Open Access
Add to BibSonomy
Abstract
We investigate a single atom cavity-QED system directly driven by a broadband squeezed light. We demonstrate how the squeezed radiation can be used to sense the presence of a single atom in a cavity. This happens by transferring one of the photons from the field in a state with an even number of photons to the atom and thereby populating an odd number of Fock states. Specifically, the presence of the atom is sensed by remarkable changing in the presence of one photon and the loss of squeezing of the cavity field. A complete study of quantum fluctuations and the excitation of multiphoton transitions is given.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Cavity quantum electrodynamics (cavity QED) is the study on the interaction between photons confined to a high-finesse cavity and quantum emitters, including atoms, quantum dots and so on, under conditions where the quantum nature of cavity photons is dominant [1]. The typical model of light-matter interaction is a single atom interacting with photons in the cavity, which is known as the Jaynes-Cummings model and provides many interesting quantum optical phenomena, including vacuum Rabi splitting [2], single photon blockade [3] and so on [4,5]. It also provides possibilities to achieve quantum information processing and communication by fully controlling the cavity QED systems [6,7].
Due to the quantum nature and extremely high coherence, atoms are good candidates for quantum computation and communication. Benefit from the development of laser cooling and trapping technology [8,9], neutral atoms can be cooled from room temperature to sub-millikelvin, and subsequently trapped at the single atom level for a long time [10]. Now, it is possible to strongly couple single or several trapped atoms to the cavity mode [11–17]. Even the ultrastrong coupling has been considered [18–20].
In experiments, detecting whether a single atom is successfully trapped inside the cavity or not is challenging. In general, one can obtain the information of intracavity by detecting the photon leaking from the cavity. Due to the intensity of leakage photons is very weak (mean photon number is less than unity), single-particle detection technology such as the homodyne detection [21] becomes a powerful tool for single atoms sensing [22]. To date, there exist several methods to detect single atoms, including direct detection of spontaneously emitted photons [23, 24], the observation of the change in the transmission spectrum [25–27] and measurement of the polarized photons via the Faraday effect [28]. With these current experimental conditions, the quantum statistical properties of photons can also be measured [29].
In this paper, we study the interaction of a single atom and the cavity directly driven by a broadband squeezed light. Our purpose is to supplement the existing methods with a one which makes use of the nonclassical light. Due to the strong coupling between the atom and cavity, the mean photon number in the cavity, the squeezed property of the cavity field and the photon number distributions of the photon leaking from the cavity will be significantly changed. Compared with the empty cavity, we show that it is possible to detect single atoms by monitoring the sideway photons, observing the photon number distribution of the one-photon state and measuring the quantum fluctuations of the cavity field. Thus the sensing cannot be achieved by injecting thermal radiation in cavity and our proposal specifically requires squeezed field in which only even Fock states are occupied. We note that the previous methods [23–28] utilize the idea of transferring the spontaneously emitted photon into an unoccupied mode which is different from the mode in which pump photons were present. The mode can, for example, be a different polarization mode [28]. In our scheme, we work with the same mode, however, we utilize the fact that the Fock state |1〉 is unoccupied and spontaneous emission leads to the population in state |1〉. Thus, there are some similarities to previous methods. Our method is the first one based on nonclassical light and thus supplements the existing methods and demonstrates the utility of nonclassical light in atomic sensing studies.
2. System model
In Fig. 1(a), we first consider a two-level atom trapped in a single-mode cavity with high Q factor, resulting in strong coupling between the cavity mode and atom. The cavity mode frequency is labeled as ωcav and the atomic resonance frequency is labeled as ωA = ωe − ωg with ħωα (α = e, g) being the energy of state |α〉. A broadband squeezed vacuum with central frequency ωsq is injected into the cavity as shown in Fig. 1(a).
Fig. 1 Sketch of the single atom cavity-QED system driven by a broadband squeezed vacuum with central frequency ωsq. The resonance frequency of this two-level atom is ωA = ωe − ωg with ħωα (α = e, g) being the energy of state |α〉. Here, γ and κ are the decay rates of the atom and cavity, respectively.
