Entangled light from Bose-Einstein condensates

We propose a method to generate entangled light with a Bose-Einstein condensate trapped in a cavity, a system realized in recent experiments. The atoms of the condensate are trapped in a periodic potential generated by a cavity mode. The condensate is continuously pumped by a laser and spontaneously emits a pair of photons of different frequencies in two distinct cavity modes. In this way, the condensate mediates entanglement between two cavity modes which leak out and can be separated and exhibit continuous variable entanglement. The scheme exploits the experimentally demonstrated strong, steady and collective coupling of condensate atoms to a cavity field.

Quantum communications can outperform their classical counterparts, for example, quantum cryptography enables secure distribution of quantum information. "Quantum correlations" or entanglement, when shared between distant parties, is a key resource for quantum communication tasks such as quantum cryptography [1], teleportation [2] and dense coding [3]. These applications provide a very strong motivation for entangled light which is widely regarded as the most ideal entity for the sharing of entanglement between genuinely distant parties [4]. Additionally, as the interface between light and matter matures as a technology [5,6], entangled light can also link up distinct matter registers of a quantum computer and thereby aid in scaling up quantum computers.
One important form of entangled light [7,8] is continuous variable (CV) entanglement between phase quadratures of two distinct modes of the light field of the type discussed in the famous Einstein-Podolsky-Rosen (EPR) paper [9]. Such entanglement has been used for quantum teleportation and has applications in quantum dense coding [10] and quantum cryptography [11]. Additionally, if the entanglement is sufficiently "narrow band" in frequencies then the quantum states of the light will efficiently interface with those of atomic ensembles [6] for applications in quantum repeaters and linking quantum registers. Thus the motivation for having entangled sources of EPR light is very strong.
The prevalent sources of EPR entangled light are non-linear crystals. It was noticed long ago that light fields produced from nonlinear crystals seem to be nonclassically correlated [12]. For non-degenerate optical parametric oscillators, Reid demonstrated that the quadrature phases of the output fields have EPR type entanglement [7] and this is indeed one of the sources in recent experiments [8]. Alternatively, the outputs of a two degenerate optical parametric oscillators are interfered to obtain EPR entangled light [13]. For such crystals, the Hamiltonian is actually phenomenologically constructed to describe the observed nonlinear processes and expressed in terms of the nonlinear susceptibility of the macroscopic crystal. This is why there has been a recent interest in deriving EPR entangled light from a more fundamental Hamiltonian, such as from the "quantized motion" of a single atom trapped in a cavity [14,15]. This provides a coherent control of the entanglement generator at the microscopic system, as opposed to a bulk crystal. In addition to this fundamental interest, such alternative sources may also have a practical interest if they can surpass the squeezing parameter (a parameter that controls the amount of entanglement in the EPR entangled light) possible from crystal sources as many of the restrictions such as the lower finesse of cavities around crystals, or the unbalanced absorption of the entangled modes while traversing the crystal, do not directly apply. Recently, the strong coupling of an atomic Bose-Einstein condensate (BEC) to a single-mode photon field of an optical cavity has been experimentally achieved [16,17]. The ultracold atoms are trapped in a periodic potential generated by a quantized field mode [18]. Since the N two-level atoms are identically coupled to the single-mode photon field which gives a collective enhancement of a factor √ N [16,17]. In fact, such strong atomphoton couplings are very useful in performing quantum information processing (QIP) before the decoherence sets in. The potential applications include long-lived quantum memory [8] and quantum network for light-matter interface [19].
In this paper, we consider an atomic BEC trapped inside an optical cavity [16]. Each atom is located at the anti-node of the quantized cavity field so that one atom per site can identically couple to the cavity field. Compared to a thermal cloud of atoms with the inhomogeneous atom-photon couplings in the cavity, we can truely apply the Tavis-Cummings model [20] to study our system. In addition, the BEC's with reduced Doppler broadening leads to much longer coherence times than that of the thermal clouds [21].
