Abstract
1. INTRODUCTION
Engineering special nonclassical quantum states are of paramount importance in quantum information science, metrology, and exploring fundamental physics [1–5]. In particular, -photon states play an essential role in a wide range of quantum technologies, including high-NOON states [6], quantum communication [7], lithography [8], spectroscopy [9,10], and biological sensing [11,12]. Methods for generating -photon states were proposed theoretically in cavity quantum electrodynamics (QED) [13–15], Rydberg atomic ensembles [16,17], and atom-coupled photonic waveguides [18–20]. Among them, -photon bundle states generated through either Mollow physics [13,14] or deterministic parametric downconversion [15] are of particular interest since they possess special statistic properties. However, due to the intrinsic weak scattering interactions between photons, the experimental realization of -photon states still remains a challenge.
On the other hand, the ability to manipulate individual phonon allows the experimental creation of -phonon states in both circuit quantum acoustodynamics [21,22] and macroscopic mechanical resonators [23–25]. These low-energy and long-lived novel phonon states could facilitate the study of the decoherence mechanisms [26] and the building of the quantum memories and transducers [27–29]. More interestingly, it was also proposed that -phonon bundle states can be generated in acoustic cavity QED by employing the Stokes processes [30] and hybrid system of nitrogen-vacancy centers and nanomechanical resonators via sideband engineering [31].
We note that existing schemes [13–15,30] for creating -quanta bundle states operate in the far dispersive regime where the frequency of the photonic/phononic mode is far detuned from the transition frequency of the two-level system. Hence the resulting steady-state photon/phonon numbers are typically very small. Moreover, in order to resolve states with distinct phonon numbers, many schemes [21–25,30] also require that the two-level system is strongly coupled to the phononic modes, which demands high-finesse acoustic cavities or mechanical resonators with long coherence time and poses a challenge to the current experiments.
Sign up for Photonics Research TOC. Get the latest issue of Photonics Research delivered right to you!Sign up now
In this work, we propose to generate -phonon bundle states by utilizing the motional degrees of freedom of a trapped alkaline-earth atom. A position-dependent clock laser is introduced to couple the atom’s center-of-mass motion to its electronic ground and the long-lived excited states, which leads to a generalized quantum Rabi model (QRM) with unprecedented tunability. We then investigate the -phonon bundle states prepared in the resonant regime where the frequency of the motional mode is in resonance with the two-level atom. We show that due to the strong anharmonicity of the energy spectrum, distinct motional -phonon bundle states can be well resolved in the atom–phonon resonance regime. Compared to the existing schemes for generating -quanta bundle states, the system proposed here has the following advantages. (i) Since our scheme operates in the resonant regime, the typical average steady-state phonon number is much larger than that of the schemes operating in the far dispersive regime. (ii) A strong coupling regime can be readily achieved here as both the motional state and the atomic internal states possess long lifetimes. (iii) In our configuration, the effective pump field is provided by the high tunable clock detuning, and our system also facilitates the study of Mollow physics without suffering heating and decoherence. Therefore, the proposed system can be used as a high-quality source for multiphonon states.
