation angle, superradiant light is emitted for.

Quantum Light on Demand Mithilesh K. Parit,1, ∗ Shaik Ahmed,2 Sourabh Singh,1 P. Anantha Lakshmi,3, † and Prasanta K. Panigrahi1 1

arXiv:1805.08642v1 [quant-ph] 22 May 2018

Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India 2 Department of Humanities and Science , MLR Institute of Technology, Dundigal, Hyderabad-500043, Telangana, India 3 School of Physics, University of Hyderabad, Hyderabad - 500046, India (Dated: May 23, 2018) The far field radiation pattern of three, dipole coupled, two level atoms is shown to yield sub and super radiant behavior, with the nature of light quanta controlled by the underlying quantum correlations. Superradiance is found to faithfully reflect the monogamy of quantum correlation and is robust against thermal effects. It persists at finite temperature with reduced intensity, even in the absence of entanglement but with non-zero quantum discord. The intensity of emitted radiation is highly focused and anisotropic in one phase and completely uniform in another, with the two phases separated by a cross over. Radiation intensity is shown to exhibit periodic variation from super to sub-radiant behavior, as a function of inter atomic spacing and observation angle, which persists up to a significantly high temperature. The precise effects of transition frequency and inter-dipole spacing on the angular spread and variations of the intensity in the uniform and non-uniform regimes are explicitly demonstrated at finite temperature. Photon-photon correlation is shown to exhibit sub and super Poissonian statistics in a parametrically controlled manner. keywords: Two-level atoms, supperradiance, Quantum Discord, Entanglement PACS numbers: 03.65.Ud , 03.67.-a, 42.50.Ar

I.

INTRODUCTION

Dicke superradiance is the coherent spontaneous emission, from a many-body system, owing its origin to the co-operative simultaneous interaction among its constituents, experiencing a common radiation field [1]. The collective behaviour of the ensemble arises from the coherent superposition and entanglement structure of the many-body wave function. The correlated structure can also show subradiant behavior due to destructive interference effect of the superposition state. It is interesting to note that some of the excited and ground states in the original study of Dicke are highly entangled [2]. In contrast, ordinary spontaneous radiation (OSR) arises from addition of incoherent radiation intensities of different constituents with random phases. Further, OSR follows exponential decay, whereas superradiance displays substantial deviation from the same. Superradiance has been extensively studied in the literature, with the identification of a phase transition, separating the coherent phase of radiation from the incoherent one [3–5]. It has attracted significant interest due to its possible applications, ranging from generation of X-ray lasers with high powers [6], short pulse generation [7] to self-phasing in a system of classical oscillators [8], to name a few. Super and subradiance have been investigated experimentally in many physical systems [9–14]. Dimitrova et al. [12] observed superradiance in Bose-Einstein condensate (BEC) in small Rayleigh

∗ †

[email protected], [email protected] [email protected], [email protected]

scattering limit. Superfluidity of BEC along the axis of ring cavity has been shown to yield supperradiant scattered photon [14]. In the context of quantum information, it is of particular interest to explore how the behavior of the radiation field gets affected for a collection of atoms, when the states are correlated quantum mechanically in different ways. This will allow optical probe of quantum correlations (QCs) and help quantify QCs present in the system. It is well understood that depending on the nature of interactions of the multi-particle system, one can realize different types of entangled states [15, 16] leading to different radiation characteristics. The atomic entangled states can find potential applications in quantum information processing [17] generating different entangled quantum states of light for quantum memories [18, 19], quantum communication [20], quantum cryptography [21, 22], to name a few. Two particle entanglement has been well characterized both for pure and mixed states [23], using von Neumann entropy and concurrence with Bell states [24] being the maximally entangled states. Recently, concurrence [25] and quantum discord [26] have been used for quantitatively characterizing entanglement governing the quantum phase transition, occurring in an antiferromagnetic spin chain consisting of weakly coupled dimers. In comparison, the three particle entanglement is much less understood. It is known to exhibit stronger QC as compared to the Bell states and shows stronger non-locality. The entanglement structure of multiparticle states of different type are yet to be completely understood.

