Photon-number dependent cavity vacuum induced transparency and single photon separation

Accepted Manuscript Photon-number dependent cavity vacuum induced transparency and single photon separation

Shuangli Fan, Hui Sun, Hongjun Zhang, Hong Guo

PII: DOI: Reference:

S0375-9601(18)30915-0 https://doi.org/10.1016/j.physleta.2018.08.027 PLA 25280

To appear in:

Physics Letters A

Received date: Revised date: Accepted date:

13 June 2018 25 August 2018 30 August 2018

Please cite this article in press as: S. Fan et al., Photon-number dependent cavity vacuum induced transparency and single photon separation, Phys. Lett. A (2018), https://doi.org/10.1016/j.physleta.2018.08.027

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Highlights • We suggest a cavity QED system to investigate photon-number dependent cavity vacuum induced transparency and magneto-optical rotation. The probe photon coupled into one cavity can be transferred to the mode of another cavity via coherent Raman scattering. Our theoretical calculations show that: • the transmission, the phase shift, as well as the vacuum Rabi splitting strongly depend upon the probe photon number coupled into the cavity; • the photon number dependent cavity vacuum induced transparency leads to photon-number dependent magneto-optical rotation (MOR); • the photon-number dependent MOR can be used to separate the single photon from higher photon number components in the direction of polarization and create a deterministic single photon source.

Photon-number dependent cavity vacuum induced transparency and single photon separation Shuangli Fana , Hui Suna,∗∗, Hongjun Zhanga , Hong Guob,∗∗ a School

of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China b State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronics Engineering and Computer Science, and Center for Quantum Information Technology, Peking University, Beijing 100871, China

Abstract We investigate photon-number dependent cavity vacuum induced transparency and magneto-optical rotation in a cavity quantum electrodynamics system, which consists of two cavities and an ensemble of Λ-type atoms. We demonstrate that the probe photon coupled into one cavity can be transferred to the mode of another cavity via coherent Raman scattering. The transmission, the phase shift, as well as the vacuum Rabi splitting therefore strongly depend upon the probe photon number coupled into the cavity. The photon number dependent cavity vacuum induced transparency can be extended into four-level tripod atoms, leading to photon-number dependent magneto-optical rotation (MOR). This can be used to separate the single photon from higher photon number components in the direction of polarization and create a deterministic single photon source. Keywords: Vacuum induced transparency, Single photon separation, Magneto-optical rotation, Cavity QED

∗ Corresponding

author author Email addresses: [email protected] (Hui Sun), [email protected] (Hong Guo)

∗∗ Corresponding

Preprint submitted to Physics Letters A

August 31, 2018

1. Introduction Electromagnetically induced transparency (EIT) is an optical coherence effect that renders an otherwise opaque medium transparent in a narrow transmission window with low absorption and steep dispersion [1, 2, 3, 4]. EIT was first 5

demonstrated in three-level atomic system, in which two electromagnetic fields excite resonantly two different transitions sharing a common state. The presence of a strong and usually classical driving field modifies the optical responses dramatically. By trapping the atoms inside an optical cavity, the dramatically enhanced

10

atom-cavity interaction [5, 6, 7] makes cavity quantum electrodynamics (QED) to be an excellent platform for investigating the strong light-matter interaction allowing for substantial narrowing of the spectral features [8, 9, 10, 11, 12, 13, 14], all-optical switching and isolation of single photon [15, 16, 17, 18], optical frequency combs [19], generation of entanglement state [20, 21, 22], holonomic

15

quantum computation [6, 23], high sensitivity magnetometry [24, 25, 26, 27], quantum nonlinear optics [28, 29, 30] and so on [31, 32, 33, 34, 35, 36]. More recently, the nonlinear interaction in optomechanical system has been suggested for precision measurement of electrical charges [37, 38]. In the strong-coupling regime of cavity QED, the classical driving field in EIT can be replaced by a

20

quantized cavity mode. Cavity EIT as well as EIT-based all-optical switching where both the probe and control fields are coupled to cavities have been observed with cold ion Coulomb crystals [11]. Furthermore, an empty cavity will induce transparency, i.e., vacuum induced transparency (VIT), which was theoretically proposed in Ref. [39], and has been demonstrated experimentally [40].

25

Different from EIT or cavity EIT [8, 9, 10], the probe photons can be transferred into cavity mode via Raman scattering, resulting to photon-number dependent group velocity [41, 42, 43] and thus can be employed for deterministic singlephoton source. By utilizing of coherent scattering, a single photon coupled into one coupled-resonant waveguide (CRW) can be converted to cavity mode of

30

another CRW, leading to quantum router of single photons [44, 45].

