A Versatile, Dynamically-Balanced Low-Noise Optical Field Manipulator Using a Coherently-Prepared Atomic Medium

Abstract

We propose a versatile dynamic optical field manipulator using coherently prepared atomic medium. We show that by locking the pump power change with the two-photon detuning a π phase shifting can be realized with unit probe fidelity in a broad two-photon detuning range. The two-photon-insensitive π phase-shift mode with significantly reduced fluctuation makes this scheme an attractive system for low-noise phase gate operations.

Control of phase and intensity of single/few photon using an atomic medium has always been an important research subject with potential applications in quantum information processing and quantum-state manipulation [1, 3]. All-optical approaches are particularly desirable because of remarkable advantages of a light field such as monochromaticity, fast propagating velocity, high spatial and temporal coherence, and propagation controllability. In the past two decades many schemes have been proposed and investigated [5–17]. Many recent studies have shown that electromagnetically induced transparency (EIT)-based schemes can lead to significant nonlinear phase shift in the classical field limit [15, 18]. Experimentally, Kerr nonlinear effects arising in an N–type EIT medium [19] and a medium with stored atomic-coherence [20] have been studied for phase control operation in the classical field limit. In a Raman gain medium, a fast, all-optical continuously controllable Kerr nonlinear phase gate was demonstrated recently [21]. In addition, fast digital signal processing based on this controllable Kerr gate operation, all-optical multi-logic gate operations and transistor functionalities using a Kerr phase gate method [22] and high-fidelity fast polarization gate operations have also been demonstrated with record low gate control and switching light powers [23].

Although both EIT- and Raman Gain-based schemes have been widely investigated in atomic media, the direct generalization of these schemes to single/few photon limit prove to be more problematic. The low fidelity due to the significant probe field attenuation in EIT media [24–26] and the large quantum noise due to the amplification of the probe field in a Raman gain medium are main obstacles that prohibit realizing a high fidelity, low noise phase shifter in single/few photon limit.

In this Letter, we propose a three-level system with coherently prepared states for phase shifting operation. This system, which rests on the process of two-wave mixing with initial atomic coherence [27–29], has a number of interesting properties. For instance, with suitable operation parameters it can act as a tow-photon-broadband phase shifter or an attenuator/amplifier with zero phase shift. Specifically, we show that by locking the pump field intensity and two-photon detuning a π phase shift can be realized with unit probe fidelity in a broad probe field frequency range. This two-photon-insensitive π phase shift can significantly reduce the phase noise associated with a Raman gain process, making it an attractive scheme for high-fidelity, low-noise phase gate operation. We also show the possibility to realize a zero-phase dynamic light attenuator/amplifier and a total transparency with zero-phase-shift in a Raman gain medium.

The scheme under investigation is a three-state atomic medium interacting with a pump and a weak probe field. The two lower states are assumed to have been coherently prepared prior to the injection of the pump and probe fields (see Fig. 1). The equations of motion for the slowly varying density matrices elements are (neglecting ρ₂₂ terms in the first order perturbation treatment)

$${\dot{\stackrel{\sim }{\rho }}}_{21}-i\delta {\stackrel{\sim }{\rho }}_{21}\approx i{\mathrm{\Omega }}_{21}{\rho }_{11}+i{\widehat{\mathrm{\Omega }}}_{23}{\stackrel{\sim }{\rho }}_{31}-{\gamma }_{21}{\stackrel{\sim }{\rho }}_{21}+{\widehat{F}}_{21},$$ (1a)

$${\dot{\stackrel{\sim }{\rho }}}_{23}-i(\delta +{\delta }_{2ph}){\stackrel{\sim }{\rho }}_{23}\approx i{\widehat{\mathrm{\Omega }}}_{23}{\rho }_{33}+i{\mathrm{\Omega }}_{21}{\stackrel{\sim }{\rho }}_{13}-{\gamma }_{23}{\stackrel{\sim }{\rho }}_{23}+{\widehat{F}}_{23},$$ (1b)