2

Recently, Wiegner et al. [2] have investigated the super and sub-radiant characteristics of an N-atom system 1 in a generalized W-states of the form √ |j, n − j >, n with j atoms in the excited state and (n − j) atoms in the ground states where the role of entanglement has been highlighted for pure states. The effect of quantum discord on super and subradiant radiation intensities in a system of X-type quantum states has also been studied [27], without taking into account finite temperature effect. Here, we carry out a systematic investigation of the sub and super radiance properties of three correlated two level atoms and explore the effects of inter atomic distance on radiation pattern considering separations for both smaller and larger than the emission wavelength. The precise effect of transition frequency and inter-dipole spacing on the nature of intensity and angular spread are explicitly demonstrated. The behavior of radiation field pattern as a function concurrence and quantum discord is probed for a physical understanding of the effect of different QCs on the emitted light. The role of QCs in producing highly collimated light as well as completely uniform illumination is illustrated. This is due to the occurrence of in two different phases of the three particle particle system, separated by a smooth cross over. The connection of entanglement on far field radiation pattern is demonstrated for two different configurations originating from the two topologically different arrangements of atoms differing in their coupling pattern and phase structure. The effect of various system parameters on the nature of radiation in the far field domain are systematically investigated. The photon-photon correlation shown to yield sub and super Poissonian characteristics as a function of system parameters. The paper is organized as follows. In Sec. II, the Hamiltonian for the system of three identical twolevel atoms interacting via dipole-dipole coupling is introduced using pseudo spin variables. This makes transparent the origin of varying QCs in different eigenstates. We present exact results at non-zero temperature for the intensity of the emitted radiation from the three two-level atoms arranged in line configuration in Sec. III. This is followed by the investigation of radiation intensity from the loop configuration with atoms placed on vertices of equilateral triangle in Sec. IV. Finally, we conclude after summarizing the obtained results and direction for future research.

II.

THEORY AND MODEL

We consider a system of three coupled identical atoms, where the excited state |ei i and the ground state |gi i, i = 1, 2, 3 are separated by an energy interval ~ω (~ = 1

in our case). The Hamiltonian for the system of three identical two-level atoms coupled through dipole-dipole interaction is given by

H=

3 3 X X ωi Siz + Ωij Si+ Sj− . i=1

(1)

i6=j=1

The first term describes the unperturbed energy of the system with the second one representing the dipole dipole interaction between the atoms, where Ωij , the dipole dipole interaction strength, is a function of the inter-atomic separation ‘d’. In the above, Si+ = |1ii h0| and Si− = |0ii h1| are the raising and lowering operators of the ith atom in the spin representation. Our system is closed and non-interactive with environment. In future, we intend to study the open system dynamics. The Hamiltonian of open system dynamics is given in [28, 29]. The two possible inequivalent configurations namely, open loop (line) and closed loop (loop) exist, for which the nature of entangled states are different. The coupling between atoms of two configurations are different giving rise to distinct differences in the resulting field intensities. The presence of the coupling Ωij between the atoms causes mixing of the energy levels leading to the creation of states with different correlations. Depending on the number of atoms that are in excited state, two distinct states are obtained in one configuration, only one atom is in the excited state and two atoms are in the excited state.

(a)

(b)

FIG. 1. (Color online) Schematic diagram of the system in the line and loop configurations with identical two-level atoms lo¯ 1 to R ¯ 3 and (b) atoms at the vertices calized at positions (a) R of an equilateral triangle. A detector is placed at position r¯ to record the photons emitted by the atoms in the far field regime.

We investigate the intensity emitted by three atom system in the far field zone i. e., ~r >> d, here d is spacing between the atoms. The detector is placed at position ~r and the positive frequency component of the electric field operator is considered. The electric field operator is

3 given by

III.

ˆ (+) = − e E

ikr

r

X

~n × (~n × p~ge )e−iφj Sˆj−

(2)

j

where ~r indicates the detector position, the unit vector ~n = ~rr and p~ge the dipole moment of the transition |ei → |gi. Here φj is the relative optical phase accumu~ j and detected at ~r. lated by a photon emitted at R We also assume p~ge to be oriented along the y direction and ~n in the x-z plane, resulting in vanishing p~ge .~n. These assumptions after normalization give rise to dimensionless expressions for the amplitude ˆ (+) = E

X

e−iφj Sˆj− ,

THE INTENSITY CHARACTERISTICS OF THE LINE-CONFIGURATION

(3)

