Strong nonlinear coupling between an ultracold atomic ensemble and a nanomechanical oscillator

Gang Chen; Yongping Zhang; Liantuan Xiao; J.-Q. Liang; Suotang Jia

doi:10.1364/OE.18.023016

Cavity optomechanics is a powerful connection between a nanomechanical oscillator and a quantized electromagnetic field induced by a cavity mode through radiation pressure [1–6]. When the cavity is intensely driven, the radiation pressure constant can be enhanced dramatically with a factor of the mean cavity amplitude. However, this novel nonlinear interaction is lost and a strong linear interaction appears. In the recent experiment, this strong linear interaction has been observed by measuring the optomechanical normal mode splitting [7]. This paves the way towards full quantum optical control of the nanomechanical oscillator. On the other hand, due to this enhancement, the entanglement [8] and the Einstein-Poldosky-Rosen channels [9] between an atomic ensemble and a nanomechanical oscillator can be well established. Moreover, the nanomechanical oscillator can be also cooled to its ground state via resonant intracavity optical gain and absorption [10]. Recently, the strong coupling between a single atom and the nanomechanical oscillator can be realized [11]. In the present paper, we still focus on the weak nonlinear interaction induced by radiation pressure and obtain a new strong nonlinear coupling between an atomic ensemble and the nanomechanical oscillator, when the cavity mode is virtually excited.

In quantum optics, the coherent nonlinear effects are indispensable for investigating and controlling the dynamics of harmonic oscillators [12]. With the exception of producing essential quantum operations such as squeezing and parametric amplification, they have been regarded as important resources for processing universal quantum information with continuous variables [13]. Thus, it is highly desirable to develop new quantum devices, especially in the solid-state system, and schemes to realize strong nonlinear couplings [14–16].

The concrete form of the new nonlinear coupling obtained in this paper is ( $J_{+}^{2} b + J_{-}^{2} b^{†}$ ), where J_± are the collective spin operators of the ultracold atoms and b^† (b) is the phonon creation (annihilation) operator of the quantized nanomechanical motion. More intriguingly, the interaction strength is enhanced largely with a factor of N, and thus, arrives at a strong coupling regime within current experimental parameters. In the large-N limit, the excitation governed by the collective spin operators J_± behaves like a harmonic oscillator degree of freedom. Thus, our obtained nonlinear coupling describes the interaction between a pair of quasiparticle (boson) and the phonon. Physically, a pair of quasiparticle is excited when a phonon is emitted and vice versa. In the meanwhile, the nonlinear optical processes with the χ⁽²⁾ term are simulated successfully.

Figure 1 is a schematic experimental-setup of our proposed system consisting of an ultracold ⁸⁷Rb atomic ensemble placed in a high-finesse optical cavity with a nanomechanical oscillator. These ultracold atoms have a transition |F = 1〉(|1〉) → |F′ = 2〉(|2〉) of the D₂ line. For a high-finesse optical cavity, the maximum coupling strength between an ultracold atom and the cavity field is given in experiment by g = 2π × 11.4 MHz [17]. Thus, the collective coupling $\sqrt{N} g$ of the order 10² GHz is far larger than the cavity field decay rate κ = 2π × 1.3 MHz and the ultracold atomic dipole decay rate γ = 2π × 3.0 MHz. Without the ultracold atomic ensemble, the dynamics of the radiation-pressure interaction between the nanomechanical oscillator and the quantized electromagnetic field is governed, in the unit h̄ = 1 hereafter, by [1]

H_{CO} = {ω a}^{†} a + ω_{M} b^{†} b - ξ a^{†} a (b^{†} + b),

where a^† (a) is the photon creation (annihilation) operator of the optical cavity with frequency ω, and ω_M is the frequency of the nanomechanical motion. The radiation-pressure interaction constant is ξ = ωΔx/l₀, where l₀ is the length of the cavity and

Δ x = 1 / \sqrt{2 M ω_{M}}

is the zero-point uncertainty with M being the mass of the nanomechanical oscillator.

Fig. 1 (Color online) Schematic diagram for our proposed triple hybrid system with the ultracold atoms, photon and phonon. When the mirror oscillates, the wavelength of the photon will be affected, and correspondingly, the interaction strength between the ultracold atoms and the photon will be changed.

