Electromagnetically induced transparency in a Y system with single Rydberg state

Optics Communications 345 (2015) 6–12

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Electromagnetically induced transparency in a Y system with single Rydberg state Xue-Dong Tian a, Yi-Mou Liu a, Xiao-Bo Yan a, Cui-Li Cui a, Yan Zhang b,n a b

College of Physics, Jilin University, Changchun 130012, PR China School of Physics, Northeast Normal University, Changchun 130024, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 25 November 2014 Received in revised form 16 January 2015 Accepted 19 January 2015 Available online 22 January 2015

We study the transmitted intensity and correlation properties of a probe field propagating through a sample of cold interacting 87Rb atoms driven into the Y level configuration with single Rydberg state. We find two electromagnetically induced transparency (EIT) windows in the transmission spectrum. One window is linear since it is immune to the incident probe field. The other window depth is sensitive to the incident probe intensity, exhibiting cooperative nonlinearity. Meanwhile, the linear window is low but the transmissivity at the nonlinear window can reach nearly 100% in case of the weak probe intensity. When two EIT windows overlap, the cooperative optical nonlinearity plays a leading role in the degenerate window. In addition, the probe propagation is affected by its two-photon correlation which is suppressed at the center of nonlinear window and is enhanced near the boundaries of two windows. The two-photon correlation is also sensitive to the probe field, and that means we can attain the strongest photon bunching (anti-bunching) effect by controlling the probe and coupling fields. & 2015 Elsevier B.V. All rights reserved.

Keywords: Rydberg atoms Van der Waals potential EIT

1. Introduction In the recent decades, much attention has been paid to the study of electromagnetically induced transparency (EIT) [1,2], a well-known phenomenon exploiting quantum destructive interference to eliminate the resonant absorption and modify the dispersion of a laser beam incident upon a coherently dressed medium with appropriate level configuration. Investigations on EIT lead to many interesting applications such as highly efficient signal light storage and retrieval [3,4,6,5], coherently induced photonic band-gaps [7,8], greatly enhanced optical nonlinearities [9–13], reduced self-focusing and defocusing [1], and quantum memory [14]. That is the reason for EIT being attractive in some fundamental quantum issues, but most of the studies on EIT were only realized in the independent atomic ensembles. Recently, the interest in studying EIT in interacting Rydberg atoms is increasing. The strong dipole–dipole interaction between atoms excited to the Rydberg state may result in the blockade effect [15–19] which prevents the excitation of two or more atoms into one Rydberg state in a mesoscopic volume. This unique property is of great technological importance in the realization of single photon switch [20], single photon transistor [21], quantum phase gate [22–26], quantum entangled state [27–29], microwave n

Corresponding author. E-mail address: [email protected] (Y. Zhang).

http://dx.doi.org/10.1016/j.optcom.2015.01.052 0030-4018/& 2015 Elsevier B.V. All rights reserved.

electrometry [30] and microwave electric field imaging [31]. The blockade effect also causes another novel phenomenon called the cooperative optical nonlinearity [32–34] which is manifested by the sensitivity of EIT window depth to the incident probe intensity. This phenomenon has been well explained with an effective method utilizing a superatom (SA) model and considering the two-photon correlation in the mean-field approximation [35]. Subsequently, some investigations on other complex level configurations [36,37] adopt this method and exhibit novel phenomena such as normal (abnormal) cooperative optical nonlinearity.However, the theory approach still has its own deficiencies: the weakprobe field approximation is adopted but large probe intensity is used in the numerical calculation. Meanwhile, some high-order components of the SA population, which play important roles in the numerical calculation under the condition of strong probe filed, are neglected. So it is essential to find a more accurate approach which has been studied in the case of three-level laddertype Rydberg atoms to avoid those deficiencies [38]. Using the improved SA model, the polarization Rydberg phase gate which is more efficient than the conventional phase gate based on crossKerr nonlinearities is realized. In this paper, we analyze a sample of cold interacting 87Rb atoms driven into a Y-type configuration with single Rydberg state adopting the improved SA model which goes beyond the weakprobe field approximation and considers all relevant collective states [38]. We find some unique phenomena in this configuration. The transmission spectrum shows that there are two EIT windows