In the line configuration, a system of three identical dipole coupled two-level atoms are placed symmetrically along a line. For simplicity, we consider all transition frequencies of three atoms to be the same, ω1 = ω2 = ω3 = ω and the nearest neighbour dipoledipole interactions Ω12 = Ω23 = Ω and Ω13 = 0. As depicted in Fig. 1(a) atoms are localized at positions R1 , R2 to R3 , with equal spacing d between adjacent atoms. For this topology the phase φj the relative optical ~ j and dephase accumulated by a photon emitted at R tected at ~r is ~ j = jkd sin θ. φj (~r) ≡ φj = k~n.R

j

resulting in the following expression for the radiated intensity at ~r

(7)

The exact expression of intensity for three atoms arranged along a line is derived by by combining Eqs. (4) to (7) and given by

E X D ˆ (−) E ˆ (+) = hSˆi+ Sˆj− iei(φi −φj ) I(~r) = E i,j

=

X

I = A (B + C + D + E + F ), with

 X hSˆi+ Sˆi− i +  hSˆi+ ihSˆj− i+

i

A=

i6=j

 X (hSˆi+ Sˆj− i − hSˆi+ ihSˆj− i) ei(φi −φj )

(4)

 ω ω e− 2T sech 2T  √  ,  2 1 + 2 cosh Tω + 8 cosh T2Ω 



i6=j

Thus, the characteristics of the intensity would depend on the incoherent terms hSˆi+ Sˆi− i, the non vanishing of the dipole moments hSˆi+ i and the QCs like hSˆi+ Sˆj− i − hSˆi+ ihSˆj− i. We take into account of the thermal effects, where at finite D E temperature, the expectation value of an observable Aˆ takes the form

ω

B = 3e T − 2e T (−2 + cos(2kdsin(θ))), √ ! 2Ω , C = 4 (2 + cos(2kdsin(θ))) cosh T √ ω T

 D = 4 e (4 + cos(2kdsin(θ))) cosh

F = −8 2 1 + e (5)

with ρˆ being the thermal density matrix operator

ρˆ =

P8 −βi i=1 |ψi i hψi | e P . 8 −βi |ψ i hψ | e Tr i i i=1

(6)

Here |ψi i is an eigenstate with i its eigenvalue. The |ψi i and i along with ρˆ is given in supplementary material for both the configurations. In the ensuing sections, we investigate the intensity pattern of the loop and line configurations as a function of the system parameters as well as inter atomic distance and observation angle.

! ,

E = 4 sin2 (kdsin(θ)), and √

D E   Aˆ = Tr ρˆAˆ ,

2Ω T

√ ω T



cos(kdsin(θ)) sinh

2Ω T

! . (8)

In an earlier work, the role of entanglement on super and subradiance behavior for the three atom system with a zero net dipole moment was shown [2]. Here, we have generalized and exhibited the presence of QCs for a three atom system. The angular intensity distribution at two different temperatures is shown in Fig. 2 for the line configuration. The periodic variation in the intensity from super to subradiant behavior is observed as a function of transition frequency and the detector angle as can be seen in Fig. 8. This reflects the subtle interference effect in the three particle system. We now probe the the role of entanglement through concurrence and QCs through quantum discord.

4 high temperatures albeit with reduced intensity. The three dimensional plot of the intensity as a function of inter atomic distance and transition frequency for two different temperatures and fixed detector angle is shown in Fig. 8. The three dimensional plot of the intensity as a function of observation angle and transition frequency for two different temperatures and fixed inter-atomic spacing are shown in Fig. 8. The blue region in the plot represents sub radiant behavior, while red regions corresponding to super radiant behavior. The variation of the inter atomic distance leads to changes in the intensity pattern. Thus, providing us yet another tool to control the far field radiation pattern. Fig. 8 clearly shows that the spacing between atoms and observation angle play a significant role for the detection of superradiant light. The intensity is maximum at inter-dipole spacing d = (2n + 1) λ2 only when observation angle θ = π2 . The intensity pattern for different combination of θ and d is shown in Fig. 9 (including θ = ± π2 and d = (2n+1) λ2 with n = 0, 1, 2, . . .). FIG. 2. (Color online) Panel a and b show the intensity variation at fixed temperature and inter-atomic spacing (d = λ2 ) as a function of observation angle and transition frequency for (a) T=0.005 and (b) T=1. Panel c and d show the intensity variation at fixed temperature and observation angle as a function of inter-atomic spacing and transition frequency for (c) T=0.005 and (d) T=1.

