Exotic quantum phase transitions in a Bose-Einstein condensate coupled to an optical cavity
A new extended Dicke model, which includes atom-atom interactions and a driving classical laser field, is established for a Bose-Einstein condensate inside an ultrahigh-finesse optical cavity. A feasible experimental setup with a strong atom-field coupling is proposed, where most parameters are easily controllable and thus the predicted second-order superradiant-normal phase transition may be detected by measuring the ground-state atomic population. More intriguingly, a novel second-order phase transition from the superradiant phase to the “Mott” phase is also revealed. In addition, a rich and exotic phase diagram is presented.
As is known, a trapped Bose-Einstein condensate (BEC) may be used to generate a macroscopic quantum object consisting of many atoms that are in the same quantum state with a longer lifetime and can be excited by either deforming the trap or varying the interactions among atoms. Thus the BEC, as a distinct macroscopic quantum system, plays an important role in the in-depth exploration of both fundamental physics and quantum device applications of many-body systems 1 (). In particular, an intriguing idea to combine the cavity quantum electrodynamics (QED) with the BEC has recently attracted significant interests both theoretically and experimentally as many exotic quantum phenomena closely related to both light and matter at ultimate quantum levels may emerge 2 (); 3 (); 4 (); 5 (); 6 (); 7 (); 8 (); 9 (); 10 (); 11 (); 12 (); 13 ().
Very recently, a so-called strong coupling of a BEC to the quantized field of an ultrahigh-finesse optical cavity was realized experimentally 14 (), which not only implies that a new challenging regime of cavity QED has been reached, where all atoms occupying a single mode of a matter-wave field that can couple identically to the photon induced by the cavity mode, but also opens a wider door to explore a variety of new quantum phenomena associated with the cavity-mediated many-body physics of quantum gas. Regrettably, the authors of Ref. 14 () ignored the important nonlinear interactions among the untracold atoms that are controllable via the Feshbach resonance technique, while these interactions are believed to have also a considerable impact on physical properties of the BEC, leading to some exotic quantum phenomena 15 ().
In this Letter, we establish an extended Dicke model with the atom-atom interactions and a driving classical laser field under the two-mode approximation. A feasible experimental setup with controllable parameters including a collective strong atom-field coupling is proposed. We illustrate how to drive a well-known second-order superradiant-normal phase transition and how to detect it experimentally. Remarkably, this superradiant phase transition was predicted in quantum optics many years ago, but has never been observed in experiments 16 (); 17 (); 18 (); 19 (); 20 (); 21 (); 22 (). More intriguingly, a novel second-order superradiant to “Mott” phase transition is also revealed. In addition, we also obtain a rich and exotic phase diagram covering phenomena from quantum optics to the BEC, which is attributed to the competition between the atom-atom and the atom-field interactions.
Our proposed experimental setup is depicted in Fig. 1. For an optical cavity with length m and the mode waist radius m, we may choose the parameters of the cavity MHz 14 (), where is the maximum single atom-field coupling strength, and are the amplitude decay rates of the excited state and the intracacity field, respectively. Such a choice implies that the system is in the strong coupling regime, and thus the long-range coherence could be well established and the quantum dissipation effect may be safely neglected. Based on a pair of coupled Gross-Pitaevskii equations for the BEC with two levels and of 23 () and under the two-mode approximation, the total Hamiltonian for the elastic two-body collisions with the interaction potential of -functional type may be written as
with ( hereafter), , , , and , where is the annihilation operator of the cavity mode with frequency ; and are the annihilation boson operators for and , respectively; with being a single magnetic trapped potential of frequencies and being the atomic mass; is the atomic resonance frequency; and with and being the intraspecies and the interspecies wave scattering lengths, respectively; with being the Rabi frequency for the introduced classical laser with a driving frequency ; and with being a interaction constant between the atom and the photon 24 ().