Under the electric dipole approximation, the Hamiltonian of this system can be expressed in the rotating frame of the central frequency of the squeezed field, which reads
where the atomic operator σij = |i〉〈j | (i, j = {e, g}) and g0 is the coupling strength between the atom and cavity. The annihilation and creation operators of the cavity photons are denoted by a and a†, respectively. Here, the detunings for the atom and cavity are defined by ΔA = ωA − ωsq and ΔC = ωcav − ωsq, respectively. The Eq. (1) gives only the coherence interaction between the atom and the cavity. The effect of the broadband squeezed radiation is accounted for by using master equation. The terms given by Eq. (3) below represent the effect of the squeezed drive.The evolution of this system is governed by the standard master equation [5], given by
where ρ is the density matrix operator. ℒA(ρ) and ℒcav(ρ) are terms describing the decay of the atom and the cavity driven by a squeezed light, respectively. In general, they have the following forms (see the Eq. 15.2.6 in [4]): and with γ being the damping rate of the atom. It is noted that κ is the half width of the mode of the cavity which interacts with almost “white” squeezed field outside. The details of coupling single mode to infinity of modes outside are given in Ref. [30] (see section 7.1). Here, N = sinh2 (r) is the photon number of the squeezed vacuum with r being the squeezing strength. M = cosh(r) sinh (r)eiϕ denotes the two-photon correlation in the injected squeezed vacuum with ϕ being the phase of the squeezed vacuum. The behavior of atom in a bad cavity (κ ≫ γ) driven by a squeezed light has been extensively discussed [31,32], where the atom-cavity coupling strength g0 ≪ γ ≪ κ. For good cavities, the effect of the squeezed light in dispersive cavities has been considered [33], and the reduction of the radiative decay of atomic coherence has been reported [34]. In this paper, we focus on the strong coupling case and show that the energy exchange between the atom and cavity will result in many appealing phenomena.3. Sensing a single atoms by squeezed drive
We first discuss one of our key results, i.e., how the squeeze drive can be used to sense a single atom in cavity. To demonstrate this we consider the special case ΔA = ΔC = 0. We solve the Eq. (2) numerically for a range of the degree of squeezing of the input field. We evaluate the mean photon number 〈a†a〉 and the photon distribution P(n) in the cavity. In the absence of the atom, these are well known
In the presence of the atom, however, the mean photon number and the photon distribution in the cavity change remarkably due to the additional transition pathway induced by the coupling between the atom and cavity.In Fig. 2 we plot 〈a†a〉 and P1 as a function of the squeezing parameter. Note that with on atom P1 = 0. It is clear that the detection of a single photon in the cavity is a sensor for the single atom in the cavity [see Fig. 2(b)]. Note that with increase in the degree of squeezing, the cavity field undergoes significant changes due to the presence of the atom. For small r, we can easily understand the process that leads to the excitation of the atom and the presence of a single photon in the cavity, i.e., The applied squeezed radiation produces at least two photons in the cavity. One of these is absorbed by the atom via the Jaynes-Cunnings coupling. The atom can then decay via spontaneous emission. The last transition can be monitored by looking at the radiation on the side of the cavity.
Fig. 2 The mean photon number 〈a†a〉 [Panel (a)] and P1 [Panel (b)] versus the squeezing parameter r. Here, the system parameters are chosen as ΔA = ΔC = 0, g0/κ = 15 and γ/κ = 1.
In Fig. 3 we show the photon distribution for arbitrary values of n. The system parameters are the same as those used in Fig. 2. Figure 3(a) shows that only the states with even number of photons can be detected (i.e., P2k+1 = 0 and P2k ≠ 0) in the absence of the atom [5]. For small r, this result can be understood by exploring the excitation pathway . With the increase of r, more and more states with even number of photons can be detected. Figure 3(b) shows how the odd number photon states get excited with the increase in the strength of the squeezing. It is obvious that larger the degree of squeezed light r is, the more the states with odd number of photons can be detected.