2. MODEL AND HAMILTONIAN
Without loss of generality, we consider a single atom trapped in a one-dimensional (1D) harmonic potential along the direction. Figure 1(a) illustrates the level structure and laser configuration of the system. An ultranarrow clock laser with wavelength drives the single-photon transition between the ground state () and the long-lived excited state () with the atomic clock transition frequency . Since the lifetime of the excited state is roughly 160 s [32], the spontaneous emission and decoherence of this state can be safely ignored. We assume that the Rabi frequency of the clock laser takes a position-dependent form, , where and are coupling strengths and is the effective laser wave vector that is tunable by varying the tilting angle of the clock laser. As shall be shown, the position-dependent Rabi coupling of is crucial to the success of our scheme and it can be experimentally generated by tailoring the clock laser by using a spatial light modulator [33–37]. We further assume that the 1D harmonic potential is state independent which can be generated by a trap laser at the “magic” wavelength . After performing a gauge transformation and [38], the resulting Hamiltonian of the system is
Figure 1.(a) Schematic of the system. A
To transfer Eq. (1) into a more familiar form, we introduce a spin rotation, , which rotates spin operators according to , , and . As a result, Hamiltonian of Eq. (1) becomes a generalized QRM,
3. GENERALIZED QUANTUM STATISTICS
For a complete description of the system, we should also take into account the dissipation of the phonons. As a result, the dynamics of the atom–phonon system is now described by the master equation
To characterize the statistic properties of the phonons, we introduce the generalized th-order correlation function,
Finally, we specify the parameters used in numerical simulations. For the atom, the single-photon recoil energy sets up the typical energy scale for the system, where is the Planck constant. In current experiments, the phonon frequency is tunable and can be up to about [32]. As a result, we realize a tunable bound between and 21.6 kHz. For convenience, we fix the value of the SOC strength in real space at such that it is of the same order as , which yields the atomic decay for the clock state [32]. We further fix the values of the decay rate and the dephasing factor at and , respectively. Now, the free parameters of the system reduce to phonon frequency , Rabi frequency , laser detuning , and SOC strength . In below, we study the statistic properties of the phonon state by varying these parameters.
4. MOTIONAL
Before presenting our results on phonon statistics, it is instructive to explore the energy spectrum of the system. In Fig. 1(b), we demonstrate the familiar level structure of a JCM () with and . The eigenenergies of the th pair of dressed states are , where and “” (“”) denotes the upper (lower) branch. Particularly, -phonon resonance occurs when the lower dressed state is tuned on resonance with the vacuum state of the system, , where is the -phonon resonance frequency. The energy spectrum for the more general case is presented in Appendix B, where should also depend on other parameters.
Let us first consider the single phonon states by fixing the phonon frequency at . Figures 2(a) and 2(b) display, respectively, the equal time second-order correlation function and steady-state phonon number in the parameter plane with . As can be seen, when and are changed, the values of and vary over a wide range. Particularly, around , the system reaches the strong sub-Poissonian statistics region with . In these regions, a considerably large number of phonons () are observed as well. In Fig. 2(c), we further plot the interval dependence of the second-order correlation function at , which shows phonon antibunching since . The evidence clearly shows that strong phonon blockade (SPB) is achieved in these regions.
Figure 2.Distributions of (a)
The SPB around can be understood by noting that the energy spectrum of the resulting JCM at is highly anharmonic in the strong coupling regime [Fig. 1(b)]. Therefore, the condition for one phonon excitation at single-phonon resonance () will block the excitation of a second phonon. At first sight, the SPB around may seem strange, as the anti-JCM realized at breaks the conservation of the number of the total excitations due to the counter-rotating terms. To explain this, we note that an anti-JCM is equivalent to a JCM under the unitary transformation , i.e., and . Consequently, an SPB should occur at . We point out that, because of the small phonon decay in the clock transition, the lifetime of the phonon blockade () in our system can be very long.
To reveal more details, we also plot, in Fig. 2(d), the distributions of and along the line on the plane. As can be seen, both and possess two local minima close to . In addition, grows rapidly from these minima when is slightly tuned away from . approaches unity for a coherent phonon state at where Hamiltonian of Eq. (2) reduces to a standard QRM.
We now turn to study the properties of phonon states in the atom–phonon resonance regime, i.e., . In addition, due to the equivalence of two SPB regions in Fig. 2(a), we shall focus, without loss of generality, on the SPB region with . The atom–phonon resonance condition can be satisfied by tuning external atom trap potential and/or power of the classical laser field in the experiment. In particular, it should be noted that our results remain qualitatively unchanged even deviating from the resonance condition, corresponding to the slight shift in the position of -phonon resonance .
Figure 3(a) shows the phonon number as a function of the phonon frequency and clock shift . In particular, the dashed lines plot the dependence of the -phonon resonance frequency obtained by numerically diagonalizing . An immediate observation is that, for a given , exhibits multiple peaks at exactly the phonon resonance frequencies , signaling the existence of possible multiphonon states. Remarkably, because our scheme operates in the resonant regime, the average phonon number of the phonon states generated here is much larger than that of the phonon states produced in the dispersive regime. In addition, there exists a large area on the plane in which the average phonon number is higher than 0.2.