$${\dot{\stackrel{\sim }{\rho }}}_{13}+i{\delta }_{2ph}{\stackrel{\sim }{\rho }}_{13}\approx i{\mathrm{\Omega }}_{12}{\stackrel{\sim }{\rho }}_{23}-i{\widehat{\mathrm{\Omega }}}_{23}{\stackrel{\sim }{\rho }}_{12}-{\gamma }_{13}{\stackrel{\sim }{\rho }}_{13}+{\widehat{F}}_{13},$$ (1c)

where δ and δ_2ph are the (large) one- and (small) two-photon detunings, Ω₂₁ = Ω_P = D₂₁E_P/ħ is the Rabi frequency of a classical pump field E_P, and ${\widehat{\mathrm{\Omega }}}_{23}=D_{23}{\widehat{E}}_p/\hslash$ is the Rabi frequency of the quantum probe field operator Ê_p. Operator ${\widehat{F}}_{\mathit{nm}}$ describes the quantum noise to the density matrix operator ${\stackrel{\sim }{\rho }}_{\mathit{nm}}$ [30].

FIG. 1.

A three-level scheme where two lower states are coherently prepared prior to the injection of a strong pump field E_P and weak quantum probe field Ê_p.

To examine the optical response of the system we first neglect quantum fluctuations. For a large one-photon detuning Eq. (1a) yields the adiabatic approximation ${\stackrel{\sim }{\rho }}_{21}\approx -{\mathrm{\Omega }}_{21}{\rho }_{11}/(\delta +{\gamma }_{21})$. Solving Eqs. (1b) and (1c) using time Fourier transform method, we obtained the Fourier transform of the density matrix element ${\stackrel{\sim }{\rho }}_{23}$ that describes the propagation dynamics of the probe field,

$${\mathcal{R}}_{23}(\omega )=\mathcal{D}(\omega )\widehat{W}(\omega )=i\widehat{W}(\omega )\frac{C(\omega )}{B(\omega )},$$

where Ŵ(ω) is the Fourier transform of the quantum probe field Ê_p, C(ω) = (−iω+iδ_2ph+γ₁₃)ρ₃₃−i|Ω₂₁|²/(δ−iγ₂₃)ρ₁₁, and B(ω) = |Ω₂₁|²+(iω−iδ_2ph−γ₁₃)(iω+iδ−γ₂₃. Physically, the first term in C represents the absorption process starting from |3〉 whereas the second term denotes the Raman gain process starting from the ground state |1〉. This dispersion function contains all the physics of a three-state Λ–scheme including EIT and Raman processes depending on the relative sizes of the detunings and pump/driving fields. In accord with the assumption of prepared states we assume in the following calculation that ρ₁₁ ≫ ρ₂₂ and ρ₃₃ ≫ ρ₂₂ (neglect pump depletion).

In time Fourier transform domain the nonlinear polarization for the probe field is given by the operator $\widehat{P}(\omega )=i\kappa {\mathcal{R}}_{23}(\omega )$ with $\kappa ={\mathcal{N}}_a{\omega }_p/(2\hslash {\epsilon }_{0}c)$ where ${\mathcal{N}}_a$ is atom density. The DC component of 𝒟(ω) is given by

$$\begin{array}{l}\mathcal{D}(0)=-\left(\frac{\text{Re}\left[B_0\right]\text{Im}\left[C_0\right]-\text{Im}\left[B_0\right]\text{Re}\left[C_0\right]}{\text{Re}{\left[B_0\right]}^2+\text{Im}{\left[B_0\right]}^2}\right)\\ +i\left(\frac{\text{Re}\left[B_0\right]\text{Re}\left[C_0\right]+\text{Im}\left[B_0\right]\text{Im}\left[C_0\right]}{\text{Re}{\left[B_0\right]}^2+\text{Im}{\left[B_0\right]}^2}\right),\end{array}$$ (2)