X.-D. Tian et al. / Optics Communications 345 (2015) 6–12

arising from the four-level EIT system. One window is low and is immune to the incident probe field intensity, since it is caused by the two-photon resonance from the ground state to the short-lived ordinary excited state. However, the other window caused by the resonance from the ground state to the long-lived Rydberg state exhibits cooperative optical nonlinearity with the probe field. Meanwhile, in this window the medium will be highly transparent if the probe field is weak enough. When two EIT windows overlap, the degenerate window becomes a sharp non-Lorenz curve and the cooperative optical nonlinearity plays a leading role. In addition, the probe field correlation is suppressed at the center of nonlinear window and is enhanced near the boundaries of two windows. The two-photon correlation is also sensitive to the probe field, and that means we can obtain the bunching (anti-bunching) photons at the sample exit and change the degree of photon correlation by controlling the probe and coupling fields.

7

Ωc and Ωd, respectively. One quantum probe field ωp ^ (z) acts on the transition |g 〉 → |m〉 with Rabi frequency Ω^ p (z) = η , p ^ and coupling strength η = ℘ ge ω p/(2= ϵ0 V ) . Here , p (z) is the local

frequencies

probe amplitude operator, V is the local probe quantum volume, and ℘ge is the electric dipole moment on the transition |g 〉 → |m〉. A pair of atoms excited to the Rydberg states |r 〉 interact with each other via the vdW potential =vdW = =C6 (n)/R6 , with C6 being the vdW coefficient and R being the atomic distance. Thus we should describe the considered Y system by Hamiltonian / = /a + =af + =vdw containing an unperturbed atomic Hamiltonian /a , an atom–field interaction Hamiltonian =af , and a vdW dipole–dipole Hamiltonian =vdW : N j j j /a = − = ∑ [Δ p σ^mm + (Δ p + Δc ) σ^ee + (Δ p + Δd ) σ^rr ], =af j N

^ σ^ j + Ω σ^ j + Ω σ^ j + H . c . ], = = − = ∑ [Ω p mg c em d rm vdW j

2. Model and equations We consider an atomic system coherently driven into a fourlevel Y-type configuration as shown in Fig. 1 (a), which has a ground state |g 〉, two ordinary excited states |m〉 and |e〉 and a highly excited Rydberg state |r 〉. This configuration could be realized in 87 Rb atoms where levels 5S1/2, 5P1/2, 6S1/2, and 60S1/2 denote the states |g 〉, |m〉, |e〉 and |r 〉 respectively. Two classical control fields ωc and ωd drive transitions |m〉 → |e〉 and |m〉 → |r 〉 with Rabi

a

N

= =∑ i
C6 (n) ^ i ^ j σrr σrr , Rij6

(1)

i

where σ^μν is the transition (μ ≠ ν ) or projection (μ = ν ) operator of the ith atom, Δp = ω p − ωmg , Δc = ωc − ωeg and Δd = ωd − ωre are the single detunings of the probe and two control fields. Considering that the probe field propagates along the z-axis and using Hamiltonian / , we obtain Heisenberg–Langevin

b

c

Fig. 1. (a) Level scheme of the Y system of interacting cold 87Rb atoms driven by two classical control fields Ωc and Ωd and a quantum probe field Ω^ p . vdW denotes van der Waals interaction experienced by a pair of cold 87Rb atoms in the Rydberg state r . Once an atom is excited to the Rydberg state, the other cold atoms in a blockade sphere degenerate from a four-level Y system into a three-level Ladder system. (b) Schematic representation of the probe field transmission in a one-dimensional sample of Y-type cold atoms, envisioned as a collection of superatoms. (c) Relevant level scheme of the superatom, composed of nSA atoms and (n SA + 1)2 collective states. When we truncate the SA level scheme to the ith-order, only (i + 1)2 levels should be considered.

