Quantum enhanced metrology is the use of quantum techniques to improve measurement precision, which has been received a great deal of attention in recent years.^[1–3] The Mach–Zehnder interferometer (MZI) and its variants can provide high precise measurements, based on which the gravitational waves were observed.^[4,5] The sensitivity of these interferometers is limited by the vacuum quantum noise injected from the unused port. To further improve measurement sensitivity, Yurke et al.^[6] introduced a new type of interferometer in which two four-wave mixers or parameter amplifiers (PAs) occupy the place of two linear beam splitters (BSs) in the traditional MZI. It is also called the SU(1,1) interferometer because it is described by the SU(1,1) group, as opposed to SU(2) for BSs. Because it can be used to improve measurement accuracy, these types of interferometers have received extensive attention both experimentally^[7–16] and theoretically.^[17–30] Kong et al.^[18] firstly proposed a simplified scheme of PA + BS, and it can also beat the standard quantum limit of phase-measurement sensitivity by a similar amount for the SU(1,1) interferometer. The improvement comes from a result of noise reduction of the interferometer because the quantum noise of the outputs of the parametric amplifier is entangled and destructive interference at the beam splitter leads to the quantum noise cancellation. The scheme of PA + BS can reduce the difficulty of the experimental implementation. However, the scheme of PA + BS requires the two outputs of the PA to be frequency degenerate and spatial modes separation. Therefore, it is worth initiating a study on how to generate a two-mode squeezed state of frequency degenerate and spatial modes separation.

The first experimental demonstration of squeezed states of light by Slusher et al.^[31] was based on four-wave mixing (4WM) in sodium vapor. Since then, many techniques for producing different types of squeezing have been explored, each with their own advantages and limitations for particular applications.^[32] Nondegenerate 4WM in a double-Λ scheme^[33] was identified as a possible scheme to generate a squeezed state or squeezed twin beams, as described in Refs. [34–41].

The generated twin beams by the 4WM process in an atomic system with higher squeezing degree were firstly realized by McCormick et al.^[40,41] based on degenerate pump fields. A single linearly polarized pump beam is crossed at a small angle with an orthogonally polarized much weaker probe beam. The 4WM process amplifies the probe and generates a quantum-correlated conjugate beam, on the other side of the pump (at a higher frequency). In this case, a pair of photons of the (single) pump is transformed, via the 4WM process, into a photon in the probe beam and a photon in the conjugate beam. By modulating the involved ground (excited) state with one (two) laser beam (beams), the gain and squeezing degree can be enhanced.^[42,43] The best initial results for two-mode intensity-difference squeezing at low frequencies seem to be ≈ 1.5 kHz^[44] to the recently reported ≈ 700 Hz^[45] or even ≈ 10 Hz.^[46] The generated entanglement between the probe and conjugate beams can realize quantum imaging.^[47,48] The cascaded 4WM can generate the quantum correlated triple beams^[49,50] and can also be used to realized SU(1,1) interferometers for highly sensitive phase measurements.^[16,51] This 4WM process also supports many spatial modes, making it possible to amplify complex two-dimensional spatial patterns.^[52–55]

Recently, a new 4WM process driven by two pump fields at a small angle was realized,^[56] where we only demonstrated the phenomena and results, without giving detailed physical explanations. Turnbull et al.^[57] developed a good theory to illustrate a 4WM process driven by a single pump field. In this paper, we describe the two pump field phase matching that can be established between the n_p > 1 and n_p < 1 by the angle adjustment, which is different from the single pump case, and describe the large bandwidth of the gain varying with the two-photon detuning. The theoretical range of phase-matching angles for achieving different regions is given.

Our paper is organized as follows. In Section 2, we describe the generation of two-mode squeezed light with two pump fields crossing at a small angle. In Section 3, the phase-matching angle range is given under the condition of two pump fields driven. In Section 4, the gain of two-mode squeezed light driven by one pump field and two degenerate pump fields is compared, and by an angle adjustment the phase matching between the n_p > 1 and n_p < 1 is described. Finally, we conclude with a summary of our results.