Fig. 3 The photon distribution for arbitrary values of n with different values of r. Panels (a) and (b) represent the empty cavity and atom-cavity system, respectively. Here, the system parameters are the same as those used in Fig. 2.
The presence of the atom also changes the phase characteristics of the field in the cavity. This is seen from the plot [Fig. 4] of the phase dependent mean value |〈aa〉|. For the empty cavity, we show that the mean value |〈aa〉| enhances significantly as the value of r increases [see the blue dashed curves in the Fig. 4]. However, the mean value |〈aa〉| will be suppressed when the atom is in the presence [see other two curves]. The reason for the suppression of |〈aa〉| is the excitation of the atom. As demonstrated by the red solid curve, the suppression of |〈aa〉| can be significantly enhanced if the atom is strongly coupled with the cavity.
Fig. 4 The phase dependent mean value |〈aa〉| versus the degree of the squeezed light r. The blue dashed curves represent the case in the absence of the atom, while other two curves represent the case in the presence of the atom. Here, we choose g0/κ = 5 for the dash-dotted curve and g0/κ = 15 for the black solid curve. Other system parameters are the same as those used in Fig. 2.
The full quantum statistical characteristics of the cavity field can be described in terms of the Wigner function of the cavity field. The Wigner function is calculated from the solution of the Eq. (2). In Fig. 5, we plot the Wigner function of the cavity field. In the absence of the atom, the cavity is excited by the two photon process as demonstrated in Fig. 3(a). Thus, the Wigner function is suppressed in one direction [see Fig. 5(a)], implying that the cavity field is squeezed. In the presence of the atom, the excitation of the atom significantly changes the Wigner function of the cavity field as shown in Figs. 5(b) and 5(c). Specifically the presence of a single photon prevents the generation of the squeezed field, resulting in the Wigner function of a circular aperture under the strong coupling condition [see Fig. 5(c)].
Fig. 5 The Wigner functions of the cavity field. Panel (a) represents the case in the absence of the atom. Panels (b) and (c) represent the cases in the presence of the atom with coupling strengths g0/κ = 5 and g0/κ = 15, respectively.
An alternate way to understand the interaction of the atom with the squeezed light in the cavity is to make a Bogoliubov transformation on the cavity field. Thus, in the absence of the atom the cavity is in vacuum state of the Bogoliubov operator b = cosh(r)a + sinh (r)a†, ϕ = 0. Then, we can rewrite the Eq. (2) as
In terms of the Bogoliubov modes, the excitation of the atom occurs via the counter rotating terms σegb†. The Hamiltonian given by Eq. (5) reduces to the Rabi model of cavity QED. Thus our work shows that we can indeed arrive at the Rabi model by starting from the Jaynes Cummings model and by pumping the system with squeezed light. Thus we show how one can realize the Rabi model. Note that considerable theoretical work [14,35–38] has been done in the context of the Rabi model and ultrastrong coupling. However so far very little progress is there in realizing the Rabi model. Clearly this alternate way of studying our system enables one to uncover the physics of the Rabi model.Finally, we study the population of the atomic excited state ρee by setting ΔC = ΔA = 0 and varying the degree of the squeezed field. In Fig. 6, we plot the population of the atomic excited state ρee as a function of the degree of the squeezed field. Here, the blue dashed curve corresponds to the case of g0/κ = 5, but the red solid curve corresponds to the case of g0/κ = 15. Other system parameters are the same as those used in Fig. 2. Clearly, the population of the atomic excited state grows quickly (i.e., the atom is excited) with the increase of the degree of the squeezed field. This characteristic implies another way of sensing the atom by detecting the emission of sideway photons once the atom is excited.
Fig. 6 The population of the atomic excited state ρee is plotted as a function of the degree of the squeezed field. Here, we set ΔC = ΔA = 0. The blue dashed curve corresponds to the case of g0/κ = 5, but the red solid curve corresponds to the case of g0/κ = 15. Other system parameters are the same as those used in Fig. 2.
