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Journal of Russian Laser Research, Volume 32, Number 3, May, 2011

AUTOCORRELATION FUNCTION FOR A TWO-LEVEL ATOM “WITH COUNTER-ROTATING TERMS”

H. Eleuch,1 Y. V. Rostovtsev,2,3 and M. Sebawe Abdalla4∗ 1 Institute

for Quantum Studies and Department of Physics and Astronomy Texas A&M University College Station, TX 77843, USA 2 Applied

Physics Group, Engineering Quad Princeton University NJ 08544, USA 3 Department

of Physics University of North Texas 1155, Union Circle 311427, Denton, TX 76203-5017, USA 4

College of Science, Mathematics Department King Saud University P.O. Box 2455, Riyadh 11451, Saudi Arabia

∗ Corresponding

author e-mail:

msebaweh @ physics.org

Abstract We study the statistics of the emitted light from a single two-level atom in a cavity. We explore the effects of counter-rotating terms. We show that the virtual processes destroy the antibunching effect in the low-frequency regime.

Keywords: two-level system, autocorrelation function.

1.

Introduction

Undoubtedly the fundamental limits of spectral resolution and sensitivity of spectroscopic techniques, as well as the information transfer and computation rates, besides the spatial resolution of optical microscopy and imaging, are determined by statistical properties of light. For the last five decades, one can see a huge number of research activities, either theoretical or experimental, that were concentrated on studying the fluctuations in the classical and quantum systems [1–15]. The first experimental work by Hanbury Brown and Twiss has shown that the correlation between the intensity fluctuations recorded at two different photodetectors can be illuminated by the same thermal light source [16]. In this experiment, photon bunching, i.e., an enhancement in the intensity–intensity correlations, has been observed. In the meantime, the quantum approach to optical coherence has been introduced in the pioneer work by Roy Glauber [17]. Also the photon antibunching has been predicted in [18]; however, the experimental realization was first reported in the resonance fluorescence [19–21]. Furthermore, the second-order correlations in fluorescence have been calculated for a two-level atomic Manuscript submitted by the authors in English first on April 23, 2011 and in final form on June 10, 2011. c 1071-2836/11/3203-0269 2011 Springer Science+Business Media, Inc.

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system; see, for example, [22–25]. In this context, we refer to the experimental work [26], where the effect of strong coupling was taken into account and it was shown that the strong coupling regime can be attained in a solid-state system. Moreover, the observation of the coherent interaction of a superconducting twolevel system with a single microwave photon (circuit QED) has been reported. More precisely, it has been established that the coherent exchange of energy between a quantized electromagnetic field and a quantum two-level system at a rate g/2π is observable if g is much larger than the decoherence rates γ and κ. This has been achieved at the exact resonance where the field and the two-level system frequencies are equal. To date, most of the theoretical work is restricted to the rotating wave approximation (RWA) where the energy nonconservation terms are neglected. However, it would be interesting to retain the nonrotating terms and to see their effects on the quantum state of radiation. This is the main target of this paper, where we introduce the experimental conditions under which the prediction of our theory can be tested. This will be seen later. The paper is organized as follows. In Sec. 2 we introduce a model of a two-level atom inside a cavity interacting with the environment. In Sec. 3, we present the expression of the autocorrelation function in the RWA. Section 4 is devoted to the study of the effect of virtual processes on the quantum statistics. Finally, our conclusions are given in Sec. 5.

2.

Model

We consider a two-level atom in a perfect cavity excited by a pump laser with frequency ω. We assume that the two levels with the fundamental states |0i (ground state) and |1i (excited state) are interacting resonantly with a single mode of the cavity field. The Hamiltonian that represents such a system in the interaction picture can be written in the form       ˆ I (t) H = ig a ˆ† σ ˆ− ei(ω0 −ω)t − a ˆσ ˆ+ e−i(ω0 −ω)t + ig a ˆ† σ ˆ+ ei(ω0 +ω)t − a ˆσ ˆ− e−i(ω0 +ω)t + i¯ ε a ˆ† − a ˆ . } (1) The first and second terms contain the rotating and counter-rotating interaction Hamiltonians, respectively, and g is the coupling parameter, given by r d d }ω g = − E0 = − , (2) } } 2ε0 V where d is the atomic dipole moment, E0 is the electric field per photon, which contains both ε0 (the vacuum permittivity) and V (the cavity volume), a ˆ† and a ˆ are the creation and annihilation operators   † satisfying the commutation relation a ˆ, a ˆ = 1, while σ ˆ± and σ ˆz are the usual spin up and spin down, and the atomic inversion Pauli’s operators obey the commutation relation [ˆ σ± , σ ˆz ] = ∓ˆ σ±

and

[ˆ σ+ , σ ˆ− ] = 2ˆ σz .