The generated frequency non-degenerate squeezed light by the 4WM process is shown in Fig. 1(a). Based on the degenerate pump fields, the twin beams with higher squeezing degree were generated,^[40,41] where the phase-matching geometry is shown in Fig. 2(a). In our scheme, one unique feature of this 4WM process is that two pump fields are applied at a small and adjustable angle, where the phase-matching geometry is shown in Fig. 2(b). The approach to generate the frequency degenerate twin beams is based on the idea of inverting the configuration from Fig. 1(a) to Fig. 1(b). For the geometry of a single pump field shown in Fig. 2(a), with a simple exchange of the roles of the pump and probe beams, single-mode squeezing is generated.^[58] To obtain the twin beams with the same frequency that are spatially nondegenerate, the geometry of two non-collinear pump fields shown in Fig. 2(b) is needed.

Fig. 1.

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	Fig. 1. Double-Λ scheme on the D1 line of 85Rb configuration for the generation of (a) the frequency non-degenerate and (b) frequency degenerate squeezed lights. The state |g,m⟩ involves the hyperfine levels |5S1/2, F = 2,3⟩. The hyperfine splitting of the excited state is not resolved due to Doppler broadening.

Fig. 2.

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Fig. 2. The geometric phase matching for (a) a single pump field and (b) two pump fields. The angle between pump fields kP1 and kP2 is θ0. The angles between the probe field kp and the pump fields kP1 and kP2 are θ1 and θ2, respectively. Here θ3 is the angle between the probe and the projected pump field. The frequency degenerate twin beams are generated with an exchange of the roles of the pump and probe beams.

In this section, we firstly theoretically describe the frequency non-degenerate and degenerate squeezed light based on a non-collinear 4WM process. As shown in Figs. 3(a) and 3(b), the double-Λ four-level processes of frequency non-degenerate and degenerate are the same except that the magnitudes of the detunings are different. Therefore, the frequency-degenerate and non-degenerate squeezed lights based on non-collinear 4WM can be described by a same set of equations.

Fig. 3.

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	Fig. 3. (a) Double-Λ scheme with two pump fields P1 and P2 for (a) the frequency non-degenerate and (b) frequency degenerate squeezed lights. States |3⟩ and |4⟩ are orthogonal linear combinations of magnetic states of the excited hyperfine levels. ΩP1 and ΩP2 are the Rabi frequencies, Δ1 (=ωP1 − ω31) and Δ2 (=ωP2 + δ − ω42) are the detunings, and δ is the two-photon detuning.

We assume that the two pump fields E_P1 and E_P2 couple the transitions |1⟩ → |3⟩ and |2⟩ → |4⟩, respectively. The probe field couples the transition |2⟩ → |3⟩, and the conjugate field couples the transition |1⟩ → |4⟩. The transitions |1⟩ → |2⟩ and |3⟩ → |4⟩ are not dipole allowed transitions. In the dipole and rotating wave approximations, the Hamiltonian of the atoms combined with the Hamiltonian of the light-atom interaction is given by

The equations for the atomic operators σ_nm (n,m = 1,2,3,4) in the Heisenberg picture are given in the Appendix. Using the atomic operators to evaluate the linear and nonlinear components of the polarization at ω_p and ω_c, the polarization of the atomic medium at a particular frequency is given by and , where N is the number density of the atomic medium. The polarizations of the medium at frequency ω_n (n = p,c) are given by

Under the condition of the slowly varying amplitude approximation, considering nearly co-propagating beams along the z axis, these field equations in the co-moving frame are written as

The number operators of the probe beam and conjugate beam are defined as and , respectively. From the above result, we define the gain of the probe beam in the 4WM process as

In this section, we describe how to obtain the optimal angles θ₁ and θ₂ between the probe field and two non-collinear pump fields when the angle θ₀ between pump fields P₁ and P₂ is given.

As shown in Fig. 4(a), when two pump fields E_P1 and E_P2 are incident at an angle, the total projection of the wavevector of the pump fields onto the z-axis is 2k_PZ and becomes smaller; i.e., 2 |k_PZ| < |k_P1| + |k_P2|. The geometric phase matching condition (GPMC) is given by

Fig. 4.

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Fig. 4. (a) The wavevectors kP1 and kP2 are projected onto the axis as 2kPZ. The angles between the kPZ and the pump fields kP1 and kP2 are θz1 and θz2, respectively. (b) The configuration in which the geometric phase matching condition (GPMC) is fulfilled. The configuration in which the effective phase matching condition (EPMC) is fulfilled for an effective index of refraction of the probe (c) np > 1 and (d) np < 1, with necessary geometric mismatches of (c) Δkz > 0 and (d) Δ kz < 0, respectively.