2

Motivated by these work, in this paper, we theoretically propose a cavity QED system for photon number-dependent cavity VIT. Although the photon number-dependent group velocity with cavity induced transparency has been widely studied [41, 42, 43], considering the situation of the probe field coupled 35

into an optical cavity to demonstrate the photon number-dependent quantum coherent effects has never been addressed, to the best of our knowledge. To do so, we consider an atom-cavity system consisting of N three-level Λ-type atoms confined in optical cavities both A and B. The atoms are coherently coupled by the probe field from input mirror of cavity A and cavity mode of

40

cavity B. In the strong coupling regime, we show that the interaction of the weak probe field with the atom-cavity system leads to a temporary transfer of probe photons coupled into the cavity A to the mode of cavity B via coherent Raman scattering. The backaction of the present atom-cavity system onto the probe field, in particular its effect onto the transmission and phase shift,

45

are strongly sensitive to the probe photon number coupled into cavity A. The photon-number dependent VIT can be extended to four-level tripod type atoms, and lead to photon number-dependent magneto-optical rotation (MOR). Our theoretical analysis and numerical calculations demonstrate that the photon number-dependent MOR supports the present atom-cavity system to separate

50

the single photon on the path from high photon number components and thus to create a deterministic single-photon source with high purity. Pure single photons produced efficiently are highly desirable for practical implementations of various quantum information processing protocols, in particular in photonbased quantum information technology [46] from quantum cryptography [47] to

55

quantum computing [48] and quantum measurement [49]. A controllable and deterministic generation of single photons is still one of the practical challenges. A number of quantum systems have been studied towards generation of a single photon [50]. For examples, semiconductor quantum dots based sources [51, 52], NV centers in diamond [53], and molecules [54] have been investigated as poten-

60

tial single-photon source. Trapping a single atom within a high-finesse optical cavity, cavity-enhanced single photons emission can be realized on demand us3

ing spontaneous four-wave mixing [55] or vacuum stimulated Raman adiabatic passage [56, 57]. The physical mechanism of the present proposal is cavity VIT, whose behaviors strongly depend upon the photon number. Furthermore, the 65

single photon component is separated from higher photon number components in the direction of polarization. This paper is organized as follows. After giving the theoretical cavity QED model and basic equations in Sec 2, we show the photon number dependent cavity VIT in Sec. 3. In Sec. 4, we extend the photon number dependent cavity

70

VIT to four-level tripod scheme, and demonstrate the photon number dependent MOR. The feasibility of separating the single photon from high photon number components is discussed. In Sec. 5, we end with some remarks.

2. Model and basic equations We consider an atom-cavity system as depicted in Fig. 1(a). N identical 75

three-level atoms in Λ-type configuration are confined in two high-finesse optical cavities (cavity A and B). We assume that the two optical cavities have symmetric mirror transmissions. The total cavity field decay rate in cavity A corresponding to the mirrors’ transmission is denoted by κA = TA /2τA with TA and τA being total transmission of mirrors and cavity round-trip time of

80

cavity A. Two mirrors of cavity B are assumed to have 100% reflectivity such that the loss rate of cavity B κB is small. States |0 and |2 are long lived ground and metastable states, and |1 is excited state. A weak probe field with carrier frequency ωp couples into cavity A through M1 , and drives the quantum  transition |0 ↔ |1 with coupling strength g = μ01 ωc /2¯h0 V with V being

85

the quantization volume. After passing through M2 , the weak probe field can be detected. The |2 ↔ |1 transition is driven by a cavity mode ˆb of cavity B, and the corresponding vacuum Rabi frequency is denoted as G. A Raman transition, which is the heart of VIT, is therefore formed by the combination of the probe field and the vacuum field [as shown in Fig. 1(b)].

90

We in addition assume that the coupling strength of the cavity modes a ˆ and

4

Figure 1: (Color online) (a) Schematic of photon number-dependent cavity vacuum induced transparency. N identical three-level Λ-type atoms (b) is confined in high-finesse optical cavities A and B. The ground and metastable states are, respectively, denoted by |0 and |2. |1 is the excited state with spontaneously decay rate 2γ. The probe field couples into cavity A through M1 , and can be detected after passing through M2 . The probe field excites the cavity mode a ˆ of cavity A to couple the transition |0 ↔ |1 with single photon Rabi frequency g. The quantum transition |2 ↔ |1 is driven by the cavity mode ˆb of cavity B with corresponding vacuum Rabi frequency G. The cavities A and B have the resonant frequencies ωc1 and ωc2 . κi (i=A, B) is the loss rates of the cavity i. Two mirrors of cavity B are assumed to have 100% reflectivity, thus the loss rate of cavity B is small.