4. Conclusion
In summary, we have studied the single atom-cavity QED system driven by a broadband squeezed light under the strong coupling regime. Due to the energy exchange between the atom and cavity, the mean photon number in the cavity, photon number distributions and quantum fluctuations of the cavity photons are significantly changed. For instance, (I) compared with the empty cavity, the mean photon number in the cavity undergoes a reduction due to the atomic spontaneous emissions, which implies an approach to sensing atoms by detecting photons leaking from the cavity or the sideway photon emissions originated from the atomic population of the excited state; (II) the mean value of |〈aa〉| can be remarkably suppressed under the strong coupling regime when the atomic resonance frequency is the same as the central frequency of the squeezed light; (III) it is possible to detect photons in the odd photon number states, especially the value of P1 if a single atom is trapped inside the cavity. Based on these interesting properties, we show that the squeezed light is a good candidate for sensing single atoms in the cavity by several potential approaches, such as monitoring the sideway photons, measuring the photon fluctuations and detecting the probability in the one-photon state. These phenomena can be realized with current experimental conditions, and the results presented in this paper may help readers to understand very wide implications of squeezed light interacting with matter.
Funding
The National Key Basic Research Special Foundation (Grant No. 2016YFA0302800); the Shanghai Science and Technology Committee (Grant No. 18JC1410900); the National Nature Science Foundation (Grant No. 11774262 and 11874287);
References
1. S. M. Dutra, Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box (John Wiley & Sons, 2005).
2. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004). [CrossRef] [PubMed]
3. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436(7047), 87–90 (2005). [CrossRef] [PubMed]
4. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997). [CrossRef]
5. G. S. Agarwal, Quantum Optics (Cambridge University Press, 2012). [CrossRef]
6. A. Imamog, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity qed,” Phys. Rev. Lett. 83(20), 4204 (1999). [CrossRef]
7. H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008). [CrossRef] [PubMed]
8. E. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59(23), 2631–2634 (1987). [CrossRef] [PubMed]
9. H. J. Metcalf and P. Straten, Laser Cooling and Trapping of Neutral Atoms (Wiley, 2007).
10. Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Förster, M. Khudaverdyan, D. Meschede, D. Schrader, and A. Rauschenbeutel, “Quantum engineering: An atom-sorting machine,” Nature 442(7099), 151 (2006). [CrossRef] [PubMed]
11. A. Neuzner, M. Köber, O. Morin, S. Ritter, and G. Rempe, “Interference and dynamics of light from a distance-controlled atom pair in an optical cavity,” Nat. Photonics 10(5), 303–306 (2016). [CrossRef]
12. S. Welte, B. Hacker, S. Daiss, S. Ritter, and G. Rempe, “Cavity carving of atomic bell states,” Phys. Rev. Lett. 118(21), 210503 (2017). [CrossRef] [PubMed]
13. S. Welte, B. Hacker, S. Daiss, S. Ritter, and G. Rempe, “Photon-mediated quantum gate between two neutral atoms in an optical cavity,” Phys. Rev. X 8(1), 011018 (2018).