Under a unitary transformation with the condition and using the Schwinger relations , , and with the Casimir invariant , Hamiltonian (1) can approximately be rewritten as
in the rotating frame, where denotes a collective coupling strength, describes the atom-atom interactions including the repulsive and attractive interactions, and with being the detuning. For a single trapped potential, we have and consider only the case of , which has the advantages that it reduces the effects of fluctuations in the total atomic number and ensures a large spatial overlap of different components of the condensate wavefunction. Thus, the parameters and can further be reduced to and . Eq. (2) is a key result, which describes the collective dynamics for the composite system and has a rich phase diagram. Here we refer this equation to as an extended Dicke model since it contains the extra laser field term (the 4-th one) and atom-atom interaction term (the 5-th one) in comparison with the standard Dicke model and its generalized version 16 (); 20 ().
A distinct property of Hamiltonian (2) lies in that all parameters can be controlled independently. For example, the effective coupling strength can be manipulated by a standard technique. The effective Rabi frequency and the detuning depend on the experimentally controllable classical laser, and especially, the detuning can vary continuously from the red to the blue detunings. The parameter ranging from the positive to the negative is determined by the wave scattering lengths via Feshbach resonance technique 15 (). For and , Hamiltonian (2) is reduced to a standard Dicke model with a second-order superradiant phase transition at the critical point 16 (); 17 (); 18 (); 19 (); 20 (); 21 (); 22 (). It should be noticed that this important prediction has never been observed in experiments. The main difficulties are likely (i) all atoms can hardly interact identically with the same quantum field; (ii) the frequencies and typically exceed the coupling strength by many orders of magnitude; (iii) it is hard to control the parameters as demanded. However, in our proposal, these difficulties could be completely overcome by using the currently available experimental techniques of BEC, as will readily be seen below.
To explore quantum phases and their transitions, we now investigate the ground-state properties of Hamiltonian (2), which can approximately be dealt with by using the Holstein-Primakoff transformation, , and with . Here we introduce two shifting boson operators and with auxiliary parameters and to describe the collective behaviors of both the atoms and the photon 19 (); 20 (); 21 (); 22 (). With the help of the boson expansion method, the scaled ground-state energy is given by with (). The critical points can be determined from the equilibrium condition and =0, which leads to two equations: and
where and are introduced as new parameters for convenience. The coefficient ( describes the intrinsic competition between the atom-atom and the atom-field interactions and gives rise to some exotic phase transitions predicted in the following.
Equation (3) contains the basic information of quantum phases and their transition. As a benchmark, we first address the simplest case that there is no nonlinear interaction among atoms, namely, . Fig. 2 shows the scaled ground-state energy and atomic population (or equivalently “magnetization”) as a function of the detuning for different Rabi frequencies (). It can be seen clearly that in the limit , this system exhibits collective excitations of both the atom and the field with macroscopic occupations (i.e., and ) for , whereas there are no such excitations for and (the solid black line). This interesting behavior typically shows the second-order superradiant phase transition in quantum optics with the critical point 16 (); 17 (); 18 (); 19 (); 20 (); 21 (); 22 (). Moreover, here we may achieve the condition that the order of magnitude of is the same as that of by controlling the detuning of the classical laser. By controlling and evaluating a partial derivative of with respect to ( or ), if a peak is detected in the derivative, which becomes sharper and shaper if becomes smaller and smaller, a second order superradiant phase transition at is signatured, even though the transition disappears at a finite Rabi frequency . In view of this, our proposed composite system with the controllable classical laser is a promising candidate for exploring cavity-induced superradiant phase transition by measuring the ground-state atomic population via the resonant absorption imaging 24 ().