ˆb to be the same for all atoms and all atoms are in the strong interaction regime with the cavity modes a ˆ and ˆb. Under dipole and rotating wave approximations, the interacting Hamiltonian reads      (i) (i) (i) (i) (i) (i) ˆ = ¯h ˆ00 + Δd σ ˆ22 − g a ˆσ ˆ10 + a ˆ† σ ˆ01 − G ˆbˆ ˆ21 Δp σ σ12 + ˆb† σ H i

ˆ† a ˆ, −¯ h Δ1 a

(1)

where Δp = ωp − (ω1 − ω0 ) is one photon detuning, and Δd = ωc2 − (ω1 − ω2 ) is 95

the detuning between the cavity mode ˆb and its corresponding transition with ωc2 being the eigenfrequency of cavity B. Δ1 = ωp − ωc1 denotes the detuning 5

(i)

between the probe field and the cavity mode a ˆ. σ ˆαβ (α, β = 0, 1, 2) are the atomic operator associated with the i-th atom. a ˆ (ˆ a† ) and ˆb (ˆb† ) are, respectively, the annihilation (creation) operators of the cavity photons in cavity A and B 100

with the conventions [ˆ a, a ˆ† ] = 1 and [ˆb, ˆb† ] = 1. The dynamical behaviors of the atomic ensemble are governed by Heisenberg-Langiven equations as following   (i) (i) (i) (i) (i) ˆ01 = id01 σ ˆ01 + igˆ a σ ˆ00 − σ ˆ11 + iGˆbˆ ∂t σ σ02 + Fˆ01 ,   (i) (i) (i) (i) (i) aσ ˆ20 + Fˆ21 , ˆ21 = id21 σ ˆ21 + iGˆb σ ˆ22 − σ ˆ00 + igˆ ∂t σ   ∂ˆb (i) ˆ (i) (i) (i) (i) σ02 = id02ˆbˆ σ02 − igˆ ˆ + F02 , aˆbˆ σ12 + iGˆbˆb† σ ˆ01 + σ ∂t ˆbˆ ∂t 02   (i) (i) (i) ˆ00 = γ σ ˆ11 + ig a ˆ† σ ˆ01 − a ˆσ ˆ10 , ∂t σ   (i) (i) (i) ˆ22 = γ σ ˆ11 + iG ˆb† σ ˆ21 − ˆbˆ σ12 . ∂t σ

(2) (3) (4) (5) (6)

Here d01 = Δp + iγ, d21 = Δd + iγ, d02 = Δp − Δd + iγ  . 2γ denotes the spontaneously decay rate of the excited level |1, and γ  is the decoherence rate between two lower states. For simplicity, we assume that γ and γ  are same 105

for all atoms, and the atoms decay from the excited state |1 to states |0 and |2 with the same rate. In the above set of coupled differential equations, the decoherence γ  is introduced in a phenomenological way. Fˆαβ are δ-correlated noise operators associated with the relaxation. The Heisenberg equations of the cavity modes a ˆ and ˆb are standardly given by a ˆ˙ = − (κA − iΔ1 ) a ˆ+i



 (i) ˆb˙ = −κB ˆb + i Gˆ σ21 ,

i

(i) gˆ σ01

 +

κA a ˆin , τ

(7) (8)

i 110

where a ˆin is the input probe field. 3. Photon number-dependent cavity vacuum induced transparency In the following, we focus our attention on the influence of the input probe photon number on the transmission and phase shift. To do so, the mean value of operator a ˆ should be derived. We assume that the vacuum Rabi frequency

6

115

of cavity B is much larger than the Rabi frequency of cavity A (i.e., G  g), and all atoms are initially prepared in ground state |0. If the two-photon resonance condition is satisfied, the two cavity modes create a dark state, and lead to transparency [42]. Let us consider the situation that the number of probe photons are much less than the number of atoms trapped in the optical

120

cavities, the populations of the states |1 and |2 are much smaller than that in the ground state |0. We therefore calculate Eqs. (2)-(4) in linear frame,  i.e., keeping only terms proportional to nin /N with nin = ˆ a†in a ˆin  being the number of input probe photon. By introducing adiabatic approximation, the mean value of intracavity probe field can be immediately obtained as √ κA τ ain , a= κA − iΔ1 − iχ

125

(9)

where χ is the susceptibility of the coherent atomic ensemble, and it is given by χ=−

g2 N , d01 − G2 (2 + n2 )/(d02 + iκB )