8

X.-D. Tian et al. / Optics Communications 345 (2015) 6–12

equations without invoking the weak probe approximation:

(|M (1) 〉 → |E (1) 〉, |M (1) 〉 → |R(1) 〉). So we have

^ ( z) = − c ∂ , ^ ∂t, p z p (z) + iNησ^eg (z),

(1) 2 2 ^ †Ω ^ Σ^MM = σ^mm [n SA Ω p p, Ω c , Ω d ],

^ p σ^gm (z) + H . c . ] + Γm σ^mm (z), ∂ t σ^gg (z) = [ − iΩ

(2) 2 2 ^ †Ω ^ Σ^MM = σ^mm [n SA Ω p p, Ω c , Ω d ] ·

^ † [σ^gg (z) − σ^mm (z)] − iΩ c σ^ge (z) − iΩ σ^gr (z), ∂ t σ^gm (z) = − (iΔ p + γgm ) σ^gm (z) − iΩ d p ^ † σ^me (z) − iΩ ⁎ σ^gm (z), ∂ t σ^ge (z) = − [i (Δ p + Δc ) + γge ] σ^ge (z) + iΩ p c

2 2 ^ †Ω ^ σ^mm [2(n SA − 1) Ω p p, 2 Ω c , 2 Ω d ],

^ † σ^mr (z) − iΩ ⁎ σ^gm (z), ∂ t σ^gr (z) = − {i [Δ p + Δd − ^s (z)] + γgr } σ^gr (z) + iΩ p d

(3) 2 2 ^ †Ω ^ Σ^MM = σ^mm [n SA Ω p p, Ω c , Ω d ] ·

^ p σ^gm (z) − iΩ c σ^me (z) − iΩ σ^mr (z) + H . c . ] − Γm σ^mm (z) + Γe σ^ee (z) + Γr σ^rr (z), ∂ t σ^mm (z) = [iΩ d

2 2 ^ †Ω ^ σ^mm [2(n SA − 1) Ω p p, 2 Ω c , 2 Ω d ] ·

^ p σ^ge (z) + iΩ ⁎ [σ^ee (z) − σ^mm (z)] − iΩ ⁎ σ^re (z), ∂ t σ^me (z) = − (iΔc + γme ) σ^me (z) + iΩ c d ^ p σ^gr (z) + iΩ ⁎ σ^er (z) + iΩ ⁎ [σ^rr (z) − σ^mm (z)], ∂ t σ^mr (z) = − {i [Δd − ^s (z)] + γgr } σ^mr (z) + iΩ c d ∂ t σ^ee (z) = [iΩ c σ^me (z) + H . c . ] − Γe σ^ee (z), ∂ t σ^er (z) = − {i [Δd − Δc − ^s (z)] + γgr } σ^er (z) + iΩ c σ^mr (z) − iΩ d⁎ σ^em (z),

2 2 ^ †Ω ^ σ^mm [3(n SA − 2) Ω p p, 3 Ω c , 3 Ω d ] (2)

i σ^μν

where σ^μν (z) is the average of all in the microvolume ΔV centered at z. Γm, Γe and Γr are the spontaneous decay rates from |m〉 to |g 〉, from |e〉 to |m〉, and from |r 〉 to |e〉, respectively. γgm, γge, γgr, γme, γmr, and γer are the coherence dephasing rates on transitions |g 〉 → |m〉, |g 〉 → |e〉, |g 〉 → |r 〉, |m〉 → |e〉, |m〉 → |r 〉, and |e〉 → |r 〉, respectively. The vdW interaction is expected to result in a conditional frequency shift s^ (z) of level |r 〉 and the expectation value of s^ (z) can be either →∞ or → (γ |Ω |2 /8(γ 2 + Δ 2 )) Σ^ (z) for the ge

atoms

confined

in

a

d

ge

blockade

RR

d

sphere

of

radius

Rb

2 ≃ 6 C6 (γge + Δd2 )/γge |Ωd |2 . The n SA = ρVSA cold atoms, in the blockade

sphere of volume VSA = 4πRb3/3, can be envisioned as a superatom (SA) containing, at most, one atom excited to the Rydberg state. Each SA can be described as in ref. [39] by the collective states:

|G〉 = |g , …, gi , …, gn SA 〉, (n SA − j) !/n SA !j! ( ∑ |mn 〉〈gn |) j |G〉,

1 ( ∑ |rn 〉〈gn |) |G〉, n SA n = 1

n SA

n SA n= 1

(n SA − i) ! (n SA − j) !/n SA n SA !i!n SA !j! n SA

n SA

( ∑ |mn 〉〈gn |)i ( ∑ |en 〉〈gn |) j ( ∑ |rn 〉〈gn |) |G〉, n= 1

n= 1

n= 1

(3)