To understand the intensity profile for the given system it is imperative to know the variation of QCs with temperature and transition frequency. In Fig. 3, panel a and b show the behavior of concurrence [30] and quantum discord [31–33] as a function of temperature for two different values of the transition frequency ω. For small transition frequency values, increasing temperature leads to reduction in the value of both concurrence and discord, with concurrence vanishes for T = 1 but discord remains non-zero. Thus, the intensity pattern at small T and small values of ω is dominant due to the QCs present in the system. The result also confirms that even for T > 1, the superradiant behavior is present albeit with reduced intensity displaying the role of quantum discord. The variation of QCs with transition frequency is shown in Fig. 3. The sharp fall in QCs with increasing transition frequency can be seen for both concurrence and quantum discord. Thus, QCs can be precisely controlled by properly tuning the temperature and the transition frequency. The magnitude of intensity decreases with increasing temperature which is due to the expected reduction in the QCs. Fig. 3 shows that at high temperatures entanglement (concurrence as a measure of entanglement) vanishes but not the quantum discord. The non-zero quantum discord reveals that some QCs are present leading to sub and super radiant behavior of the intensity at

FIG. 3. (Color online) QCs (Concurrence (blue) and Discord (red)) for the bipartite system AB. Panel a and b show the variation of QCs as function of temperature for different transition frequencies. Panel c and d show the variation of QCs as function of transition frequency for different temperatures. Expectedly, at high temperatures entanglement vanishes with non-zero quantum discord.

Fig. 9 depicts that the super and subradiant nature of radiation can be found at all inter-dipole distances and observation angles (except θ = nπ). Which shows the variation of intensity as a function of observation angle and spacing between atoms at fixed transition frequency (ω = 1) for different temperatures. Fig. 9(a) depicts the behavior of sub and superradiant radiation

5 due to high QCs present at “ω = 1 and T = 0.005” (see Fig. 3). Fig. 9(b) shows the subradiant behavior due to low QCs at “ω = 1 and T = 1” (see Fig. 3). This result is important because it reveals the property that for a particular combination of spacing between atoms and observation angle, superradiant light is emitted for quantum mechanically correlated systems.

FIG. 4. (Color online) The variation of intensity is shown as a function of observation angle and spacing between atoms at fixed transition frequency (ω = 1) for different temperatures of magnitude (a) T = 0.005 and (b) T = 1 is shown here.

correlation is presented in supplementary material.

IV.

THE INTENSITY CHARACTERISTICS OF THE LOOP-CONFIGURATION

The far-field intensity pattern corresponding to the loop configuration shows entirely different behaviour from that of the line-configuration discussed in the previous section. In the loop configuration, the schematic of which is shown in Fig. 1(b), the relative optical phase ~ j and detected at accumulated by a photon emitted at R ~r is given by ~ 1 = kd sin θ, φ1 (~r) ≡ φ1 = k~n.R

(9)

~ 2 = 2kd sin θ φ2 (~r) ≡ φ2 = k~n.R

(10)

~3 = φ3 (~r) ≡ φ3 = k~n.R

√ 3kd sin θ + 3kd cos θ . 2

(11)

The contour plot of intensity as function of QCs is depicted in Fig. 5 for T = 0.005. Panel a and b show relation of intensity with monogamy score [34, 35] of negativity (τ1:23 ). Negativity is an another measure of entanglement [36]. Panel c and d show the variation of intensity with quantum discord (QD). It is evident from the Fig. 5 that the superradiance faithfully reflect the property of monogamy.

FIG. 6. (Color online) Panel a and b show the intensity variation at fixed temperature and inter-atomic spacing (d = λ2 ) as a function of observation angle and transition frequency for (a) T=0.005 and (b) T=1. Panel c and d show the behavior of intensity at fixed temperature and observation angle as a function of inter-atomic spacing and transition frequency for (c) T=0.005 and (d) T=1.

FIG. 5. (Color online) Panel a and b show intensity variation with respect to monogamy score of negativity while panel c and d depict the radiation intensity with quantum discord at T = 0.005.