(3)

The last term in Eq. (1) is the external field, where ε¯ is the amplitude of the intra-cavity pump field. The time-dependent exponential terms contain both the field frequency ω and the atomic frequency difference ω0 . In the present work, we restrict our consideration to the resonance case where ω = ω0 .

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As one can see, it is not an easy task to handle such kind of complicated Hamiltonian; therefore, to simplify the matter, we adopt the master equation technique. In this case, we can write the master equation of the density operator for the present system in the form h i h i dˆ ρ =g a ˆ† σ ˆ− − a ˆσ ˆ+ , ρˆ + g a ˆ† σ ˆ+ e2iωt − a ˆσ ˆ− e2iωt , ρˆ dt   γ +¯ ε[ˆ a† − a ˆ, ρˆ] + (2ˆ σ− ρˆσ ˆ+ − σ ˆ+ σ ˆ− ρˆ − ρˆσ ˆ+ σ ˆ− ) + κ 2ˆ aρˆa† − a ˆ† a ˆρˆ − ρˆa ˆ† a ˆ , (4) 2 where γ and κ are respectively the atomic and photonic dissipation rates. The full analytic solution of this equation is too cumbersome and may not be obtained. However, it is noted that, in the weak excitation regime, the terms (2ˆ σ− ρˆσ ˆ+ ) and (2ˆ aρˆa† ) are small compared with the other terms in the master equation. Therefore, it is convenient for us to use this approximation and to neglect these two terms [27]. In this case and after some elementary manipulations of the commutators, the master equation takes the form ! !†  ˆ im ˆ im dˆ ρ H H 1 ˆ ˆ† , = ρˆ + ρˆ = Him ρˆ − ρˆH (5) im dt i} i} i} ˆ im is the nonunitary Hamiltonian operator defined as where H   γ ˆ im H = ε¯(ˆ a† − a ˆ) + g(a+ σ ˆ− − aˆ σ+ ) + g a ˆ† σ ˆ+ e2iωt − a ˆσ ˆ− e−2iωt − σ ˆ+ σ ˆ− − κˆ a† a ˆ. i~ 2

(6)

To determine the time evolution of the state |Ψ(t)i, we have to solve the nonunitary Schr¨ odinger equation d |Ψ(t)i Him |Ψ(t)i = i~ . (7) dt In the case of a weak excitation regime, we can regard the state |Ψ(t)i as a linear superposition of the atomic state and the field state, such that |ψ(t)i = |0, 0i + A10 (t) |1, 0i + A01 (t) |0, 1i + A11 (t) |1, 1i + A20 (t) |2, 0i ,

(8)

where Aij (t), (i = 0, 1, 2; j = 0, 1) are time-dependent functions to be determined. In the state labeled |n, mi, n refer to the photon number and m to the atomic level. It should be noted that, since the effect of the field is to increase the photon number within the cavity, therefore and without loss of generality, we can neglect the term ε¯a ˆ from Eq. (6). This will be taken into consideration when we derive the time evolution of the functions Aij (t). It is well known that for a weak excitation, the states with zero, one, or two quanta of excitation are sufficient to give a full description of the system. The differential equations that describe the time evolution of the functions Aij (t) are given by dA10 = gA01 − κA10 + ε¯, dt dA01 γ = − A01 − gA10 , dt 2 γ  √ dA11 = ε¯A01 − 2gA20 − + κ A11 + ge2iωt , dt 2 √ dA20 √ = 2¯ εA10 − 2κA20 + 2gA11 . dt

(9) (10) (11) (12)

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It is worth pointing out that the differential equations of the amplitudes A10 , A01 , and A20 contain constant coefficients that reflect the rotating wave approximation terms. In the meantime, we observe that Eq. (11) for the amplitude A11 contains the term ge2iωt , which reflects the existence of the counterrotating terms. Since the pump field ε0 is partially transmitted by the cavity, an effective field influences √ the atom (ε = κε0 ). In this case and for the sake of simplicity, we define a dimensionless quantity ε=

ε¯ ε0 =√ . κ κ

(13)

It should be noted that for large values of frequencies (the average ge2iωt → 0), we obtain the same dynamic equation as in the RWA case where the nonresonant terms in this range of frequencies have no effect.