14. A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, “Ultrastrong coupling between light and matter,” Nat. Rev. Phys. 1(1), 19–40 (2019). [CrossRef]
15. J. Chen, D. Konstantinov, and K. Mølmer, “Adiabatic preparation of squeezed states of oscillators and large spin systems coupled to a two-level system,” Phys. Rev. A 99(1), 013803 (2019). [CrossRef]
16. J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the jaynes-cummings ladder and observing its nonlinearity in a cavity qed system,” Nature 454(7202), 315–318 (2008). [CrossRef] [PubMed]
17. J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Dressed collective qubit states and the tavis-cummings model in circuit qed,” Phys. Rev. Lett. 103(8), 083601 (2009). [CrossRef] [PubMed]
18. X. Gu, A. F. Kockum, A. Miranowicz, Y. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Reports 718, 1–102 (2017). [CrossRef]
19. R. F. Ribeiro, L. A. Martínez-Martínez, M. Du, J. Campos-Gonzalez-Angulo, and J. Yuen-Zhou, “Polariton chemistry: controlling molecular dynamics with optical cavities,” Chem. Sci. 9(30), 6325–6339 (2018). [CrossRef] [PubMed]
20. Q. H. Chen, Y. F. Xie, and L. Duan, “Generalized quantum rabi model with both one- and two-photon terms: A concise analytical study,” Phys. Rev. A 99(1), 013809 (2019). [CrossRef]
21. P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67(4), 043806 (2003). [CrossRef]
22. H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley, 2004). [CrossRef]
23. N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single atoms in a microscopic dipole trap,” Nature 411(6841), 1024–1027 (2001). [CrossRef] [PubMed]
24. S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic delivery of a single atom,” Science 293(5528), 278–280 (2001). [CrossRef] [PubMed]
25. I. Teper, Y. J. Lin, and V. Vuletić, “Resonator-aided single-atom detection on a microfabricated chip,” Phys. Rev. Lett. 97(2), 023002 (2006). [CrossRef] [PubMed]
26. S. Nußmann, K. Murr, M. Hijlkema, B. Weber, A. Kuhn, and G. Rempe, “Vacuum-stimulated cooling of single atoms in three dimensions,” Nat. Phys. 1(2), 122–125 (2005). [CrossRef]
27. K. M. Fortier, S. Y. Kim, M. J. Gibbons, P. Ahmadi, and M. S. Chapman, “Deterministic loading of individual atoms to a high-finesse optical cavity,” Phys. Rev. Lett. 98(23), 233601 (2007). [CrossRef] [PubMed]
28. M. Terraciano, R. O. Knell, D. Norris, J. Jing, A. Fernández, and L. Orozco, “Photon burst detection of single atoms in an optical cavity,” Nat. Phys. 5(7), 480–484 (2009). [CrossRef]
29. S. Barzanjeh, D. DiVincenzo, and B. Terhal, “Dispersive qubit measurement by interferometry with parametric amplifiers,” Phys. Rev. B 90(13), 134515 (2014). [CrossRef]
30. D. F. Walls and G. J. Milburn, Quantum Optics (Springer Science & Business Media, 2007).
31. P. R. Rice and L. M. Pedrotti, “Fluorescent spectrum of a single atom in a cavity with injected squeezed vacuum,” J. Opt. Soc. Am. B 9(11), 2008–2014 (1992). [CrossRef]
32. C. Gardiner, “Inhibition of atomic phase decays by squeezed light: A direct effect of squeezing,” Phys. Rev. Lett. 56(18), 1917–1920 (1986). [CrossRef] [PubMed]
33. S. Kono, Y. Masuyama, T. Ishikawa, Y. Tabuchi, R. Yamazaki, K. Usami, K. Koshino, and Y. Nakamura, “Nonclassical photon number distribution in a superconducting cavity under a squeezed drive,” Phys. Rev. Lett. 119(2), 023602 (2017). [CrossRef] [PubMed]
34. K. Murch, S. Weber, K. Beck, E. Ginossar, and I. Siddiqi, “Reduction of the radiative decay of atomic coherence in squeezed vacuum,” Nature 499(7456), 62–65 (2013). [CrossRef] [PubMed]
35. T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nat. Phys. 6(10), 772–776 (2010). [CrossRef]
36. S. De Liberato, C. Ciuti, and I. Carusotto, “Quantum vacuum radiation spectra from a semiconductor microcavity with a time-modulated vacuum rabi frequency,” Phys. Rev. Lett. 98(10), 103602 (2007). [CrossRef] [PubMed]
37. T. Jaako, Z. L. Xiang, J. J. Garcia-Ripoll, and P. Rabl, “Ultrastrong-coupling phenomena beyond the dicke model,” Phys. Rev. A 94(3), 033850 (2016). [CrossRef]
38. W. Qin, A. Miranowicz, P. B. Li, X. Y. Lü, J. Q. You, and F. Nori, “Exponentially enhanced light-matter interaction, cooperativities, and steady-state entanglement using parametric amplification,” Phys. Rev. Lett. 120(9), 093601 (2018). [CrossRef] [PubMed]