We consider the BEC is continuously driven by an external laser and spontaneously emits photons with the two different frequencies in pair. Hence, the BEC acts as a medium to mediate the entanglement between the two cavity modes. The two quantum-correlated light modes are emitted through a one-sided mirror as shown in Fig.  1. Since the ultracold atoms have long coherence times [22], this can provide a robust way in generating the entangled light. We will show that the degree of entanglement depends on the ratio of the decay rate of the cavity and the effective Rabi frequency. This means that the degree of entanglement can be controlled by adjusting the atom-photon coupling strength. We consider a two-component condensate trapped inside a cavity in which the atoms are trapped in a onedimensional optical lattice as shown in Fig. 1. A classical laser, with frequency ω L , is used to pump the two internal states |1 and |2 to a higher level |3 . Then, the two different quantized light fields with frequencies ω 1 and ω 2 are spontaneously emitted due to large detuning [23]. The Hamiltonian is written as where ω j1 are the energy splitting between the states |j k and |1 k . The frequencies of the two modes satisfy the two-photon Raman resonance condition 2ω L = ω 1 + ω 2 [23] so that we can write ω 1,2 = ω L ±ν. These two modes must satisfy the boundary condition of the cavity. It is instructive to work in the rotating frame by using the unitary transformation The transformed Hamiltonian reads as where ∆ = ω 31 − ω L and δ j = ω j − ω L . For ∆ ≫ Ω j , λ j , this enables us to adiabatically eliminate the upper level |3 .
The effective Hamiltonian is given by . The parametersω and g j are ω 21 + (Ω 2 1 − Ω 2 2 )/∆ and λ j Ω j /∆ respectively. We note that C = J z + a † 1 a 1 − a 2 a † 2 is a constant of motion. We consider that the low-lying collective excitations in the condensates involve throughout the dynamics. Then, we can approximate the angular momentum operator as a harmonic oscillator as [24]: The Hamiltonian can be rewritten as We assume ω ′ = (ω − ν) ≫ g j √ N such that the low excitations approximation is valid. In the large detuning limit, we can write H eff = χ 1 a † 1 a 1 + χ 2 a 2 a † 2 + χ(a † 1 a † 2 + a 1 a 2 ), (5) where χ j = g 2 j N/ω ′ and χ = g 1 g 2 N/ω ′ . For convenience, we represent the Hamiltonian in terms of the operators [25]: These operators satisfy the commutation relations [K 3 , K ± ] = ±K ± and [K + , K − ] = −2K 3 . Thus, the Hamiltonian is represented in the form as (we have ignored the constant term): We consider the initial state is the vacuum state and thus the time evolution of two-mode state is |Ψ(τ ) = e −iH eff τ |0, 0 . According to the operator ordering theorem [25], we have e −iH eff τ = e ΓK+ e lnΓK3 e ΓK− , The parameters γ,γ, β 2 are −iχτ , −i(χ 1 + χ 2 )τ /2 and γ 2 /4−γ 2 respectively. Therefore, the state can be readily obtained as Clearly, this state is a entangled state in which the two cavity modes are entangled in pair.
We have shown that the entanglement between the two cavity modes can be produced inside the cavity. However, it is necessary to detect the entangled light out of the cavity. The entangled light is emitted through the mirror and splits into the two different frequency components via a prism as shown in Fig 1. They are measured via a homodyne detection. The difference in the photon current is then recorded. The resulting squeezing spectrum can be obtained by the spectrum analyser [26].
To evaluate the entangled light outside the cavity, we thus have to take account of the input-output theory [26]. The Langvein equations of motion for the system are given by [27] The output fields are a jo = a jin + 2κ j a j . We assume that the radiative noise from the cavity is much larger than the noise coming from the BEC and also the input noise source is in vacuum. Now we study the squeezing spectrum by transforming to the Fourier space as [15]: where The squeezing spectrum can be defined as In our case, we found that S θ + (ω) = S θ − (ω). This can be shown that the two output modes are entangled if S θ ± (ω) < 1 [15,28].