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When the prepared ultracold atomic ensemble is sent into the high-finesse optical cavity, the Hamiltonian for this triple-hybrid-system, in the two-mode approximation, is written as

H_{T} = H_{CO} + g (a^{†} S_{-} + {a S}_{+}) + ω_{0} S_{z} - {qS}_{z}^{2},

where

S_{+} = \sum_{i = 1}^{N} c_{i, 2}^{†} c_{i, 1}, S_{-} = \sum_{i = 1}^{N} c_{i, 1}^{†} c_{i, 2}, S_{z} = \frac{1}{2} \sum_{i = 1}^{N} (c_{i, 2}^{†} c_{i, 2} - c_{i, 1}^{†} c_{i, 1})

with

c_{i, 1} (c_{i, 1}^{†})

and

c_{i, 2} (c_{i, 2}^{†})

being the annihilation (creation) operators for the ith atomic states |1〉 and |2〉, respectively. The collective spin operators in Eq. (3) satisfy the SU(2) angular-momentum commutation relations [S_z, S_±] = ±S_± and [S₊, S₋] = 2S_z, where the Casimir invariant is S² = N(N/2 + 1)/2. The atom-photon interaction strength is given by

g = - μ ɛ sin {k x}_{0}

with k = 2πm/l₀ (m = 1, 2, 3...) being wave vector and

ɛ = \sqrt{ω / ɛ_{0} V}

, where μ is the electric-dipole transition matrix element, x₀ is the position of center of mass for the ultracold atomic ensemble, ɛ₀ is the dielectric constant in vacuum, V is the volume of the cavity [12]. The effective atomic resonant frequency is given by

ω_{0} = ω_{2} - ω_{1} + \frac{1}{2} (N - 1) (η_{2} - η_{1}) + ω_{12}

with

ω_{l} = \int d^{3} r {ϕ_{l}^{*} (r) [- \frac{\nabla^{2}}{{2 m}_{R}} + V_{l} (r)] ϕ_{l} (r) (l = 1, 2)

and

η_{l} = \frac{{4 π ρ}_{l}}{m_{R}} \int d^{3} r {| ϕ_{l} (r) |}^{4}

, where ϕ_l(r) is the macroscopic wave function of the ultracold atoms, V_l(r) is the single magnetic trap potential with frequency ω_i(i = x, y, z), m_R and ω₁₂ are the mass and resonance frequency of the ultracold atoms, respectively, and ρ_l is the intraspecies s–wave scattering length. The collision-induced long-range interaction among the ultracold atoms is given by

q = \frac{1}{2} [(η_{1} + η_{2}) - 2 χ_{1, 2}],

where

χ_{1, 2} = \frac{{4 π ρ}_{1, 2}}{m_{R}} \int d^{3} r {| ϕ_{1} (r) |}^{2} {| ϕ_{2} (r) |}^{2}

with ρ_1,2(= ρ_2,1) being the interspecies s–wave scattering length. In experiment, this collision-induced interaction can be adjustable by using the Feshbach resonance technique [18].

With a pump laser of cavity mode

H_{d - ph} (t) = Ω_{p} [a^{†} exp (- i ω_{p} t) + a exp (i ω_{p} t)]

and a driving laser of the ultracold atoms

H_{d - at} (t) = Ω_{a} [S_{+} exp (- i ω_{a} t) + S_{-} exp (i ω_{a} t)],

where Ω_p (Ω_a) and ω_p (ω_a) are the magnitude and frequency of the pump (driving) laser, the total time-dependent Hamiltonian for our proposal is given by

\tilde{H} (t) = H_{T} + H_{d - p h} (t) + H_{d - at} (t) .