The plots and analytical expression of photon-photon

The intensity profile for the loop configuration is shown in Fig. 10 for two different temperatures. As evident from the figure the intensity emitted is nearly independent of the observation angle. This implies that the radiation characteristic is nearly isotropic. An interesting jump in the intensity can be seen for the lower temperature (see Fig. 10(a)) which is analogous to a quantum

6 switch. These quantum switches have important consequences in quantum computing, in the implementation of quantum gates. This switch like features disappear with increasing temperature due to thermal smearing as seen in Fig. 10(b).

V.

CONCLUSION

In conclusion, we have obtained the exact expression of the resulting intensity in the far field domain for three atoms placed along a line. The analytical expression of photon-photon correlation is also derived and given in supplementary material. The effect of QCs on three dipole coupled atomic system is shown to yield sub and super-radiant behavior of light controlled by transition frequency, inter-atomic distance, temperature, dipole coupling, and observation angle. The radiative behavior shows dramatic variation as a function of concurrence, quantum discord, and monogamy score of negativity revealing the roles of distinct QCs, thereby providing

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an optical probe for study of quantum characteristics of the emitting sources. The shareability of QCs in a multiparty system is recorded through monogamy, affect the radiation intensity. Thus, superradiance can find the secure communication and multiparty quantum protocols. Extending the study from two to three atoms, has opened up a host of exciting newer possibilities. It will be exciting to explore further extension of this analysis such as hyperradiance and radiation intensity of open system dynamics.

VI.

ACKNOWLEDGEMENTS

M. K. Parit acknowledges Dr. Chiranjib Mitra for discussion and Department of Science and Technology, New Delhi, India for providing the DST-INSPIRE fellowship during his stay at IISER Kolkata.

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7 SUPPLEMENTARY INFORMATION: QUANTUM LIGHT ON DEMAND A.

The model

We consider a system of three coupled identical atoms, where the excited state |ei i and the ground state |gi i, i = 1, 2, 3 are separated by an energy interval ~ω (~ = 1 in our case). The Hamiltonian for the system of three identical two-level atoms coupled through dipole-dipole interaction is given by

H=

3 3 X X ωi Siz + Ωij Si+ Sj− . i=1

i6=j=1

The schematic representation of the two configurations shown below.

FIG. 7. (Color online) Schematic representation of Line(open loop) and Loop (closed loop) configurations for three qubits coupled via dipole-dipole interaction.

In the next sections, we have calculated the eigenvalues of the Hamiltonian and their corresponding eigenstates for two configurations. The analytical expression of intensity and photon-photon correlation of three atoms arranged in the line configuration is given.

B.

Photon-photon correlation in line configuration

At thermal equilibrium, the quantum state of a three atom system is a weighted superposition of all the eigenstates. By diagonalizing the Hamiltonian H (Ω12 = Ω23 = Ω and Ω13 = 0), we can obtain all the eigenvalues i and their corresponding eigenstates |ψi i. The eigenvalues i , in the line configuration, are √ √ −3ω ω ω ω ; 2 = − 2Ω − ; 3 = − ; 4 = 2Ω − 2 2 2 2 √ √ ω ω ω 3ω 5 = − 2Ω + ; 6 = ; 7 = 2Ω + ; 8 = 2 2 2 2 1 =

(12)

and the corresponding eigenstates |ψi i of the system, are given by i √ 1h |ψ1 i = |g1 g2 g3 i; |ψ2 i = |e1 g2 g3 i − 2|g1 e2 g3 i + |g1 g2 e3 i 2 i i √ 1h 1 h |ψ3 i = √ |g1 g2 e3 i − |e1 g2 g3 i ; |ψ4 i = |e1 g2 g3 i + 2|g1 e2 g3 i + |g1 g2 e3 i 2 2 i i √ 1h 1 h |ψ5 i = |e1 e2 g3 i − 2|e1 g2 e3 i + |g1 e2 e3 i ; |ψ6 i = √ |g1 e2 e3 i − |e1 e2 g3 i 2 2 i √ 1h |ψ7 i = |e1 e2 g3 i + 2|e1 g2 e3 i + |g1 e2 e3 i ; |ψ8 i = |e1 e2 e3 i. 2 (13)

8 Considering the standard basis {|g1 g2 g3 i, |e1 g2 g3 i, |g1 e2 g3 i, |g1 g2 e3 i, |e1 e2 g3 i, |e1 g2 e3 i, |g1 e2 e3 i , |e1 e2 e3 i} of the system, one can obtain the thermal density matrix of the form