with Re[B₀] = |Ω₂₁|² + δ_2phδ + γ₃₁γ₂₃, Im[B₀] = δ_2phγ₂₃−δγ₁₃, Re[C₀] = γ₁₃ρ₃₃ − γ₂₃Xρ₁₁ and Im[C₀] = δ_2phρ₃₃ + δρ₁₁ where $X={\left|{\mathrm{\Omega }}_{21}\right|}^2/({\delta }^2+{\gamma }_{21}^2)$. Since iκ𝒟(0) = iκRe[𝒟(0)] − κIm]𝒟(0)] we see that Im[𝒟(0)] gives the gain or loss whereas Re[𝒟(0)] gives the phase. Equation (2) has several interesting properties.

1. Two-photon Broad-band Phase Shifter

Setting κIm[𝒟(0)]=0 by making Re[C₀]Re[B₀]+Im[C₀]Im[B₀]=0 in Eq. (2), we obtain

$${\gamma }_{23}{\rho }_{11}{X}^2+{\gamma }_{13}({\rho }_{11}-{\rho }_{33})X-{\gamma }_{23}\frac{{\delta }_{2ph}^2+{\gamma }_{13}^2}{{\delta }^2+{\gamma }_{23}^2}{\rho }_{33}=0.$$ (3)

Clearly, Eq. (3) has no meaningful solution if ρ₃₃=0, signifying the importance of coherently prepared states.

Equation (3) yields a solution for X as a function of δ_2ph, which “locks the pump with the two-photon detuning”. When this dynamic-locking condition is enforced we will always have κIm[𝒟(0)]=0. Consequently, zero gain/loss to the probe field everywhere in the region of interest (i.e., the two-photon detuning δ_2ph) can be achieved. This “|Ω_P|²-δ_2ph-locking” is the key operation condition that allows a significant suppression of the atom-light-interaction-noise in a gain medium.

In the case where maximum atomic coherence is created prior to the injection of the probe and pump fields, i.e., ρ₁₁ = ρ₃₃ = ρ₃₁ = 0.5, Eq. (3) gives

$$X=\frac{{\left|{\mathrm{\Omega }}_{21}\right|}^2}{{\delta }^2+{\gamma }_{23}^2}=\sqrt{\frac{{\delta }_{2ph}^2+{\gamma }_{13}^2}{{\delta }^2+{\gamma }_{23}^2}.}$$ (4)

Using Re[C₀] = −Im[C₀]Im[B₀]/Re[B₀] from the requirement of Im[𝒟(0)]=0 we immediately obtain the phase shift per unit propagation distance as ϕ = κRe[𝒟(0)],

$$\varphi =-\kappa \frac{\text{Im}\left[C_0\right]}{\text{Re}\left[B_0\right]}=-\kappa \frac{{\delta }_{2ph}{\rho }_{33}+\delta {\rho }_{11}X}{({\delta }^2+{\gamma }_{23}^2)X+{\delta }_{2ph}\delta +{\gamma }_{23}{\gamma }_{13}}.$$ (5)

Consider the operation condition that δ_2phδ dominates the denominator (i.e., δX < δ_2ph) then Eq. (5) gives

$${\varphi }_0=\kappa \text{Re}[\mathcal{D}(0)]\approx -\frac{\kappa }{\delta }{\rho }_{33}.$$

This is a remarkably simple result indicating a typical one-photon transition type of phase shift with a flat zero-gain/loss dispersion, the situation that cannot be achieved with a one-photon process. In addition, since in a typical operation only δ_2ph is scanned, thus we have achieved a constant phase shift over the entire region of the operation (i.e., insensitive to δ_2ph). Indeed, these features can never be achieved by any simple three-state EIT scheme or Raman gain scheme alone.