On the other hand, the nonlinear interactions among atoms controlled by Feshbach resonance technique play an important role for the ground-state properties. Fig.3 plots a zero-temperature phase diagram for the atom-atom interaction strength and the detuning with a Rabi frequency in the framework of mean field. The Table lists the corresponding ranges of the mean intracavity photon number , the atomic population , and the “susceptibility” for three different quantum phases. In the case of the repulsive interaction , the critical point becomes , which implies that an effective atom-field interaction is enhanced, while in the weak attractive interaction case , the effective interaction is suppressed. However, the basic features of the superradiant phases remain. In particular, in the case of , this system exhibits a novel second-order phase transition from the superradiant to the “Mott” phases (Red line) 25 (). The relevant physics can be intuitively understood as following. In an optical cavity with , the cavity mode is only weakly or virtually excited, and the energy term is therefore nearly equal to . If , the ground-state properties are governed by the energy . The effective potential in the Landau-Ginzburg theory is a double-well potential with the photon-assisted Josephson tunneling, which means that this system is located at the superradiant phase. If , the energy is dominant and the corresponding effective potential is a single-well potential with no internal Josephson tunneling, leading to the same atomic numbers for the two levels , which may be referred to as the “Mott” phase 26 (). Also, when is decreased, a second-order phase transition from the “Mott” to the “superfluid” phases (Blue line) occurs at the critical point . In the so-called “superfluid” case, the effective potential is another double-well potential with the internal Josephson tunneling induced by the attractive interaction 26 (). It should be pointed out that these three different phases can be distinguished experimentally by measuring the atomic population and the “susceptibility” . In the limit , this predicted second-order phase transition from the superradiant to the “Mott” phases becomes a direct transition from the superradiant to the “superfluid” phases with the same order at the critical point and .
Although the second-order superradiant phase transition disappears in the strong attractive interaction , another interesting phase transition (from the phase with nonzero macroscopic occupation of the level 1 to that of the level 2) in the “superfluid” regime emerges when the detuning changes from negative to positive (i.e., from the red to the blue detunings). Fig. 4 shows the scaled atomic population versus for different s. We see that a novel first-order “superfluid” phase transition occurs at , and moreover this first-order phase transition exists until (Red dashed line). For , it becomes a second-order phase transition with the same critical point. For , no phase transition has been seen by varying .
We now estimate the energy scales for the parameters in Hamiltonian (2) to address the experimental feasibility. Under the two-mode approximation, the wavefunctions of the macroscopic condensate states for the single magnetic trap may roughly be approximated by with , and . Hence, the atom-atom interaction strength can be estimated by . For the typical values Hz, nm, nm, and kg, the energy scale of is about MHz with , which ensures that the error (the order of ) for determining the ground-state properties by means of the Holstein-Primakoff transformation is very low. The effective coupling strength MHz for MHz 14 () is indeed in the strong coupling regime. The energy scale for is about MHz for MHz 14 (), which can be adjusted by controlling the frequency of photon. These energy scales for and imply that the intrinsic competition between the atom-atom and atom-field interaction should be taken into account seriously in the BEC coupled to the optical cavity. Also note that the aforementioned condition is well satisfied once is tuned around since MHz 23 ().
Finally, we elaborate briefly how to probe the predicted phase transitions experimentally. From the condition with MHz and MHz, we can immediately evaluate the maximum of the scaled mean intracavity photon number and find it to be much less than the critical intracavity photon number . Therefore, one is able to perform the transmission spectroscopy measurement with a weak probe laser to obtain the ground-state energy spectrum and atomic population since different quantum phases are, in general, characterized by their specific dispersion relations. The transmission (of this probe laser through the cavity) versus the detuning may be monitored and/or detected by counting photons out of the cavity. Only when the probe laser frequency matches a system in resonance, the corresponding transmission is anticipated 27 ().
In summery, we have established an extended Dicke model and designed a feasible experimental setup with controllable parameters. An exotic phase diagram has been obtained, which covers various phenomena from quantum optics to the BEC and reveals particularly several novel quantum phase transitions.
We thank S. L. Zhu, Y. Li, D. L. Zhou, L. B. Shao, and Z. Y. Xue for helpful discussions. This work was supported by the RGC of Hong Kong under Grant No. HKU7051/06P, the URC fund of HKU, the NSFC under Grant Nos. 10429401, 10775091 and 10704049, and the State Key Program for Basic Research of China (No. 2006CB921800).
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