Firstly, we take attention to the SA projection operator Σ^RR which is indispensable for attaining the final probe polarizability conditioned upon the Rydberg excitation. In previous studies [35– 37], by considering the weak-probe and low-intensity limit, it is assumed that fewer than two atoms are excited out of level |g 〉 on average in a blockade sphere, in which case we can think that Σ^ RR

is only contributed by state |R(1) 〉 and can be written as Σ^RR = |R(1) 〉〈R(1) | [35]. However, in more general cases, other states (|E (1) R(1) 〉, |M (1) R(1) 〉……) also play important roles in evaluating Σ^ . RR

We

can attain the steady-state atomic populations † † † σ^mm (Ω^ p Ω^ p , |Ωc |2 , |Ωd |2), σ^ee (Ω^ p Ω^ p , |Ωc |2 , |Ωd |2), σ^rr (Ω^ p Ω^ p , |Ωc |2 , |Ωd |2) by numerical calculation of the Heisenberg–Langevin equations, and it is easy to understand that the populations are functions of † 2 Ω^ Ω^ , |Ω |2 and |Ω . When we calculate the first-, second- and p

p

c

d

third-order populations of the Rydberg state, we should change the probe and coupling coefficients (Fig. 1 (c)), e.g, the probe (coupling) coefficient is n Ω^ (Ω , Ω ) on transition |G〉 → |M (1) 〉 SA

2 2 ^ †Ω ^ σ^mm [2(n SA − 1) Ω p p, 2 Ω c , 2 Ω d ] · 2 2 ^ †Ω ^ σ^rr [3(n SA − 2) Ω p p, 3 Ω c , 3 Ω d ]

p

c

(4)

Similarly, we can attain the required atomic polarizations of three-level Ladder system (α3) and four-level Y system (α4) by numerical calculation of the Heisenberg–Langevin equations. Thus the total probe polarization should be written as

(n SA − j) !/n SA n SA !j! ( ∑ |en 〉〈gn |) j ( ∑ |rn 〉〈gn |) |G〉, =

2 2 ^ †Ω ^ = σ^mm [n SA Ω p p, Ω c , Ω d ] ·

n= 1

n= 1

|M (i) E (j) R (1) 〉

(3) Σ^RR = Σ^ M (2) R (1) + Σ^ M (1) E (1) R (1) + Σ^ E (2) R (1)

2 2 ^ †Ω ^ σ^rr [2(n SA − 2) Ω p p, 3 Ω c , 3 Ω d ].

(n SA − j) !/n SA n SA !j! ( ∑ |mn 〉〈gn |) j ( ∑ |rn 〉〈gn |) |G〉 n= 1 n SA

|E (j) R (1) 〉 =

2 2 ^ †Ω ^ σ^rr [(n SA − 1) Ω p p, 2 Ω c , Ω d ],

2 2 ^ †Ω ^ σ^ee [(n SA − 1) Ω p p, 2 Ω c , Ω d ] ·

n SA

n SA

|M (j) R (1) 〉 =

2 2 ^ †Ω ^ + σ^ee [n SA Ω p p, Ω c , Ω d ] ·

2 2 ^ †Ω ^ + σ^ee [n SA Ω p p, Ω c , Ω d ] ·

n SA

=

2 2 ^ †Ω ^ σ^rr [2(n SA − 1) Ω p p, 2 Ω c , 2 Ω d ]

2 2 ^ †Ω ^ σ^rr [2(n SA − 2) Ω p p, 2 Ω c , 2 Ω d ]

(n SA − j) !/n SA !j! ( ∑ |en 〉〈gn |) j |G〉, n= 1

|R (1) 〉

2 2 ^ †Ω ^ = σ^mm [n SA Ω p p, Ω c , Ω d ] ·

2 2 ^ †Ω ^ σ^mm [(n SA − 1) Ω p p, 2 Ω c , Ω d ] ·

n= 1 n SA

|E (j) 〉 =

(2) Σ^RR = Σ^ M (1) R (1) + Σ^ E (1) R (1)

2 2 ^ †Ω ^ + σ^ee [n SA Ω p p, Ω c , Ω d ] ·

n SA

|M (j) 〉 =

(1) 2 2 ^ †Ω ^ Σ^RR = σ^rr [n SA Ω p p, Ω c , Ω d ],

d

α^ (z) = α3 Σ^RR (z) + α4 [1 − Σ^RR (z)].