(10)

in which n2 = ˆb†ˆb is the photon number in cavity mode ˆb. In the above derivations, we insert Eq. (8) into the right side of Eq. (4), and use the commute (i) (j) (i) relation [ˆb, ˆb† ] = 1 and σ ˆkl σ ˆmn = σ ˆkn δij δlm . The transmitted probe field from √ the cavity A through M2 , which is given by at = κ1 τ a, reads

teiφ = 130

at κA . = ain κA − iΔ1 − iχ

(11)

Introducing the excitation number operator as ˆ =a ˆ + ˆb†ˆb + N ˆ† a



(i)

σ ˆ11 ,

(12)

i

ˆ , H] ˆ = 0, which means that the number it can be immediately calculated that [N of quanta in the present cavity QED system is conserved. If the spectrum of the input probe field is within the transparency window, all atoms will remain in the dark state. The population of the excited state |1 can be safely ignored, 135

one immediately has n1 + n2 = nin . 7

(13)

Here n1 = ˆ a† a ˆ is the photon number in cavity mode a ˆ. The above relation can also be derived by inserting Eqs. (7) and (8) into Eqs. (5) and (6) in adiabatic limit and combining with the transparency conditions. Elimiting σ ˆ01 adiabatically and neglecting the Langiven noise, Eqs. (2) immediately yields G2 n2 = g 2 N n1 . 140

(14)

Thus the photon number in cavity mode ˆb n2 can be easily derived as n2 =

g2 N nin . g 2 N + G2

(15)

If the number of atoms is large such that g 2 N  G2 , almost all input probe photons can be converted to the cavity mode ˆb via coherent Raman scattering process. Thus the photon number dependent susceptibility can be rewritten as χ−

d01 −

G2 (1

g2 N . + nin )/(d02 + iκB )

(16)

Equations (11) and (16) clearly exhibit that the spectrum of the present 145

atom-cavity system strongly depends upon the input probe photon number. In order to see this more clearly, it is instructive to investigate the effective width of the transparency window and phase shift. By expanding the transmission around two-photon resonance, the width of the central VIT feature can be calculated. Recalling the condition γγ   G2  g 2 N , one finds that the probe

150

transmission is Lorentzian shaped [12] 2 1 , T = t2 ∝ κVIT − iΔp

(17)

in which κVIT means the effective half width of the transparency window, and it is given by κVIT = γ  + κB + κA

G2 (1 + nin ) . g2 N

(18)

Starting from Eq. (11), the photon-number-dependent phase shift φ with Δd = 0 can be calculated tan φ = −

g 2 N Δp , g 2 N γ + κA (Δp2 + γ 2 ) 8

(19)

(a) 1.0

(c)1.0 nin=1 nin=2 nin=3 nin=4

0.6 0.4

0.6 0.4 0.2

0.2 0 -0.3 -0.2 -0.1

0

0.1

0.2

0 -0.3 -0.2 -0.1

0.3

1.0

1.0

Phase shift

(d)1.5

Phase shift

(b)1.5

0.5 0

-0.5

-0.2 -0.1

0

0.1

0.2

0.1

0.2

0.3

0.5 0

-0.5

nin=1 nin=2 nin=3 nin=4

-1.0

0

Probe detuning Δp (MHz)

Probe detuning Δp (MHz)

-1.5 -0.3

nin=1 nin=2 nin=3 nin=4

0.8

Transmission

Transmission

0.8

nin=1 nin=2 nin=3 nin=4

-1.0 -1.5 -0.3 -0.2 -0.1

0.3

Probe detuning Δp (MHz)

0

0.1

0.2

0.3

Probe detuning Δp (MHz)

Figure 2: (Color online) The evolutions of the cavity transmitted intensity ratios t2 (a) and phase shift φ (b) versus the probe detuning Δp with different input probe photon number. √ The parameters are as G = 10.0 MHz, g N = 160.0 MHz, Δd = 0 MHz, γ = 3.0 MHz, γ  = 3.0 kHz. The loss rates of cavities A and B are κA = 3.0 MHz, κB = 3.0 kHz (left column) and κB = 30.0 kHz (right column).

155

with Δp = Δp −

G2 (1 + nin ) . Δp

(20)

The intensity ratio of the cavity-transmitted probe field t2 and phase shift φ as functions of probe detuning Δp with different input probe photon number nin are depicted in Figs. 2 (a) and 2(b), respectively. In order to see the influence of the loss rate of cavity B on cavity VIT, and we take κB = 3.0 kHz (left 160

column) and κB = 30.0 kHz (right column). Other experimentally realistic 9

√ parameters of the present QED system are G = 10 MHz, g N = 160 MHz, γ = 3.0 MHz, γ  = 3.0 kHz, κA = 3.0 MHz. The cavity mode of cavity B is assumed to be resonant with its corresponding transition, i.e., Δd = 0. As shown in Figs. 2(a) and 2(c), the linewidth of the standard cavity VIT 165

spectral peak at Δp = 0 are much smaller than that of the bare cavity when √ g N  G and the natural linewidth of state |1. While, different from the conventional cavity EIT with classical control field, the line width of the cavity VIT spectral peak is strongly dependent upon the input probe photon number. Within the transparency peak, the phase shift of the transmitted probe field