Using the conservation of energy condition ω_p + ω_c = ω_P1 + ω_P2 = 2ω₀, where ω₀ is the frequency of the pump field, and considering θ_z1 = θ_z2 = θ₀/2, equation (17) is written as

When n_p > 1, the EPMC of Eq. (18) is established to require θ₃ > θ₀/2, which means that the GPMC of Eq. (15) cannot be satisfied and will occur Δk_z > 0, as shown in Fig. 4(c). Considering θ₁ = θ₂ = θ and using the law of cosines we obtain the angle requirement between the probe field and the pump fields

If n_p < 1, similarly, the EPMC of Eq. (18) requires θ₃ < θ₀/2 and imposes Δk_z < 0, as shown in Fig. 4(d). In addition, the generated probe and conjugate beams have separate directions, which requires that the angle θ₃ > 0. Furthermore, using the minimum value of the refractive index min(n_p) according to Eq. (18), we obtain

For degenerate case, the form of Eqs. (15)–(17) is the same except for the magnitude of the wave vectors. With two strong beams with the frequency of the probe and conjugate beams, and along the direction of them, and a week beam having the frequency and direction of the previous pump, we can generate the frequency degenerate and spatial nondegenerate twin beams by changing the detunings of Δ₁ and Δ₂. Similar to the non-degenerate case, if n_p + n_c = 2, then GPMC Δk_z = 0 will be satisfied. If n_p + n_c > 2 or n_p + n_c < 2, then the corresponding GPMC Δk_z > 0 or Δk_z < 0 will also be obtained.

In this section, we numerically analyze the gain of the frequency non-degenerate squeezed light, and describe the phase matching between the n_p > 1 and n_p < 1 by an angle adjustment.

In our non-degenerate experiment,^[56] the state |g,m⟩ (or state |1,2⟩) involves the hyperfine levels |5S_1/2, F = 2,3⟩, where the hyperfine splitting of the ground state is ω₂₁ = 2π × 3.035 GHz, and the excited state |e⟩ (or state |3,4⟩) is |5P_1/2⟩ has an excited state decay rate of γ = 2π × 5.75 MHz. The pump field is blue-detuned approximately 1 GHz to the D1 line of Rb-85 5S_1/2 → 5P_1/2. The powers of the pump fields E_P1 and E_P2 are set to 350 mW, and their waists at the crossing point are 622 μm and 596 μm, respectively. The Rabi frequencies of Ω_P1 and Ω_P2 are Ω_P1 ≃ 28γ and Ω_P2 ≃ 30γ for an effective electric dipole d = 1.47 × 10⁻²⁹ Cm.^[62] The atomic number density of Rb-85 at 125 °C is approximately N ≃ 4.5 × 10¹⁸ m⁻³.

Figures 5(a) and 5(b) show the direct susceptibilities χ_pp and χ_cc for the probe and conjugate fields as a function of the two-photon detuning δ/γ, and we obtain that χ_cc is far less than χ_pp due to large detuning. In Fig. 5(a), when δ < 0, the real part of χ_pp is effectively responsible for the index of refraction of the probe for the single pump field case.^[59,57] However, for our two-pump-fields input case, the phase matching condition can also be satisfied on the other side δ > 0.

Fig. 5.

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Fig. 5. The direct and cross susceptibilities for the probe and conjugate fields as a function of the two-photon detuning δ/γ. The solid lines are the real parts, and the dashed lines are the imaginary parts. The excited state decay rate is γ = 2π × 5.75 MHz, and γc = 0.5γ. The hyperfine splitting of the ground state is ω21 = 2π × 3.035 GHz. The detuning of pump 1 is Δ1 = 174γ. The Rabi frequencies of ΩP1 and ΩP2 are ΩP1 = 28γ and ΩP2 = 30γ, respectively.

According to Eq. (12), we plot the probe gain G_p as a function of the two-photon detuning δ and the geometric phase mismatch Δk_z in the presence of a single pump field and two pump fields, as shown in Fig. 6. One can see that the maximum gains are obtained on the side δ < 0 with Δk_z > 0, for both the cases. When Δk_z < 0, the probe gain G_p does not exist for the single pump field input case and only occurs for the two-pump-fields input case. The bandwidth of the probe gain is relatively large due to the slow change in the refractive index with the two-photon detuning.

Fig. 6.