Figure 3.(a) Distribution of
To explore the statistic properties of the phonon emissions, we first map out, in Fig. 3(b), on the parameter plane around . As can be seen, phonon blockade is realized for the whole parameter space covered by Fig. 3(b). There even exists a large area on the plane such that the strong phonon blockade condition, say , is satisfied. Then combined with requirement of large phonon emission number, say , our system can be used as a high-quality single motional phonon source operating in a parameter regime that is easily accessible to current experiments. To further quantify the quality of the single-phonon states around , we introduce , which measures the fraction of -phonon states among the total emitted phonons. In Fig. 3(c), we plot the distribution of on the plane. As can be seen, for the parameter region of our interest, nearly 100% of phonon emission is of the single-phonon nature. It should also be noted that alone is not sufficient to judge the quality of the single-phonon source, because, by comparing Figs. 3(b) and 3(c), the main feature of is inconsistent with that of when is small.
Finally, we study the properties of the emitted phonon in the multiphonon resonance regime. As shown in Fig. 3(a), only for sufficiently large pumping, i.e., and 0.094 for two- and three-phonon resonances, respectively, the phonon numbers around and become significant (). We shall then explore the statistic properties of the multiphonon resonances in these regions. Figure 4 summarizes the main properties of the motional multiphonon states. Due to the similarity between the two- and three-phonon states, we shall only discuss the two-phonon emissions. In Fig. 4(a), we map out the third-order correlation function in the two-phonon resonance regime of the plane. As can be seen, there exists a large parameter regime with in which the -phonon emission is blockaded. Meanwhile, the minimum value of is achieved at , where the highest phonon numbers for the multiphonon resonance is reached. This indicates that at the strongest three-phonon blockade, we have the highest phonon number.
Figure 4.Statistic properties of motional two-phonon (left column) and three-phonon (right column) states. (a) and (b) show the distributions of
To further confirm the bundle-emission nature of the phonon states, we plot, in Fig. 4(c), the typical interval dependence of the correlation functions and . As can be seen, the criteria for motional two-phonon bundle states, and , are indeed satisfied. Another observation from Fig. 4(c) is that the decay times for both and are proportional to , which indicates that the decay of the bunching for single phonon and the decay of the antibunching for the separated bundles of phonons are of the same time scale for the two-phonon bundle states. The two-phonon nature of the emission is also demonstrated by the phonon-number distribution shown in Fig. 4(e). Indeed, for the two-phonon emission case, becomes negligibly small for . We point out that, as shown in the right panels of Fig. 4, the three-phonon emission possesses similar statistic properties as those of two-phonon emission. The observed phonon probability for steady-state -phonon bundle states exhibits a monotonical decreasing distribution with increasing for , arising in dynamical processes of bundle emissions [13]. In general, the -phonon bundle states contain the various dynamical processes of emissions in Fock state with a distinguishable short temporal window . Thus, the multiphonon state () can be directly extracted with a high phonon probability by choosing a very short temporal window [13,30]. Moreover, we should note that the generated motional -phonon bundle states are essentially different from the experimentally observed -quanta blockade, i.e., two-photon blockade [52], where the quantum statistic for the latter only characterizes the single photon but not for separated bundles of photons with satisfying -photon bunching and -photon antibunching as well.
We remark that the underlying reason that -phonon bundle states are well-resolved in the parameters space is the strong anharmonicity of the energy spectrum such that the condition is satisfied for any . This is in striking contrast to the proposed -photon bundle emission utilizing Mollow physics [53] in which the th-order process of quantum states is used [13,14]. In addition, the whole process must be operated in the far dispersive regime and under a strong pump field, which, in terms of our model, requires and , simultaneously. Finally, we check that our numerical results are not affected when we include the weak atomic decay of the clock state () and phonon dephasing (), albeit the strong damping of and could induce a significant decoherence for realization of nonclassical quantum states [54–56].