170

has a linear shape of normal dispersion. The most important is that the slope of the dispersion is sensitive to the photon number coupled into the cavity, and it is therefore possible to modulate the phase shift by controlling the probe photon number [see Fig. 2(b)]. These photon-number dependent transmission

175

and dispersion are totally different from the usual behaviors of cavity EIT. √ In the collective strong-coupling regime, i.e., g N > κA , γ, except the mode at zero detuning, one can calculate the other two modes corresponding to vacuum Rabi splitting peaks by using Eqs. (11) and (16). Their positions are given by  Δp = ± g 2 N + G2 (1 + nin ).

(21)

As did in Refs. [10, 12], we ignore the atomic absorption represented by γ in 180

the above derivation. The dependence of vacuum Rabi splitting upon the input probe photon number nin is plotted in Fig. 3 with the same parameter conditions in Fig. 2. The dashed curves in Figs. 3(a) and 3(b) are given by Eq. (21). The photon-number-dependent vacuum Rabi splitting is totally different from that corresponding to the two-leve ensemble [10] and three-level ensemble with

185

classical control field [12].

10

100

(b)

80

0.8

60

0.6

60

0.6

40

0.4

40

0.4

20

0.2

20

0.2

n

n

in

0.8

−2

0

2

Δ

−2 8

x 10

p

(c)

100

80

in

(a)

0

2

Δ

8

x 10

p

40

40

(d)

35

35

0.8

0.8 30

30 25 in

0.6

20

n

n

in

25

0.4

15 10

0.2

5 −1

0.6

20 0.4

15 10

0.2

5 −0.5

0

Δp

0.5

1

−1

7

x 10

−0.5

0

Δp

0.5

1 7

x 10

Figure 3: (Color online) The cavity transmitted intensity ratios t2 versus the probe detuning Δp and input probe photons nin with κB = 3.0 kHz (a) and κB = 30.0 kHz (b). The dashed curves in (a) and (b) are given by Eq. (21). The details in the boxes of (a) and (b) are, respectively, shown in (c) and (d). All other parameters are same as those in Fig. 2.

4. Photon number-dependent magneto-optical rotation and single photon separation As demonstrated in the above section, both the transmission and dispersion of the atom-cavity system under consideration are sensitive to the input 190

probe photon number. By placing the atom-cavity system in magnetic field, the behaviors of magneto-optical rotation (MOR) such as the rotation angle of the weak probe polarization direction after transmitting through and/or reflecting by the cavity system depend upon the input probe photon number. Thus, the natural question is whether we can separate single photon component from

195

Fock-state to create a deterministic single photon source via photon-numberdependent MOR. In this section, we investigate the photon number dependent

11

Figure 4: (Color online) (a) Atom-cavity system and (b) four-level tripod configuration for single photon separation. The magnetic field along the cavity axis defines the quantization axis and lifts the degeneracy of the ground states with mF = ±1, producing an energy shift by an amount ¯ hδ. Single sided cavity A is assumed to support both the left and right circular-polarized cavity modes {ˆ a+ , a ˆ− }. Two cavity modes are degenerate with resonant frequency ωc1 . The horizontal-polarized coherent probe field couples into cavity A through mirror M1 and its left (σ+ ) and right (σ− )-polarized components, respectively, drive the quantum transitions |− ↔ |1 and |+ ↔ |1 with equal coupling strength g. The output probe field reflected off the cavity A is detected by photon detector. The transition |2 ↔ |1 is driven by cavity B field ˆb with corresponding vacuum Rabi frequency G. All parameters of cavity A and B are same as those in Fig. 1 except that cavity A is single sided optical cavity.