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Fig. 6. Theoretical output probe gain Gp as a function of the two-photon detuning δ/γ and the geometrical phase match Δkz with (a) a single pump field and (b) two pump fields. Here the excited state decay rate is γ = 2π × 5.75 MHz, the decoherence rate is γc = 0.5γ, the atom density is N = 4.5 × 1018 m−3, the length of the medium is L = 12.5 mm and the pump Rabi frequencies are ΩP1 = 28γ and ΩP2 = 30γ.

The theoretical output probe gain G_p as a function of the two-photon detuning δ/γ and the probe-pump angle θ with (a) a single pump field and (b) two pump fields is shown in Fig. 7, where we consider k_P1 = k_P2 = k_P and θ₁ = θ₂ = θ. It can be seen that for a single pump field, the gain and 4WM process are on the δ < 0 side as the angle θ increases due to phase matching. For the case of two pump fields, the gain and 4WM process can be achieved on the left side (line L₂) or the right side (line L₀ or line L₁) by choosing the angle θ between the probe field and the pump fields when the angle θ₀ is given.

Fig. 7.

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Fig. 7. Theoretical output probe gain Gp as a function of the two-photon detuning δ/γ and the probe-pump angle θ with (a) a single pump field and (b) two pump fields. The area intersecting the dashed line is the area selected by our experimental parameters. The angle between pump fields P1 and P2 is θ0/2 = 0.615°. The angles between the probe field and the pump fields P1 and P2 are θ1 = θ2 = θ = 0.861°.

The area intersecting the dashed line L₁ in Fig. 7(b) is the area selected by our experimental parameters, where θ₀/2 = 0.615°, and θ = 0.861°. According to the minimum value min(Re(χ_pp)) = −1.609 × 10⁻⁴ in Fig. 6, using Eq. (21), we obtain

Here θ₀ has a fundamental effect on wavevector matching in the new 4WM process, thus opening up a region in which high-intensity-difference squeezed light can be obtained over a wide bandwidth with low loss and moderate gain. The gain curve in Fig. 7(b) shifts upward as the angle θ₀ increases, because the minimum value of the angle θ is greater than the angle θ₀/2.

The gain of the probe field and the squeezing as a function of the two-photon detuning δ is shown in Fig. 8, where the square represents the experimental data from Ref. [56] and the solid line is a theoretical simulation. The theoretical simulations and experimental data of the gain of the probe field are in good agreement, as shown in Fig. 8(a). In the experiment, the intensity-difference-squeezed light for the nondegenerate case with quantum noise almost 7 dB reduction was generated.^[56] The squeezing degree is affected by the spatial mode mismatch, optical absorption by the atomic system, optical loss in the light path, and atomic decoherence. Figure 8(b) shows the theoretical simulations and experimental data of squeezing, where the theoretical squeezing curve is reduced by 0.56 times and the agreement is not very good because these effects are not included in our model in order to clarify the physics picture concisely. The bandwidth of the gain is relatively larger than that for the single pump field case, which is advantageous for realizing wide-bandwidth frequency-degenerate and nondegenerate intensity-squeezed light. These light fields can be widely used in quantum information and other fields.

Fig. 8.

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	Fig. 8. The gain of the probe field (a) and the squeezing (b) as a function of the two-photon detuning δ. The parameters are as follows: γ = 2π × 5.75 MHz, γc = 0.5γ, N = 4.5 × 1018 m−3, L = 12.5 mm, ΩP1 = 28γ, ΩP2 = 30γ, θ0/2 = 0.615°, and θ1 = θ2 = 0.861°.

We have studied that a two-mode squeezed light is generated from a 4WM process driven by two pump fields crossing a small angle, where the twin beams are generated with a new phase matching condition. Different from 4WM realized by a single pump field where the gain peak can only be achieved on the δ < 0 side, the new 4WM process is implemented from the δ < 0 side to the δ > 0 side by an angle adjustment. The refractive index of the corresponding probe field n_p can be converted from n_p > 1 to n_p < 1, which can also be used to convert between slow light^[63] and fast light.^[64] On the n_p < 1 side, the refractive index n_p changes slowly with the two-photon detuning δ over a large range, which leads to a relatively large gain bandwidth. With an exchange of the roles of the pump and probe beams, the frequency degenerate and spatial nondegenerate twin beams can be generated. This type of twin beams can be combined and interfered directly on the beam splitter, and can be applied in quantum information and quantum metrology.