(5)

The physical reason is that if the SA at position z contains a Rydberg excitation, then the state |r 〉 is decoupled from state |m〉 due to a large frequency shift s^ (z) → ∞, and the SA will behave like a three-level Ladder system; otherwise the SA will behave like a four-level Y system. Then we pay attention to transmission properties of the probe field. It is clear that the propagation dynamics ^ † (z) , ^ (z) 〉 should be determined by of probe intensity I (z) = 〈, p

p

p

examining the random Rydberg excitation of all SAs one after the ^ (z) = 0 in the steady state in Eq. (2), the other. By setting ∂t , p expectation value of the probe intensity obeys the equation:

^ † (z) , ^ (z) ∂z , p p

^ † (z) Im[α^ (z)] , ^ (z) =−κ , p p

(6)

where κ = ρ0 ω p |℘ ge |2 /(= ϵ0 cγe ) denotes the resonant absorption coefficient. Note that solving Eq. (6) requires factorizing out Im[α^ (z)] from 〈…〉 in the mean-field sense. In this process we should † † substitute 〈Ω^ (z) Ω^ (z) 〉g (2) (z) for Ω^ (z) Ω^ (z) to preserve p

p

p

p

p

X.-D. Tian et al. / Optics Communications 345 (2015) 6–12

a

e

b

f

c

g

d

h

9

(1) (2) (3) (1) (2) (3) Fig. 2. Mean values of SA population operators Σ^MM (a), Σ^MM (b), Σ^MM (c), Σ^MM (d), Σ^RR (e), Σ^RR (f), Σ^RR (g), Σ^RR (h) versus probe detuning Δp /2π for the first few collective states attained with our accurate SA model. Ωc /2π = Ωd /2π = 2.5 MHz , Δc /2π = 2 MHz , Δd /2π = − 2 MHz , Γm/2π = 5.75 MHz , Γe/2π = 2.3 MHz , Γr /2π = 1 KHz , g p2 (0) = 1. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

realizations to examine the transmitted intensity Ip(L) and transmitted correlation g p(2) (L) of a probe field propagating through a sample. In each realization, we first divide the propagation distance L into L/(2Rb ) intervals and then determine whether Σ^ (z) → 1 or Σ^ (z) → 0 within each interval via Monte Carlo

a

RR

RR

sampling.

b

3. Numerical results and discussion In this section, we will implement numerical calculations adopting practical parameters relative to cold 87Rb atoms such as and Γe/2π = 2.3 MHz , Γm/2π = 5.75 MHz, Γr /2π = 1 KHz C6/2π = 1.4 × 1011 s−1μm corresponding to the vdW interactions. The atomic density is ρ (z) = ρ0 exp[ − (z − z0 )2/2σρ2 ] with peak

Fig. 3. Probe field transmission Ip (L)/Ip (0) (a) and correlation function g p(2) (L)/g p(2) (0) (b) versus the probe detuning, for different input intensities. Relevant parameters are the same as in Fig. 2. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

the important information on the two-particle quantum correlation due to the conditioned Rydberg excitation. The field intensity is now coupled to its two-photon correlation ^ † (z) , ^ † (z) , ^ (z) , ^ (z) 〉/〈, ^ † (z) , ^ (z) 〉2, and we know that g (2) (z) = 〈, p

p

p

p

p

p

p

only the nonlinear optical process is related to Rydberg excitation modifies g p(2) (z) which consequently obeys the equation:

∂ z g p(2) (z) = − κ〈Σ^RR (z) 〉 Im(α2 − α3 ) g p(2) (z)

(7)

With given incident field intensity Ip (0) and its correlation function g p(2) (0), we consider a stochastic procedure to integrate the coupled equations (4)–(6) after replacing operators with their expectation values. Then we average over several independent

ρ0 = 1.32 × 107 mm−3 and half-with σρ = 0.7 mm . We set the control field Ωc /2π = Ωd /2π = 2.5 MHz in Figs. 2–4. When the detunings of the coupling field are not large, the corresponding blockade radius is Rb ≃