5. CONCLUSIONS
Based upon the currently available techniques in experiments, we have proposed to generate motional -phonon bundle states using a trapped alkaline-earth atom driven by a clock laser. Since our system works in the resonant regime, the steady-state phonon number contained in the bundle states is 3 orders of magnitude larger than those obtained in the earlier theoretical schemes operating in the far dispersive regime. Moreover, the quality of the -phonon bundle states is also demonstrated by the strong antibunching for the separated bundles of phonons and bunching for single phonons. Finally, we emphasize that the nonclassical nature of the long-lived motional -phonon bundle states can be quantum-state transferred to the photonic mode by applying a readout cavity field with phonon–photon beam-splitter interaction [57,58]. Moreover, as the scheme mitigates the laser-induced heating, it could inspire an interesting opportunity of exploration of Dicke phase transition for the external motional modes [59], superradiances from the clock transition [60], and novel quantum states of matters hindered by heating [61–63]. Our proposal for trapped single atoms could be equivalently applied to the hybrid spin–mechanical systems [64,65]. In particular, the parameter in our model is outside the Lamb–Dicke regime with , which could provide a versatile platform for exploring long-lived mesoscopic entanglement for trapped atoms [66]. Furthermore, we could expect that the proposed system provides versatile applications in quantum metrology limited by decoherence [50,67] and in fundamental tests of quantum physics [68].
Acknowledgment
Acknowledgment. We are grateful to Yue Chang for insightful discussions.
APPENDIX A: MODEL HAMILTONIAN
We present the derivation of the generalized QRM of phonon by utilizing optical clock transition in an ultracold single atom. For specificity, we consider an optical clock transition frequency of is in a single ultracold atom, which includes a ground-state and an excited-state . Here the is the ground state and is an exceptionally long-lived electronic state (160?s). The single atom is resonantly coupled by a linearly -polarized classical plane-wave laser with the frequency and wavelength , which is propagating in the plane making an angle with respect to the axis. As a result, the spontaneous emission and decoherence of the excited state can be safely ignored, which is of paramount importance to realizations of motional -phonon states in our model.
In addition, the single two-level atom is confined in a spin-dependent one-dimensional harmonic trap generated by a -polarized laser at the “magic” wavelength , where is the mass of the atom and is the trap frequency. Now, it can be read out that the Hamiltonian for the internal states of an atom under the rotating-wave approximation reads
After performing the gauge transformations and [
To gain more insight, we introduce the position-momentum representation, , with denoting the annihilation operator of the bosonic phonon mode with harmonic oscillator frequency . Then the Hamiltonian of Eq.?(
APPENDIX B: ENERGY SPECTRA FOR GENERALIZED QRM
We present the details on the derivation of the energy spectrum for the generalized QRM. By introducing a gauge transformation of the spin rotation , the Hamiltonian Eq.?(
For fixing and , the Hamiltonian Eq.?(
Figure 5.Typical energy spectrum of Hamiltonian Eq. (
Figure 6.(a) Energy spectrum of Hamiltonian Eq. (
APPENDIX C: THE EFFECT OF DECAY AND DEPHASING
We present the numerical results of the antibunching of the phonon with different atomic decay and dephasing. Figures?
Figure 7.
Figure 8.
References
[1] L. M. Duan, M. D. Lukin, J. I. Cirac, P. Zoller. Long-distance quantum communication with atomic ensembles and linear optics. Nature, 414, 413-418(2001).
[2] V. Giovannetti, S. Lloyd, L. Maccone. Quantum metrology. Phys. Rev. Lett., 96, 010401(2006).
[3] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Milburn. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys., 79, 135-174(2007).
[4] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, P. Treutlein. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys., 90, 035005(2018).
[5] D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, S. Pirandola. Quantum-enhanced measurements without entanglement. Rev. Mod. Phys., 90, 035006(2018).
[6] I. Afek, O. Ambar, Y. Silberberg. High-noon states by mixing quantum and classical light. Science, 328, 879-881(2010).
[7] H. J. Kimble. The quantum internet. Nature, 453, 1023-1030(2008).
[8] M. D’Angelo, M. V. Chekhova, Y. Shih. Two-photon diffraction and quantum lithography. Phys. Rev. Lett., 87, 013602(2001).