3.

Quantum Statistics for the RWA Case

Now we turn our attention to some statistical properties of the system; more precisely, we discuss the antibunching case. In the case of the RWA and in the stationary regime, we obtain for the first excited state εγκ 2g A10 = 2 = α, A01 = − α. (14) 2g + γκ γ The stationary amplitudes for the two excited states are given by A11 = and

−2gεκ(γ + 2κ)α γ[2g 2 + κ(γ + 2κ)]

(15)

εκ(γ 2 + 2κγ − 4g 2 )α A20 = √ . 2γ(2g 2 + γκ + 2κ 2 )

(16)

hΨ| a ˆ† a ˆ |Ψi = |A10 |2 + |A11 |2 + 2|A20 |2 ≈ α2 .

(17)

The average photon number reads

Now let us determine the autocorrelation function that is defined by

†  a ˆ (t)ˆ a† (t + τ )ˆ a(t + τ )ˆ a(t) (2) g (τ ) = . 2 hˆ a† a ˆi

(18)

Because the photon statistics is linked directly to the autocorrelation function at τ = 0, we have to determine the expression of the autocorrelation function in this delay time. We note that after emission of a photon the reduced state becomes √ a ˆ |Ψi 2 |Φ(0)i = p = |0, 0i + A20 |1, 0i + |0, 1i , (19) † α hΨ| a ˆ a ˆ |Ψi and consequently the autocorrelation function can be expressed as follows: g (2) (0) =

272

2 |A20 |2 . (α)4

(20)

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From Eqs. (14) and (16) together with Eq. (20), after some algebra, we obtain g (2) (0) =

(2g 2 + γκ)2 [−4g 2 + γ(γ + 2κ)]2 . γ 4 [2g 2 + κ(γ + 2κ)]2

(21)

For the case of a single two-level atom, the same expression has also been obtained in [10, 28]. It is interesting to point out that the autocorrelation function shows antibunching whatever the values of g, γ and κ. This means that the antibunching can always be detected as long as the atom and field are correlated. However, the total antibunching occurs for the case where 4g 2 = γ(γ + 2κ) with g (2) (0) = 0. Furthermore, antibunching also exists if the coupling parameter satisfies the inequality  2 2   g g 2 1 γ  2 < + + 1. (22) γκ κ 2 κ For the bad cavity where γ  κ and g  κ, we can define C = g 2 /γκ ∼ 1 to represent the atomic cooperativity can be detected from the above inequality √ coefficient [29,30]. In this case, the1antibunching √ 1 for C < 2 2. However, for the case where C = 2 2, the total antibunching can also be reported.

4.

Quantum Statistics Beyond the RWA

We devote this section to considering the effect of the counter-rotating term on the behavior of the autocorrelation function. In this case, we have to examine the effect of the parameters that are involved in the system. In the presence of the counter-rotating term (beyond the rotating wave approximation), we can write Aij = (Aij )0 + A¯ij , (23) where (Aij )0 is the amplitude for RWA at a steady state. The evolution of the time-dependent amplitude for one quantum excitation can be written as dA¯10 = g A¯01 − κ A¯10 , dt

γ dA¯01 = − A¯01 − g A¯10 . dt 2

(24)

Also, the time evolution of the functions A¯11 (t) and A¯20 (t) is governed by the equations dA¯11 dt dA¯20 dt

γ  √ = − 2g A¯20 − + κ A¯11 + ge2iωt , 2 √ = −2κ A¯20 + 2g A¯11 .

(25)

For A¯ij (0) = 0, we have A¯10 (0) = 0 and A¯01 (0) = 0. The solution of this system is dominated by oscillation with double frequency of the pump; thus, we can write A¯11 = A11p e2iωt ,

A¯20 = A20p e2iωt .