In the rotating frame, the time-dependent Hamiltonian (9) becomes the time-independent case H̃_R = U^†(t)H̃(t)U(t) – iU^†(t)U̇(t) with U(t) = exp[−i(ω_pa^†a + ω_aS_z)t], namely,

\begin{array}{l} {\tilde{H}}_{R} = Δ_{p} a^{†} a + ω_{M} b^{†} b - ξ a^{†} a (b^{†} + b) + Δ_{a} S_{z} + \\ g (a^{†} S_{-} + a S_{+}) + Ω_{p} (a^{†} + a) + 2 Ω_{a} S_{x} - q S_{z}^{2}, \end{array}

where Δ_p = ω – ω_p – ω_a and Δ_a = ω₀ – ω_p – ω_a. In the following discussions, we mainly focus on the dispersive interaction (Δ = |ω – ω₀| >> g) between the atom and the photon, and the steady state of cavity field a_ss = – i(gS₋ + Ω_p)/[(κ + iΔ_p) – iξ (b^† + b)], which is obtained by solving a set of quantum Langevin equations [19]. In the dispersive limit, Hamiltonian (10) is rewritten in terms of the Fröhlich’s transformation as

H_{R} = U_{R}^{†} {\tilde{H}}_{R} U_{R} = exp (- F) {\tilde{H}}_{R} exp (- F)

, where F is the operator satisfying

H_{I} + \frac{1}{2} [H_{0}, F] = 0

with

H_{0} = Δ_{p} a^{†} a + ω_{M} b^{†} b - ξ a^{†} a (b^{†} + b) + Δ_{a} S_{z} + Ω_{p} (a^{†} + a) + 2 Ω_{a} S_{x} - q S_{z}^{2}

and H_I = g(a^†S₋ + aS₊) [20]. After a careful calculation, the operator F is given by

F = \frac{g}{Δ} (a^{†} S_{-} - a S_{+})

. Thus, by using the Baker-Campbell-Hausdorff formula

exp [- F] {\tilde{H}}_{R} exp [F] = {\tilde{H}}_{R} + [{\tilde{H}}_{R}, F] + \frac{1}{2!} [[{\tilde{H}}_{R}, F], F] + \dots

, it can be found approximately that

U_{R}^{†} Δ_{p} a^{†} a U_{R} ≃ Δ_{p} a^{†} a + \frac{ω_{p} g}{Δ} (a^{†} S_{-} + a S_{+}) + Δ_{p} {(\frac{g}{Δ})}^{2} (S^{2} - S_{z}^{2}) - 2 Δ p {(\frac{g}{Δ})}^{2} S_{z} a^{†} a

,

U_{R}^{†} ω_{M} b^{†} b U_{R} = ω_{M} b^{†} b

,

U_{R}^{†} ξ a^{†} a (b^{†} + b) U_{R} ≃ [ξ a^{†} a + \frac{ξ g}{Δ} (a^{†} S_{-} + a S_{+})] (b^{†} + b) + ξ {(\frac{g}{Δ})}^{2} (S^{2} - S_{z}^{2}) (b^{†} + b)

,

U_{R}^{†} Δ_{a} S_{z} U_{R} ≃ Δ_{a} S_{z} - \frac{ω_{a} g}{Δ} (a^{†} S_{-} + a S_{+}) - Δ_{a} {(\frac{g}{Δ})}^{2} (S^{2} - S_{z}^{2}) + 2 Δ_{a} {(\frac{g}{Δ})}^{2} S_{z} a^{†} a

,

U_{R}^{†} g (a^{†} S_{-} + a S_{+}) U_{R} ≃ g (a^{†} S_{-} + a S_{+}) + \frac{2 g^{2}}{Δ} (S^{2} - S_{z}^{2}) - \frac{4 g^{2}}{Δ} S_{z} a^{†} a

,

U_{R}^{†} Ω_{p} (a^{†} + a) U_{R} ≃ Ω_{p} (a^{†} + a) - \frac{2 g Ω}{Δ} S_{x}

, and

U_{R}^{†} (Ω_{a} S_{x} - q S_{z}^{2}) U_{R} ≃ 2 Ω_{a} S_{x} - q S_{z}^{2} + [(Ω_{a} S_{x} - q S_{z}^{2}), \frac{g}{Δ} (a^{†} S_{-} - a S_{+})]

. Since in the dispersive limit the cavity mode only supports the excitation of virtual photons, all terms with the photons are eliminated. Finally, we have