6

2 C6 (γge + Δd2 )/γge |Ωd |2 ≃ 6.5 μm and each SA contains

n SA ≃ 15 atoms on average. The length of the medium containing 100 SAs is L ¼1.3 mm. The two-photon correlation of the incident probe field is g p2 (0) = 1. In Fig. 2, we plot the collective state population as a function of probe detuning Δp /2π for Ω p (0)/2π = 0.01 MHz (red triangle); 0.1 MHz (blue circle); 0.5 MHz (black square) and 1 MHz (green triangle), with Δc /2π = 2 MHz and Δd /2π = − 2 MHz. From the red, blue, and black curves we can see that high-order components of the SA population are negligible except for the first-order one when the probe field is weak. From the green-triangle curves we find that the maximal mean value of n SA Σ^MM (n SA Σ^RR ) is 2.6832 (2.9074), such that there exist 2.6832 (2.9074) atoms in state |m〉 (|r 〉) in each SA and the contributions of the first, second and third order components are respectively 2.1795 (2.8368), 0.4171 (0.3585) and 0.0865 (0.0856). So we attain a conclusion that in the case of strong probe field, the approximate SA model is not

10

X.-D. Tian et al. / Optics Communications 345 (2015) 6–12

a

c

d

e

b

f

(1)

(2)

(3)

Fig. 4. Probe field transmission Ip (L)/Ip (0) (a), correlation function g p(2) (L)/g p(2) (0) (b)and mean values of SA population operators Σ^RR (c), Σ^RR (d), Σ^RR (e) and Σ^RR (f) versus the probe detuning, for different input intensities. Relevant parameters are all the same as in Fig. 3 except Δc = Δd = 0 MHz . (2)

accurate enough for neglecting a lot of collective states (Σ^MM , (3) Σ^MM…).

However, for both weak and strong probe fields, our numerical calculations are really reliable when we truncate the SA level scheme to the third-order. In Fig. 3, we examine the intensity and correlation properties of the probe field at the sample exit with the same parameters in Fig. 2. It is clear that two EIT windows are generated in the transmission spectrum, which arise from the two-photon resonances from state |g 〉 to state |e〉 and state |r 〉. We can find that

a

the EIT window centered at Δp /2π = − 2 MHz (caused by twophoton resonance from state |g 〉 to state |e〉) is low and rarely depends on the probe intensity, so this window is linear. However, the other window centered at Δp /2π = 2 MHz (caused by twophoton resonance from state |g 〉 to state |r 〉) whose depth varies obviously with the probe field intensity exhibits cooperative nonlinearity. As shown by the red-triangle curve in Fig. 3(a), the medium is highly transparent if the probe field is properly weak. The physical reason of the differences between the two windows is that the state |e〉 is a short-lived ordinary excited state while the

e

d

b

e

f

(1) (2) (3) Fig. 5. Probe field transmission Ip (L)/Ip (0) (a), correlation function g p(2) (L)/g p(2) (0) (b) and mean values of SA population operators Σ^RR (c), Σ^RR (d), Σ^RR (e) and Σ^RR (f) versus the probe detuning, for different input intensities. Relevant parameters are all the same as in Fig. 3 except Δc /2π = 10 MHz , Δd /2π = − 10 MHz and Ωc /2π = Ωd /2π = 4.7 MHz .

X.-D. Tian et al. / Optics Communications 345 (2015) 6–12

a

b

Fig. 6. Correlation function g p(2) (L)/g p(2) (0) versus the probe intensity at Δp = ( − Δd + Δd2 + 4|Ωd |2 )/2 (a) and the center of EIT window (b). In the red curve, relevant parameters are all the same as in Fig. 3; in the blue curve, the parameters are all the same as in Fig. 4; and in the green curve, the parameters are all the same as in Fig. 5. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