In Fig. 2, we plotted the phase shift and the gain or loss dispersion of the probe field as a function of the two-photon detuning. To show a constant π-phase shift with flat zero loss or gain dispersion, we choose the system parameters as σ₁₁ = σ₃₃ = 0.5, κ = 10¹¹ s⁻¹cm⁻¹, γ₂₁/2π = 6 MHz, γ₂₃/2π = 6 MHz, γ₃₁/2π = 10 kHz, δ/2π = −1 GHz, Ω₂₁/2π = 30 MHz and L = 1 cm. It is clear to see that there exist a wide range of two-photon detuning (corresponding to a probe field with broad frequency domain) to realize a π-phase shift with zero gain or loss by locking the pump field intensity and the one-photon detuning simultaneously. It is worthy to point out that this flat zero gain/loss dispersion contributes to extremely low quantum noise fluctuation, which will be discussed in detail in the following section.

FIG. 2.

Probe phase shift κRe[𝒟(0)]L (red solid-line) and the loss/gain κIm[𝒟(0)]L (blue dashed-line) as a function of the two-photon detuning δ_2ph. A flat zero gain/loss dispersion with a constant π-phase shift can be achieved by maintaining the locking condition Eq. (4).

2. Dynamic Optical Field Attenuator/Amplifier With Zero Phase Shift

From Eq. (2), taking Re[B₀]Im[C₀] − Im[B₀]Re[C₀] = 0 we obtain Re[𝒟(0)]=0 and

$$\delta {\rho }_{11}{X}^2+{\delta }_{2ph}X+\delta {\rho }_{33}\frac{{\delta }_{2ph}^2+{\gamma }_{13}^2}{{\delta }^2+{\gamma }_{23}^2}=0.$$ (6)

The dynamic attenuation coefficient is then given by

$$\alpha =\kappa \text{Im}[\mathcal{D}(0)]=\kappa \frac{{\delta }_{2ph}{\rho }_{33}+\delta {\rho }_{11}X}{{\delta }_{2ph}{\gamma }_{23}-\delta {\gamma }_{13}},$$ (7)

where X is given by the solution of Eq. (6). Thus, one achieves a phase-insensitive optical field attenuation or amplification depending on the choice of parameters.

In Fig. 3 we show the probe phase shift and loss/gain dispersion (δ/2π = −100 MHz, Ω₂₁/2π = 10 MHz, other parameters are the same as in Fig. 2). Here, two specific probe field frequencies, corresponding to two δ_2ph at which dynamic probe field absorption and amplification can be achieved with zero phase shift.

FIG. 3.

(color online) Probe phase shift κRe[𝒟(0)]L (red solid-line) and the loss/gain κIm[𝒟(0)]L (blue dashed-line) versus the two-photon detuning. The vertical dashed- and solid-lines indicate significant probe attenuation and amplification without phase change.

3. Dynamic Probe Field Transparency

Probe transparency in a resonant medium relies on EIT configurations where both one- and two-photon detunings are zero, resulting in the loss of tunability unless a strong pump field is used to create a large transparency window. It is possible, however, to achieve a total transparency with a weak pump in our scheme. It is seen from Eq. (2) that by making $X={\left|{\mathrm{\Omega }}_{21}\right|}^2/({\delta }^2+{\gamma }_{23}^2)=-{\rho }_{33}{\delta }_{2ph}/{\rho }_{11}\delta$ and −δ_2ph/δ = γ₁₃/γ₂₃, then C₀=0 can be achieved. This is a total transparency without phase shift (see Fig. 4). Clearly, this is possible only when ρ₃₃ ≠ 0.

FIG. 4.

(color online) Probe phase shift κRe[𝒟(0)]L (red solid-line) and the loss/gain κIm[𝒟(0)]L (blue dashed-line) versus the two-photon detuning. The vertical dashed-line denotes the two-photon-detuning at which a total transparency with zero phase shift can be achieved. Ω₂₁/2π = 30 MHz and δ/2π = −37.5 MHz. Other parameters are the same as in Fig. 2.