MOR in the atom-cavity system, and apply it in separation of single photon from higher photon number components. The atom-cavity system and atomic configuration for single photon separa200

tion are shown in Figs. 4(a) and 4(b). We assume that the single sided optical cavity A supports the degenerated left and right circular-polarized cavity modes  is applied along the probe wave vector to lift the {ˆ a+ , a ˆ− }. A magnetic field B degeneracy of the ground state Zeeman sublevels |± corresponding to magnetic unmber mF = +1 and mF = −1 by amount δ = mF gL μB B, where gL

12

205

and μB = 1.4 MHz/G being Lande g-factor and Bohr magneton. A horizontal polarized probe field with carrier frequency ωp can be decomposed into two circularly polarized components with equal amplitudes but opposite handness and difference phases. The probe components of σ+ and σ− couple the quantum transitions |− ↔ |1 and |+ ↔ |1, respectively. The probe field reflected by

210

the atom-cavity system can be collected by detectors. Considering the symmetry of the atomic configuration (dipole moments and C-G cofficients corresponding to the quantum transitions |± ↔ |1), we assume that the cavity-atom coupling for both circular components are same, and denote them by g. The cavity mode ˆb drives the |2 ↔ |1 transition. The corresponding vacuum Rabi frequency

215

is denoted as G. Following the standard processes in the above section, one obtains the following set of coupled differential equations   (i) (i) (i) (i) (i) ˆ++ − σ a+ σ ˆ+1 = id+1 σ ˆ+1 + igˆ a− σ ˆ11 + igˆ ˆ+− ∂t σ

(i)

∂t σ ˆ−1

(i) +iGˆbˆ σ+2 + Fˆ+1 ,   (i) (i) (i) (i) ˆ−− − σ a− σ = id−1 σ ˆ−1 + igˆ a+ σ ˆ11 + igˆ ˆ−+ (i) +iGˆbˆ σ−2 + Fˆ−1 ,

  ∂ˆb (i) (i) (i) (i) (i) ˆ + Fˆ+2 , ∂t ˆbˆ a−ˆbˆ ˆ+1 + σ σ+2 = id+2ˆbˆ σ+2 − igˆ σ12 + iGˆbˆb† σ ∂t +2   ∂ˆb (i) (i) (i) (i) (i) ∂t ˆbˆ σ−2 = id−2ˆbˆ σ−2 − igˆ σ12 + iGˆbˆb† σ ˆ + Fˆ−2 , a+ˆbˆ ˆ−1 + σ ∂t −2   (i) κA a ˆ˙ + = − (κA − iΔ1 ) a a ˆ+in , ˆ+ + i gˆ σ−1 + τ i   (i) κA ˙a ˆ− = − (κA − iΔ1 ) a a ˆ−in , ˆ− + i gˆ σ+1 + τ i  (i) ˆb˙ = −κB ˆb + i Gˆ σ21 ,

(22)

(23) (24) (25) (26) (27) (28)

i

in which d±1 = Δp ∓ δ + iγ, d±2 = Δp − Δd ∓ δ + iγ  with Δp = ωp − [ω1 − (ω+ + ω− )/2] being the probe detuning. Starting from the above set of equations in adiabatic limit, the reflected two probe components by cavity A 220

can be calculated immediately, and they read R

r± eiφ± =

κA − 1. κA − iΔ1 − iχ± 13

(29)

Here, the photon number-dependent susceptibilities χ± are given by χ± = −

g2 N , 2 [d±1 − G2 (1 + n2 )/(d±2 + iκB )]

(30)

with n2 =

g2 N nin . 2G2 + g 2 N

(31)

In the above derivation, we define the conversed exciton operator as ˆ =a N ˆ†+ a ˆ+ + a ˆ†− a ˆ− + ˆb†ˆb +



(i)

σ ˆ11 .

(32)

i

√ ˆ = (ˆ According to the relation H σ+ + σ ˆ− )/ 2, we assume that n+ = n− = nin /2 225

with n± = ˆ a†± a ˆ±  being average photon numbers with cavity modes a ˆ± . Figure 5 exhibits the dependences of the intensity ratio r2 (top row) and polarization rotation angles φR (bottom row) of the probe field reflected by cavity A as functions of input probe photon number nin with different value √ of g N . Both r2 and φR are given by Eqs. (29) and (30). As usual, the

230

T,R polarization rotation angles are defined as φT,R = (φT,R + − φ− )/2. We assume

that the input horizontal polarized probe field is resonant with the cavity mode a ˆ± and its corresponding transitions without magnetic field, i.e. Δp = 0. The energy shift induced by magnetic field is taken as δ = 30 kHz (B  43 mG for D1 line of rubidium atoms). In the left (right) column, we choose the 235

loss rate of cavity B as κB = 3.0 kHz (κB = 30.0 kHz). As same those in Fig. 2, we take κA = 3.0 MHz, γ = 3.0 MHz, γ  = 3.0 kHz, G = 10.0 MHz, √ Δd = 0 MHz. The value of g N is chosen as 100 MHz, 160 MHz, 200 MHz, and 400 MHz. As a result of cavity-enhanced VIT, the stronger collective coupling between atomic ensemble and cavity modes a ˆ± , the higher reflectivity [as shown