H_{R} = ω_{M} b^{†} b + λ_{0} (S^{2} - S_{z}^{2}) (b^{†} + b) + Δ_{a} S_{z} + Ω S_{x} - v S_{z}^{2},

where λ₀ = ξ (g/Δ)² denotes the indirect nonlinear interaction between the atomic ensemble and a nanomechanical oscillator by the exchange of virtual photons, Ω = 2Ω_a – 2gΩ_p/Δ is the effective Rabi frequency for the ultracold atoms, and v₀ = p + q with p = g2/Δ is the total long-range atom-atom interaction. It has been demonstrated that, when the photon-assisted long-range interaction p has a competition with the collision-induced interaction q, some exotic quantum phenomena have been predicted [21, 22]. Hamiltonian (11) shows clearly that, although no energy exchanges from the atomic ensemble, the photon has still acted as a bus, generating long-range atom-atom interactions including the terms

λ_{0} (S^{2} - S_{z}^{2})

and

p S_{z}^{2}

.

By using Feshbach resonance technique and controlling the pump laser of cavity mode, the weaker interaction v (≃ 0) can be considered, and thus, the term $v S_{z}^{2}$ in Hamiltonian (11) is neglected. Moreover, with a rotation of coordinates S_x → J_z and S_z → −J_x, Hamiltonian (11) with Δ_a = 0 becomes $H_{RC} = ω_{M} b^{†} b + λ_{0} (J^{2} - J_{x}^{2}) (b^{†} + b) + Ω J_{z}$ . Near the resonant condition, the terms with high frequencies of Hamiltonian H_RC in the interaction picture vary rapidly and therefore average to zero over time scales on the order of 1/ω_M for a weak λ₀. Thus, Hamiltonian H_RC becomes

H = ω_{M} b^{†} b - λ (J_{+}^{2} b + J_{-}^{2} b^{†}) + Ω J_{z}

with λ = λ₀/4. Hamiltonian (12) possesses, to our knowledge, a new nonlinear coupling (

J_{+}^{2} b + J_{-}^{2} b^{†}

) in quantum optics and condensed matter physics. This result is only governed by the collective excitation of the ultracold atoms since (σ₊)² = (σ₋)² = 0 for a single ultracold atom, where σ_± are the Pauli spin operators. Since this coupling describes the interaction with the phonon field of the nanomechanical motion, our proposal setup is regarded as a mechanical cavity quantum electrodynamic system at the mesoscopic scale. Based on Hamiltonians (11) or (12), the coherent nonlinear effect at the mesoscopic scale can be achieved by controlling the pump laser and the driving laser. It should be noted that parameter Ω is not the atomic resonant frequency, but is dependent of both the controllable lasers.

It is interesting that, the interaction strength for the obtained nonlinear coupling ( $J_{+}^{2} b + J_{-}^{2} b^{†}$ ) is enhanced collectively with a factor of the atomic number N, and thus arrives at a strong limit within current experimental parameters. These results will be demonstrated by calculating the well-known vacuum Rabi splitting [23]. In the limit of λN >> κ_b, where κ_b = ω_M/2Q is the phonon decay rate with Q being the quality factor, the susceptibility for an external field with frequency ϖ is derived from the master equation of Hamiltonian (12) given by

\frac{\partial ρ}{\partial t} = - i [H, ρ] - κ_{b} (b^{†} b ρ - 2 b ρ b^{†} + ρ b^{†} b) .

In the secular approximation the diagonal elements of ρ satisfy the Pauli-type master equation, whereas the corresponding off-diagonal elements satisfy: ∂ρ_ij/∂t = −i(E_i – E_j)ρ_ij – Γ_ijρ_ij, where Γ_ij is the damping factor and E_l (l = i, j) is the eigenvalue in terms of the eigenequation H_F |ψ_l〉 = E_l |ψ_l〉. Due to Hamiltonian (12) conserves J² and J_z + 2b^†b, the eigenfunctions have following simple form:

{\begin{array}{r} | ψ_{+} 〉 = cos θ | - \frac{N}{2} + 2, 0 〉 + sin θ | - \frac{N}{2}, 1 〉 \\ | ψ_{-} 〉 = - sin θ | - \frac{N}{2} + 2, 0 〉 + cos θ | - \frac{N}{2}, 1 〉 \end{array}

with E_± ≃ Ω(−N/2 + 2) – δ/2 ± d, where

tan θ = (- δ + d) / 2 \sqrt{2} N λ

with

d = \sqrt{δ^{2} + 8 N^{2} λ^{2}} / 2

.