[9] J. C. López Carreño, C. Sánchez Muñoz, D. Sanvitto, E. del Valle, F. P. Laussy. Exciting polaritons with quantum light. Phys. Rev. Lett., 115, 196402(2015).
[10] K. E. Dorfman, F. Schlawin, S. Mukamel. Nonlinear optical signals and spectroscopy with quantum light. Rev. Mod. Phys., 88, 045008(2016).
[11] W. Denk, J. H. Strickler, W. W. Webb. Two-photon laser scanning fluorescence microscopy. Science, 248, 73-76(1990).
[12] N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, C. Xu.
[13] C. S. Muñoz, E. del Valle, A. G. Tudela, K. Müller, S. Lichtmannecker, M. Kaniber, C. Tejedor, J. J. Finley, F. P. Laussy. Emitters of
[14] C. S. Muñoz, F. P. Laussy, E. del Valle, C. Tejedor, A. González-Tudela. Filtering multiphoton emission from state-of-the-art cavity quantum electrodynamics. Optica, 5, 14-26(2018).
[15] Y. Chang, A. González-Tudela, C. Sánchez Muñoz, C. Navarrete-Benlloch, T. Shi. Deterministic down-converter and continuous photon-pair source within the bad-cavity limit. Phys. Rev. Lett., 117, 203602(2016).
[16] P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, H. P. Büchler. Scattering resonances and bound states for strongly interacting Rydberg polaritons. Phys. Rev. A, 90, 053804(2014).
[17] M. F. Maghrebi, M. J. Gullans, P. Bienias, S. Choi, I. Martin, O. Firstenberg, M. D. Lukin, H. P. Büchler, A. V. Gorshkov. Coulomb bound states of strongly interacting photons. Phys. Rev. Lett., 115, 123601(2015).
[18] A. González-Tudela, V. Paulisch, D. E. Chang, H. J. Kimble, J. I. Cirac. Deterministic generation of arbitrary photonic states assisted by dissipation. Phys. Rev. Lett., 115, 163603(2015).
[19] J. S. Douglas, T. Caneva, D. E. Chang. Photon molecules in atomic gases trapped near photonic crystal waveguides. Phys. Rev. X, 6, 031017(2016).
[20] A. González-Tudela, V. Paulisch, H. J. Kimble, J. I. Cirac. Efficient multiphoton generation in waveguide quantum electrodynamics. Phys. Rev. Lett., 118, 213601(2017).
[21] B. A. Moores, L. R. Sletten, J. J. Viennot, K. W. Lehnert. Cavity quantum acoustic device in the multimode strong coupling regime. Phys. Rev. Lett., 120, 227701(2018).
[22] L. R. Sletten, B. A. Moores, J. J. Viennot, K. W. Lehnert. Resolving phonon Fock states in a multimode cavity with a double-slit qubit. Phys. Rev. X, 9, 021056(2019).
[23] P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, R. Van Laer, A. H. Safavi-Naeini. Resolving the energy levels of a nanomechanical oscillator. Nature, 571, 537-540(2019).
[24] R. Manenti, A. F. Kockum, A. Patterson, T. Behrle, J. Rahamim, G. Tancredi, F. Nori, P. J. Leek. Circuit quantum acoustodynamics with surface acoustic waves. Nat. Commun., 8, 975(2017).
[25] Y. Chu, P. Kharel, T. Yoon, L. Frunzio, P. T. Rakich, R. J. Schoelkopf. Creation and control of multi-phonon Fock states in a bulk acoustic-wave resonator. Nature, 563, 666-670(2018).
[26] M. Arndt, K. Hornberger. Testing the limits of quantum mechanical superpositions. Nat. Phys., 10, 271-277(2014).
[27] M. J. A. Schuetz, E. M. Kessler, G. Giedke, L. M. K. Vandersypen, M. D. Lukin, J. I. Cirac. Universal quantum transducers based on surface acoustic waves. Phys. Rev. X, 5, 031031(2015).