240

in Figs. 5(a) and 5(c)] could be realized associated with larger polarization rotation of cavity-reflected probe field [as depicted in Figs. 5(b) and 5(d)]. In the present atom-cavity system, the probe photon coupled into cavity A can be converted into photons with cavity B mode ˆb via coherent Raman scattering process, the reflection r2 therefore depends upon the input probe photon number

245

nin , as well as the rotation angles φR [see Figs. 5(a)-5(d)]. The intensity ratio 14

(c) 1

(a) 1

1/2

gN =100 1/2 gN =160 gN1/2=200 gN1/2=400

0.8

1/2

MHz MHz MHz MHz

gN =100 1/2 gN =160 gN1/2=200 gN1/2=400

0.8

MHz MHz MHz MHz

0.6

r2

r2

0.6 0.4

0.4

0.2

0.2

0 0

5

10 nin

15

0 0

20

(d)1.5

1

1

10 nin

15

20

φ

φ

R

R

(b)1.5

5

0.5

0.5 1/2

gN =100 1/2 gN =160 1/2 gN =200 1/2 gN =400

0 0

5

10 nin

1/2

MHz MHz MHz MHz

15

gN =100 1/2 gN =160 1/2 gN =200 1/2 gN =400

0 0

20

5

10 nin

15

MHz MHz MHz MHz

20

Figure 5: (Color online) The intensity ratios of r 2 (top row) and Faraday rotation angles φR (bottom row) given by Eqs. (29) and (30) versus the photon number of input probe field nin √ with different value of g N . The probe detuning and energy shift produced by magnetic field are, respectively, taken as Δp = 0 and δ = 30 kHz. Two values of κB are considered, i.e., κB = 3.0 kHz (left column) and κB = 30.0 kHz (right column). All other parameters for the calculations are same as those in Fig. 2.

of cavity-reflected probe field r2 varies rapidly within small input probe photon √ number regime (for example nin ≤ 3). If we fix the value of g N , for example √ g N = 160 MHz, the probability of the probe reflected by cavity A decreases from r2  0.56 with nin = 1 to r2  0.38 with nin = 2 with κB = 30.0 kHz. 250

When we increase the input probe photon number much more, the values of r2 is close to 0. This is an natural result of photon number-dependent cavity VIT. The photon number-dependent MOR in the present atom-cavity system

15

(b) 1

φ21 R φ32 R φ43 R

2

r2(nin=1) r2(nin=2) r2(nin=3)

0.1

0 0

r

R

φm,m+1

(a) 0.2

0.5

1 ζ

1.5

0.5

0 0

2

0.5

1 ζ

1.5

2

R Figure 6: (Color online) The differences of rotation angle φR 21 (solid curve), φ32 (dashed curve), 2 φR 43 (longdashed curve) given by Eq. (33) and the refection r given by Eq. (34) corresponding

to different input probe photon number nin as functions of ζ.

Figure 7: (Color online) Schematic of cavity chain for single photon separation.

makes it is possible to separate single probe photon from higher photon number components in the direction of polarization. We assume that the probe field 255

is initially in a single-mode coherent state, which can be expanded as a single ∞ mode superposition of Fock states as |α = n=1 αn |n. It is instructive to investigate the difference of the Faraday rotation angle between m and m + 1 components associated with the reflection. We apply the resonance condition for the probe laser and the cavity (Δ1 = Δp = Δd = 0) and derive the analytical

260

results with γ  = 0 and κB = 0, i.e., negelecting the decoherence between three 16

lower states and decay rate of cavity B. Considering the situation g 2 N  G2 , n2 can be replaced by nin approximately. Under the conditions of the strong collective coupling and weak magnetic field, i.e., g 2 N  κ1 γ and G2  γδ, the intensity ratios of the cavity-reflected probe field can be simplified as 2 r2 = r± 265



1 , 1 + (1 + nin )2 ζ 2

(33)

in which the parameter ζ is defined as ζ = κG2 /(δg 2 N ). The above relation suggests that it is possible to detect single photon in the cavity-reflected probe field. The Faraday rotation angle difference between components with m and m + 1 components after reflected by cavity A is given by φR m,m+1

φR (nin = m + 1) − φR (nin = m)   ζ .  tan−1 1 + (m + 1)(m + 2)ζ 2

=

(34)