Following the standard procedure [23], the imaginary part of the quantum mechanical susceptibility is given by

Im [χ (ϖ)] = \frac{Γ_{-} {cos}^{2} θ}{π {Γ_{-}^{2} + {[ϖ - 2 Ω - δ / 2 - d]}^{2}}} + \frac{Γ_{+} {sin}^{2} θ}{π {Γ_{+}^{2} + {[ϖ - 2 Ω - δ / 2 + d]}^{2}}},

where the effective damping factors Γ_± are not important for our arguments. Since the absorption spectrum is proportional to Im[χ (ϖ)], it has a doublet structure in resonant case δ = 0 with peaks at

ϖ - 2 Ω \pm \sqrt{2} N λ .

Eq. (16) shows that the interaction strength for (

J_{+}^{2} b + J_{-}^{2} b^{†}

) is proportional to the atomic number N, and therefore, reaches a strong coupling regime in current experimental parameters as will be shown below.

By means of the experimental parameters M = 4 × 10⁻¹⁰ kg, ω_M = 3 × 10⁵ Hz, l₀ = 2.5 × 10⁻² m, and ω = 1.8 × 10¹⁵ Hz [S. Gigan, et. al., Nature (London) 444, 67 (2006)], the zero-point uncertainty is $Δ x = \sqrt{\bar{h} / 2 M ω_{M}} = 6.6 \times 10^{- 16} m$ and the light-pressure constant is ξ = ωΔx/l₀ = 48 Hz. If g/Δ = 10⁻¹, the nonlinear interaction strength is λ = (ξ/4)(g/Δ)² = 0.12 Hz. In fact, when the length is chosen as l₀ = 1.76 × 10⁻⁴ m, the prepared ultracold atoms can be sent experimentally into the ultrahigh-finesse cavity with the frequency 2.4 × 10¹⁵ Hz [17]. In this case, the light-pressure constant becomes ξ = 9.1 × 10³ Hz, and correspondingly, the nonlinear interaction strength turns into λ = 23.0 Hz. On the other hand, when k_BT/Q ≪ κ_b ≪ ω_M, the thermal decoherence rate of the nanomechanical oscillator is small compared to the cavity decay rate, and is thus not taken into account [24].

For thousands of ultracold atoms, the collective atom-phonon coupling strength $\sqrt{2} N λ$ is smaller than the total decay rate (κ_b + γ) ≃ 2π × 3.0 MHz. It should be noticed that the factor $\sqrt{2}$ also originates from the second-order transition between states |−N/2 + 2, 0〉 and |−N/2, 1〉. In this regime, the vacuum Rabi splitting disappears and the phonon Purcel effect with a single Lorentzian line occurs [25]. With the increasing of the atomic number N, the effective atom-phonon coupling strength $\sqrt{2} N λ$ will be enhanced rapidly. When the atomic number is chosen as N = 8.4 × 10⁵ used in experiment [17], this effective coupling strength can reach $\sqrt{2} N λ = 2 π \times 4.4 MHz$ (λ = 23.0 Hz), which is larger than the total decay rate (κ_b + γ). Thus, the vacuum Rabi splitting $\bar{d} = 2 d = 2 \sqrt{2} N λ$ with equal line widths given by (γ + κ_b)/2 is achieved and coherent dynamics governed by Hamiltonian (12) occurs [26]. It implies that strong coupling for the effective atom-phonon interaction is achieved.

Finally, we simulate the nonlinear optical processes with the χ⁽²⁾ term from Hamiltonian (12) based on the collective excitation of the ultracold atoms. For a large atomic number N, the collective spin operators are mapped by means of the Holstein-Primakoff transformation into the boson operators [27],

J_{+} = \sqrt{N} d^{†}, J_{-} = \sqrt{N} d, J_{z} = (d^{†} d - \frac{N}{2}) .