[28] A. Noguchi, R. Yamazaki, Y. Tabuchi, Y. Nakamura. Qubit-assisted transduction for a detection of surface acoustic waves near the quantum limit. Phys. Rev. Lett., 119, 180505(2017).
[29] P. Arrangoiz-Arriola, E. A. Wollack, M. Pechal, J. D. Witmer, J. T. Hill, A. H. Safavi-Naeini. Coupling a superconducting quantum circuit to a phononic crystal defect cavity. Phys. Rev. X, 8, 031007(2018).
[30] Q. Bin, X.-Y. Lü, F. P. Laussy, F. Nori, Y. Wu.
[31] X.-L. Dong, P.-B. Li. Multiphonon interactions between nitrogen-vacancy centers and nanomechanical resonators. Phys. Rev. A, 100, 043825(2019).
[32] S. Kolkowitz, S. Bromley, T. Bothwell, M. Wall, G. Marti, A. Koller, X. Zhang, A. Rey, J. Ye. Spin–orbit-coupled fermions in an optical lattice clock. Nature, 542, 66-70(2017).
[33] M. C. Beeler, R. A. Williams, K. Jiménez-Garca, L. J. LeBlanc, A. R. Perry, I. B. Spielman. The spin hall effect in a quantum gas. Nature, 498, 201-204(2013).
[34] D. Palima, C. A. Alonzo, P. J. Rodrigo, J. Glückstad. Generalized phase contrast matched to Gaussian illumination. Opt. Express, 15, 11971-11977(2007).
[35] M. Pasienski, B. DeMarco. A high-accuracy algorithm for designing arbitrary holographic atom traps. Opt. Express, 16, 2176-2190(2008).
[36] J. G. Lee, B. J. McIlvain, C. Lobb, W. Hill. Analogs of basic electronic circuit elements in a free-space atom chip. Sci. Rep., 3, 1034(2013).
[37] A. L. Gaunt, Z. Hadzibabic. Robust digital holography for ultracold atom trapping. Sci. Rep., 2, 721(2012).
[38] Y. Deng, J. Cheng, H. Jing, C.-P. Sun, S. Yi. Spin-orbit-coupled dipolar Bose-Einstein condensates. Phys. Rev. Lett., 108, 125301(2012).
[39] Y.-J. Lin, K. Jiménez-Garca, I. B. Spielman. Spin-orbit-coupled Bose-Einstein condensates. Nature, 471, 83-86(2011).
[40] I. I. Rabi. On the process of space quantization. Phys. Rev., 49, 324-328(1936).
[41] M.-J. Hwang, R. Puebla, M. B. Plenio. Quantum phase transition and universal dynamics in the Rabi model. Phys. Rev. Lett., 115, 180404(2015).
[42] M. Liu, S. Chesi, Z.-J. Ying, X. Chen, H.-G. Luo, H.-Q. Lin. Universal scaling and critical exponents of the anisotropic quantum Rabi model. Phys. Rev. Lett., 119, 220601(2017).
[43] D. Lv, S. An, Z. Liu, J.-N. Zhang, J. S. Pedernales, L. Lamata, E. Solano, K. Kim. Quantum simulation of the quantum Rabi model in a trapped ion. Phys. Rev. X, 8, 021027(2018).
[44] V. Galitski, I. B. Spielman. Spin–orbit coupling in quantum gases. Nature, 494, 49-54(2013).
[45] D. F. Walls, G. J. Milburn. Quantum Optics(2007).
[46] F. Beaudoin, J. M. Gambetta, A. Blais. Dissipation and ultrastrong coupling in circuit QED. Phys. Rev. A, 84, 043832(2011).
[47] A. Ridolfo, M. Leib, S. Savasta, M. J. Hartmann. Photon blockade in the ultrastrong coupling regime. Phys. Rev. Lett., 109, 193602(2012).
[48] R. J. Glauber. The quantum theory of optical coherence. Phys. Rev., 130, 2529-2539(1963).
[49] H. J. Carmichael. Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations(2013).
[50] F. Wolf, C. Shi, J. C. Heip, M. Gessner, L. Pezzè, A. Smerzi, M. Schulte, K. Hammerer, P. O. Schmidt. Motional Fock states for quantum-enhanced amplitude and phase measurements with trapped ions. Nat. Commun., 10, 2929(2019).