R Figures 6(a) and 6(b) exhibit the dependences of φR 21 (solid curve), φ32 270

2 (dashed curve), φR 43 (longdashed curve) and the ratio of reflection r upon the

parameter ζ according to Eqs. (34) and (33). ζ = 0 means G = 0, the atomic configuration can be simplified as two two-level scheme, the probe field is therefore reflected totally. No probe photon converts from mode a ˆ to ˆb, the Faraday rotation angle φR does not sensitive to the probe photon number coupled into 275

cavity A. Increasing the value of ζ, the transparency channel can be opened gradually, the reflection of the probe field decreases [see Figs. 5(a)]. Since the Raman scatter coherently by the atomic ensemble, the Faraday rotation angle φR becomes sensitive to the input probe photon number nin . When we increase R the value of ζ, φR 21 increases and reaches its maximum φ21 = 0.2 rad at ζ  0.4,

280

and then decreases. With ζ = 0.4, the ratio of reflection r2 decreases rapidly with the increase of the probe photon number nin [see Fig. 6(b)]. In order to separate single probe photon from higher photon components in the polarization direction, the difference of Faraday rotation angle φR 21 has to be of the order of π/2. The above analysis show that the value of φR 21 is not large

285

enough, a cavity chain as shown in Fig. 7 is therefore needed. In Fig. 8, we depict the numerical results of the relative probabilities of different photon number 17

(c) 1

p1 p2 p3 p4 p5

0.5

0 1

Probabilities

Propabalities

(a) 1

2

3

4

5

6

7

0.5

0 1

8

p1 p2 p3 p4 p5

2

3

4

n

6

7

8

(d) 1

Probabilities

Probablities

(b) 1

p1 p2 p3 p4 p5

0.5

0 1

5 n

2

3

4

5

6

7

0.5

0 1

8

n

p1 p2 p3 p4 p5

2

3

4

5

6

7

8

n

Figure 8: (Color online) The relative probabilities of different photon number components |i pi (i = 1−5) after reflecting by cavity A as functions of n with κB = 3.0 kHz (left column) and κB = 30.0 kHz (right column). Two situations are considered: (top row) based on photon number-dependent cavity VIT; (bottom row) based on photon number dependent cavityenhanced MOR. The probe field is assumed to be initially in coherent state with |α|2 = 4. √ The parameters for numerical calculations are Δp = 0, g N = 160.0 MHz, κA = 3.0 MHz, and the other parameters are taken as γ = 3.0 MHz, γ  = 3.0 kHz, G = 10.0 MHz, Δd = 0 MHz.

components pi (i = 1−5) in the cavity-reflected light as functions of the number of cavity system n. The top row and bottom row are, respectively, calculated based on photon number dependent cavity VIT without MOR and photon290

number dependent cavity MOR. To be specific let us assume that the probe field is initial in coherent state with |α|2 = 4. We take the loss rate of single-sided cavity A and B as κA = 3.0 MHz, κB = 3.0 kHz (left column) and κB = 30.0 kHz

18

(right column). In Fig. 8, we neglect all propagation and uncoupling loss for the √ √ sake of simplicity. The probe detuning Δp and g N are Δp = 0 and g N = 295

160 MHz (with this set of parameters, one has ζ  0.4). All other parameters use in numerical calculation are taken as γ = 3.0 MHz, γ  = 3.0 kHz, G = 10.0 MHz, Δd = 0 MHz. By utilizing of photon number-dependent cavity VIT, by increasing the number of cavity scheme, the probability p1 increases, and reaches 0.9538 after reflecting by eight cavity schemes [see Fig. 8(c)]. By using

300

of photon number-dependent MOR, the probability of separating single photon from higher photon-number components of a few-photon probe field can be enhanced dramatically. After reflecting by five cavity schemes, the probability p1 can be enhanced to 0.9975 [see Figs. 8(d)]. The present atom-cavity system could be used to create a deterministic single-photon source.

305

5. Conclusions In conclusion, we suggest a cavity QED scheme to investigate the dependence of transmission spectrum upon the photon number coupled into the cavity. The atomic ensemble is confined in two optical cavities, whose modes couple their corresponding quantum transitions. Assuming adiabaticity and an infinite

310

homogeneous medium, we derived general expressions for the dependence of transmission, reflection, phase shift and Faraday rotation angle upon the photon number. Our study demonstrates that the photon coupled into cavity A can be transferred to be mode ˆb via the coherent Raman scattering, leading to the dependence of the transmission and dispersion in the central peak and the positions

315

of vacuum Rabi splitting peaks on the photon number. By placing the atomcavity system in magnetic field, the photon number-dependent dispersion in the central peak results in the photon number-selected MOR when a weak linearly polarized probe field reflected by the cavity. The photon number-dependent MOR can be used to separate single probe photon from higher photon-number

320

components of a few-photon probe field and create a deterministic single photon source. Our numerical calculations show that the efficiency of achieving single

19

photon with 0.9975 is feasible. We believe that the photon number-dependent MOR can be of significance to quantum information processing.

Acknowledgements 325

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