 ρ11 0 0 0 0 0 0 0  0 ρ22 ρ23 ρ24 0 0 0 0   0 ρ32 ρ33 ρ34 0 0 0 0 1   0 ρ42 ρ43 ρ44 0 0 0 0 ρABC (T ) =  Z  0 0 0 0 ρ55 ρ56 ρ57 0  0 0 0 0 ρ ρ ρ 0  65 66 67  0 0 0 0 ρ ρ ρ 0 75 76 77 0 0 0 0 0 0 0 ρ88

          

(14)

where the partition function Z is given by

Z=e

3ω 2kB T

+e

−3ω 2kB T

+e

√ ( ω − 2Ω) 2 kB T

+e



√ ( ω + 2Ω) 2 kB T

+e

√ ( ω + 2Ω) 2 kB T

+e



√ ( ω − 2Ω) 2 kB T

ω

−ω

+ e 2kB T + e 2kB T .

The non-vanishing density matrix elements of ρABC (T ) are given by −3ω



ρ11 = e 2kB T ;

ρ88 = e 2kB T

√ √ " ( ω −√2Ω) # # ( ω + 2Ω) ( ω + 2Ω) ω 2 2 2 1 1 kB T 2kB T kB T kB T = e e +e + 2e ; ρ33 = +e 4 2 √ √ √ " " # # ( ω − 2Ω) ( ω + 2Ω) ( ω + 2Ω) 1 − 2k T 1 −( ω −√2Ω)kB T − 2k T − 2kω T − 2k T B B B B e 2 e +e + 2e ; ρ66 = +e = 4 2 √ √ √ √ " ( ω − 2Ω) " ( ω − 2Ω) # # ( ω + 2Ω) ( ω + 2Ω) ω 2 2 2 2 1 1 e kB T = √ e kB T − e kB T ; ρ24 = + e kB T − 2e 2kB T 4 2 2 √ √ " ( ω +√2Ω) # # " ( ω +√2Ω) ω ( − 2Ω) ( ω − 2Ω) 1 1 − 2k T − 2k T − 2k T − ω − 2k T B B B B e = √ e −e ; ρ57 = +e − 2e 2kB T . 4 2 2

"

ρ22 = ρ44

ρ55 = ρ77

ρ23 = ρ34

ρ56 = ρ67

√ ( ω − 2Ω) 2 kB T

(15)

From the above description, one observes that the system at high temperature is perfectly separable. However, for intermediate temperatures, the system is in a mixed state and we have investigated the intensity pattern and photon-photon correlation of such a system. We have used kB = 1 in our calculation. In the line configuration intensity (I = hE − E + i) is given by

I = hE − E + i = A (B + C + D), with  ω ω e− 2T sech 2T  √  , A=  2 1 + 2 cosh Tω + 8 cosh T2Ω 



ω

B = 3e T −2e T (−2+cos(2kdsin(θ))), √

! 2Ω C = 4 2 + cos(2kdsin(θ) + e (4 + cos(2kdsin(θ)))) cosh , and T √ ! √ ω  2Ω D = 4 sin2 (kdsin(θ)) − 8 2 1 + e T cos(kdsin(θ)) sinh . T ω T



(16)

9 The term hE − E − E + E + i is given by hE − E − E + E + i = N1 (N2 + N3 ), with e2kdsin(θ)+

N1 =



2 1 + 2 cosh

√ ω− 2Ω 2T

ω T



sech

+ 8 cosh

ω 2T √

 2Ω T

 ,

√  √  2 2Ω cos(kdsin(θ))+2 (2+cos(2 kdsin(θ))), N2 = −4 2 −1 + e T

N3 = e

√ 2 2Ω T







and



4 + 3e T + 2cos(2 kdsin(θ)) +4e

2Ω T

sin2 (kdsin(θ)). (17)

Now, the photon-photon correlation is given by g (2) (0) =

hE − E − E + E + i hE − E + ihE − E + i

(18)

FIG. 8. (Color online) Panel a and b show the variation of photon-photon correlation at fixed temperature and inter-atomic spacing (d = λ2 ) as a function of observation angle and transition frequency for (a) T=0.005 and (b) T=1. Panel c and d show the intensity variation at fixed temperature and observation angle as a function of inter-atomic spacing and transition frequency for (c) T=0.005 and (d) T=1.