It means that the collective excitation governed by J_± behaves like a harmonic oscillator degree of freedom. Thus, Hamiltonian (12) becomes

H_{CM} = ω_{M} b^{†} b - N λ [{(d^{†})}^{2} b + d^{2} b^{†}] + Ω d^{†} d

Hamiltonian (18) describes an interaction between the phonon and the pair of quasiparticle (boson) excited from the ultracold atomic ensemble. Again, this interaction is also enhanced collectively with a factor of N, and thus, arrives at the strong coupling regime in the current experimental parameters. Physically, this interaction represents that a pair of quasiparticle is excited when a phonon is emitted and vice versa. In the position-momentum coordinates such as $x_{b} = \sqrt{1 / 2 ω_{M}} (b^{†} + b)$ and $x_{d} = \sqrt{1 / 2 Ω} (d^{†} + d)$ , Hamiltonian (18) has been regarded as a promising resource for processing quantum information with continuous variables under strong coupling [13].

On the other hand, Hamiltonian (18) shows that the nonlinear optical processes with the χ⁽²⁾ term are simulated successfully, which means that the essential quantum operations such as squeezing and parametric amplification are implemented in our proposal hybrid solid-state and atomic systems [12]. For example, when the nanomechanical motion is treated classically, Hamiltonian (18), in the interaction picture and the condition 2Ω – ω_M = ±(2n + 1)π with n = 1, 2, 3,..., is given by

H_{int} = N λ λ_{b} [{(d^{†})}^{2} exp (- i φ) + d^{2} exp (i φ)],

where λ_b and φ are the real amplitude and phase of the nanomechanical oscillator. The Heisenberg equation of motion for Hamiltonian (19) is evaluated as ḋ = −iΩ_dd^† exp(−iφ) and d^† = iΩ_dd exp(iφ), where Ω_d = 2Nλλ_b is the effective Rabi frequency. By solving these first-order different equations, we have

{\begin{matrix} d (t) = d (0) cosh (u) - i d^{†} (0) sinh (u) exp (- i φ) \\ d^{†} (t) = d^{†} (0) cosh (u) + i d (0) sinh (u) exp (i φ) \end{matrix},

where d(0) and d^†(0) are the initial conditions, u = Ω_dt is the effective squeeze parameter. If the initial state is prepared in a vacuum state, the variances in the two quadratures

{\begin{array}{l} X_{1} = \frac{1}{2} (d^{†} + d) ≃ \frac{J_{x}}{\sqrt{N}} \\ X_{2} = \frac{1}{2 i} (d - d^{†}) ≃ \frac{J_{y}}{\sqrt{N}} \end{array}

are given, if φ = π/2, by [12]

{\begin{matrix} {(Δ X_{1})}_{t}^{2} = \frac{1}{4} exp (- 2 u) \\ {(Δ X_{2})}_{t}^{2} = \frac{1}{4} exp (2 u) \end{matrix}

It shows that the quasiparticle operators or the spin angular momentums are squeezed fully for any atomic number. It should be emphasized that this squeezing only occurs in the quadratic terms in Hamiltonian (18), but fails in the well-known Dicke model with the linear interaction.

In summary, we have put forward a new nonlinear coupling between the ultracold atomic ensemble and the nanomechanical oscillator by the exchange of virtual photons. In the large atomic number, our proposal hybrid system has also described the interaction between a pair of quasiparticle and the phonon. With the factor of the atomic number, both interactions have reached the strong coupling regime in current experimental parameters. In addition, we have also simulated the nonlinear optical processes with the χ⁽²⁾ term, based on the collective excitation for the ultracold atomic ensemble.

Acknowledgments

We thank Profs. Qian Niu, Jinwu Ye, and Jing Zhang, as well as Dr. Zheng-Yuan Xue, for helpful discussions and suggestions. This work was supported by the 973 Program under Grant No. 2006CB921603, the National Natural Science Foundation of China under Grant Nos. 10704049, 10775091, 10934004, 60978001, 60978018, and 11074154, and ZJNSF under Grant No. Y6090001.

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Strong nonlinear coupling between an ultracold atomic ensemble and a nanomechanical oscillator

Abstract

Acknowledgments

References and links

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