We investigate the squeezing spectrum S θ (ω) for the regime of ω ′ ≫ g 1,2 √ N . For simplicity, we take g 1,2 = g and κ 1,2 = κ. In Fig. 3 (a), we plot the squeezing spectrum S 0 (ω) as a function of ω (in units of g) for the different decay rates κ. It shows that the squeezing occurs at the negative frequency domain whereas the unsqueezing occurs at the positive frequency domain. We also plot the out-of-phase squeezing spectrum S π/2 (ω) in The squeezing spectrums S 0 (ω) and S π/2 (ω) are plotted as a function of the frequency ω/g, where g1 = g2 = g, ω = 10 4 g and N = 10 4 . The solid, dash and dotted lines represent the different values of κ = 10, 5, 2.5 (in units of g) respectively. Fig. 3 (b). In contrast, the squeezing(unsqueezing) occurs at the positive(negative) frequency domain. Apart from that, we can see that a larger squeezing compensates a narrower range of frequencies as shown in Fig. 3 (a) and (b).
We study the maximal degree of squeezing attainable in the system for the values of cavity decay and the number of atoms. The minimal values of squeezing for the spectrum S 0 (ω) as a function of κ (in units of g) as shown in Fig. 4. The degree of squeezing increases significantly as the cavity decay parameter κ decreases. The nearly perfect squeezing can be obtained for in the limit of κ approaching to zero. In the inset of Fig. 4, the minimal values of squeezing is plotted as a function of log 10 N . This means that the strong squeezing can be attained as the number of atoms increases.
For ω ′ ≫ g 2 1,2 N , the approximated analytical expression of the squeezing spectrum S(ω) can be found as The approximate solution Eq. (16) is compared with the numerical solution in Fig 5. This shows that it is a very good approximation in the limit of this large detuning. We now briefly examine the range of parameters available for our scheme if the setups of some recently performed experiments are directly used. If one literally uses the parameters from the recent Brennecke et. al. experiment [16] then one can have λ j = 2π × 10.6 ≈ 67 MHz (what we say below holds for both j = 1, 2). Typically, Ω j , can be made even larger as it is proportional to the strength of the external laser field, so we assume it to be ηλ j , where η is a numerical factor which can be varied between 1 and 4. The detuning ∆ 2 >> λ 2 j , Ω 2 j , λ j Ω j is required for the adiabatic elimination of the level |3 . So we choose ∆ ∼ 10λ j . Thus the effective Rabi frequency g j ∼ λ j Ω j /∆ = 6.7η MHz. The cavity decay κ = 2π × 1.3 ≈ 8.1MHz. Thus the ratio κ/g j can be made to vary between 1 and 0.3 by varying η between 1and 4. The energy splitting ω ′ ∼ ∆ 12 is around 10 GHz (which means ω ′ , when expressed in terms of g j is 10 3 − 10 4 g j ). Though the number of atoms N can be up to 2 × 10 5 [16], we restrict the number to about ∼ 10 4 which is also a number in typical experiments, so that ω ′ >> g j √ N is fulfilled and the BEC is not excited. We noted that the cavity decay rate κ can be adjusted because κ = πc/(2LF ) depends on the length L and the finesse F of the cavity [29]. Hence, the different extent of squeezing can be observed in experiment by varying the parameters κ and g.
We have studied the entangled light generation with the cavity-BEC system in which the atoms are continuously driven by the external laser field. The entangled light can emit from the cavity through the one-side mirror and then the entangled light can be measured via the homodyne detection. We have shown that the degree of entanglement can be controlled by adjusting the strength of atom-photon couplings. Our scheme to generate entangled light can be realized with the current experimental technology [16,17].