4. Low Noise Probe Field Phase shifter

We now examine the quantum noise characteristics of the probe field when a large phase shift is realized (see sec.1). The Maxwell equation for the probe field Rabi frequency in the slowly varying amplitude and phase approximation is given by

$$\frac{\partial {\widehat{\mathrm{\Omega }}}_p}{\partial z}+\frac{1}{c}\frac{\partial {\widehat{\mathrm{\Omega }}}_p}{\partial t}=i\kappa {\stackrel{\sim }{\rho }}_{23}+\frac{D_{23}}{\hslash }\widehat{F}(z,t),$$ (8)

where the Langevin-like vacuum fluctuation noise operator is given by (define D₀ = (ω + d₁₃)(ω + d₂₃) − |Ω₂₁|²),

$$\widehat{F}(z,t)={\widehat{b}}_{13}(t){\widehat{F}}_{13}(z,t)+{\widehat{b}}_{23}(t){\widehat{F}}_{23}(z,t).$$ (9)

where b₁₃ = Ω₂₁/D₀, b₂₃ = −(ω + d₁₃)/D₀.

Equation (8) can be solved by formal integration over z to yield the delayed output probe field [30–32],

$$\widehat{W}(L,\omega )=\widehat{W}(0,\omega )e^{-\mathrm{\Lambda }L}+\frac{D_{23}}{\hslash }{\displaystyle {\int }_0^L{e}^{-\mathrm{\Lambda }(L-s)}}\mathcal{F}(s,\omega )ds,$$ (10)

where Λ = Λ(ω) = −iκ𝒟(ω) and $\mathcal{F}(s,\omega )$ is the Fourier transform of Eq. (9). Using Eq. (10) and applying the quantum regression theorem [33, 34], the probe field amplitude noise spectrum in the Fourier domain is

$$S_X(L,\omega )=S_{X1}(L,\omega )+S_{X2}(L,\omega )+S_{X3}(L,\omega ),$$ (11)

where lengthy expressions of S_XJ(L, ω) (J=1,2,3) can be obtained using procedures illustrated in Ref.[30–32]. In Eq. (11), S_X1(L, ω) arises from the amplitude noise spectrum of the input probe S_X(0, ω), S_X2(L, ω) represents the contribution of the phase noise spectrum of the input probe S_Y (0, ω) via the phase–to–amplitude noise conversion [35, 36], and S_X3(L, ω) arises from atomic noise due to random decay process. The probe field phase noise spectrum can be calculated similarly,

$$S_Y(L,\omega )=S_{Y1}(L,\omega )+S_{Y2}(L,\omega )+S_{Y3}(L,\omega ).$$ (12)

In Fig. 5 we plot probe amplitude and phase noise spectra versus δ_2ph at the exit of the medium. The input field is in 3-dB squeezed state with quadrature components S_X(0, ω) = 0.5 and S_Y (0, ω) = 2. The probe field acquires negligible additional quantum noise in a broad δ_2ph region where zero loss/gain is achieved with a π phase shift. It is in this broad two-photon detuning region that a high-fidelity, low-noise phase-gate operation may be realized.

FIG. 5.

(color online) Plot of amplitude (S_X(L, ω = 0), blue dashed-line) and phase (S_Y (L, ω), red dot-dashed-line) noise spectra versus δ_2ph un the condition of π phase shift (see Fig. 1). Black-solid curve: Normalized probe field using Eq.(8).

In conclusion, We have investigated a novel three-state two-wave mixing scheme with coherently prepared states. The new scheme has many intriguing properties and may operated in several modes as a versatile π-phase shifter or a zero-phase attenuator/amplifier. By locking the pump excitation with the two-photon detuning a π phase shifting can be maintained with negligible additional operational quantum noise and also a unit probe fidelity in a broad two-photon detuning range which are very attractive virtual of the scheme that may lead to the realization of a low-noise phase gate operation.