[51] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, H. J. Kimble. Photon blockade in an optical cavity with one trapped atom. Nature, 436, 87-90(2005).
[52] C. Hamsen, K. N. Tolazzi, T. Wilk, G. Rempe. Two-photon blockade in an atom-driven cavity QED system. Phys. Rev. Lett., 118, 133604(2017).
[53] B. R. Mollow. Power spectrum of light scattered by two-level systems. Phys. Rev., 188, 1969-1975(1969).
[54] A. Auffèves, D. Gerace, J.-M. Gérard, M. F. Santos, L. C. Andreani, J.-P. Poizat. Controlling the dynamics of a coupled atom-cavity system by pure dephasing. Phys. Rev. B, 81, 245419(2010).
[55] D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, J. Vučković. Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity. Phys. Rev. Lett., 104, 073904(2010).
[56] D. Englund, A. Majumdar, M. Bajcsy, A. Faraon, P. Petroff, J. Vučković. Ultrafast photon-photon interaction in a strongly coupled quantum dot-cavity system. Phys. Rev. Lett., 108, 093604(2012).
[57] V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, H. Wang. Storing optical information as a mechanical excitation in a silica optomechanical resonator. Phys. Rev. Lett., 107, 133601(2011).
[58] S. Zahedpour, J. K. Wahlstrand, H. M. Milchberg. Quantum control of molecular gas hydrodynamics. Phys. Rev. Lett., 112, 143601(2014).
[59] C. Hamner, C. Qu, Y. Zhang, J. Chang, M. Gong, C. Zhang, P. Engels. Dicke-type phase transition in a spin-orbit-coupled Bose–Einstein condensate. Nat. Commun., 5, 4023(2014).
[60] M. A. Norcia, J. R. K. Cline, J. A. Muniz, J. M. Robinson, R. B. Hutson, A. Goban, G. E. Marti, J. Ye, J. K. Thompson. Frequency measurements of superradiance from the strontium clock transition. Phys. Rev. X, 8, 021036(2018).
[61] N. Q. Burdick, Y. Tang, B. L. Lev. Long-lived spin-orbit-coupled degenerate dipolar Fermi gas. Phys. Rev. X, 6, 031022(2016).
[62] X. Zhou, J.-S. Pan, Z.-X. Liu, W. Zhang, W. Yi, G. Chen, S. Jia. Symmetry-protected topological states for interacting fermions in alkaline-earth-like atoms. Phys. Rev. Lett., 119, 185701(2017).
[63] F. Iemini, L. Mazza, L. Fallani, P. Zoller, R. Fazio, M. Dalmonte. Majorana quasiparticles protected by F2 angular momentum conservation. Phys. Rev. Lett., 118, 200404(2017).
[64] P.-B. Li, Y. Zhou, W.-B. Gao, F. Nori. Enhancing spin-phonon and spin-spin interactions using linear resources in a hybrid quantum system. Phys. Rev. Lett., 125, 153602(2020).
[65] Y. Zhou, B. Li, X.-X. Li, F.-L. Li, P.-B. Li. Preparing multiparticle entangled states of nitrogen-vacancy centers via adiabatic ground-state transitions. Phys. Rev. A, 98, 052346(2018).
[66] M. J. McDonnell, J. P. Home, D. M. Lucas, G. Imreh, B. C. Keitch, D. J. Szwer, N. R. Thomas, S. C. Webster, D. N. Stacey, A. M. Steane. Long-lived mesoscopic entanglement outside the Lamb-Dicke regime. Phys. Rev. Lett., 98, 063603(2007).
[67] D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, D. J. Wineland. Generation of nonclassical motional states of a trapped atom. Phys. Rev. Lett., 76, 1796-1799(1996).
[68] M. G. Kozlov, M. S. Safronova, J. R. C. López-Urrutia, P. O. Schmidt. Highly charged ions: optical clocks and applications in fundamental physics. Rev. Mod. Phys., 90, 045005(2018).
Set citation alerts for the article
Please enter your email address