(26)

Therefore, after straightforward calculations, we arrive at A¯11 =

2g 2

+ γκ +

2g(κ + iω) e2iωt + 2i[(γ/2) + 3κ]ω − 4ω 2

2κ 2

(27)

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and

√

2g 2 e2iωt . 2g 2 + γκ + 2κ 2 + 2i[(γ/2) + 3κ)]ω − 4ω 2 The nonstationary autocorrelation function in this case is A¯20 =

2 |A20 (t)|2 g (2) (t, τ = 0) ≈  2 , 2 |A10 (t)|

(28)

(29)

which can be written as g (2) (t, τ = 0) =

(B1 (t) − B2 (t) cos 2ωt − B3 (t) sin 2ωt) , B4 (t)

(30)

where we have used the abbreviations n  2  B1 = (2g 2 + γκ)2 4 2g 2 + κ(γ + 2κ) 16g 8 + 16g 6 γκ − 8g 2 γε4 κ 4 (γ + 2κ) + γ 2 ε4 κ 4 (γ + 2κ)2   2 +4g 4 κ 2 (γ 2 + 4ε4 κ 2 ) − ω 2 ε4 κ 4 (16g 2 − γ 2 − 4γκ − 20κ 2 ) 4g 2 + γ 2 + 2γκ  4  √ 2 + 2εκω 4g 2 − γ(γ + 2κ) , (31) B2 = 16g 2 ε2 κ 2 (2g 2 + γκ)3 [4g 2 − γ(γ + 2κ)][2g 2 + κγ + 2κ 2 ][2g 2 + κγ + 2κ 2 − ω 2 ], 2 2

2

2

3

2

2

2

(32)

2

B3 = 8g ε κ ω(γ + 6κ)(2g + γκ) [4g − γ − 2γκ][2g + κγ + 2κ ], (33)  4 4 4 2 2 2 2 2 2 2 2 4 B4 = γ ε κ [2g + κ(γ + 2κ)] 4[2g + κ(γ + 2κ)] + (γ + 4γκ + 20κ − 16g )ω + 4ω . (34) The average of the autocorrelation function, which corresponds to the measured autocorrelation function, is defined by Z D E B1 1 T (2) (2) g (0) = = g (t, τ = 0) dt, (35) B4 T 0 ω where T = 2π/ω. In Fig. 1, we show the average autocorrelation function against the frequency ω and the normalized parameter κ1 = κ/g for ε = 0.1 as well as for γ = 2κ. In fact, these sets of parameters fit well with the value of the autocorrelation function in RWA, where g (2) (0) = 0. From Fig. 1, we realize that for low values of the frequency ω and large values of κ1 the autocorrelation function shows antibunching. However, with decrease of ω the value g (2) (0) increases, and consequently the antibunching is destroyed by the antiresonant term. For very high frequency, the effect of the antiresonant terms is negligible, and then Fig. 1. Average autocorrelation function total antibunching can be seen. In fact, there is an inter- hg (2) (0)i vs cavity dissipation κ and laser fre1 esting interplay between the rotating, counter-rotating wave quency ω for ε = 0.1, g1 = 1, and γ1 = 2κ1 . approximation and optical polarization. The Hamiltonian for the right-left circular polarization light looks exactly like RWA, and for linear polarization it has antiresonant terms [31]. So even working at the same frequency, but modifying cavity modes by switching their polarization, we can see a different quantum statistical behavior for g (2) (0).

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5.

Journal of Russian Laser Research

Conclusions

In this work, we have considered the problem of a two-level atom in a cavity taking into account the effects of nonrotating terms. We have derived the time evolution of the wave function using the quantum trajectory method. The autocorrelation function was discussed for two different cases — in the absence and in the presence of the counter-rotating terms. It has been shown that the quantum statistics of the emitted field depends crucially on the frequency regime. The virtual processes destroy the nonclassical effect in the low-frequency regime. Finally, in recent experimental work where the strong coupled regime was examined, the photon decay rate was determined as κ = 1.6π MHz, the photon dispersion rate as γ = 1.4π MHz, while the Rabi frequency, which depends on the coupling parameter, was taken as gRabi ≈ 11.6 MHz. For more details, see [26].

Acknowledgments M.S.A. extends his appreciation to the Deanship of Scientific Research at the King Saud University for funding the work through the Research Group Project No. RGP/VPP/101.

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