The Figures (8) and (9) show the photon statistics (g 2 (0)) of the three atoms placed along the line. The value of g (0) < 1 shows sub Poissonian bahavior and g 2 (0) > 1 shows super Poissonian bahavior. 2

10

FIG. 9. (Color online) The variation of photon-photon correlation as a function of observation angle and spacing between atoms at fixed transition frequency (ω = 1) for different temperatures of magnitude (a) T = 0.005 and (b) T = 1 is shown here.

C.

Photon-photon correlation in loop configuration

In this section, we present the eigenvalues, eigenstates, and non-vanishing terms of thermal density matrix of dipole coupled two-level atoms, placed at the vertices of the equilateral triangle. The eigenvalues of the Hamiltonian with Ω12 = Ω23 = Ω13 = Ω are given by 1 =

−3ω ω ω ω ; 2 = −Ω − ; 3 = −Ω − ; 4 = 2Ω − 2 2 2 2

5 =

ω ω ω 3ω − Ω; 6 = − Ω; 7 = + 2Ω; 8 = 2 2 2 2

and the corresponding eigenstates |φi i of the system are given by i 1 h |φ2 i = √ |g1 e2 g3 i − |e1 g2 g3 i 2 i i 1 h 1 h |φ3 i = √ |g1 g2 e3 i − |e1 g2 g3 i ; |φ4 i = √ |e1 g2 g3 i + |g1 e2 g3 i + |g1 g2 e3 i 2 3 i i 1 h 1 h |φ5 i = √ |e1 g2 e3 i − |e1 e2 g3 i ; |φ6 i = √ |g1 e2 e3 i − |e1 e2 g3 i 2 2 i 1 h |φ7 i = √ |e1 e2 g3 i + |e1 g2 e3 i + |g1 e2 e3 i ; |φ8 i = |e1 e2 e3 i. 3

|φ1 i = |g1 g2 g3 i;

11 One can obtain the temperature dependent density matrix elements and the non-vanishing density matrix elements thus obtained are listed below. −3ω



ρ11 = e 2kB T ; ρ88 = e 2kB T ω ω ω ( ω +Ω) 2 1 ( 2 −2Ω) 1 ( 2 +Ω) 1 ( 2 −2Ω) ρ22 = e kB T + e kB T ; ρ33 = ρ44 = e kB T + e kB T 3 3 2 ω ω ( ω +2Ω) ( ω −Ω) 2 2 1 − ( 2k +2Ω) 1 1 − ( 2 −Ω) − − BT ρ55 = e + e kB T ; ρ66 = ρ77 = e kB T + e kB T 3 3 2 ω ( ω +Ω) ( ω −2Ω) 2 2 1 ( 2k−2Ω) 1 1 ρ23 = ρ24 = e B T − e kB T ; ρ34 = e kB T 3 2 3 ω ω ( ω −Ω) 2 1 − ( 2k +2Ω) 1 1 − ( 2 +2Ω) − BT ρ56 = ρ57 = e − e kB T ; ρ67 = e kB T 3 2 3 (19) The partition function for this system is given by 3ω

−3ω

Z = e 2kB T + e 2kB T + e

( ω −2Ω) 2 kB T

+e



( ω +2Ω) 2 kB T

+ 2e

( ω +Ω) 2 kB T

+ 2e

−( ω −Ω) 2 kB T

.

At high temperature, the density matrix reduces to a mixed state, which can not be written in terms of product states.

FIG. 10. (Color online) Panel a and b show the variation of photon-photon correlation at fixed temperature and inter-atomic spacing (d = λ2 ) as a function of observation angle and transition frequency for (a) T=0.005 and (b) T=1. Panel c and d show the intensity variation at fixed temperature and observation angle as a function of inter-atomic spacing and transition frequency for (c) T=0.005 and (d) T=1.

The Figures (10) and (11) show the photon statistics (g 2 (0)) of the three atoms placed along the line. The value of g (0) < 1 shows sub Poissonian bahavior and g 2 (0) > 1 shows super Poissonian bahavior. 2

12

FIG. 11. (Color online) The variation of photon-photon correlation as a function of observation angle and spacing between atoms at fixed transition frequency (ω = 1) for different temperatures of magnitude (a) T = 0.005 and (b) T = 1 is shown here.