state |r 〉 is a long-lived Rydberg state whose blockade effect causes the cooperative nonlinearity. Then we can get better EIT effect when the excited state in the Y system is the Rydberg state. Moreover, as shown in Fig. 3(b), the probe photons leaving from the sample exit exhibit a remarkable anti-bunching effect ( g p(2) (L) < 1) at the nonlinear window but a bunching effect ( g p(2) (L) > 1) near the boundaries of two EIT windows, especially the nonlinear one. Fig. 4 shows that when Δc /2π = Δd /2π = 0 MHz , the two EIT windows will overlap and are almost indistinguishable. As the degeneracy results from the overlap of the linear and nonlinear windows, it is accessible that the cooperative optical nonlinearity plays a leading role in the degenerate window and the window becomes a sharp non-Lorenz curve. We also plot the relevant order populations of the Rydberg state, and from Fig. 4(f), we can see that the Rydberg state population is larger than that of the condition in Fig. 2. In addition, the two-photon correlation has a symmetric structure and the probe photon bunching (antibunching) effect is similar to Fig. 3. As we set the detuning Δd large enough, the Rydberg blockade radius Rb will increase, so that we need to set the coupling field Rabi frequencies as Ωc /2π = Ωd /2π = 4.7 MHz to ensure that the blockade radius does not change when we consider the large detunings Δc /2π = 10 MHz, and Δd /2π = − 10 MHz. In Fig. 5, we plot probe field transmission Ip (L)/Ip (0) (a); correlation function

g p(2) (L)/g p(2) (0) (b) and relevant order Rydberg state populations versus the probe detuning, for different input intensities. When the detunings of the coupling fields are very large, the population of the Rydberg state is large, too. Comparing Figs. 3(b), 4(b) and 5 (b), we can find that the two-photon correlation reaches the maximum at the position Δp = ( − Δd + Δd2 + 4|Ωd |2 )/2 and the maximum is greatly enhanced in Fig. 5. For further study of the photon bunching (anti-bunching) effect, we plot the correlation at

Δp = ( − Δd + Δd2 + 4|Ωd |2 )/2 as a function of the probe intensity in Fig. 6(a). It is clear that we can get a maximum of the twophoton correlation with the changing of the probe intensity, and the position of the maximum is relative to the coupling fields. We also plot two-photon correlation at the center of the nonlinear EIT window (Δp = − Δd ) as a function of the probe intensity in Fig. 6 (b) and get the position of the minimum value of the two-photon

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correlation in different conditions. It is also confirmed via the green-square curve in Fig. 6(a) that the photon bunching effect is more obvious under the condition of large detunings. Fig. 6 (b) shows that the photon anti-bunching effect is more obvious when the detunings of the coupling fields are not very large. All of the conclusions above are obtained under conditions Ωc = Ωd , Δc + Δd = 0 and g p2 (0) = 1. In general, Rabi frequencies and detunings of the control fields just affect widths (depths) and positions of the EIT windows, respectively. So we choose the equal Rabi frequencies and equal or opposite detunings of two control fields to make two EIT windows appear in symmetrical positions and have almost the same width, and it does not lose the generality. As to the case of g p2 (0) > 1 or g p2 (0) < 1, we can obtain qualitatively similar results in the presence of some quantitative differences. In the reference [37], Yan et al. have reported a detailed analysis about how the correlation function of the incident probe field affects the steady optical response of interacting Rydberg atoms in the EIT regime.

4. Conclusions To conclude, we have studied the spectra of intensity transmission and two-photon correlation of a probe field propagating through a sample of cold 87Rb atoms interacting via the vdW potential. We find that two EIT windows would appear in the transmission spectrum and one window is sensitive to the incident probe intensity, which is called the nonlinear response. The linear EIT window which is caused by the two-photon resonance from the ground state to the short-lived ordinary excited state is low, while the transmissivity of the probe field can reach nearly 100% within the nonlinear window caused by the two-photon resonance from the ground state to the long-lived Rydberg state when the probe intensity is weak enough, so we can get better EIT when the excited state is the Rydberg state in a Y system. In addition, we can overlap the two EIT windows via controlling the two coupling fields, on which condition, the optical nonlinearity plays a leading role in the degenerate window and the window becomes a sharp non-Lorenz curve. If the incident probe field is in a coherent state ( g p2 (0) = 1), one may obtain the bunching (near the boundaries of two EIT windows) and anti-bunching (at the nonlinear window) photons at the sample exit. We can also attain the strongest photon bunching (anti-bunching) effect by controlling the probe and coupling fields and this is important in the field of the quantum information.

Acknowledgments This work is supported by the National Natural Science Foundation of China (11247005), the Fundamental Research Funds for Central Universities of China (12QNJJ006), the Postdoctoral Scientific Research Program of Jilin Province (RB201330), and the China Postdoctoral Science Foundation Grant (2013T60316).

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